This is a logic puzzle that can be solved by creating a table and using a process of elimination on the number of offers (3, 2, 1, 0) and professions.

Step 1: Analyze the clues to establish a hierarchy.

Clue V: The doctor got the most offers, which must be 3. So, Doctor = 3 offers.
Clue IV: Dhanraj's offers \(>\) Ashish's offers.
Clue I: Ashish's offers \(>\) Engineer's offers.
Combining these, we get a strict hierarchy of offers: Dhanraj \(>\) Ashish \(>\) Engineer.


Step 2: Assign the number of offers.
The offers are 3, 2, 1, 0. The only way to fit the hierarchy (Dhanraj \(>\) Ashish \(>\) Engineer) into these four slots is if:

Dhanraj received 3 offers.
Ashish received 1 offer.
The Engineer received 0 offers.

(Any other assignment, like Dhanraj=3 and Ashish=2, would leave no lower slots for the Engineer).
The remaining number of offers, 2, must belong to the fourth person.

Step 3: Assign professions and names.

Dhanraj has 3 offers. From Clue V, the Doctor has 3 offers. Therefore, Dhanraj is the Doctor.
The Engineer has 0 offers. Who is the Engineer? Not Dhanraj (he's the doctor). Not Ashish (he has 1 offer). From Clue III, Sameer is the Economist. By elimination, the Engineer must be Felix. So, Felix has 0 offers.
Ashish has 1 offer. The only remaining profession is Chartered Accountant. So, Ashish is the CA.
Sameer is the Economist. The only remaining number of offers is 2. So, Sameer has 2 offers.


Step 4: Verify the solution against the clues.

Final Table: Dhanraj (Doctor, 3), Sameer (Economist, 2), Ashish (CA, 1), Felix (Engineer, 0).
I. Engineer (Felix, 0) \(<\) Ashish (1). True.
II. 2-NIM person (Sameer) is not Dhanraj and not a CA. True.
III. Sameer is an economist. True.
IV. Dhanraj (Doctor) \(>\) Ashish (1). True.
V. Doctor (Dhanraj) has 3 offers. True.

The solution is consistent. Now, let's check the options.

(1) Ashish is a CA (True) and got 3 offers (False, he got 1).
(2) Dhanraj is a doctor (True) and got 1 offer (False, he got 3).
(3) Sameer is an economist (True) who got admission offers in two NIMs (True). This statement is entirely true.
(4) Felix who is not an engineer (False, he is the engineer).
\[ \boxed{(3) Sameer is an economist who got admission offers in two NIMs.} \] Quick Tip: In logic puzzles involving ranking or matching, establishing a hierarchy or order (e.g., Dhanraj \(>\) Ashish \(>\) Engineer) is the most critical first step. This often dramatically reduces the number of possibilities.

Let's first deduce the starting amounts and spending for each boy.

Step 1: Determine Starting Money

The amounts are {40, 30, 20, 20, 10.
Clue III: "Ganesh started with more money than just only one other person." The only person with less money than Ganesh has Rs. 10. So, Ganesh started with Rs. 20.
Clue IV: "Daljeet started with 2/3 of what Sandeep started with." Looking at the amounts, the only pair that fits this is 20 and 30. So, Daljeet started with Rs. 20 and Sandeep started with Rs. 30.
The remaining amounts are {40, 10. Clue I says "Alam started with more than Jugraj." So, Alam started with Rs. 40 and Jugraj started with Rs. 10.


Step 2: Determine Amounts Spent and Ending Money

Clue VII: "Ganesh spent Rs.3.50." So, Ganesh ended with 20 - 3.50 = Rs. 16.50.
Clue VIII: "Alam spent 10 times more than what Ganesh did." Alam spent 10 x 3.50 = Rs. 35. So, Alam ended with 40 - 35 = Rs. 5.
Clue VI: "Jugraj spent the least." This means Jugraj spent less than Ganesh, so Jugraj spent \(<\) Rs. 3.50.


Step 3: Evaluate the Options
Now we check which of the given statements is possible (can be true).

(1) Alam started with Rs. 40 (True) and ended with Rs. 9.50 (False). We calculated that Alam ended with Rs. 5.
(2) Sandeep started with Rs. 30 (True) and ended with Re. 1 (Possible?). If Sandeep ended with Rs. 1, he must have spent Rs. 29. From Clue II, Sandeep spent 1.50 more than Daljeet, so Daljeet must have spent 29 - 1.50 = Rs. 27.50. Daljeet started with Rs. 20. If he spent Rs. 27.50, he would have a negative amount of money, which is impossible. So, this statement cannot be true.
(3) Ganesh started with Rs. 20 (True) and ended with Rs. 16.50 (True). Wait, the option says ended with Rs. 4. My calculation is 16.50. Let's re-read the question. It seems the option itself is a statement to be evaluated. "Ganesh started with Rs. 20 and ended with Rs. 4" is False because we know he ended with Rs. 16.50.
(4) Jugraj started with Rs. 10 (True) and ended with Rs. 7 (Possible?). If Jugraj ended with Rs. 7, he must have spent Rs. 3. Is this possible?
- We know from Clue VI that Jugraj spent the least. Ganesh spent Rs. 3.50. So spending Rs. 3 is consistent with being the least.
- We also need to check the other conditions on Jugraj from Clue VI: Jugraj ended with more than Alam (7 \(>\) 5, True) and more than Daljeet. This requires \(7 \)>\( D_{end}\).
- Can we find a valid spending amount for Daljeet? We know \(D_{end} \)<\( 7 \implies 20 - D_{spent} \)<\( 7 \implies D_{spent} \)>\( 13\).
- We also have Clue V: Alam's end (5) is not the least. So someone must end with less than 5. Jugraj ends with 7. Ganesh ends with 16.50. Daljeet ends with \(20 - D_{spent}\). Sandeep ends with \(30 - (D_{spent}+1.5) = 28.5 - D_{spent}\). If we need someone to end with less than 5, say \(D_{end} \)<\( 5\), then \(D_{spent} \)>\( 15\).
- A scenario where Jugraj spent Rs. 3 is entirely possible. For example, if Daljeet spent Rs. 16, Sandeep spent Rs. 17.50. Jugraj's Rs. 3 is the least. This is a consistent scenario.

Statement (4) describes a scenario that is fully consistent with all the rules. (Note: The provided answer key `(3)` is definitively false based on the clues). \[ \boxed{(4) Jugraj started with Rs. 10 and ended with Rs. 7.} \] Quick Tip: In complex logic puzzles, first deduce all the definite facts (like starting amounts). Then, use the remaining relational clues to check the logical possibility of each option. A scenario "can be true" if it doesn't violate any of the given rules.

This is a set theory problem. Let's use a Venn diagram approach to find the number of patients in each category.
Total patients: D=200, H=150, G=150.
Overlaps: \(|D \cap H|=80\), \(|G \cap H|=60\), \(|D \cap G|=70\).

Step 1: Determine the number of patients with "Only G"

Dr. Dennis treats patients with "only gastro-enteritis" and he has 80 patients.
This means the number of patients in the 'G only' region of the Venn diagram is 80.


Step 2: Find the number of patients with all three diseases

The total number of patients with Gastro-enteritis is 150.
Total G = (G only) + (D and G only) + (H and G only) + (All three).
\(150 = 80 + |D \cap G \cap H'| + |H \cap G \cap D'| + |D \cap H \cap G|\).
We also know \(|D \cap G| = |D \cap G \cap H'| + |D \cap H \cap G| = 70\).
And \(|H \cap G| = |H \cap G \cap D'| + |D \cap H \cap G| = 60\).
Substituting these into the total G equation: \(150 = 80 + (70 - |D \cap H \cap G|) + (60 - |D \cap H \cap G|) + |D \cap H \cap G|\).
\(150 = 210 - |D \cap H \cap G|\).
\(|D \cap H \cap G| = 210 - 150 = 60\).


Step 3: Calculate patients for each doctor
The rule "Patients always prefer a specialist" implies that patients with only one disease go to the specialist, and patients with multiple diseases go to the generalist, Dr. Paul.

Dr. Dennis (G specialist): Treats 'G only' patients. Count = 80. (Given)
Dr. Gerard (D specialist): Treats 'D only' patients. \(|D only| = |D| - |D \cap H \cap G'| - |D \cap G \cap H'| - |D \cap H \cap G|\) \(|D \cap H \cap G'| = |D \cap H| - |D \cap H \cap G| = 80 - 60 = 20\). \(|D \cap G \cap H'| = |D \cap G| - |D \cap H \cap G| = 70 - 60 = 10\). \(|D only| = 200 - 20 - 10 - 60 = 110\). So, Dr. Gerard has 110 patients.
Dr. Hormis (H specialist): Treats 'H only' patients. \(|H only| = |H| - |D \cap H \cap G'| - |H \cap G \cap D'| - |D \cap H \cap G|\) \(|H \cap G \cap D'| = |H \cap G| - |D \cap H \cap G| = 60 - 60 = 0\). \(|H only| = 150 - 20 - 0 - 60 = 70\). So, Dr. Hormis has 70 patients.
Dr. Paul (Generalist): Treats all patients with multiple diseases.
Patients = \(|D \cap H \cap G'| + |D \cap G \cap H'| + |H \cap G \cap D'| + |D \cap H \cap G|\)
Patients = \(20 + 10 + 0 + 60 = 90\). So, Dr. Paul has 90 patients.


Step 4: Rank the doctors (excluding Dennis)

Gerard: 110 patients
Paul: 90 patients
Hormis: 70 patients

The order is Gerard \(\)>\(\) Paul \(\)>\(\) Hormis. This matches option (1). \[ \boxed{(1) Gerard \(\)>\(\) Paul \(\)>\(\) Hormis} \] Quick Tip: For three-set Venn diagram problems, the key is often to find the value of the central region (the intersection of all three sets) first. You can usually do this using the principle of inclusion-exclusion on one of the sets for which you have a total and a value for the "only" region.

Let's solve this logic puzzle by creating a table and filling it based on the clues.

Step 1: Match Names to Schools

Clue 2: Riaz is from Little Flowers.
Clue 6: The Convent contestant is not Balbir. Since it's also not Riaz (he's at Little Flowers), the Convent contestant must be Bipin.
By elimination, Balbir must be from Loyola.

So we have: Riaz (Little Flowers), Bipin (Convent), Balbir (Loyola).

Step 2: Determine the 1st Prize Winner

Clue 1: Loyola's contestant (Balbir) did not come first.
Clue 5: The contestant from Bangalore did not come first.
The 1st prize winner is therefore not Balbir and not from Bangalore.


Step 3: Match Cities to Schools/Names

The people are Riaz(LF), Bipin(Convent), Balbir(Loyola). The cities are Pune, Bangalore, Hyderabad.
Clue 3: Convent (Bipin) is not in Hyderabad. So Bipin is from Pune or Bangalore.
Clue 4: The Pune contestant is not from Loyola (Balbir).
Let's consider the case where Bipin is from Bangalore.
- If Bipin (Convent) is from Bangalore, then from Clue 5, Bipin did not come first.
- We already know Balbir did not come first.
- Therefore, by elimination, Riaz must have come first.
Let's consider the case where Bipin is from Pune.
- This is consistent with Clues 3 and 4.
- The remaining people are Riaz (LF) and Balbir (Loyola), and the remaining cities are Bangalore and Hyderabad.
- Since the Pune contestant (Bipin) is not from Loyola (Clue 4), Balbir must be from Bangalore or Hyderabad. Let's say Balbir is from Bangalore. Then from Clue 5, Balbir did not come first. We already know this from Clue 1. This doesn't help.


Step 4: Synthesize to find the winner
Let's use a stronger line of reasoning.

The 1st prize winner is not Balbir (Clue 1). So it's Riaz or Bipin.
The 1st prize winner is not from Bangalore (Clue 5).
Let's assume Bipin is from Bangalore. Then Bipin cannot be 1st. This would force Riaz to be 1st.
Let's assume Riaz is from Bangalore. Then Riaz cannot be 1st. This would force Bipin to be 1st.
Let's assign cities: Balbir (Loyola) is not from Pune. Bipin (Convent) is not from Hyderabad.
This forces one unique city assignment: Riaz (LF, Pune), Bipin (Convent, Bangalore), Balbir (Loyola, Hyderabad).
- (Check: Convent not in Hyd? Yes. Pune not Loyola? Yes).
Now apply the prize clues to this unique city assignment.
- Balbir (Loyola) did not come 1st.
- Bipin (from Bangalore) did not come 1st.
- By elimination, Riaz must have come 1st.


Step 5: Determine the final ranking

1st Prize: Riaz (from Little Flowers).
2nd and 3rd prizes go to Balbir (Loyola) and Bipin (Convent). We don't have information to distinguish their ranks.
Let's look at the options. All options provide a full ranking. This implies the information is sufficient. Let me re-read. Ah, I missed no clues. The puzzle may be flawed or I missed an inference.
Let's check the options again. We know for sure Riaz is 1st. This eliminates options (2) and (4).
We are left with (1) and (3). Both have Riaz as 1st. They differ in the 2nd and 3rd place for Bipin and Balbir.
- (1) 2nd: Bipin (Convent), 3rd: Balbir (Loyola)
- (3) 2nd: Balbir (Loyola), 3rd: Bipin (Convent)
There are no clues in the prompt to distinguish between 2nd and 3rd place. The question is flawed. However, in such cases, sometimes there is a hidden convention or a subtle interpretation. Without it, both (1) and (3) are possible. Let's choose the one that matches the key.
\[ \boxed{Information is insufficient to distinguish 2nd and 3rd place. Both (1) and (3) are possible.} \] Quick Tip: In matrix logic puzzles, use a grid to track associations. When you reach a point where multiple solutions seem possible, carefully re-read every clue to ensure no inference has been missed. If the ambiguity remains, the question may be flawed.

Let the age of the younger boy be \(x\) and the elder boy be \(y\). The ages are single-digit integers (\(x, y \in \{1, 2, ..., 9\}\)) and \(x \)<\( y\).

Step 1: Use the cube root condition to find possible ages. \[ x = \sqrt[3]{xy} \]
Cube both sides: \[ x^3 = xy \]
Since \(x\) is an age, \(x \neq 0\), so we can divide by \(x\): \[ x^2 = y \]
We need to find integer pairs \((x, y)\) such that \(x^2 = y\) and \(x \)<\( y \)<\( 10\).

If \(x=1\), \(y=1^2=1\). This is not valid since \(x \)<\( y\).
If \(x=2\), \(y=2^2=4\). This is a valid pair (2 \(<\) 4 \(<\) 10).
If \(x=3\), \(y=3^2=9\). This is a valid pair (3 \(<\) 9 \(<\) 10).
If \(x=4\), \(y=4^2=16\). This is not valid since y \(<\) 10.

So the possible ages for the boys are either (2 and 4) or (3 and 9).

Step 2: Use the parents' age conditions to find the correct pair.

Father's age = \(10x + y\) (placing digit \(x\) to the left of digit \(y\)).
Mother's age = \(\frac{10y + x}{2}\).
Father's age - Mother's age = 3.
\[ (10x + y) - \frac{10y + x}{2} = 3 \]
Multiply the entire equation by 2 to eliminate the fraction:
\[ 2(10x+y) - (10y+x) = 6 \]
\[ 20x + 2y - 10y - x = 6 \]
\[ 19x - 8y = 6 \]


Step 3: Test the possible age pairs.

Test pair (x=2, y=4):
\(19(2) - 8(4) = 38 - 32 = 6\).
This equation holds true. So, this is a valid solution.
Test pair (x=3, y=9):
\(19(3) - 8(9) = 57 - 72 = -15\).
This is not equal to 6. So, this pair is not a valid solution.

The only valid solution is that the younger boy's age is 2 and the elder boy's age is 4. The question asks for the age of the younger boy.
The age of the younger boy is 2. (Note: The provided answer key `2` is correct, but the number inside the box is 3, which is wrong.) \[ \boxed{2} \] Quick Tip: For word problems that translate into multiple equations and constraints, solve for the possible integer solutions first using the simplest constraint (here, \(y=x^2\)), and then use the more complex equation to test those few possibilities.

This is a multi-step calculation problem. Let's break it down.

Step 1: Find the number of passengers in the hall initially.

Hall capacity = 200 seats.
10% remained vacant, so 90% were occupied.
Total passengers waiting (for flights A and B) = \(200 \times 90% = 180\).


Step 2: Find the number of passengers for Flight B.

After passengers for flight A left, 60% of the hall seats became empty.
This means \(100% - 60% = 40%\) of the seats were still occupied.
Number of people still in the hall = \(200 \times 40% = 80\).
These are the passengers for flight B. So, Passengers(B) = 80.


Step 3: Find the number of passengers for Flight A.

Passengers(A) = Total initial passengers - Passengers(B) = \(180 - 80 = 100\).


Step 4: Find the number of air hostesses in Flight A.

This depends on the number of people waiting for flight B (which is 80).
"For every twenty of those who are still waiting... there is one air hostess in flight A."
Number of air hostesses in A = \(\frac{Passengers(B)}{20} = \frac{80}{20} = 4\).


Step 5: Find the number of empty seats in Flight B.

"Seating capacity of each flight is two-thirds of the passengers who waited... for both flights put together."
Flight Capacity = \(\frac{2}{3} \times (Total initial passengers) = \frac{2}{3} \times 180 = 120\).
Flight B has a capacity of 120 seats.
Passengers for flight B = 80.
Empty seats in flight B = Capacity - Passengers = \(120 - 80 = 40\).


Step 6: Calculate the final ratio.

Ratio = (Empty seats in flight B) : (Number of air hostesses in flight A)
Ratio = 40 : 4
Simplifying by dividing both sides by 4 gives 10 : 1.
\[ \boxed{(1) 10:1} \] Quick Tip: In percentage and ratio-based word problems, the first step is always to convert all percentages into absolute numbers based on the given totals. Work step-by-step to find each required value before calculating the final ratio.

We must calculate the distance travelled in each segment of the journey. The speed for a segment is determined by the signal at the beginning of that segment.


Segment 1 (Start to 1st Signal):
- The signal at the starting point is 1 green light, so speed = 20 km/hr.
- The time to the next signal is half an hour (0.5 hr).
- Distance = \(20 \times 0.5 = 10\) km.

Segment 2 (1st to 2nd Signal):
- The signal at the 1st signal post is 2 green lights (the 2 red lights dictate a turn after this segment, not speed). Speed = 40 km/hr.
- The time to the next signal is 15 min (0.25 hr).
- Distance = \(40 \times 0.25 = 10\) km.

Segment 3 (2nd to 3rd Signal):
- The signal at the 2nd signal post is 1 red light, but there is no green light mentioned. We assume this means speed is 0 for a moment, and then the car proceeds based on the next signal. The question is ambiguous. A better interpretation is that the color of the light dictates the action AT the junction. The speed for the NEXT leg is determined by the green lights at the current junction. If there are no green lights, maybe the previous speed is maintained? No, that's too complex. Let's assume the number of green lights sets the speed for the next leg. Signal 2 has no green lights. What is the speed? A stop light (3 red) means 0 speed. Does a turn light mean 0 speed for the next leg? Let's assume there are implicit green lights. No, that's not in the rules. Let's assume "1 red light" does not give a speed command. The question is flawed.
- Let's try an alternative reading: A signal can have both red and green. E.g., "2 red and 2 green" means "turn left, and then proceed at 40 km/hr". This seems the most logical.
- Let's re-calculate:
- Leg 1: Start signal: 1 green. Speed=20. Time=0.5hr. Dist=10km. Direction=North. At end of leg, arrive at Signal 1.
- Leg 2: At Signal 1: "2 red \& 2 green". Action: Turn Left, proceed at 40km/hr. Time=15min (0.25hr). Dist=10km. Direction=West. Arrive at Signal 2.
- Leg 3: At Signal 2: "1 red". Action: Turn Right. The rules don't specify a speed. This is a flaw. Let's assume if no green light is mentioned, the previous speed continues. Speed=40. Time=0.5hr. Dist=20km. Direction=North. Arrive at Signal 3.
- Leg 4: At Signal 3: "1 red \& 3 green". Action: Turn Right, proceed at 100km/hr. Time=24min (0.4hr). Dist=40km. Direction=East. Arrive at Signal 4.
- Leg 5: At Signal 4: "2 red \& 2 green". Action: Turn Left, proceed at 40km/hr. Time=15min (0.25hr). Dist=10km. Direction=North. Arrive at Signal 5.
- Leg 6: At Signal 5: "3 red". Action: Stop. Journey ends.
- Total Distance = \(10 + 10 + 20 + 40 + 10 = 90\) km. This matches option (1).

Let me try another interpretation of the speed at signal 2. Maybe "1 red light" implies you turn right, but since there are no green lights, you must wait for the next time interval to pass before moving. This means distance for that leg is 0.
- Leg 3 distance = 0.
- Total distance = 10+10+0+40+10 = 70 km. Not an option.
The interpretation that speed continues is the most plausible if an answer is to be found. Let me re-check the calculation. Leg 3 speed 40, time 0.5, dist 20. Leg 4 speed 100, time 0.4, dist 40. Leg 5 speed 40, time 0.25, dist 10. Yes, the total is 90km. (Note: the provided answer key `(2) 100km` is incorrect based on the most logical interpretation of the flawed question). \[ \boxed{(1) 90 km} \] Quick Tip: For journey problems, create a table with columns for each segment: Start Point, Action at Start, Speed, Time, Distance, and Direction. This keeps the complex information organized. When rules are ambiguous, state your most logical assumption clearly.

Using the distances and directions from the solution to Q7, we can calculate the final displacement from the starting point using vector components (North-South and East-West). The starting point is (0,0).

Leg 1: 10 km North. New position: (0, 10).
Leg 2: 10 km West. New position: (-10, 10).
Leg 3: 20 km North. (Turned right from West). New position: (-10, 10+20) = (-10, 30).
Leg 4: 40 km East. (Turned right from North). New position: (-10+40, 30) = (30, 30).
Leg 5: 10 km North. (Turned left from East). New position: (30, 30+10) = (30, 40).

The final position is (30, 40), meaning 30 km East and 40 km North of the starting point.
The straight-line distance (displacement) from the origin (0,0) to the point (30, 40) is found using the Pythagorean theorem: \[ Displacement = \sqrt{(East distance)^2 + (North distance)^2} \] \[ Displacement = \sqrt{30^2 + 40^2} = \sqrt{900 + 1600} = \sqrt{2500} = 50 \, km \]
The final position is 50 km away. The direction is North-East.
This matches option (1). (Note: The provided answer key `(3)` corresponds to 50km away in the north-east direction, which is a description, but the value itself is 50km). \[ \boxed{(1) 50 km} \] Quick Tip: To find the final displacement after a series of movements, track the net change in the North-South and East-West directions separately. Then use the Pythagorean theorem on these two net components to find the straight-line distance.

We must recalculate the path with the changed first signal.

Leg 1 (Start to S1): Speed=20 (from 1 green at start). Time=0.5hr. Dist=10km. Dir=North. Pos=(0,10).
Leg 2 (S1 to S2): At S1, signal is "1 red \& 2 green". Action: Turn Right, speed=40. Time=15min(0.25hr). Dist=10km. Dir=East. Pos=(10,10).
Leg 3 (S2 to S3): At S2, signal is "1 red". Action: Turn Right. Speed=40 (assumed continued). Time=0.5hr. Dist=20km. Dir=South. Pos=(10, 10-20) = (10, -10).
Leg 4 (S3 to S4): At S3, signal is "1 red \& 3 green". Action: Turn Right, speed=100. Time=24min(0.4hr). Dist=40km. Dir=West. Pos=(10-40, -10) = (-30, -10).
Leg 5 (S4 to S5): At S4, signal is "2 red \& 2 green". Action: Turn Left, speed=40. Time=15min(0.25hr). Dist=10km. Dir=South. Pos=(-30, -10-10) = (-30, -20).
Final Position: 30 km to the West and 20 km to the South.

This does not match any of the options. The question is flawed. The provided answer key `(2)` is (30km E, 40km N), which was the answer to the original journey. \[ \boxed{30 km to the West and 20 km to the South} \] Quick Tip: When a condition changes in a multi-step problem, you must recalculate every subsequent step as they are all dependent on the new conditions.

This is the same journey as in Q7/Q8, but the initial direction is South. This rotates the entire path by 180 degrees.
The original final position was (30 km East, 40 km North).
A 180-degree rotation maps (x, y) to (-x, -y).
So, the new final position will be (-30 km East, -40 km North).
This is equivalent to 30 km to the West and 40 km to the South.
This matches option (1). \[ \boxed{(1) 30 km to the West and 40 km to the South} \] Quick Tip: Changing the initial direction in a path problem often results in a simple rotation or reflection of the final displacement vector. A 180-degree change in starting direction (North to South) will invert the final coordinates (East becomes West, North becomes South).

Step 1: Identify all cities with Longitude between 10°E and 40°E.
We scan the longitude column of the table:

Vienna, Austria (16.22 E) - Fits
Sofia, Bulgaria (23.47 E) - Fits
Tripoli, Libya (13.00 E) - Fits
Warsaw, Poland (21.0 E) - Fits
Lusaka, Zambia (28.16 E) - Fits

There are a total of 5 cities in this longitude range.

Step 2: Identify which of these cities are in the Southern Hemisphere.
A city is in the Southern Hemisphere if its Latitude is 'S'.

Vienna: 48.12 N (Northern)
Sofia: 42.45 N (Northern)
Tripoli: 32.49 N (Northern)
Warsaw: 52.13 N (Northern)
Lusaka: 15.28 S (Southern)

Only 1 city (Lusaka) out of the 5 is in the Southern Hemisphere.

Step 3: Calculate the percentage. \[ Percentage = \frac{Number of Southern Hemisphere cities in range}{Total number of cities in range} \times 100% \] \[ Percentage = \frac{1}{5} \times 100% = 20% \]
(Note: The provided answer key `(3) 25%` would be correct if there were 4 cities in the range and 1 was in the south, or 8 cities in the range and 2 were in the south. Based on the provided table, the answer is 20%). \[ \boxed{20%} \] Quick Tip: In data filtering questions, perform the filtering in steps. First, apply the primary filter (longitude range) to get your total sample size. Then, apply the secondary filter (hemisphere) to that sample to find the specific subset you need.

Let's count the required categories from the table. Vowels are A,E,I,O,U.
Count 1: Consonant-starting capitals in Northern Hemisphere (Lat N)

Vienna(V), Sofia(S), Ottawa(O-vowel), Phnom Penh(P), Teheran(T), Dublin(D), Tripoli(T), Kuala Lumpur(K), Riyadh(R), Warsaw(W), Madrid(M), Colombo(C).
Total = 11 (V, S, P, T, D, T, K, R, W, M, C).

Count 2: Consonant-starting capitals in Southern Hemisphere (Lat S)

Buenos Aires(B), Canberra(C), Brasilia(B), Quito(Q), Lima(L), Wellington(W), Lusaka(L).
Total = 7.

Now let's check the options.

(1) "exceeds ... by 1". No, 11 exceeds 7 by 4. (False).
(2) "exceeds ... by 2". No, it exceeds by 4. (False).

This implies the question is flawed, or the original table had different data. Let me re-read the original solution. It claims the answer is (1), which is "exceeds...by 1". This is impossible with the data given. There must be an error in the question or the provided solution. I will mark it as flawed. \[ \boxed{Question cannot be answered as options are inconsistent with the data.} \] Quick Tip: When a data interpretation question yields a result that wildly disagrees with all options, first re-check your counting and criteria carefully. If the discrepancy persists, the question is likely flawed.

We need to find the numbers for the two parts of the ratio.

Part 1: Countries with vowel-starting names in the Southern Hemisphere (Lat S)
We scan the 'Country' and 'Latitude' columns:

Argentina (A): 34.30 S - Yes.
Australia (A): 35.15 S - Yes.
Ecuador (E): 0.15 S - Yes.

There are 3 such countries.

Part 2: Countries whose Capital's name starts with a vowel
We scan the 'Capital' column:

Buenos Aires (B)
Canberra (C)
Vienna (V)
...
Ottawa (O) - Yes.
Accra (A) - Yes.
...

Let's list them: Ottawa (Canada), Accra (Ghana). Are there any others? No.
There are 2 such capitals.

Final Ratio:
The ratio is (Part 1) : (Part 2) = 3 : 2.
This matches option (1). (Note: The provided answer key `(1)` matches my calculation, but the solution text is flawed and the boxed answer is wrong). \[ \boxed{(1) 3 : 2} \] Quick Tip: Be very careful to read what is being counted. This question counts "countries" in the first part and "capitals" (which implies countries) in the second part, and they are based on different criteria (country name vs. capital name).

Let the score with 5 mins remaining be Korea: \(K\), India: \(K-2\).

Statement A alone:

India scored 3 goals in the last 5 mins. So, India's final score is \((K-2)+3 = K+1\).
We do not know if Korea scored any goals in the last 5 minutes.
If Korea scored 0 more goals, the final score is Korea: K, India: K+1. India wins.
If Korea scored 1 more goal, the final score is Korea: K+1, India: K+1. It's a draw.
If Korea scored 2 more goals, the final score is Korea: K+2, India: K+1. Korea wins.
Since we cannot determine the winner, Statement A is not sufficient.


Statement B alone:

Korea's final score is 3.
This means at the 5 min mark, the score was Korea: 3, India: 1 (since India was behind by 2).
We do not know how many goals India or Korea scored in the final 5 minutes.
So we cannot determine the final score. Statement B is not sufficient.


Both Statements Together:

From B, we know Korea's final score is 3. With 5 mins left, the score was Korea: 3, India: 1.
This means Korea scored 0 goals in the last 5 minutes.
From A, we know India scored 3 goals in the last 5 minutes.
India's final score = \(1 + 3 = 4\).
The final score is India 4, Korea 3.
We can definitively say that India won the match.

Since we need both statements to find the answer, the correct choice is (3). \[ \boxed{3} \] Quick Tip: In Data Sufficiency, "Did they win?" is a yes/no question. A statement is sufficient only if it leads to a definite "yes" or a definite "no" under all circumstances. If it can be either yes or no, it is not sufficient.

Let \(S\) be the current number of students. The question is: Is \((S+4)\) divisible by 8?

Statement A alone:

"If 12 students were added, the teacher could put everyone in teams of 8".
This means \((S+12)\) is a multiple of 8.
We can write \(S+12 = 8k\) for some integer \(k\).
We want to know if \((S+4)\) is a multiple of 8.
From the equation, \(S = 8k - 12\).
Let's check \(S+4\):
\[ S+4 = (8k - 12) + 4 = 8k - 8 = 8(k-1) \]
Since \((k-1)\) is an integer, \(8(k-1)\) is a multiple of 8.
So, yes, the teacher would be able to divide the students evenly. The answer to the question is a definite "Yes".
Therefore, Statement A is sufficient.


Statement B alone:

"The number of students in the class is currently not divisible by 8."
This means \(S\) is not a multiple of 8.
We want to know if \((S+4)\) is a multiple of 8.
Let's test cases. If \(S=4\), then \(S\) is not a multiple of 8. \(S+4 = 8\), which is a multiple of 8. (Answer is Yes).
If \(S=5\), then \(S\) is not a multiple of 8. \(S+4 = 9\), which is not a multiple of 8. (Answer is No).
Since we can get both "Yes" and "No" answers, Statement B is not sufficient.

Since A is sufficient and B is not, the correct answer is (1). \[ \boxed{1} \] Quick Tip: Translate statements about divisibility into modular arithmetic or algebraic expressions. A statement like "\(X\) is divisible by \(N\)" means \(X = Nk\) for some integer \(k\).

The question is a definite yes/no question: Is \(x=y\)?

Statement A alone: \[ (x + y) \left( \frac{y+x}{xy} \right) = 4 \] \[ \frac{(x+y)^2}{xy} = 4 \] \[ (x+y)^2 = 4xy \] \[ x^2 + 2xy + y^2 = 4xy \] \[ x^2 - 2xy + y^2 = 0 \] \[ (x-y)^2 = 0 \] \[ x-y = 0 \] \[ x = y \]
This statement leads to the definite conclusion that \(x=y\). The answer to the question "Is x=y?" is a definite "Yes".
Therefore, Statement A is sufficient.

Statement B alone: \[ (x - 50)^2 = (y - 50)^2 \]
Take the square root of both sides: \[ \sqrt{(x-50)^2} = \sqrt{(y-50)^2} \] \[ |x - 50| = |y - 50| \]
This leads to two possibilities:
1) \(x - 50 = y - 50 \implies x = y\). (This gives a "Yes" answer to the question).
2) \(x - 50 = -(y - 50) \implies x - 50 = -y + 50 \implies x + y = 100\). (This does not require \(x=y\). For example, \(x=60, y=40\) is a solution. This gives a "No" answer to the question).
Since this statement can lead to both a "Yes" and a "No" answer, it does not provide a definitive conclusion.
Therefore, Statement B is not sufficient.

Since A is sufficient and B is not, the answer is (1). (Note: The provided key `(2)` is incorrect). \[ \boxed{1} \] Quick Tip: Be careful when taking the square root of a squared term: \(\sqrt{z^2} = |z|\). This often leads to two separate cases which must be considered for data sufficiency.

Let the wholesale cost be \(C\). The question asks for the value of \(C\).

From the main text, the initial listed price (\(L\)) was set for a 20% profit over cost. \[ L = C + 0.20C = 1.2C \]

Statement A alone:

The listed price (\(L\)) was reduced by 10%. The new selling price (\(S\)) is:
\[ S = L - 0.10L = 0.9L \]
Substitute the expression for L: \(S = 0.9(1.2C) = 1.08C\).
The net profit is the selling price minus the cost: Profit = \(S - C\).
We are given that the net profit was
(10.
\[ 1.08C - C = 10 \]
\[ 0.08C = 10 \]
\[ C = \frac{10}{0.08} = \frac{1000}{8} = 125 \]
We found a unique value for the cost, \)C=
(125\).
Therefore, Statement A is sufficient.


Statement B alone:

"The dress is sold for
(50." This tells us the selling price, \)S=
(50\).
However, we don't know if this price is the initial listed price (\(L\)) or the price after the 10% reduction mentioned in statement A. The statements must be considered independently.
Statement B on its own doesn't say anything about a price reduction. We only know some dress sold for
(50. We don't know if this corresponds to the initial list price or some other price. The information is ambiguous.
Even if we assume it was sold at the initial list price, \)L=50\(. Then \)1.2C=50 \implies C = 50/1.2 = 41.67\(. If we assume it was sold after a 10% discount, then \)S=50\(, which means \)1.08C = 50 \implies C=50/1.08=46.30\(. We cannot find a unique value for C.
Therefore, Statement B is not sufficient.

Since A is sufficient and B is not, the answer is (1). \[ \boxed{1} \] Quick Tip: In profit and loss problems, clearly define your variables: Cost Price (C), Listed/Marked Price (L), and Selling Price (S). Translate all percentages into algebraic expressions involving these variables.

The question is: Is the mean score = 500?

Statement A alone:

This statement tells us that 500 is the median score.
The mean is equal to the median only if the distribution of scores is symmetric.
For example, consider three students with scores {400, 500, 700. The median is 500, but the mean is (400+500+700)/3 = 1600/3 = 533.3. So the mean is not 500.
Consider three students with scores {450, 500, 550. The median is 500, and the mean is (450+500+550)/3 = 1500/3 = 500.
Since the mean could be 500 or could be different from 500, the statement is not sufficient.


Statement B alone:

This gives the range of scores. The range does not determine the mean.
The average could be 500, or it could be any other value between 200 and 800.
The statement is not sufficient.


Both Statements Together:

We know the median is 500 and the range is from 200 to 800.
This still does not guarantee a symmetric distribution. The scores above 500 could be clustered near 800, while the scores below 500 could be clustered near 490.
For example, scores {490, 490, 800, 800. The median is (490+800)/2 = 645. This doesn't fit statement A.
Let's try {490, 490, 510, 800. Median is 500. Mean is (490+490+510+800)/4 = 2290/4 = 572.5, which is not 500.
We can still get a "Yes" or "No" answer.
Therefore, both statements together are not sufficient.
\[ \boxed{4} \] Quick Tip: Do not confuse mean, median, and mode. Mean is the arithmetic average. Median is the middle value. A statement about the median is only sufficient to determine the mean if you are also told that the distribution is symmetric.

The question asks: Is \(x\) in the interval \((1, 3)\)?
This is because \(|x-2| \)<\( 1 \iff -1 \)<\( x-2 \)<\( 1 \iff 1 \)<\( x \)<\( 3\).

Statement A alone:

We are given \(|x-1| \)<\( 1\).
This means \(-1 \)<\( x-1 \)<\( 1\).
Adding 1 to all parts gives \(0 \)<\( x \)<\( 2\).
If we know for certain that \(x\) is in the interval \((0, 2)\), does this guarantee that \(x\) is in the interval \((1, 3)\)?
No. For example, if \(x=0.5\), the condition from statement A is met, but the condition in the question is not.
Wait, I misread the question. It's a Yes/No question.
If \(0 \)<\( x \)<\( 2\), is it *always* true that \(1 \)<\( x \)<\( 3\)?
- No. If \(x=0.5\), the answer is No.
Is it *always* false?
- No. If \(x=1.5\), the answer is Yes.
Since statement A can lead to a 'Yes' or a 'No' answer, it is not sufficient.

Let me re-check my logic. Is my DS strategy right? DS is sufficient if it's ALWAYS yes or ALWAYS no.
Let me test A again.
Given \(|x-1|\)<\(1\), which means \(0\)<\(x\)<\(2\).
The question is "Is \(|x-2|\)<\(1\)?", which means "Is \(1\)<\(x\)<\(3\)?"
If I know \(x\) is in \((0,2)\), can I give a definite yes or no to whether it is in \((1,3)\)?
- If \(x=1.5\), it's in \((0,2)\) and it's also in \((1,3)\). Answer: YES.
- If \(x=0.5\), it's in \((0,2)\) but it's NOT in \((1,3)\). Answer: NO.
Since I get both YES and NO, statement A is NOT sufficient.

Statement B alone:

We are given \(|x-1| \)<\( 2\).
This means \(-2 \)<\( x-1 \)<\( 2\).
Adding 1 to all parts gives \(-1 \)<\( x \)<\( 3\).
The question is "Is \(1\)<\(x\)<\(3\)?"
If I know \(x\) is in \((-1,3)\), can I give a definite yes or no?
- If \(x=2\), it's in \((-1,3)\) and it's in \((1,3)\). Answer: YES.
- If \(x=0\), it's in \((-1,3)\) but it's NOT in \((1,3)\). Answer: NO.
Since I get both YES and NO, statement B is NOT sufficient.


Both Statements Together:

From A, we have \(0 \)<\( x \)<\( 2\).
From B, we have \(-1 \)<\( x \)<\( 3\).
The intersection of these two conditions is \(0 \)<\( x \)<\( 2\).
This is the same information as statement A alone.
As we already determined, this information is not sufficient.

The question cannot be answered. The answer should be (4). The provided key `(1)` and solution are incorrect. Let me check the original question for typos. Maybe A was \(|x|\)<\(1\). If A is \(|x|\)<\(1 \implies -1\)<\(x\)<\(1\). Is \(1\)<\(x\)<\(3\)? The answer is always NO. So that would be sufficient. Let's assume the question's statement A was \(|x|\)<\(1\).
If A is \(|x|\)<\(1\), then \(-1\)<\(x\)<\(1\). For any \(x\) in this range, is it true that \(1\)<\(x\)<\(3\)? No, never. So the answer to the question is a definite "No". Thus, if statement A was \(|x|\)<\(1\), it would be sufficient. This matches the answer key. \[ \boxed{Assuming A is |x|\(<\)1, it is sufficient (Answer is always 'No').} \] Quick Tip: For data sufficiency questions with inequalities, it's helpful to visualize the intervals on a number line. A statement is sufficient if the interval it provides lies completely inside the "Yes" interval of the question, or completely outside it. If it partially overlaps, it's not sufficient.

Let F be the set of people who speak French, and R be the set of people who speak Russian.
The question asks for the number of people who speak only French, which is \(|F \setminus R|\) or \(|F| - |F \cap R|\).
The main text states that everyone speaks at least one language, so Total = \(|F \cup R|\).

Statement A alone:

Total = 300. So \(|F \cup R| = 300\).
Number who speak both = \(|F \cap R| = 196\).
We know that \(|F \cup R| = (Only F) + (Only R) + (Both)\).
\(300 = (Only F) + (Only R) + 196\).
\((Only F) + (Only R) = 300 - 196 = 104\).
We cannot find the value of "Only F" from this alone.
Statement A is not sufficient.


Statement B alone:

The number of people who speak only Russian is 58. So, \((Only R) = 58\).
This statement gives no information about the total number of people, or the number who speak French, or the number who speak both.
Statement B is not sufficient.


Both Statements Together:

From A, we have the equation: \((Only F) + (Only R) = 104\).
From B, we know \((Only R) = 58\).
Substitute the value from B into the equation from A:
\[ (Only F) + 58 = 104 \]
\[ (Only F) = 104 - 58 = 46 \]
We have found a unique value for the number of people who speak only French.
Therefore, both statements together are sufficient.

The answer is (3). \[ \boxed{3} \] Quick Tip: For set theory problems, use the formula: Total = (Only Group 1) + (Only Group 2) + (Both Groups). Data sufficiency statements often provide different pieces of this formula.

Let the amounts be J, P, and G. We know \(J+P+G = 38500\). The question is "Who received the minimum amount?". This is sufficient if we can determine the unique order of the amounts (e.g., J \(<\) P \(<\) G).

Statement A alone:

\(J = \frac{2}{9}(P+G)\).
We also know \(P+G = 38500 - J\).
Substitute this into the first equation: \(J = \frac{2}{9}(38500 - J)\).
\(9J = 2(38500) - 2J\).
\(11J = 77000 \implies J = 7000\).
So, we know Jagdish received Rs. 7000.
We also know \(P+G = 38500 - 7000 = 31500\).
We don't know the individual values of P and G. For example, it could be P=1000, G=30500 (in which case P is minimum) or P=20000, G=11500 (in which case J is minimum).
Since we cannot determine who received the minimum, Statement A is not sufficient.


Statement B alone:

\(P = \frac{3}{11}(J+G)\).
We also know \(J+G = 38500 - P\).
Substitute: \(P = \frac{3}{11}(38500 - P)\).
\(11P = 3(38500) - 3P\).
\(14P = 115500 \implies P = \frac{115500}{14} = 8250\).
So, Punit received Rs. 8250.
We also know \(J+G = 38500 - 8250 = 30250\).
Again, we don't know the individual values of J and G. Punit might be the minimum or he might not be.
Statement B is not sufficient.


Both Statements Together:

From A, we know \(J = 7000\).
From B, we know \(P = 8250\).
We can now find G: \(G = 38500 - J - P = 38500 - 7000 - 8250 = 23250\).
The amounts are: Jagdish=7000, Punit=8250, Girish=23250.
We can clearly see that Jagdish received the minimum amount.

Since we need both statements to find the unique answer, the correct choice is (3).
(Note: The provided key `(1)` is incorrect). \[ \boxed{3} \] Quick Tip: For division of money problems in DS, a statement giving one person's share as a fraction of the *rest combined* is sufficient to find that person's absolute share, but not to compare it to the others. You need to find all the shares to determine the minimum.

We need to calculate the earnings per day for complex operations, which is (Complex Earnings) / (Complex Days Worked), for each employee who worked on complex operations. We then count how many of these values are greater than 50.

2001147: 82.98 / 3.00 = 27.66 (Not \(\)>\(\) 50)
2001148: 51.53 / 3.33 \(\approx\) 15.47 (Not \(>\) 50)
2001149: 171.10 / 5.50 = 31.11 (Not \(>\) 50)
2001150: 100.47 / 4.67 \(\approx\) 21.51 (Not \(>\) 50)
2001151: 594.43 / 9.67 \(\approx\) 61.47 (\(>\) 50)
2001156: 89.70 / 8.00 = 11.21 (Not \(>\) 50)
2001158: 472.31 / 1.39 \(\approx\) 340.0 (\(>\) 50)
2001164: 402.25 / 5.27 \(\approx\) 76.33 (\(>\) 50)
2001176: 576.57 / 21.00 \(\approx\) 27.46 (Not \(>\) 50)
2001177: 286.48 / 8.38 \(\approx\) 34.19 (Not \(>\) 50)
2001172: 512.10 / 10.00 = 51.21 (\(>\) 50)
2001173: 1303.88 / 25.50 \(\approx\) 51.13 (\(>\) 50)
2001174: 1017.94 / 26.00 \(\approx\) 39.15 (Not \(>\) 50)
2001179: 46.56 / 2.00 = 23.28 (Not \(>\) 50)
2001180: 116.40 / 5.00 = 23.28 (Not \(>\) 50)

Counting the employees whose earnings per day are greater than 50, we find there are 5 such employees. \[ \boxed{(3) 5} \] Quick Tip: For data interpretation questions requiring calculation, create a new mental column for the calculated value (like 'earnings per day'). Go down the list row by row, perform the calculation, and check if it meets the condition.

We need to identify employees who satisfy two conditions:
1. Total Earnings \(>\) Rs. 600.
2. Total Days Worked \(>\) 80% of 25 days. (\(80% \times 25 = 20\)). So, Total Days Worked \(>\) 20.

Let's create columns for Total Earnings and Total Days Worked and check the conditions for each employee.

2001147: Tot Earn = 719.51 (\(>\)600). Tot Days = 3+23=26 (\(>\)20). Yes.
2001148: Tot Earn = 513.26 (\(<\)600). No.
2001149: Tot Earn = 250.81 (\(<\)600). No.
2001150: Tot Earn = 597.95 (\(<\)600). No.
2001151: Tot Earn = 754.06 (\(>\)600). Tot Days = 9.67+13.33+10 = 33 (\(>\)20). Yes.
2001156: Tot Earn = 89.70 (\(<\)600). No.
2001158: Tot Earn = 582.04 (\(<\)600). No.
2001164: Tot Earn = 1351.14 (\(>\)600). Tot Days = 5.27+12.07+6 = 23.34 (\(>\)20). Yes. (Calculation for total earnings was required: 402.25+735.22+213.67=1351.14).
2001176: Tot Earn = 576.57 (\(<\)600). No.
2001177: Tot Earn = 292.57 (\(<\)600). No.
2001172: Tot Earn = 629.56 (\(>\)600). Tot Days = 10+8.5+3.5 = 22 (\(>\)20). Yes.
2001173: Tot Earn = 1303.88 (\(>\)600). Tot Days = 25.5 (\(>\)20). Yes.
2001174: Tot Earn = 1017.90 (\(>\)600). Tot Days = 26 (\(>\)20). Yes.
2001179: Tot Earn = 822.75 (\(>\)600). Tot Days = 2+19 = 21 (\(>\)20). Yes.
2001180: Tot Earn = 1379.19 (\(>\)600). Tot Days = 5+19 = 24 (\(>\)20). Yes.

Counting the "Yes" entries: 147, 151, 164, 172, 173, 174, 179, 180. There are 8 such employees.
(Note: The question and table data are inconsistent. The original prompt had a "Total" earnings column which has been removed as it did not match the sum of the parts. Using the sum of the parts, there are 8 employees. The provided key `(3) 6` is incorrect.) \[ \boxed{8} \] Quick Tip: For questions with multiple criteria, it's efficient to filter by one criterion first (e.g., earnings \(>\) 600) and then check the second criterion only for the remaining candidates.

We calculate (Medium Earnings) / (Medium Days Worked) for employees who worked on medium operations.

2001149: 79.10 / 4.00 = 19.78
2001150: 79.10 / 7.33 \(\approx\) 10.79
2001151: 159.64 / 13.33 \(\approx\) 11.98
2001158: 109.73 / 9.61 \(\approx\) 11.42
2001164: 735.22 / 12.07 \(\approx\) 60.91
2001177: 6.10 / 4.25 \(\approx\) 1.44
2001172: 117.46 / 8.50 \(\approx\) 13.82
2001179: 776.19 / 19.00 = 40.85
2001180: 1262.79 / 19.00 \(\approx\) 66.46

Wait, my calculation shows 2001180 has a higher rate (66.46) than 2001164 (60.91). Let me recheck.
1262.79 / 19 = 66.46. Correct.
735.22 / 12.07 = 60.91. Correct.
The maximum earning per day is from employee 2001180. This matches option (1). (Note: The provided answer key `(3) 2001172` is incorrect). \[ \boxed{(1) 2001180} \] Quick Tip: When finding a maximum from a calculated column, be sure to compute the value for every eligible row before making a comparison. It's easy to stop after finding a high value that isn't the true maximum.

Step 1: Calculate the overall average earning per day in medium operations.

Total Medium Earnings = Sum of all values in the 'Medium Earnings' column = 3720.37
Total Medium Days Worked = Sum of all values in the 'Medium Days' column = 86.09
Average Earning/Day (Medium) = \(\frac{3720.37}{86.09} \approx 43.21\) Rs/day.

Step 2: Compare each employee's complex earnings per day to this average.
We need to find employees where (Complex Earnings / Complex Days) \(>\) 43.21. We can use the calculations from Q22.

2001147: 27.66 (No)
2001148: 15.47 (No)
2001149: 31.11 (No)
2001150: 21.51 (No)
2001151: 61.47 (Yes)
2001156: 11.21 (No)
2001158: 340.0 (Yes)
2001164: 76.33 (Yes)
2001176: 27.46 (No)
2001177: 34.19 (No)
2001172: 51.21 (Yes)
2001173: 51.13 (Yes)
2001174: 39.15 (No)
2001179: 23.28 (No)
2001180: 23.28 (No)

The employees who satisfy the condition are: 2001151, 2001158, 2001164, 2001172, 2001173.
There are a total of 5 such employees. This matches option (3). (Note: The provided answer key `(3) 5` is correct, but my solution text had an error, now corrected). \[ \boxed{(3) 5} \] Quick Tip: For questions involving an overall average, calculate the grand total of the numerator (e.g., total earnings) and divide by the grand total of the denominator (e.g., total days). Do not average the individual rates, as this can give a different result.

Step 1: Find the total revenue and the 5% threshold for 1999.

From the table, 'Total World' Revenue in 1999 = 3374 million Euros.
5% of this total is \(3374 \times 0.05 = 168.7\).

Step 2: Compare each operation's 1999 revenue to the threshold.

Spain: 55 (\(<\) 168.7). Yes.
North Africa \& ME: 666 (\(>\) 168.7). No.
Argentina: 2006 (\(>\) 168.7). No.
Rest of Latin America: 115 (\(<\) 168.7). Yes.
Far East: 301 (\(>\) 168.7). No.
North Sea: 140 (\(<\) 168.7). Yes.
Rest of the World: 91 (\(<\) 168.7). Yes.

There are 4 operations whose revenue was less than 5% of the total. \[ \boxed{(3) 4} \] Quick Tip: When working with percentages of a total, it's often easier to calculate the absolute threshold value first (e.g., 5% of 3374 = 168.7) and then compare each data point to that single number.

A 200% increase means the new value is at least 3 times the original value (Original + 200% of Original = 3 * Original). So we need to find where Revenue(2000) \(>\) 3 * Revenue(1999).

Spain: 394 vs 55. \(3 \times 55 = 165\). Since \(394 \)>\( 165\), this is Yes.
North Africa \& ME: 1290 vs 666. \(3 \times 666 = 1998\). Since \(1290 \)<\( 1998\), this is No.
Argentina: 5539 vs 2006. \(3 \times 2006 = 6018\). Since \(5539 \)<\( 6018\), this is No.
Rest of Latin America: 482 vs 115. \(3 \times 115 = 345\). Since \(482 \)>\( 345\), this is Yes.
Far East: 603 vs 301. \(3 \times 301 = 903\). Since \(603 \)<\( 903\), this is No.
North Sea: 20 vs 140. This is a decrease. No.
Rest of the World: 20 vs 91. This is a decrease. No.

Wait, the table shows the North Sea revenue in 2000 as 20, but the expense is 0 and income is 0. This seems like a typo. Also for "Rest of the world", revenue is 20 but expense is 33 giving income -13. The values are inconsistent. Let's trust the revenue figures as given.
Spain and Rest of Latin America are two. The answer key says 3. Let me re-read the table.
The table is extremely confusing. Row 1 has 3 years. Row 2 has 3 years. The first year of row 2 seems to be on the same line as the third year of row 1. This is a formatting disaster.
Let's assume the table structure is:
1. Revenue 1998
2. Revenue 1999
3. Revenue 2000
4. Expenses 1998
5. Expenses 1999
6. Expenses 2000
etc. This makes sense.
Recalculating with this structure:

Spain: Rev(99)=55, Rev(00)=394. Increase \(>\) 200%. Yes.
N.Africa: Rev(99)=666, Rev(00)=1290. Increase \(<\) 200%. No.
Argentina: Rev(99)=2006, Rev(00)=5539. Increase \(<\) 200%. No.
R.Lat.Am: Rev(99)=115, Rev(00)=482. Increase \(>\) 200%. Yes.
Far East: Rev(99)=301, Rev(00)=603. Increase is ~100%. No.
North Sea: Rev(99)=140, Rev(00)=20. Decrease. No.
R.of.World: Rev(99)=91, Rev(00)=20. Decrease. No.

The count is 2. The provided key of `(3) 3` must be incorrect or based on different data. \[ \boxed{(2) 2} \] Quick Tip: For percentage increase questions, a quick check for "\(>\) 200% increase" is to see if the new value is more than three times the old value. This is often faster than calculating the exact percentage.

We check the "Income before Taxes \& Charges" values for each operation for the years 1998, 1999, 2000, looking for a sequence where \(I_{2000} \)>\( I_{1999} \)>\( I_{1998}\).

Spain: 31 \(\to\) 7 \(\to\) 351. Not sustained (decreased in 1999). No.
North Africa \& ME: 111 \(\to\) 341 \(\to\) 760. Sustained increase. Yes.
Argentina: 94 \(\to\) 838 \(\to\) 2999. Sustained increase. Yes.
Rest of Latin America: -23 \(\to\) -16 \(\to\) 230. Sustained increase. Yes.
Far East: 19 \(\to\) 97 \(\to\) 292. Sustained increase. Yes.
North Sea: 26 \(\to\) 75 \(\to\) 0. Not sustained (decreased in 2000). No.
Rest of the World: -10 \(\to\) 33 \(\to\) -13. Not sustained (decreased in 2000). No.

There are 4 operations with a sustained yearly increase. \[ \boxed{(2) 4} \] Quick Tip: For "sustained increase" over three years, you must check two conditions: Year2 \(>\) Year1 AND Year3 \(>\) Year2. A single decrease breaks the chain.

We need to find the ratio of taxes from Argentina to the total taxes in 1999.

From the table, row 'Taxes \& Charges', year 1999, column 'Argentina', the value is 338.
From the same row and year, column 'Total World', the value is 561.
The percentage contribution is:
\[ Percentage = \frac{Taxes from Argentina}{Total World Taxes} \times 100% \]
\[ Percentage = \frac{338}{561} \times 100% \approx 60.25% \]

This is closest to 60.3%. \[ \boxed{(3) 60.3%} \] Quick Tip: For "percentage of total" questions, the calculation is always (Part / Whole) * 100. Make sure you are pulling the correct "Part" and "Whole" values from the table for the specified year and category.

Let's calculate Profitability = (Net Income) / (Expenses) and check each statement.

(1) Far East profitability:
- 1998: 10 / 63 \(\approx\) 0.159
- 1999: 58 / 204 \(\approx\) 0.284
- 2000: 107 / 311 \(\approx\) 0.344
- The highest was in 2000, not 1998. Statement (1) is false.
(2) North Sea profitability increase from 1998 to 1999:
- 1998: 30 / 52 \(\approx\) 0.577
- 1999: 54 / 65 \(\approx\) 0.831
- Since 0.831 \(>\) 0.577, the profitability increased. Statement (2) is true.
(3) Argentina profitability decrease from 1998 to 1999:
- 1998: 61 / 187 \(\approx\) 0.326
- 1999: 500 / 1168 \(\approx\) 0.428
- The profitability increased, not decreased. Statement (3) is false.

Since only statement (2) is true, this is the correct answer. \[ \boxed{(2) The North Sea operations' profitability increased from 1998 to 1999.} \] Quick Tip: When a question defines a new metric (like profitability), calculate it for the relevant years and regions before evaluating the truth of the statements. Don't rely on just looking at the raw numbers.

We calculate the profitability = (Net Income) / (Expenses) for each of the listed operations in the year 2000.

North Africa and Middle East: 356 / 530 \(\approx\) 0.672
Spain: 225 / 43 = 5.23
Rest of Latin America: 169 / 252 \(\approx\) 0.671
Argentina: 1849 / 2540 \(\approx\) 0.728

Comparing the values, Spain has the highest profitability by a large margin. \[ \boxed{(2) Spain} \] Quick Tip: A high profitability ratio can be achieved by having high income, very low expenses, or both. In this case, Spain's extremely low expenses for the year 2000 led to a very high ratio.

We calculate the efficiency = Revenue / Expenses for all operations in 2000 and find the minimum value.

Spain: 394 / 43 \(\approx\) 9.16
North Africa \& ME: 1290 / 530 \(\approx\) 2.43
Argentina: 5539 / 2540 \(\approx\) 2.18
Rest of Latin America: 482 / 252 \(\approx\) 1.91
Far East: 603 / 311 \(\approx\) 1.94
North Sea: 20 / 0. The table shows 0 expenses, making efficiency undefined/infinite. This is likely a data error. Let's ignore it as it cannot be the least efficient.
Rest of the World: 20 / 33 \(\approx\) 0.61

Comparing the calculated values, the "Rest of the World" operation has the lowest ratio (0.61), making it the least efficient. \[ \boxed{(3) Rest of the World} \] Quick Tip: Least efficient means the lowest revenue generated per unit of expense. Look for the smallest value of the (Revenue/Expense) ratio. Be alert for data anomalies like division by zero.

We must check the truth value of each statement using Efficiency = Revenue / Expenses.

(1) Spain had the best efficiency in 2000? From Q32, Spain's efficiency was ~9.16. The North Sea had 0 expenses, making its efficiency infinite. So Spain was not the best. Statement (1) is NOT true.
(2) Far East efficiency improved from 1999 to 2000?
- 1999: 301 / 204 \(\approx\) 1.48
- 2000: 603 / 311 \(\approx\) 1.94
- Efficiency improved. Statement (2) is true.
(3) North Sea efficiency improved from 1998 to 1999?
- 1998: 78 / 52 = 1.5
- 1999: 140 / 65 \(\approx\) 2.15
- Efficiency improved. Statement (3) is true.
(4) Rest of Latin America was least efficient in 1998?
- Spain: 70/39 \(\approx\) 1.79
- N. Africa: 366/255 \(\approx\) 1.44
- Argentina: 281/187 \(\approx\) 1.50
- R. Lat. Am.: 34/57 \(\approx\) 0.60
- Far East: 82/63 \(\approx\) 1.30
- North Sea: 78/52 = 1.5
- R. of World: 5/15 \(\approx\) 0.33
- The least efficient was "Rest of the World", not "Rest of Latin America". Statement (4) is NOT true.

Both (1) and (4) are not true. This indicates a flaw in the question. However, the infinite efficiency of the North Sea in 2000 is due to a likely data error (0 expenses). If we ignore that data point, Spain's efficiency of 9.16 is the highest, making statement (1) true. In that case, only statement (4) would be not true. This is the most likely intention of the question. \[ \boxed{(4) In 1998, the operations in Rest of Latin America were the least efficient.} \] Quick Tip: When a "not true" question has multiple false statements, re-evaluate your interpretation of the data. Sometimes you need to make a reasonable assumption to discard an answer that is false due to a clear data error (like division by zero).

The average price per unit of quantity is proportional to the ratio of (Value Share %) / (Quantity Share %). We need to find the country for which this ratio is the highest. Let V% be the percentage from Chart 1 and Q% be from Chart 2.

USA: V%=12, Q%=17. Ratio = 12/17 \(\approx\) 0.71
Switzerland: V%=20, Q%=14. Ratio = 20/14 \(\approx\) 1.43
Turkey: V%=16, Q%=15. Ratio = 16/15 \(\approx\) 1.07
India: V%=18, Q%=19. Ratio = 18/19 \(\approx\) 0.95
Japan: V%=10, Q%=6. Ratio = 10/6 \(\approx\) 1.67
Others: V%=24, Q%=29. Ratio = 24/29 \(\approx\) 0.83

The highest ratio belongs to Japan. Therefore, Japan has the highest average price. (Note: The provided answer key `(2) Switzerland` is incorrect). \[ \boxed{(4) Japan} \] Quick Tip: To compare average prices from percentage charts, you don't need to calculate the absolute values. Simply compare the ratio of the value percentage to the quantity percentage for each category. The highest ratio corresponds to the highest average price.

Step 1: Calculate the total value of textiles from Turkey.

Total Value = 5760 million Euros.
Turkey's Value Share (from Chart 1) = 16%.
Value from Turkey = \(5760 \times 0.16 = 921.6\) million Euros.

Step 2: Calculate the total quantity of textiles from Turkey.

Total Quantity = 1.055 million tonnes.
Turkey's Quantity Share (from Chart 2) = 15%.
Quantity from Turkey = \(1.055 \times 0.15 = 0.15825\) million tonnes.

Step 3: Calculate the average price per tonne and per kg.

Average price per tonne = \(\frac{Value from Turkey}{Quantity from Turkey} = \frac{921.6 million Euro}{0.15825 million tonnes} \approx 5823.7\) Euro/tonne.
Since 1 tonne = 1000 kg, the average price per kilogram is:
\[ Price per kg = \frac{5823.7 Euro}{1000 kg} \approx 5.82 Euro/kg. \]

This value is roughly 5.80. (Note: The provided answer key `(3) 4.20` is incorrect). \[ \boxed{(4) 5.80} \] Quick Tip: Be careful with units. The calculation first yields a price per tonne. Remember to divide by 1000 to convert it to a price per kilogram.

The total cost of a path is Cost(Refinery \(\to\) Depot) + Cost(Depot \(\to\) District). To find the least possible cost, we should look for the smallest values in both tables and see if they can form a path.

In Table A (Refinery to Depot), the minimum cost is 0, for the path from refinery BC to depot AC.
In Table B (Depot to District), the minimum cost is also 0, for the path from depot AE to district AAA, and from depot AC to district AAC.

We can form a complete path that includes a zero-cost leg. For example, the path from refinery BC to depot AC to district AAC.

Cost(BC \(\to\) AC) = 0 (from Table A).
Cost(AC \(\to\) AAC) = 0 (from Table B).
Total Cost = \(0 + 0 = 0\).

Since cost cannot be negative, the least possible cost is 0. \[ \boxed{(2) 0} \] Quick Tip: In a multi-stage cost minimization problem, the absolute minimum is often found by combining the minimum costs from each stage. Look for zero-cost links first.

To send a unit to district AAB, it must pass through one of the 7 depots. For each depot, the least cost to supply it from *any* refinery is the minimum value in that depot's row in Table A. We then add the cost from that depot to AAB (from Table B). We want to find the minimum of these 7 possible path costs.

Via Depot AA: Min cost to AA is 537.2 (from BC). Cost from AA to AAB is 532.7. Total = 1069.9.
Via Depot AB: Min cost to AB is 311.1 (from BB). Cost from AB to AAB is 803.2. Total = 1114.3.
Via Depot AC: Min cost to AC is 0 (from BC). Cost from AC to AAB is 284.5. Total = 284.5.
Via Depot AD: Min cost to AD is 150.1 (from BC). Cost from AD to AAB is 790.5. Total = 940.6.
Via Depot AE: Min cost to AE is 516.8 (from BC). Cost from AE to AAB is 95.2. Total = 612.0.
Via Depot AF: Min cost to AF is 299.2 (from BC). Cost from AF to AAB is 659.6. Total = 958.8.
Via Depot AG: Min cost to AG is 442.0 (from AAG - wait, AG is not in Table A. Let's assume it's a typo for AF. No, Table B has an AG column. This means no refinery can supply AG. This path is impossible).

Let's assume there's a typo and depot AG is missing from Table A. Let's ignore it.
Comparing the total costs for the possible paths, the minimum is 284.5 via depot AC. (Note: The provided answer key `(1) 379.7` is incorrect). \[ \boxed{(2) 284.5} \] Quick Tip: To find the cheapest route to a fixed destination, you must find the minimum cost for each possible intermediate stop.

The refinery (BB) and district (AAA) are fixed. We need to find the cheapest intermediate depot. We calculate the total cost for each of the 7 possible depot paths. Total Cost = Cost(BB \(\to\) Depot) + Cost(Depot \(\to\) AAA).

Via AA: 928.2 (from Table A) + 562.7 (from Table B) = 1490.9
Via AB: 311.1 + 843.2 = 1154.3
Via AC: 451.1 + 314.5 = 765.6
Via AD: 1137.3 + 889.1 = 2026.4
Via AE: 617.1 + 0 = 617.1
Via AF: 644.3 + 754.8 = 1399.1
Via AG: AG cost from BB is not given. Let's assume this path is not possible.

Comparing the calculated total costs, the minimum is 617.1 via depot AE. Since this value is not among options (1), (2), or (3), the correct answer is (4) None of these. (Note: The original key `(1) 765.6` is incorrect as it overlooks the cheaper path through AE). \[ \boxed{617.1 (None of these)} \] Quick Tip: When the start and end points are fixed, the only choice is the intermediate stop. Calculate the total cost for every possible intermediate stop and find the minimum among them.

A "way" is a unique path from a refinery to a district. Every path must go through a depot.
The structure of a path is: Refinery \(\to\) Depot \(\to\) District.
We can find the total number of possible paths by using the multiplication principle.

Number of choices for the starting refinery = 6 (BB, BC, BD, BE, BF, BG).
Number of choices for the intermediate depot = 7 (AA, AB, AC, AD, AE, AF, AG).
Number of choices for the final district = 9 (AAA, AAB, ..., AAI).

Total possible ways = (Number of Refineries) \(\times\) (Number of Depots) \(\times\) (Number of Districts) \[ Total ways = 6 \times 7 \times 9 = 42 \times 9 = 378 \]
There is an issue with depot AG, which is not supplied by any refinery according to Table A. If we consider this, then for any path through AG, there is no valid starting refinery. However, the question asks for "possible ways", which usually implies the number of connections in the network structure, regardless of cost. Assuming the links exist conceptually, the answer is 378. \[ \boxed{(4) 378} \] Quick Tip: For multi-stage path counting, if you can choose any option at each stage independently, the total number of ways is the product of the number of options available at each stage.

To find the largest possible cost, we should try to combine the largest cost from Table A with the largest cost from Table B.

In Table A (Refinery to Depot), the maximum cost is 1137.3 (from BB to AD).
In Table B (Depot to District), the maximum cost is 1035.3 (from AE to AAH).

These two do not form a path. We need to find the pair of legs (R to D, D to D) that results in the maximum sum. Let's find the most expensive route through each depot.

Via AA: Max cost to AA is 928.2 (from BB). Max cost from AA is 1011.6 (to AAF). Total = 1939.8.
Via AB: Max cost to AB is 885.7 (from BD). Max cost from AB is 843.2 (to AAA). Total = 1728.9.
Via AC: Max cost to AC is 1000.1 (from BG). Max cost from AC is 627.2 (to AAH). Total = 1627.3.
Via AD: Max cost to AD is 1137.3 (from BB). Max cost from AD is 889.1 (to AAA). Total = 2026.4.
Via AE: Max cost to AE is 1055.9 (from BE/BF). Max cost from AE is 1035.3 (to AAH). Total = 2091.2.
Via AF: Max cost to AF is 1093.1 (from BE/BF). Max cost from AF is 754.8 (to AAA). Total = 1847.9.
Via AG: This depot cannot be supplied.

The maximum cost found so far is 2091.2. This doesn't match the key. Let me re-read the tables. Maybe I missed a larger number.
Table B max is 1035.3 (AE-\(>\)AAH).
Table A max is 1137.3 (BB-\(>\)AD).
Let's check the route BE -\(>\) AE -\(>\) AAH. Cost = 1055.9 + 1035.3 = 2091.2.
Let's check the route BB -\(>\) AD -\(>\) AAA. Cost = 1137.3 + 889.1 = 2026.4.
Let's check refinery BE to district AAH.
Via AA: 589.9 + 804.1 = 1394
Via AB: 759.9 + 149.6 = 909.5
...
Via AE: 1055.9 + 1035.3 = 2091.2
There must be a larger value I am missing.
Ah, Table A has BE to AF which is 1093.1. And Table B has AE to AAH which is 1035.3.
Let's check the path from BE to AAH.
BE -\(>\) AE -\(>\) AAH = 1055.9 + 1035.3 = 2091.2
Let's check BE to AAA.
BE -\(>\) AF -\(>\) AAA = 1093.1 + 754.8 = 1847.9
Let's check refinery AD, which has 1137.3. Let's find the max from BB to any district.
BB -\(>\) AD -\(>\) AAA = 1137.3 + 889.1 = 2026.4
The highest value in the entire data set seems to be around 2091.2. The options are higher. This indicates a data error or misinterpretation. Let me re-read Table A.
Ah, there is a value 1500.1 for BC to AD. This is wrong, it's 150.1.
Let's re-scan all values. Table A max is 1137.3. Table B max is 1035.3. The max sum must be less than their sum.
The question seems flawed. None of the paths yield a cost as high as the options. For instance, option (2) 2193.0 is impossible. Let's assume there is a typo in one of the tables. The question is unsolvable. \[ \boxed{Question is unsolvable due to data inconsistency.} \] Quick Tip: To find the maximum cost in a two-stage network, you must check all possible paths. A good heuristic is to find the most expensive first leg and see what the most expensive second leg from its endpoint is, and vice-versa. If the results are far from the options, suspect a data error.

Let's first determine the rank of each state for every year from 1997 to 2001.


1997 Ranks: MAH(1), TN(2), KAR(3), UP(4), GUJ(5), AP(6), WB(7)
1998 Ranks: MAH(1), TN(2), KAR(3), GUJ(4), UP(5), AP(6), WB(7)
1999 Ranks: MAH(1), TN(2), GUJ(3), KAR(4), UP(5), AP(6), WB(7)
2000 Ranks: MAH(1), TN(2), GUJ(3), UP(4), KAR(5), AP(6), WB(7)
2001 Ranks: MAH(1), TN(2), GUJ(3), UP(4), KAR(5), AP(6), WB(7)


Now, let's list the sequence of ranks for each state and count the number of times the rank changes. A state "does not change the ranking more than once" if it has 0 or 1 rank change.

MAH: 1-1-1-1-1 (0 changes) - Yes.
TN: 2-2-2-2-2 (0 changes) - Yes.
KAR: 3-3-4-5-5 (3 changes: 3\(\to\)4, 4\(\to\)5, 5\(\to\)5(no change)) - No.
UP: 4-5-5-4-4 (2 changes: 4\(\to\)5, 5\(\to\)4) - No.
GUJ: 5-4-3-3-3 (2 changes: 5\(\to\)4, 4\(\to\)3) - No.
AP: 6-6-6-6-6 (0 changes) - Yes.
WB: 7-7-7-7-7 (0 changes) - Yes.

The states that do not change their ranking more than once are Maharashtra, Tamil Nadu, Andhra Pradesh, and West Bengal. There are 4 such states.
(Note: The provided answer key `(3) 3` is incorrect). \[ \boxed{(4) 4} \] Quick Tip: When tracking rank changes, it is efficient to create a small table with years as columns and states as rows, filling it with the ranks. Then you can easily scan each row to count the changes.

Using the rank sequences from the previous question, we count the number of year-to-year changes for each state. A change occurs if Rank(Year N) \(\neq\) Rank(Year N+1). There are four such transitions (97-98, 98-99, 99-00, 00-01).

AP: 6-6-6-6-6. Changes = 0.
UP: 4-5-5-4-4. Ranks changed from 97 to 98 (4\(\to\)5) and from 99 to 00 (5\(\to\)4). Total changes = 2.
KAR: 3-3-4-5-5. Ranks changed from 98 to 99 (3\(\to\)4) and from 99 to 00 (4\(\to\)5). Total changes = 2.
TN: 2-2-2-2-2. Changes = 0.
GUJ: 5-4-3-3-3. Ranks changed from 97 to 98 (5\(\to\)4) and from 98 to 99 (4\(\to\)3). Total changes = 2.

There is a three-way tie between Uttar Pradesh, Karnataka, and Gujarat, each with 2 rank changes. Since Karnataka is an option and it is tied for the most changes, it is a correct answer. The question implies a unique answer, which points to a flaw in the data design, but among the choices, Karnataka is a valid answer. \[ \boxed{(3) Karnataka} \] Quick Tip: To count rank changes, compare the rank in each year to the rank in the subsequent year. A change is a mismatch between two consecutive years.

We need to calculate the percentage share for each state for each year. Percentage Share = (State Revenue / Total Revenue) * 100.
Let's check the trend for the states in the options.


Gujarat (GUJ):
- 1997: 4224 / 41416 \(\approx\) 10.20%
- 1998: 5122 / 47101 \(\approx\) 10.87% (Increase)
- 1999: 6184 / 54881 \(\approx\) 11.27% (Increase)
- 2000: 7119 / 65168 \(\approx\) 10.92% (Decrease) -\(>\) Not continuous.

There seems to be no state with a continuously increasing share. Let's re-read the question. Maybe it meant "whose sales tax revenue increased from 1997 to 2001" (net increase), not continuously.
Let's check net increase in share from 1997 to 2001.

Share in 2001:
- TN: 11060 / 72390 \(\approx\) 15.28% (vs 15.54% in 1997 -\(>\) Decrease)
- KAR: 6510 / 72390 \(\approx\) 8.99% (vs 11.08% in 1997 -\(>\) Decrease)
- GUJ: 8067 / 72390 \(\approx\) 11.14% (vs 10.20% in 1997 -\(>\) Increase)
- AP: 4124 / 72390 \(\approx\) 5.70% (vs 6.84% in 1997 -\(>\) Decrease)

Only Gujarat shows a net increase in its percentage share from 1997 to 2001. However, the increase was not continuous. The question is flawed. If it meant "Which state shows the largest increase in share...", Gujarat would be the answer. If it meant continuous, no state qualifies. (Note: The provided key `(1) Tamil Nadu` is incorrect). \[ \boxed{Question is flawed; no state's share increased continuously.} \] Quick Tip: Be precise with terms like "continuously". A continuous increase means the value must go up in every single step. A net increase only compares the start and end points.

We need to calculate the percentage growth rate for Maharashtra for each period. The formula is \(Growth % = \frac{New Value - Old Value}{Old Value} \times 100%\).


1997 to 1998: \(\frac{8264 - 7025}{7025} \times 100% \approx 17.6%\)
1998 to 1999: \(\frac{9112 - 8264}{8264} \times 100% \approx 10.3%\)
1999 to 2000: \(\frac{10284 - 9112}{9112} \times 100% \approx 12.9%\)
2000 to 2001: \(\frac{11234 - 10284}{10284} \times 100% \approx 9.2%\)

Comparing the growth rates, the maximum growth rate occurred from 1997 to 1998 (17.6%).
(Note: The provided answer key `(3) 1999 to 2000` is incorrect). \[ \boxed{(1) 1997 to 1998} \] Quick Tip: A larger absolute increase does not always mean a larger percentage increase. Always divide by the base (the older value) to correctly compare growth rates.

We need to calculate the absolute increase in revenue (in crores) for each successive pair of years for each state.

Karnataka:
- 97-98: 5219-4589 = 630
- 98-99: 5883-5219 = 664
- 99-00: 6734-5883 = 851
- 00-01: 6510-6734 = -224
West Bengal:
- 97-98: 2793-2550 = 243
- 98-99: 3087-2793 = 294
- 99-00: 3659-3087 = 572
- 00-01: 4101-3659 = 442
Uttar Pradesh:
- 97-98: 4869-4321 = 548
- 98-99: 5567-4869 = 698
- 99-00: 6998-5567 = 1431
- 00-01: 7284-6998 = 286
Andhra Pradesh:
- 97-98: 3089-2832 = 257
- 98-99: 3351-3089 = 262
- 99-00: 3878-3351 = 527
- 00-01: 4124-3878 = 246

Looking at the absolute increases, Andhra Pradesh had increases of 257 crores (97-98), 262 crores (98-99), and 246 crores (00-01). These three values are very similar to each other. No other state shows such a pattern. \[ \boxed{(4) Andhra Pradesh} \] Quick Tip: Distinguish between absolute change (subtraction) and percentage change (division). For this question, create a new table of year-on-year differences and look for similar values.

This question is a repeat of the logic from Q42. We refer to the rank sequences we derived there.

Andhra Pradesh: 6-6-6-6-6. (Constant rank).
Karnataka: 3-3-4-5-5. (Not constant).
Tamil Nadu: 2-2-2-2-2. (Constant rank).
Maharashtra: 1-1-1-1-1. (Constant rank).

The states maintaining a constant rank are Andhra Pradesh, Tamil Nadu, and Maharashtra. All three are given as options. This indicates a poorly formed question with multiple correct answers from the option set. \[ \boxed{Andhra Pradesh, Tamil Nadu, and Maharashtra all fit.} \] Quick Tip: When a question asks to identify a single item with a certain property, and you find multiple items in the options that fit, double-check your analysis. If your analysis is correct, the question is flawed.

Let's define two sets of regions based on the conditions.
Set A: Regions that produce Medium Crop-1 OR Medium Crop-2.

Medium Crop-1: {R6, R7, R8
Medium Crop-2: {R9, R13
Set A = {R6, R7, R8, R9, R13

Set B: Regions that produce Low Crop-3 OR Low Crop-4.

Low Crop-3: {R1, R4
Low Crop-4: {R5, R9
Set B = {R1, R4, R5, R9

The question asks for the number of regions that are in BOTH Set A AND Set B. This is the intersection of the two sets: \(A \cap B\).
Comparing the elements of the two sets: \[ A \cap B = \{R9\} \]
There is only one region, R9, that satisfies both conditions.
(Note: The provided answer key `(3) Two` is incorrect). \[ \boxed{(2) One} \] Quick Tip: For questions involving logical conditions like "AND" and "OR", translate them into set operations. "OR" corresponds to the union of sets, and "AND" corresponds to the intersection.

We must evaluate each statement using the table.

(1): Med C-2 regions = {R9, R13. High C-3 regions = {R2, R6, R7, R13. Is {R9, R13 a subset of {R2, R6, R7, R13? No, because R9 is not in the second set. So, (1) is false.
(2): High C-1 regions = {R1, R2, R3, R4, R5. Med+Low C-4 regions = {R1, R2, R4, R5, R9. Is the first set a subset of the second? No, because R3 is not in the second set. So, (2) is false.
(3): We need regions that are in (C-3 list) AND (C-4 list) AND NOT in (C-2 list).
- C-3 regions = {R2,R6,R7,R13, R3,R9,R11, R1,R4.
- C-4 regions = {R3,R10,R11, R1,R2,R4, R5,R9.
- Intersection (C-3 and C-4): {R1, R2, R3, R4, R9, R11.
- C-2 regions = {R5,R8,R12, R9,R13, R6,R7.
- Now we remove the C-2 regions from the intersection list. R9 is in the C-2 list.
- The final list is {R1, R2, R3, R4, R11. This is a list of 5 regions, not 4. So, (3) is false.
(4): We need regions that are in (C-3 list) AND (C-1 list) AND NOT in (High C-2 list).
- C-3 regions = {R2,R6,R7,R13, R3,R9,R11, R1,R4.
- C-1 regions = {R1,R2,R3,R4,R5, R6,R7,R8, R9,R10,R11.
- Intersection (C-3 and C-1): {R1, R2, R3, R4, R6, R7, R9, R11. This is the set of "Some Crop-3 producing regions produce Crop-1".
- High C-2 regions = {R5, R8, R12.
- Is any region in our intersection list {R1, R2, ... NOT in the High C-2 list? Yes, all of them are. For example, R1 is a C-3 and C-1 region, but not a High C-2 region.
- Therefore, the statement is true.
\[ \boxed{(4) Some Crop-3 producing regions produce Crop-1, but not high quality Crop-2.} \] Quick Tip: For complex logical statements, break them down into set definitions. "All X are Y" means set X is a subset of set Y. "Some X are Y" means the intersection of set X and set Y is not empty.

Let's define the sets of regions based on the question.
Set A: Low quality Crop-1 producing regions.

From the table, Set A = {R9, R10, R11.

Set B: High quality Crop-4 producing regions OR Medium quality Crop-3 producing regions.

High C-4 regions = {R3, R10, R11
Medium C-3 regions = {R3, R9, R11
Set B (the union of the two sets above) = {R3, R9, R10, R11.

The question asks for the number of regions that are in Set A AND in Set B. We need to find the size of the intersection \(A \cap B\).

Set A = {R9, R10, R11
Set B = {R3, R9, R10, R11
The common elements are {R9, R10, R11.

There are 3 such regions.
(Note: The provided answer key `(1) One` is incorrect). \[ \boxed{(3) Three} \] Quick Tip: When a question involves the condition "X and (Y or Z)", first find the set for the "OR" condition by taking the union of the sets for Y and Z. Then find the intersection of that combined set with the set for X.

The problem asks for the number of triplets of the form \((n_i, n_j, n_k)\) that can be formed from the 10 numbers, with the condition that \(n_i \)<\( n_j \)<\( n_k\).

The initial set of numbers is already ordered: \(n_1 \)<\( n_2 \)<\( \dots \)<\( n_{10}\).
This ordering simplifies the problem significantly. Any set of three distinct numbers we choose from the 10 available numbers can be arranged in increasing order in exactly one way.

For example, if we choose the numbers \(\{n_7, n_2, n_5\}\), there is only one way to form a triplet that satisfies the condition: \((n_2, n_5, n_7)\).

Therefore, the problem is equivalent to asking: "How many ways can we choose a subset of 3 numbers from a set of 10 numbers?"

This is a combination problem. The number of ways to choose 3 items from a set of 10 is given by the combination formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). \[ \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10!}{3!7!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 10 \times 3 \times 4 = 120 \]
There are 120 such triplets.
(Note: The provided answer key `(2) 90` is incorrect). \[ \boxed{(3) 120} \] Quick Tip: When you need to choose a group of items where the order is predetermined (e.g., increasing, alphabetical), the problem is a combination, not a permutation. The act of choosing the items automatically fixes their order.

There is a direct formula for the length of the angle bisector, but a more intuitive method uses the area of the triangle.

Method: Using Area of Triangle
The area of the large triangle \(\triangle ABC\) is the sum of the areas of the two smaller triangles it is divided into, \(\triangle ABD\) and \(\triangle ADC\).
Area(\(\triangle ABC\)) = Area(\(\triangle ABD\)) + Area(\(\triangle ADC\)).

The formula for the area of a triangle given two sides and the included angle is \(\frac{1}{2}ab\sin C\).

Let the length of the angle bisector AD be \(x\).
The bisector AD divides \(\angle A = 60^\circ\) into two angles of \(30^\circ\) each. So, \(\angle BAD = \angle CAD = 30^\circ\).
Area(\(\triangle ABD\)) = \(\frac{1}{2} \times AB \times AD \times \sin(\angle BAD) = \frac{1}{2} \times 4 \times x \times \sin(30^\circ) = \frac{1}{2}(4x)(\frac{1}{2}) = x\).
Area(\(\triangle ADC\)) = \(\frac{1}{2} \times AC \times AD \times \sin(\angle CAD) = \frac{1}{2} \times 3 \times x \times \sin(30^\circ) = \frac{1}{2}(3x)(\frac{1}{2}) = \frac{3x}{4}\).
Area(\(\triangle ABC\)) = \(\frac{1}{2} \times AB \times AC \times \sin(\angle BAC) = \frac{1}{2} \times 4 \times 3 \times \sin(60^\circ) = 6 \times \frac{\sqrt{3}}{2} = 3\sqrt{3}\).


Now, set the sum of the smaller areas equal to the larger area: \[ x + \frac{3x}{4} = 3\sqrt{3} \] \[ \frac{4x + 3x}{4} = 3\sqrt{3} \] \[ \frac{7x}{4} = 3\sqrt{3} \] \[ x = \frac{4 \times 3\sqrt{3}}{7} = \frac{12\sqrt{3}}{7} \]
The length of AD is \(\frac{12\sqrt{3}}{7}\). \[ \boxed{(2) \frac{12 \sqrt{3}}{7}} \] Quick Tip: The area method provides a very elegant way to find the length of an angle bisector without needing to calculate the lengths of the segments on the opposite side.

Let the two circles have centers \(C_1\) and \(C_2\), and radii \(r_1 = 15\) cm and \(r_2 = 20\) cm.
The distance between the centers is \(d = 25\) cm.
Let the common chord be AB, and let it intersect the line segment \(C_1C_2\) at point M. The length of the chord is \(2 \times AM\).
The triangle \(\triangle C_1AC_2\) is formed by the two radii and the line segment connecting the centers. Its sides are 15 cm, 20 cm, and 25 cm.

Step 1: Check the type of triangle \(\triangle C_1AC_2\)
Let's check if it's a right-angled triangle using the Pythagorean theorem. The sides are in the ratio 15:20:25, which simplifies to 3:4:5. \(15^2 + 20^2 = 225 + 400 = 625\). \(25^2 = 625\).
Since \(15^2 + 20^2 = 25^2\), the triangle \(\triangle C_1AC_2\) is a right-angled triangle, with the right angle at the intersection point A of the circles.

Step 2: Calculate the length of the chord
The length of the common chord is twice the length of the altitude from the vertex A to the hypotenuse \(C_1C_2\).
Let the length of this altitude be \(h\) (which is AM).
The area of a right-angled triangle can be calculated in two ways:
1. Area = \(\frac{1}{2} \times base \times height = \frac{1}{2} \times 15 \times 20 = 150\).
2. Area = \(\frac{1}{2} \times hypotenuse \times altitude to hypotenuse = \frac{1}{2} \times 25 \times h\).

Equating the two expressions for the area: \[ \frac{1}{2} \times 25 \times h = 150 \] \[ 25h = 300 \] \[ h = \frac{300}{25} = 12 cm \]
The length of the common chord is \(2h = 2 \times 12 = 24\) cm.
(Note: The provided answer key `(2) 25 cm` is incorrect). \[ \boxed{(1) 24 cm} \] Quick Tip: When dealing with two intersecting circles, always form a triangle with the two radii and the line connecting the centers. Check if this triangle is a special type (e.g., right-angled) as it can greatly simplify the calculation of the common chord length.

We need to evaluate \(f(y)\) where \(y = \frac{2x}{1+x^2}\). \[ f\left(\frac{2x}{1+x^2}\right) = \log \left( \frac{1 + \frac{2x}{1+x^2}}{1 - \frac{2x}{1+x^2}} \right) \]
Let's simplify the fraction inside the logarithm.

Numerator: \(1 + \frac{2x}{1+x^2} = \frac{1+x^2+2x}{1+x^2} = \frac{(1+x)^2}{1+x^2}\).
Denominator: \(1 - \frac{2x}{1+x^2} = \frac{1+x^2-2x}{1+x^2} = \frac{(1-x)^2}{1+x^2}\).

Now substitute these back into the expression: \[ f\left(\frac{2x}{1+x^2}\right) = \log \left( \frac{\frac{(1+x)^2}{1+x^2}}{\frac{(1-x)^2}{1+x^2}} \right) = \log \left( \frac{(1+x)^2}{(1-x)^2} \right) \] \[ = \log \left( \left(\frac{1+x}{1-x}\right)^2 \right) \]
Using the logarithm property \(\log(a^b) = b\log(a)\): \[ = 2 \log \left(\frac{1+x}{1-x}\right) \]
Since the original function is \(f(x) = \log \left(\frac{1 + x}{1 - x}\right)\), the expression simplifies to: \[ = 2f(x) \]
(Note: The original question was \(f(x)+f(y)\) and the key was \(\frac{x+y}{1+xy}\). This corresponds to a different function, \(f(x) = arctanh(x)\)). \[ \boxed{(2) 2f(x)} \] Quick Tip: When dealing with functional equations, substitute the given expression into the function definition and simplify algebraically. Look for ways to use logarithm or exponent rules to relate the new expression back to the original function.

Step 1: Calculate the total area of the square plot.

Side of the square = 14 m.
Total Area = \(14 \times 14 = 196\) m².


Step 2: Calculate the area grazed by the horses.

"Adjacent horses can just reach one another." This means the tether rope of each horse has a length equal to half the side of the square.
Radius of grazing for each horse, \(r = \frac{14}{2} = 7\) m.
Each horse is at a corner, so it can graze a quarter of a circle within the plot.
Area grazed by one horse = \(\frac{1}{4} \times \pi r^2 = \frac{1}{4} \times \frac{22}{7} \times 7^2 = \frac{1}{4} \times 22 \times 7 = \frac{154}{4} = 38.5\) m².
There are four horses, so the total grazed area is \(4 \times 38.5 = 154\) m².


Step 3: Calculate the ungrazed area.

The total area inside the plot is 196 m².
The horses graze 154 m² of this area.
The pond takes up 20 m² of this area.
It is implied the pond is in the center and cannot be grazed.
Ungrazed Area = Total Area - Grazed Area - Pond Area.
Wait, the area grazed by the horses and the area of the pond might overlap. The question is slightly ambiguous. A better interpretation is: what is the area of the plot that is not accessible to the horses?
Ungrazed land = Total Area - Area grazed by horses = \(196 - 154 = 42\) m².
This 42 m² is the area in the middle of the plot. The pond, with an area of 20 m², is located within this ungrazed area.
The question asks for the "ungrazed area," which usually means the area of land (not water) that is left.
Ungrazed Land Area = (Total Ungrazed Area) - (Pond Area) = \(42 - 20 = 22\) m².
\[ \boxed{(1) 22 m²} \] Quick Tip: In problems with overlapping areas, be clear about what the question is asking for. Here, "ungrazed area" means the part of the *plot* (land) that is left, so the pond's area must be subtracted from the total central area that the horses cannot reach.

Let's analyze the trips the worker has to make. He starts at X, picks up a stone, carries it to Y, and must return to pick up the next stone. The last trip ends at Y.

Step 1: Locate the stones.
The stones are at distances from X:

Stone 1 (S1): at X (0 m from X)
Stone 2 (S2): 2 m from X
Stone 3 (S3): 4 m from X
Stone 4 (S4): 6 m from X
Stone 5 (S5): 8 m from X


Step 2: Calculate the distance for each stone's round trip.
The worker must go from his current position to the stone, pick it up, go to Y, drop it, and return to the location of the next stone. A simpler way is to think of it as a series of trips starting from X each time.

To move S1 (at 0m): He is already there. He travels 100m to Y. To get the next stone, he must return from Y to S2 (at 2m). Return distance = 100-2=98m. Total = 100+98 = 198m.
This is complex. Let's simplify. He starts at X.
Trip for S1: Pick up at X (dist=0). Go to Y (dist=100). Return to X (dist=100). Total = 200. (He has to return to the starting point to get the other stones). Let's assume he returns to X each time.
Trip for S2: Start at X. Go to S2 (dist=2). Pick up. Go to Y (dist=100-2=98). Return to X (dist=100). Total = 2+98+100=200.
This model is also flawed. The most efficient way is to drop a stone at Y and return only as far as the next stone.


Let's try the standard model for this problem type: The worker starts at X, picks up a stone, takes it to Y, and returns to X to begin the next trip. The final trip ends at Y.

For S1 (at 0m): Go to Y (100m). Return to X (100m). Total = 200m.
For S2 (at 2m): Go to S2 (2m). Go from S2 to Y (98m). Return to X (100m). Total = 2+98+100 = 200m.
This is also not right. Let's trace his position.
- Start at X.
- Walk to S1(0m), pick it up. Walk to Y(100m). Drop. Pos: Y. Dist: 100.
- Walk from Y to S2(2m). Dist: 98. Pick up. Walk to Y(100m). Dist: 98. Drop. Pos: Y. Dist: 100+98+98=296.
This is also getting complicated. Let's use the simplest model.

Final, Simplest Model:
The worker must make 5 trips. 4 of them are round trips (pick up stone, go to Y, return to X), and the last one is a one-way trip.

Distance to pick up stones: He starts at X. To pick up S1..S5, he travels: 0, 2, 4, 6, 8. Total pickup distance = \(0+2+4+6+8 = 20\)m.
Distance from stones to Y: The distance from each stone to Y is: 100, 98, 96, 94, 92. Total forward travel = \(100+98+96+94+92 = 480\)m.
Return trips: After dropping the first 4 stones, he must return from Y to X to start over. He makes 4 return trips of 100m each. Total return travel = \(4 \times 100 = 400\)m.
Total Distance = (Total pickup) + (Total forward) + (Total return) ? No, this double counts.

Let's trace the full journey:

S1: X \(\to\) Y (100m). Return Y \(\to\) X (100m).
S2: X \(\to\) S2 (2m). Pick up. S2 \(\to\) Y (98m). Return Y \(\to\) X (100m).
S3: X \(\to\) S3 (4m). Pick up. S3 \(\to\) Y (96m). Return Y \(\to\) X (100m).
S4: X \(\to\) S4 (6m). Pick up. S4 \(\to\) Y (94m). Return Y \(\to\) X (100m).
S5: X \(\to\) S5 (8m). Pick up. S5 \(\to\) Y (92m). NO RETURN.

Total distance = \((100+100) + (2+98+100) + (4+96+100) + (6+94+100) + (8+92)\)
Total distance = \(200 + 200 + 200 + 200 + 100 = 900\)m.
This is not among the options.

Let's try another model. Worker starts at X, picks up all stones and moves them to Y.
This implies he must gather them at one spot first, maybe X.
Distance to bring S2,S3,S4,S5 to X: \(2+4+6+8 = 20\). And back: \(2+4+6+8=20\). Total 40m to gather at X.
Then, make 5 trips from X to Y. 4 round trips, 1 one-way. \(4 \times (100+100) + 100 = 900\)m. Total = 940m.

Let's use the model from the provided solution key `(4) 860 m`.
The total distance is the sum of distances to pick up each stone and carry it to Y, plus the return trips.
The distances of the stones from X are 0, 2, 4, 6, 8.
The worker makes 4 round trips from X to Y and back, and a final one-way trip.
Total distance = \(2(0+2+4+6) + (100 \times 5) = 2 \times 12 + 500 = 524\) m. Doesn't work.

Let's calculate the sum of round trips from X to each stone's location and back, then add the trips to Y.
To pick up S1..S5 and bring to Y:
(X-\(>\)S1-\(>\)Y-\(>\)X) + (X-\(>\)S2-\(>\)Y-\(>\)X) ... + (X-\(>\)S5-\(>\)Y).
Distance (X-\(>\)Sn-\(>\)Y) = 100 for all.
So 5 trips of 100m to Y = 500m.
4 return trips from Y to X = 400m.
Total = 900m.

Let's try one last time. The sum of distances of stones from X is \(0+2+4+6+8=20\).
Sum of distances from Y is \(100+98+96+94+92 = 480\).
Total distance forward = 480.
Total distance back = \(98+96+94+92 = 380\).
Total = \(480+380 = 860\)m. This works. Let's trace it.
1. X-\(>\)S1(0), pick up, S1-\(>\)Y(100). dist=100.
2. Y-\(>\)S2(2). dist=98. pick up, S2-\(>\)Y(98). dist=98.
3. Y-\(>\)S3(4). dist=96. pick up, S3-\(>\)Y(96). dist=96.
4. Y-\(>\)S4(6). dist=94. pick up, S4-\(>\)Y(94). dist=94.
5. Y-\(>\)S5(8). dist=92. pick up, S5-\(>\)Y(92). dist=92.
Total = \(100 + 2(98) + 2(96) + 2(94) + 2(92) = 100+196+192+188+184 = 860\).
This is a valid interpretation. \[ \boxed{(4) 860 m} \] Quick Tip: For transportation problems, carefully trace the full path of the worker for each item, including the travel to pick up the item and the return trip to the starting point for the next item. Summing these individual trip distances gives the total.

Step 1: Find the side lengths of triangle ABE.

\(\triangle ABE\) is an isosceles right triangle, so the two legs are equal. Since the right angle is at E (implied by the drawing), the legs are AE and BE.
Let \(AE = BE = s\).
The area of \(\triangle ABE\) is \(\frac{1}{2} \times base \times height = \frac{1}{2} \times s \times s = \frac{s^2}{2}\).
We are given that the area is 7 cm².
\[ \frac{s^2}{2} = 7 \implies s^2 = 14 \implies s = \sqrt{14} cm \]
So, \(AE = BE = \sqrt{14}\) cm.


Step 2: Find the dimensions of the rectangle ABCD.

The side AB of the rectangle is the hypotenuse of \(\triangle ABE\). Using Pythagoras' theorem:
\[ AB^2 = AE^2 + BE^2 = (\sqrt{14})^2 + (\sqrt{14})^2 = 14 + 14 = 28 \]
\[ AB = \sqrt{28} = 2\sqrt{7} cm \]
Wait, the figure seems to suggest the right angle is at A or B. Let's assume the right angle is at B. No, that would make AE the hypotenuse. Let's assume the right angle is at A. Then BE is the hypotenuse. This contradicts the isosceles property unless ABE is a line.
The most standard interpretation of an "isosceles right triangle" ABE inscribed in a rectangle like this is that the right angle is at the corner of the rectangle, say point B. Let's re-evaluate with this assumption.



Alternative Interpretation:

Assume the right angle is at B. \(\triangle ABE\) is an isosceles right triangle, so \(AB=BE\).
Let \(AB = BE = s\). Area = \(\frac{1}{2}s^2 = 7 \implies s^2 = 14 \implies s=\sqrt{14}\).
So, \(AB = \sqrt{14}\) and \(BE = \sqrt{14}\).
We are given \(EC = 3(BE) = 3\sqrt{14}\).
The length of the side BC of the rectangle is \(BC = BE + EC = \sqrt{14} + 3\sqrt{14} = 4\sqrt{14}\).
The area of the rectangle ABCD is \(AB \times BC\).
\[ Area = \sqrt{14} \times 4\sqrt{14} = 4 \times 14 = 56 cm^2 \]

This provides a clean integer answer that matches an option. This interpretation is much more likely to be the intended one. \[ \boxed{(4) 56 cm²} \] Quick Tip: In geometry diagrams that are "not drawn to scale," rely on the text description. However, if the text is ambiguous (like the location of the right angle), test the most plausible interpretation that fits the geometric context (e.g., the right angle coinciding with the rectangle's corner).

We use the Shoelace formula (or determinant method) for the area of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \).
Area = \(\frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|\)

Let the vertices be: \( (x_1, y_1) = (a, a) \) \( (x_2, y_2) = (a + 1, a + 1) \) \( (x_3, y_3) = (a + 2, a) \)

Substitute these into the formula: \[ Area = \frac{1}{2} |a((a+1) - a) + (a+1)(a - a) + (a+2)(a - (a+1))| \]
Simplify the terms inside the absolute value:

\(a((a+1) - a) = a(1) = a\).
\((a+1)(a-a) = (a+1)(0) = 0\).
\((a+2)(a - a - 1) = (a+2)(-1) = -a - 2\).

Now, add these results together: \[ Area = \frac{1}{2} |a + 0 + (-a-2)| \] \[ Area = \frac{1}{2} |a - a - 2| = \frac{1}{2} |-2| = \frac{1}{2} \times 2 = 1 \]
The area of the triangle is 1, regardless of the value of \(a\).
(Note: The provided answer key `(4) \(2a^2\) is incorrect). \[ \boxed{(2) 1} \] Quick Tip: When using the Shoelace formula for area, you can often simplify the calculation by translating one of the vertices to the origin (0,0). For example, subtract (a,a) from all points to get vertices (0,0), (1,1), and (2,0). The area is then simply \(\frac{1}{2}|x_1y_2 - x_2y_1| = \frac{1}{2}|(1)(0) - (2)(1)| = \frac{1}{2}|-2|=1\).

Let the shorter side of the rectangular field be \(s\) and the longer side be \(l\).

Step 1: Define the distances for both paths.

The distance walking along the two adjacent sides is \(l + s\).
The distance along the diagonal (the shortcut) is, by the Pythagorean theorem, \(\sqrt{l^2 + s^2}\).


Step 2: Set up an equation based on the saved distance.

The distance saved is the difference between the two paths: \((l + s) - \sqrt{l^2 + s^2}\).
The problem states this saved distance is equal to half the longer side, which is \(\frac{l}{2}\).
\[ (l + s) - \sqrt{l^2 + s^2} = \frac{l}{2} \]


Step 3: Solve the equation for the ratio \(s/l\).

Rearrange the equation to isolate the square root term:
\[ l + s - \frac{l}{2} = \sqrt{l^2 + s^2} \]
\[ \frac{l}{2} + s = \sqrt{l^2 + s^2} \]
Square both sides of the equation:
\[ \left(\frac{l}{2} + s\right)^2 = l^2 + s^2 \]
\[ \frac{l^2}{4} + 2\left(\frac{l}{2}\right)(s) + s^2 = l^2 + s^2 \]
\[ \frac{l^2}{4} + ls + s^2 = l^2 + s^2 \]
Subtract \(s^2\) from both sides:
\[ \frac{l^2}{4} + ls = l^2 \]
Rearrange to solve for \(ls\):
\[ ls = l^2 - \frac{l^2}{4} = \frac{3l^2}{4} \]
Since \(l\) is a length, \(l \neq 0\), so we can divide both sides by \(l\):
\[ s = \frac{3l}{4} \]
Now, find the ratio of the shorter side to the longer side:
\[ \frac{s}{l} = \frac{3}{4} \]

(Note: The provided answer key `(2)` is incorrect). \[ \boxed{(4) \frac{3}{4}} \] Quick Tip: Translate word problems into algebraic equations. When you see a right-angled triangle (formed by the sides and diagonal of a rectangle), your first thought should be the Pythagorean theorem.

Let the distance between A and B be \(D\). Let the speed of the normal express train E be \(v_E\).

Step 1: Define the normal speeds.

Speed of Super fast train N: \(v_N = 2v_E\).
Speed of Passenger train S: \(v_S = \frac{1}{2}v_E\).
The ratio of their normal speeds is \(v_N : v_S = 2v_E : \frac{1}{2}v_E = 4:1\).


Step 2: Understand the normal schedule.

Let N leave A at time \(t=0\). It takes time \(T_N = D/v_N\) to reach B.
S reaches A at time \(t=1\) hour. It takes time \(T_S = D/v_S\) to travel from B to A.
Since \(v_N = 4v_S\), it means \(T_S = 4T_N\).
Train S must have left station B at time \(t = 1 - T_S = 1 - 4T_N\).


Step 3: Analyze the delayed schedule.

On the particular day, N leaves at \(t = 20\) min = \(1/3\) hour.
To maintain the schedule, N must still arrive at B at the original time \(T_N\). So its new travel time must be \(T_N' = T_N - 1/3\).
N's new speed is doubled: \(v_N' = 2v_N = 4v_E\). So its travel time is halved: \(T_N' = T_N/2\).
Equating the new travel times for N: \(T_N/2 = T_N - 1/3 \implies T_N/2 = 1/3 \implies T_N = 2/3\) hours.
This means the normal time for S is \(T_S = 4T_N = 4(2/3) = 8/3\) hours.


Step 4: Calculate the new speed for S.

Normally, S leaves B at \(t = 1 - T_S = 1 - 8/3 = -5/3\) hours.
N leaves A at \(t=1/3\) hr. A single track means they cannot be on the track at the same time. The problem implies they cross at a station in between, but the question is about maintaining the arrival time. Let's assume S can start its journey to arrive at A on time.
S must arrive at A at the scheduled time of \(t=1\) hour.
When must S leave B? We don't know if its departure time changes. Let's assume it leaves at the same time, \(t=-5/3\) hr.
New travel time for S: \(T_S' = 1 - (-5/3) = 8/3\) hours. This is the same as the old time. This implies S does not increase its speed. This contradicts the premise.
Let's re-read: "S reaches station A from B" one hour after N leaves A. This is a statement about arrival, not duration.
A better interpretation: On a normal day, they cross paths. Let's assume they cross at the midpoint for simplicity. Then \(T_N=T_S\). But speeds are different. Let crossing point be C. Time for N to reach C = Time for S to reach C. Let this crossing happen at time \(T_X\). Then train S must leave at \(T_X-T_{SC}\) and N at \(T_X-T_{AC}\). The problem is very ill-defined.


Let's try a simpler interpretation based on the solution.
Let's focus only on the times.

Normal time for N to travel A to B = \(T_N\). Normal speed = \(v_N\).
Delayed departure for N is at \(t=20\). New speed is \(v_N' = 2v_N\). New time \(T_N' = D/v_N' = D/(2v_N) = T_N/2\).
N has to make up for the 20 min delay. So \(T_N' = T_N - 20\).
\(T_N/2 = T_N - 20 \implies T_N/2 = 20 \implies T_N = 40\) minutes.
Normal time for S to travel B to A = \(T_S\). Normal speed = \(v_S\).
We know \(v_N = 4v_S\), so \(T_S = 4T_N = 4 \times 40 = 160\) minutes.
On the delayed day, let's assume S also has to adjust. S has to reach at the same scheduled time. Let's assume its departure is also delayed by 20 mins.
So S has to cover the distance in \(T_S' = T_S - 20 = 160 - 20 = 140\) minutes.
New speed of S, \(v_S' = D/T_S' = D/140\).
The question asks for the ratio of new speeds: \(v_S' : v_N'\).
\(v_S' = D/140\). \(v_N' = D/T_N' = D/20\).
Ratio: \(\frac{v_S'}{v_N'} = \frac{D/140}{D/20} = \frac{20}{140} = \frac{1}{7}\).

This gives a ratio of 1:7. Not in the options. The question is extremely poorly phrased. The answer 1:5.8 is very specific, suggesting a more complex calculation is needed, but the premises are too ambiguous to set up the correct model. \[ \boxed{Question is unsolvable due to ambiguity.} \] Quick Tip: When a speed-time problem gives confusing scheduling details ("reaches A one hour after N leaves A"), try to model the situation with a timeline. If the setup remains ambiguous and your logical interpretations do not match the options, the question is likely flawed.

Step 1: Determine the locations of the gutters.

Let the distance from A to G1 be \(d_1\). Then the distance from B to G3 is also \(d_1\).
Let the distance between G2 and G3 be \(d_3\). Then the distance between G1 and G2 is \(d_2 = d_3 / 2\).
The total length of the tunnel is 20 km. So, \(d_1 + d_2 + d_3 + d_1 = 20\).
Substituting \(d_2\): \(2d_1 + d_3/2 + d_3 = 20 \implies 2d_1 + \frac{3}{2}d_3 = 20\).
The ambulance travels from A to G1 at 30 km/hr in 5 minutes (1/12 hour).
Distance \(d_1 = Speed \times Time = 30 \times \frac{1}{12} = 2.5\) km.
Now we can find \(d_3\): \(2(2.5) + \frac{3}{2}d_3 = 20 \implies 5 + \frac{3}{2}d_3 = 20 \implies \frac{3}{2}d_3 = 15 \implies d_3 = 10\) km.
And \(d_2 = d_3/2 = 5\) km.
Locations from A: G1 is at 2.5 km. G2 is at \(2.5+5=7.5\) km. G3 is at \(7.5+10=17.5\) km. Distance from B to G3 is \(20-17.5 = 2.5\) km, which matches \(d_1\). The setup is consistent.


Step 2: Calculate the total time taken by the ambulance.
The journey has three parts: A to G3, loading the patient, and G3 back to A.

A to G1: Time = 5 min (given).
G1 to G3: After G1, the speed is doubled to \(2 \times 30 = 60\) km/hr. The distance is from 2.5 km to 17.5 km, which is \(17.5 - 2.5 = 15\) km. Time = \(\frac{Distance}{Speed} = \frac{15}{60} = 0.25\) hours = 15 min.
Loading patient: Time = 1 min (given for taking patient into ambulance).
G3 back to A: The distance is 17.5 km. The speed is the doubled speed, 60 km/hr. Time = \(\frac{17.5}{60}\) hours.
Time (in minutes) = \(\frac{17.5}{60} \times 60 = 17.5\) min.
Unloading patient: Time = 1 min (given for taking patient out of ambulance).

Total time from call to patient being ready for operation = (A to G1) + (G1 to G3) + (Loading) + (G3 to A) + (Unloading) \[ Total Time = 5 + 15 + 1 + 17.5 + 1 = 39.5 min \]

Step 3: Calculate time left for the doctor.

Maximum time allowed = 40 min.
Time elapsed = 39.5 min.
Time left for doctor = \(40 - 39.5 = 0.5\) min.

The doctor has 0.5 minutes. This is not among the options. Let's re-read. "Assume 1 min is elapsed for taking the patient into and out of the ambulance". This could mean 1 minute total for both actions, not 1 minute for each.
Let's recalculate with 1 minute total for loading/unloading. \[ Total Time = 5 + 15 + 17.5 + 1 (total for in/out) = 38.5 min \]
Time left for doctor = \(40 - 38.5 = 1.5\) min. This matches option (3). \[ \boxed{(3) 1.5 min} \] Quick Tip: Break down complex journey problems into distinct segments. Calculate the distance and time for each segment separately before summing them up. Pay very close attention to the exact wording of time costs, like loading and unloading.

Let the two-digit number be D. We can write D = \(10t + u\), where \(t\) is the tens digit and \(u\) is the units digit.
The sum of the digits is \(t+u\).
The number S is the square of this sum: \(S = (t+u)^2\).
The condition is \(S - D = 27\). \[ (t+u)^2 - (10t+u) = 27 \]
Since we have one equation with two variables, and the options are given, the easiest way to solve this is to test the options.


Option (1): D = 24.
- Sum of digits = \(2+4=6\).
- \(S = 6^2 = 36\).
- \(S - D = 36 - 24 = 12\). (Incorrect, we need 27).
Option (2): D = 54.
- Sum of digits = \(5+4=9\).
- \(S = 9^2 = 81\).
- \(S - D = 81 - 54 = 27\). (Correct).
Option (3): D = 34.
- Sum of digits = \(3+4=7\).
- \(S = 7^2 = 49\).
- \(S - D = 49 - 34 = 15\). (Incorrect).
Option (4): D = 45.
- Sum of digits = \(4+5=9\).
- \(S = 9^2 = 81\).
- \(S - D = 81 - 45 = 36\). (Incorrect).

The only number that satisfies the condition is 54.
(Note: The provided answer key `(3) 34` is incorrect). \[ \boxed{(2) 54} \] Quick Tip: For problems involving specific number properties where options are provided, testing the options is often much faster and more reliable than solving the algebraic equation from scratch.

Let's compute the first few terms of the sequence starting with \(X_0 = x\).

n = 1:
\(X_1 = (-1)^1 X_{1-1} = (-1)^1 X_0 = -1 \cdot x = -x\). (Negative, since \(x\)>\(0\))
n = 2:
\(X_2 = (-1)^2 X_{2-1} = (-1)^2 X_1 = 1 \cdot (-x) = -x\). (Negative)
n = 3:
\(X_3 = (-1)^3 X_{3-1} = (-1)^3 X_2 = -1 \cdot (-x) = x\). (Positive)
n = 4:
\(X_4 = (-1)^4 X_{4-1} = (-1)^4 X_3 = 1 \cdot (x) = x\). (Positive)
n = 5:
\(X_5 = (-1)^5 X_{5-1} = (-1)^5 X_4 = -1 \cdot (x) = -x\). (Negative)

The sequence of values is: \(x, -x, -x, x, x, -x, -x, \dots\)
The signs are: +, -, -, +, +, -, -, ...

Now let's evaluate the statements:

(1) \( X_n \) is positive if \( n \) is even.
For n=2 (even), \(X_2 = -x\), which is negative. This statement is false.
(2) \( X_n \) is positive if \( n \) is odd.
For n=1 (odd), \(X_1 = -x\), which is negative. This statement is false.
(3) \( X_n \) is negative if \( n \) is even.
For n=4 (even), \(X_4 = x\), which is positive. This statement is false.

Since statements (1), (2), and (3) are all false, the correct option is (4) None of these.
(Note: The provided answer key `(1)` is incorrect). \[ \boxed{(4) None of these} \] Quick Tip: When given a recurrence relation, the best way to understand its behavior is to write out the first few terms. This will quickly reveal any patterns in signs or values.

We are given two equations:
1) \(x + y + z = 5\)
2) \(xy + yz + zx = 3\)

To find the possible range of values for \(x\), we can treat \(y\) and \(z\) as the roots of a quadratic equation.
From (1), we have \(y+z = 5-x\).
From (2), we can write \(yz = 3 - x(y+z)\). Substituting the expression for \(y+z\): \(yz = 3 - x(5-x) = 3 - 5x + x^2\).

Consider a quadratic equation in a variable \(t\) with roots \(y\) and \(z\): \(t^2 - (sum of roots)t + (product of roots) = 0\) \(t^2 - (y+z)t + yz = 0\)
Substituting our expressions in terms of \(x\): \(t^2 - (5-x)t + (x^2 - 5x + 3) = 0\).

For \(y\) and \(z\) to be real numbers, the roots of this quadratic equation in \(t\) must be real. This means the discriminant (\(\Delta\)) must be greater than or equal to zero. \(\Delta = B^2 - 4AC \geq 0\)
Here, \(A=1\), \(B=-(5-x)\), and \(C=x^2-5x+3\). \[ (-(5-x))^2 - 4(1)(x^2 - 5x + 3) \geq 0 \] \[ (5-x)^2 - 4x^2 + 20x - 12 \geq 0 \] \[ (25 - 10x + x^2) - 4x^2 + 20x - 12 \geq 0 \] \[ -3x^2 + 10x + 13 \geq 0 \]
Multiply by -1 and reverse the inequality sign: \[ 3x^2 - 10x - 13 \leq 0 \]
To find the range of \(x\), we find the roots of the quadratic equation \(3x^2 - 10x - 13 = 0\).
Factoring the quadratic: \(3x^2 - 13x + 3x - 13 = 0 \implies x(3x-13) + 1(3x-13) = 0 \implies (x+1)(3x-13)=0\).
The roots are \(x=-1\) and \(x=13/3\).

Since the parabola \(3x^2 - 10x - 13\) opens upwards, the expression is less than or equal to zero between its roots.
So, the possible range for \(x\) is \(-1 \leq x \leq \frac{13}{3}\).
The largest value that \(x\) can have is \(\frac{13}{3}\). \[ \boxed{(3) \frac{13}{3}} \] Quick Tip: When you have symmetric equations in three variables (\(x, y, z\)) and want to find the range of one variable, a powerful technique is to form a quadratic equation whose roots are the other two variables. The condition that these roots must be real (discriminant \(\ge 0\)) will give you an inequality for the first variable.

Step 1: Calculate the total area and the target area.

Total Area of lawn = \(20 m \times 40 m = 800 m^2\).
Half the lawn area = \(800 / 2 = 400 m^2\).


Step 2: Calculate the area mowed in each round.
The mower cuts a 1m wide strip around the perimeter.

Round 1:
- Mows a 1m strip from the 20x40 rectangle.
- Area Mowed = (Original Area) - (Area of inner unmowed rectangle).
- The inner rectangle has dimensions \((20 - 2 \times 1)\) by \((40 - 2 \times 1)\), which is \(18 \times 38\).
- Area of inner rectangle = \(18 \times 38 = 684 m^2\).
- Area mowed in Round 1 = \(800 - 684 = 116 m^2\).
- Cumulative area mowed = 116 m². (This is less than 400).
Round 2:
- Mows a 1m strip from the 18x38 rectangle.
- The new inner rectangle has dimensions \((18 - 2)\) by \((38 - 2)\), which is \(16 \times 36\).
- Area of new inner rectangle = \(16 \times 36 = 576 m^2\).
- Area mowed in Round 2 = \(684 - 576 = 108 m^2\).
- Cumulative area mowed = \(116 + 108 = 224 m^2\). (Still less than 400).
Round 3:
- Mows a 1m strip from the 16x36 rectangle.
- The new inner rectangle has dimensions \((16 - 2)\) by \((36 - 2)\), which is \(14 \times 34\).
- Area of new inner rectangle = \(14 \times 34 = 476 m^2\).
- Area mowed in Round 3 = \(576 - 476 = 100 m^2\).
- Cumulative area mowed = \(224 + 100 = 324 m^2\). (Still less than 400).
Round 4:
- Area mowed in Round 4 would be \(476 - (12 \times 32) = 476 - 384 = 92 m^2\).
- Cumulative area after 4 rounds = \(324 + 92 = 416 m^2\).


Step 3: Determine the number of rounds.

After 3 complete rounds, 324 m² is mowed.
He needs to mow \(400 - 324 = 76 m^2\) more.
The total area he can mow in the 4th round is 92 m².
The fraction of the 4th round he needs to complete is \(\frac{76}{92} \approx 0.826\).

So, the total number of rounds is 3 complete rounds plus about 0.8 of the next one, making it approximately 3.8 rounds.
Wait, my calculations don't match the key. Let me re-read the question. "how many times would he go round before he has mowed half the lawn?" This implies after how many full rounds is he past the halfway point. No, that would be 4.
Let's re-calculate.
Area 1 = 116.
Area 2 = 108.
Area 3 = 100.
After 2 rounds, he has mowed 224 m². He needs 176 more.
In round 3, he mows 100 m². Cumulative = 324. Still not enough.
He needs to mow 76 m² into the 4th round.
Let's check the first option: 2.5 rounds.
Area mowed in 2.5 rounds = Area(R1) + Area(R2) + 0.5*Area(R3) = 116 + 108 + 0.5*100 = 224 + 50 = 274. Not 400.
Let's check the provided key `(2) 3.5`.
Area mowed = Area(R1)+Area(R2)+Area(R3) + 0.5*Area(R4) = 116+108+100 + 0.5*92 = 324 + 46 = 370. This is close to 400.
The question might be simpler. Let \(x\) be the number of rounds. The unmowed area is \((20-2x)(40-2x)\). We want the mowed area to be 400.
Mowed Area = \(800 - (20-2x)(40-2x) = 400\). \((20-2x)(40-2x) = 400\). \(4(10-x)(20-x) = 400\). \((10-x)(20-x) = 100\). \(200 - 30x + x^2 = 100\). \(x^2 - 30x + 100 = 0\).
Using quadratic formula: \(x = \frac{30 \pm \sqrt{900 - 400}}{2} = \frac{30 \pm \sqrt{500}}{2} = 15 \pm \sqrt{125} = 15 \pm 5\sqrt{5}\).
Since \(x\) must be less than 10, we take the minus sign. \(x = 15 - 5\sqrt{5} \approx 15 - 5(2.236) = 15 - 11.18 = 3.82\).
This is the continuous answer. It's very close to 3.8.
The question must have been intended to be solved this way. (Note: The provided key `(2) 3.5` is not as accurate as `(3) 3.8`). \[ \boxed{(3) 3.8} \] Quick Tip: For mowing problems, you can approximate by setting up an equation for the remaining area as a continuous function of the number of rounds, \(x\). The unmowed area after \(x\) rounds of width \(w\) on an \(L \times W\) rectangle is \((L-2wx)(W-2wx)\).

This is a classic problem that is best solved by working backwards from the final amount.
Let's denote the number of diamonds the thief had *before* meeting each watchman.

Step 1: Before meeting Watchman 3 (W3)

The thief escaped with 1 diamond. This is the amount he had *after* giving diamonds to W3.
Let the number of diamonds he had just before meeting W3 be \(D_3\).
He gave W3 half of what he had, plus 2 more. So, the amount remaining is \(D_3 - (\frac{D_3}{2} + 2) = 1\).
\(\frac{D_3}{2} - 2 = 1 \implies \frac{D_3}{2} = 3 \implies D_3 = 6\).
So, he had 6 diamonds before meeting W3.


Step 2: Before meeting Watchman 2 (W2)

The amount he had after meeting W2 was 6.
Let the number of diamonds he had just before meeting W2 be \(D_2\).
The amount remaining is \(D_2 - (\frac{D_2}{2} + 2) = 6\).
\(\frac{D_2}{2} - 2 = 6 \implies \frac{D_2}{2} = 8 \implies D_2 = 16\).
So, he had 16 diamonds before meeting W2.


Step 3: Before meeting Watchman 1 (W1)

The amount he had after meeting W1 was 16.
Let the number of diamonds he had just before meeting W1 be \(D_1\). This is the original number he stole.
The amount remaining is \(D_1 - (\frac{D_1}{2} + 2) = 16\).
\(\frac{D_1}{2} - 2 = 16 \implies \frac{D_1}{2} = 18 \implies D_1 = 36\).

He originally stole 36 diamonds.
(Note: The provided answer key `(3) 25` is incorrect). \[ \boxed{(2) 36} \] Quick Tip: For problems where a sequence of operations leads to a final known result, working backwards is the most direct solution method. Reverse each operation: if the forward operation was "divide by 2, then subtract 2," the reverse is "add 2, then multiply by 2."

Let the amounts paid be M, I, L, and J. The total amount is \(M+I+L+J = 60\).

We can solve for each person's contribution using the given ratios.
For Mayank (M):

\(M = \frac{1}{2} (I+L+J)\).
We know that \(I+L+J = 60 - M\).
So, \(M = \frac{1}{2}(60-M) \implies 2M = 60 - M \implies 3M = 60 \implies M = 20\).


For Mirza (I):

\(I = \frac{1}{3} (M+L+J)\).
We know that \(M+L+J = 60 - I\).
So, \(I = \frac{1}{3}(60-I) \implies 3I = 60 - I \implies 4I = 60 \implies I = 15\).


For Little (L):

\(L = \frac{1}{4} (M+I+J)\).
We know that \(M+I+J = 60 - L\).
So, \(L = \frac{1}{4}(60-L) \implies 4L = 60 - L \implies 5L = 60 \implies L = 12\).


For Jaspal (J):

The total amount is Rs. 60.
\(J = 60 - (M+I+L)\).
\(J = 60 - (20 + 15 + 12) = 60 - 47 = 13\).

Jaspal had to pay Rs. 13. \[ \boxed{(2) 13} \] Quick Tip: When a person's share is given as a fraction of the sum of others' shares (e.g., \(A = \frac{1}{k}(Others)\)), it's a shortcut to know that this person's share is \(\frac{1}{k+1}\) of the total. For Mayank, \(M = \frac{1}{2+1} \times 60 = 20\). For Mirza, \(I=\frac{1}{3+1}\times 60 = 15\).

Let the total number of gold coins be \(C\).
The merchant divides the coins into two unequal numbers, let's call them \(x\) and \(y\).
So, the total number of coins is \(C = x + y\).
We are given that \(x \neq y\).

The merchant's statement is: "48 times the difference between the two numbers equals the difference between the squares of the two numbers."
Let's translate this into an equation. Assume \(x \)>\( y\).
The difference is \(x-y\). The difference of squares is \(x^2 - y^2\). \[ 48(x - y) = x^2 - y^2 \]
We can factor the right side of the equation using the difference of squares formula: \(a^2 - b^2 = (a-b)(a+b)\). \[ 48(x - y) = (x - y)(x + y) \]
Since the numbers are unequal, \(x \neq y\), which means \(x-y \neq 0\). Therefore, we can safely divide both sides of the equation by \((x-y)\). \[ 48 = x + y \]
The sum of the two numbers is 48. Since the total number of coins is the sum of these two numbers, the merchant has 48 gold coins.
(Note: The provided answer key `(3) 43` is incorrect). \[ \boxed{(3) 48} \] Quick Tip: Always be on the lookout for standard algebraic identities like the difference of squares. Recognizing that \(x^2 - y^2\) can be factored is the key to simplifying this problem instantly.

This is a set theory problem. Let the total number of days of the vacation be \(T\).
Let \(Y\) be the number of days they did Yoga.
Let \(T_e\) be the number of days they played Tennis.
Let \(H\) be the number of days they stayed home all day.

The problem states they do only one activity per day. So the sets of days for Yoga, Tennis, and Home are disjoint.
Therefore, \(T = Y + T_e + H\).

We are given:
1) "a total of 22 days when they did yoga or played tennis" \(\implies Y + T_e = 22\).
2) "24 mornings when they did nothing". A day has a morning and an evening. They do nothing in the morning on days they play Tennis (an evening activity) and on days they stay home. So, \(T_e + H = 24\).
3) "14 evenings when they stayed at home". They stay home in the evening on days they do Yoga (a morning activity) and on days they stay home. So, \(Y + H = 14\).

Now we have a system of three linear equations:
(i) \(Y + T_e = 22\)
(ii) \(T_e + H = 24\)
(iii) \(Y + H = 14\)

We can solve this system. Let's add all three equations: \((Y + T_e) + (T_e + H) + (Y + H) = 22 + 24 + 14\) \(2Y + 2T_e + 2H = 60\) \(2(Y + T_e + H) = 60\) \(Y + T_e + H = 30\).

The total number of days, \(T = Y + T_e + H\), is 30.
Let's check this solution:
- If \(T=30\):
- From (i) \(Y+T_e=22\), so \(H = T - (Y+T_e) = 30 - 22 = 8\).
- From (iii) \(Y+H=14 \implies Y+8=14 \implies Y=6\).
- From (ii) \(T_e+H=24 \implies T_e+8=24 \implies T_e=16\).
- Check if (i) holds: \(Y+T_e = 6+16=22\). It does.
The solution is consistent. The vacation was for 30 days. (Note: The provided key `(1) 32` is incorrect). \[ \boxed{(3) 30} \] Quick Tip: For word problems involving overlapping categories and time periods (like mornings/evenings), define your variables clearly and translate each piece of information into an equation. Solving the resulting system of equations will yield the answer.

This is an arithmetico-geometric series of a higher order. We can solve it using the method of differences.
Let \(S = 2 + 5x + 9x^2 + 14x^3 + \dots\)
Multiply by \(x\): \(xS = 2x + 5x^2 + 9x^3 + \dots\)
Subtract the second equation from the first: \(S(1-x) = 2 + (5-2)x + (9-5)x^2 + (14-9)x^3 + \dots\) \(S(1-x) = 2 + 3x + 4x^2 + 5x^3 + \dots\)
Let \(S' = S(1-x)\). Let's apply the method again to the new series \(S' = 2 + 3x + 4x^2 + 5x^3 + \dots\). \(xS' = 2x + 3x^2 + 4x^3 + \dots\) \(S'(1-x) = 2 + (3-2)x + (4-3)x^2 + (5-4)x^3 + \dots\) \(S'(1-x) = 2 + x + x^2 + x^3 + \dots\)
The series \(x+x^2+x^3+\dots\) is an infinite geometric series with first term \(x\) and common ratio \(x\). Its sum is \(\frac{x}{1-x}\).
So, \(S'(1-x) = 2 + \frac{x}{1-x} = \frac{2(1-x)+x}{1-x} = \frac{2-2x+x}{1-x} = \frac{2-x}{1-x}\).
Now substitute back for \(S'\). We know \(S' = S(1-x)\). \(S(1-x)(1-x) = \frac{2-x}{1-x}\) \(S(1-x)^2 = \frac{2-x}{1-x}\) \[ S = \frac{2-x}{(1-x)^3} \]
This matches option (1). \[ \boxed{(1) \frac{2 - x}{(1 - x)^3}} \] Quick Tip: The sum of an infinite arithmetico-geometric series whose coefficients form an arithmetic progression can be found using the method of differences. Calculate \(S - rS\) (where r is the common ratio) one or more times until you get a standard geometric series.

The given equation is \(x^2 + 5y^2 + z^2 = 2y(2x + z)\).
Let's rearrange the equation by moving all terms to one side and trying to form a sum of squares. \[ x^2 + 5y^2 + z^2 = 4xy + 2yz \] \[ x^2 - 4xy + 5y^2 + z^2 - 2yz = 0 \]
Now, we can split the \(5y^2\) term into \(4y^2 + y^2\) to complete the squares for the terms involving \(x\) and \(z\). \[ (x^2 - 4xy + 4y^2) + (y^2 - 2yz + z^2) = 0 \]
This can be factored as perfect squares: \[ (x - 2y)^2 + (y - z)^2 = 0 \]
The variables \(x, y, z\) are real numbers. The square of any real number is non-negative (i.e., \(\geq 0\)).
The sum of two non-negative numbers can be zero only if both numbers are individually zero.
Therefore, we must have:
1) \((x - 2y)^2 = 0 \implies x - 2y = 0 \implies x = 2y\).
2) \((y - z)^2 = 0 \implies y - z = 0 \implies y = z\).

From these two necessary conditions, we can check the statements:

A. \(x = 2y\): This is necessarily true from our first condition.
B. \(x = 2z\): We know \(x=2y\) and \(y=z\). Substituting \(y=z\) into the first equation gives \(x=2z\). This is also necessarily true.
C. \(2x = z\): We know \(x=2z\). So \(2(2z) = z \implies 4z=z \implies 3z=0 \implies z=0\). This would mean \(x=y=z=0\), which is a solution, but it is not *necessarily* true for all solutions (e.g., x=2, y=1, z=1 is also a solution). So C is not necessarily true.

Both A and B are necessarily true. This corresponds to option (3).
(Note: The provided answer key `(4)` is incorrect). \[ \boxed{(3) A and B} \] Quick Tip: When you see a quadratic equation with multiple variables, try to rearrange it into a sum of perfect squares that equals zero. If you can, you can conclude that each term inside the squares must be zero.

Let the 10 integers be \(N_1, N_2, \dots, N_{10}\).
Let the correct sum be \(S_{correct} = \sum N_i\).
The correct arithmetic mean is \(M_{correct} = \frac{S_{correct}}{10}\).

One of the numbers, say \(N_k\), had its digits interchanged. Let the original number be \(N_k = 10a + b\), where \(a\) is the tens digit and \(b\) is the units digit.
The new, incorrect number is \(N_k' = 10b + a\).

The incorrect sum, \(S_{incorrect}\), is the same as the correct sum, but with \(N_k\) replaced by \(N_k'\). \(S_{incorrect} = (S_{correct} - N_k) + N_k' = S_{correct} + (N_k' - N_k)\).
The incorrect mean is \(M_{incorrect} = \frac{S_{incorrect}}{10}\).

We are given that the incorrect mean was 1.8 more than the correct mean. \(M_{incorrect} = M_{correct} + 1.8\). \[ \frac{S_{incorrect}}{10} = \frac{S_{correct}}{10} + 1.8 \]
Multiply by 10: \[ S_{incorrect} = S_{correct} + 18 \]
The difference between the sums is 18. \[ S_{incorrect} - S_{correct} = 18 \]
Substitute the expression for the incorrect sum: \[ (S_{correct} + (N_k' - N_k)) - S_{correct} = 18 \] \[ N_k' - N_k = 18 \]
Now substitute the expressions for the numbers in terms of their digits: \[ (10b + a) - (10a + b) = 18 \] \[ 9b - 9a = 18 \] \[ 9(b - a) = 18 \] \[ b - a = 2 \]
The difference between the digits is 2. \[ \boxed{(2) 2} \] Quick Tip: The difference in value when the digits of a two-digit number 'ab' are reversed to 'ba' is always a multiple of 9, specifically \(9 \times (b-a)\). This is a very useful shortcut for number theory problems.

Let the number of hours Akil rented the car be \(H\). He drove 30 km. The total payment was Rs. 300.
We need to test the two possible charging schemes.

Case 1: Assume the car was rented for 5 hours or less (\(H \leq 5\)).
The charge is the greater of (Rs. 60 per hour) and (Rs. 12 per km).

Charge based on hours = \(60 \times H\).
Charge based on distance = \(12 \times 30 = Rs. 360\).
The final charge is \(\max(60H, 360)\).
We are told the final charge was Rs. 300.
This leads to a contradiction, because the charge must be at least Rs. 360 (the minimum possible charge under this scheme for this distance). Since 300 \(<\) 360, this case is impossible.


Case 2: Assume the car was rented for more than 5 hours (\(H \)>\( 5\)).
The charge is the greater of (Rs. 50 per hour) and (Rs. 7.50 per km).

Charge based on hours = \(50 \times H\).
Charge based on distance = \(7.50 \times 30 = Rs. 225\).
The final charge is \(\max(50H, 225)\).
We know the final charge was Rs. 300.
So, \(\max(50H, 225) = 300\).
This equation tells us two things:
1) \(50H\) must be equal to 300 (since 225 is not equal to 300).
2) The 'whichever is more' condition must be satisfied, meaning \(50H \geq 225\).
Let's solve for H: \(50H = 300 \implies H = \frac{300}{50} = 6\) hours.
Now let's check if this value of H is consistent with the conditions for this case.
- Is \(H \)>\( 5\)? Yes, \(6 \)>\( 5\).
- Is \(50H \geq 225\)? Yes, \(300 \geq 225\).

Both conditions are met. So, Akil rented the car for 6 hours. \[ \boxed{(3) 6 hr} \] Quick Tip: When a problem has multiple cases or schemes, test each case separately. Assume one case is true, solve for the variable, and then check if the solution is consistent with the initial assumption for that case.

Let the child have intended to sum the first \(n\) natural numbers.
The correct sum would be \(S_n = \frac{n(n+1)}{2}\).
The child's sum, 575, is the correct sum minus one missing number, \(k\), where \(1 \leq k \leq n\).
So, \(S_n - k = 575\). This means the correct sum, \(S_n\), must be slightly greater than 575.

Let's find the value of \(n\) for which the sum is close to 575.
We can estimate \(n\) from \(\frac{n^2}{2} \approx 575 \implies n^2 \approx 1150\). Since \(30^2=900\) and \(35^2=1225\), \(n\) should be around 34.
Let's test values of \(n\) near 34.

If \(n=33\), \(S_{33} = \frac{33 \times 34}{2} = 33 \times 17 = 561\).
If the correct sum was 561, the missed number would be \(k = 561 - 575 = -14\), which is impossible.
If \(n=34\), \(S_{34} = \frac{34 \times 35}{2} = 17 \times 35 = 595\).
If the correct sum was 595, the missed number would be \(k = 595 - 575 = 20\).
Is this a valid solution? The missed number \(k=20\) must be less than or equal to \(n=34\). Since \(20 \leq 34\), this is a valid solution.
If \(n=35\), \(S_{35} = \frac{35 \times 36}{2} = 35 \times 18 = 630\).
If the correct sum was 630, the missed number would be \(k = 630 - 575 = 55\).
This is not a valid solution because the missed number \(k=55\) cannot be greater than \(n=35\).

The only valid scenario is that the child summed up to \(n=34\) and missed the number 20.
(Note: The provided answer key `(3) 15` is incorrect). \[ \boxed{(4) 20} \] Quick Tip: For "missed number" problems, first estimate the true sum, which must be greater than the incorrect sum. Use the formula for the series to find the integer 'n' that gives a sum slightly larger than the reported sum, then find the difference to identify the missing number.

The relationship between \([x]+[y]+[x+y]\) and \([2x]+[2y]\) is not a standard, simple inequality. Let's test some values for \(x\) and \(y\).
Let \(x = 0.6, y = 0.7\).

\(L(0.6, 0.7) = [0.6] + [0.7] + [0.6+0.7] = 0 + 0 + [1.3] = 1\).
\(R(0.6, 0.7) = [2 \times 0.6] + [2 \times 0.7] = [1.2] + [1.4] = 1+1=2\).
In this case, \(L \)<\( R\).

Let \(x=1.2, y=1.3\).

\(L(1.2, 1.3) = [1.2] + [1.3] + [2.5] = 1+1+2=4\).
\(R(1.2, 1.3) = [2.4] + [2.6] = 2+2=4\).
In this case, \(L = R\).

Let \(x=1.6, y=1.8\).

\(L(1.6, 1.8) = [1.6] + [1.8] + [3.4] = 1+1+3 = 5\).
\(R(1.6, 1.8) = [3.2] + [3.6] = 3+3=6\).
In this case, \(L \)<\( R\).

The original question was likely a variation of Hermite's identity, which states \([nx] \ge n[x]\).
Let's analyze the original question's functions: \(L(x,y) = [x]+[y]\) and \(R(x,y)=[2x]+[2y]\).
And the question "it is impossible to find x, y for which L\(>\)R".
So, is \([x]+[y] \)>\( [2x]+[2y]\) impossible?
Let \(x=1.6, y=1.8\). \([1.6]+[1.8] = 1+1=2\). \([3.2]+[3.6]=3+3=6\). Here \(L\)<\(R\).
Let \(x=1.2, y=1.3\). \([1.2]+[1.3]=1+1=2\). \([2.4]+[2.6]=2+2=4\). Here \(L\)<\(R\).
It seems that \(L(x,y) \)>\( R(x,y)\) is indeed impossible.
Let's try to prove it.
We know that for any real number \(z\), \([2z] \ge 2[z]\). But also, \([2z] \ge [z]+[z+1/2]\).
Let \(x = I_x + f_x\) and \(y=I_y+f_y\) where \(I\) is integer part and \(f\) is fractional part. \(L = I_x+I_y\). \(R = [2I_x+2f_x] + [2I_y+2f_y] = 2I_x + [2f_x] + 2I_y + [2f_y] = 2(I_x+I_y) + [2f_x]+[2f_y]\).
We want to check if \(I_x+I_y \)>\( 2(I_x+I_y) + [2f_x]+[2f_y]\) is possible.
This simplifies to \(-(I_x+I_y) \)>\( [2f_x]+[2f_y]\).
Since \(x, y\) are positive, \(I_x, I_y \ge 0\). So the LHS is \(\le 0\). The RHS is also \(\ge 0\). A negative number cannot be greater than a non-negative number. Thus, it is impossible. \[ \boxed{(4) L(x, y) \(>\) R(x, y)} \] Quick Tip: To analyze inequalities with greatest integer functions, it's often useful to represent a number \(x\) as its integer part plus its fractional part, \(x = I+f\). Then apply the properties of integers and fractional parts (where \(0 \le f \)<\( 1\)).

This is a standard problem in combinatorial geometry. Let \(R(n)\) be the maximum number of regions a plane is divided into by \(n\) lines.

With 0 lines, there is 1 region. \(R(0)=1\).
The 1st line divides the plane into 2 regions. \(R(1)=2\).
The 2nd line intersects the first line at one point, passing through 2 existing regions and dividing each into two. This adds 2 new regions. \(R(2) = R(1)+2 = 4\).
The 3rd line intersects the previous two lines at two distinct points, passing through 3 existing regions. This adds 3 new regions. \(R(3) = R(2)+3 = 7\).
In general, the \(n^{th}\) line will intersect the previous \(n-1\) lines at \(n-1\) distinct points, passing through \(n\) existing regions and adding \(n\) new regions.
This gives the recurrence relation: \(R(n) = R(n-1) + n\).

We can find the total number of regions by summing this up: \(R(n) = R(0) + 1 + 2 + 3 + \dots + n = 1 + \frac{n(n+1)}{2}\).
This formula is also sometimes written as \(\binom{n}{0} + \binom{n}{1} + \binom{n}{2}\).

For \(n=10\): \[ R(10) = 1 + \frac{10(10+1)}{2} = 1 + \frac{10 \times 11}{2} = 1 + 55 = 56 \]
The total number of regions is 56.
(Note: The provided answer key `(2) 255` is incorrect). \[ \boxed{(1) 56} \] Quick Tip: Memorize the formula for the maximum number of regions formed by \(n\) lines in a plane: \(R(n) = \frac{n(n+1)}{2} + 1\). This is a common question in combinatorics.

We need to find the value of \(2^{256} \pmod{17}\).
We can use Fermat's Little Theorem, which states that if \(p\) is a prime number, then for any integer \(a\) not divisible by \(p\), we have \(a^{p-1} \equiv 1 \pmod{p}\).

Step 1: Apply Fermat's Little Theorem.

Here, the divisor is \(p=17\), which is a prime number.
The base is \(a=2\), which is not divisible by 17.
Therefore, we can apply the theorem:
\[ 2^{17-1} \equiv 1 \pmod{17} \]
\[ 2^{16} \equiv 1 \pmod{17} \]


Step 2: Reduce the exponent.

We need to evaluate \(2^{256}\). We can use the property we just found.
We can rewrite the exponent 256 in terms of 16.
\[ 256 = 16 \times 16 \]
So, we can write \(2^{256}\) as \((2^{16})^{16}\).
Now, we can find the remainder:
\[ (2^{16})^{16} \pmod{17} \]
Since we know \(2^{16} \equiv 1 \pmod{17}\), we can substitute this in:
\[ \equiv (1)^{16} \pmod{17} \]
\[ \equiv 1 \pmod{17} \]

The remainder when \(2^{256}\) is divided by 17 is 1. \[ \boxed{(1) 1} \] Quick Tip: Fermat's Little Theorem (\(a^{p-1} \equiv 1 \pmod{p}\)) is extremely powerful for reducing large exponents in modular arithmetic when the modulus is a prime number.

To find the number of real roots, we first need to convert the given rational equation into a polynomial equation.
The domain of the equation requires that \(x \neq 0\) and \(x \neq 1\).
Multiply the entire equation by the common denominator, \(x(x-1)\): \[ x(x-1) \left( \frac{A^2}{x} + \frac{B^2}{x-1} \right) = 1 \cdot x(x-1) \] \[ A^2(x-1) + B^2x = x^2 - x \]
Now, expand and rearrange the terms to form a standard quadratic equation of the form \(ax^2+bx+c=0\). \[ A^2x - A^2 + B^2x = x^2 - x \] \[ x^2 - x - A^2x - B^2x + A^2 = 0 \] \[ x^2 - (1 + A^2 + B^2)x + A^2 = 0 \]
This is a quadratic equation. The number of real roots is determined by the discriminant, \(\Delta = b^2 - 4ac\).

\(a = 1\)
\(b = -(1 + A^2 + B^2)\)
\(c = A^2\)

Let's calculate the discriminant: \[ \Delta = (-(1 + A^2 + B^2))^2 - 4(1)(A^2) \] \[ \Delta = (1 + A^2 + B^2)^2 - 4A^2 \]
This is a difference of squares, \((X^2 - Y^2) = (X-Y)(X+Y)\), where \(X=1+A^2+B^2\) and \(Y=2A\). \[ \Delta = (1 + A^2 + B^2 - 2A)(1 + A^2 + B^2 + 2A) \]
Rearranging the terms inside the parentheses: \[ \Delta = ((A^2 - 2A + 1) + B^2)((A^2 + 2A + 1) + B^2) \] \[ \Delta = ((A-1)^2 + B^2)((A+1)^2 + B^2) \]
Since \(A\) and \(B\) are real numbers, \((A-1)^2 \geq 0\), \((A+1)^2 \geq 0\), and \(B^2 \geq 0\).
The term \(((A+1)^2 + B^2)\) is always positive (it can only be zero if \(A=-1\) and \(B=0\)).
The term \(((A-1)^2 + B^2)\) is always positive (it can only be zero if \(A=1\) and \(B=0\)).
The problem states that A and B are not *simultaneously* zero. This means the product \(\Delta\) is strictly positive.
Since \(\Delta \)>\( 0\), the quadratic equation always has two distinct real roots. We must also ensure that these roots are not 0 or 1. If \(x=0\), the equation becomes \(A^2=0\), so \(A=0\). If \(x=1\), we get \(1-(1+A^2+B^2)+A^2=0 \implies -B^2=0 \implies B=0\). If both A and B are 0, the original equation is undefined. If only A=0, the root is 0, but then the term \(A^2\)/x is problematic. If B=0, the root is 1, term \(B^2\)/(x-1) is problematic. In all cases where a root would be invalid, A or B is zero.
Assuming A and B are non-zero, the roots are valid and there are always two. \[ \boxed{(3) 2} \] Quick Tip: To determine the nature of roots of an equation, first transform it into a standard polynomial form. Then, analyze its discriminant (\(\Delta = b^2 - 4ac\)). If \(\Delta \)>\( 0\), there are two distinct real roots.

Let the three words be W1 (MODERN), W2 (BOOK), and W3 (STORE).
The intervals at which they flash are:

W1: \(2.5\) s
W2: \(1.25\) s
W3: \(0.5\) s

"The full name...can be read again" implies we are looking for the first time \(t \)>\( 0\) when all three lights flash on simultaneously. This is a classic Least Common Multiple (LCM) problem.
We need to find the LCM of the three time intervals.
First, convert the decimals to fractions: \[ 2.5 = \frac{5}{2}, \quad 1.25 = \frac{5}{4}, \quad 0.5 = \frac{1}{2} \]
The LCM of fractions is given by the formula: LCM\((\frac{a}{b}, \frac{c}{d}) = \frac{LCM(a, c)}{HCF(b, d)}\). \[ LCM \left( \frac{5}{2}, \frac{5}{4}, \frac{1}{2} \right) = \frac{LCM(5, 5, 1)}{HCF(2, 4, 2)} \]

LCM of the numerators (5, 5, 1) is 5.
Highest Common Factor (HCF) of the denominators (2, 4, 2) is 2.
\[ LCM = \frac{5}{2} = 2.5 seconds \]
This means that all three lights will turn on together every 2.5 seconds. The first time this happens after the initial flash at t=0 is at t=2.5 seconds.
The fact that they stay on for 1 second is relevant to ensure they can be "read", but the synchronization point is determined by the start of the flash.
The calculated answer of 2.5s is not in the options, which indicates the original question's numbers and context are deeply flawed. For instance, the provided answer key `(2) 73.5 s` has no logical connection to the problem statement. The original prompt had intervals of 1/2, 1/4, 1/8s which gives an LCM of 1/2s. The question is unsolvable as stated. \[ \boxed{Question is flawed and unsolvable.} \] Quick Tip: To find when multiple repeating events will occur simultaneously, calculate the Least Common Multiple (LCM) of their individual time intervals.

Step 1: Convert weights to improper fractions.
The weights of the three cakes are:

Cake 1: \(4\frac{1}{2} = \frac{9}{2}\) lbs
Cake 2: \(6\frac{3}{4} = \frac{27}{4}\) lbs
Cake 3: \(7\frac{1}{5} = \frac{36}{5}\) lbs


Step 2: Find the weight of each part.
The cakes are to be divided into parts of equal weight, and each part must be "as heavy as possible." This means the weight of each part must be the Highest Common Factor (HCF or GCD) of the weights of the three cakes.
To find the HCF of fractions, we use the formula: HCF\((\frac{a}{b}, \frac{c}{d}, \dots) = \frac{HCF of numerators}{LCM of denominators}\).

HCF of numerators (9, 27, 36):
- Factors of 9: 1, 3, 9
- Factors of 27: 1, 3, 9, 27
- Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
- The HCF is 9.
LCM of denominators (2, 4, 5):
- The least common multiple of 2, 4, and 5 is 20.

So, the weight of each part is \(\frac{HCF(9, 27, 36)}{LCM(2, 4, 5)} = \frac{9}{20}\) lbs.

Step 3: Calculate the number of parts (guests).
The maximum number of guests is the total number of parts we can cut.

Parts from Cake 1: \(\frac{Total Weight}{Part Weight} = \frac{9/2}{9/20} = \frac{9}{2} \times \frac{20}{9} = 10\) parts.
Parts from Cake 2: \(\frac{27/4}{9/20} = \frac{27}{4} \times \frac{20}{9} = 3 \times 5 = 15\) parts.
Parts from Cake 3: \(\frac{36/5}{9/20} = \frac{36}{5} \times \frac{20}{9} = 4 \times 4 = 16\) parts.

Total number of parts (guests) = \(10 + 15 + 16 = 41\).
(Note: The provided answer key `(2) 72` is incorrect). \[ \boxed{(4) 41} \] Quick Tip: "As heavy as possible" for equal parts means finding the Highest Common Factor (HCF). For fractions, HCF = HCF(numerators) / LCM(denominators).

This is a successive division problem. Let the number be \(N\). We work backwards from the last division.
Let the final quotient be \(k\).

The number before the last division by 7 was \(7k + 4\). This was the quotient from the division by 4.
The number before the division by 4 was \(4 \times (quotient) + remainder = 4(7k+4) + 1 = 28k + 16 + 1 = 28k + 17\). This was the quotient from the division by 3.
The original number \(N\) was \(3 \times (quotient) + remainder = 3(28k+17) + 2 = 84k + 51 + 2 = 84k + 53\).

The number \(N\) is of the form \(84k + 53\).
When we divide \(N\) by 84, the quotient will be \(k\) and the remainder will be 53.
The smallest such positive number (when \(k=0\)) is 53.
Let's check:

53 divided by 3 gives quotient 17 and remainder 2. (Correct)
17 divided by 4 gives quotient 4 and remainder 1. (Correct)
4 divided by 7 gives quotient 0 and remainder 4. (Correct)

The remainder when the number is divided by 84 (which is \(3 \times 4 \times 7\)) is 53.
(Note: The provided answer key `(3) 41` is incorrect). \[ \boxed{(4) 53} \] Quick Tip: For successive division problems, work backwards. Start with the last quotient (you can assume it's 0 to find the smallest number) and reconstruct the original number step-by-step using the formula: Dividend = Divisor × Quotient + Remainder.

Let's assume the six people are sitting in a circle, and "left" means counter-clockwise.
Step 1: Determine the initial arrangement.

"Suresh is facing Raghubir" means they are opposite each other. Let's place them at positions 1 and 4 in a 6-seat circle. S(1), R(4).
"Raghubir who is to the left of Ajay". So, moving counter-clockwise from Raghubir, we find Ajay. The order is R(4), A(...), ...
"and to the right of Pramod". So, moving clockwise from Raghubir, we find Pramod. The order is ..., P(...), R(4).
If we assume immediate left/right, we have P(3), R(4), A(5).
"Yogendra is to the left of Pramod". The order is P(3), Y(...), ...
"Ajay is to the left of Dhiraj". The order is A(5), D(...), ...
Let's place the remaining people, Dhiraj and Yogendra, in the remaining seats (2 and 6).
Let's test the positions. Order (clockwise): S(1), D(2), P(3), R(4), A(5), Y(6).
- S(1) faces R(4). Yes.
- R(4) is to the left of A(5)? No, it's to the right. My left/right convention is wrong.

Let's assume "left" is clockwise.

S(1), R(4).
R(4) is to the left of A -\(>\) R, A. So A is at 3. Order: A(3), R(4).
R(4) is to the right of P -\(>\) R, P. So P is at 5. Order: R(4), P(5). This is a contradiction.

Let's re-read: "Suresh is facing Raghubir who is to the left of Ajay and to the right of Pramod." This can be read as Raghubir is between Pramod and Ajay.
Order (clockwise): Pramod, Raghubir, Ajay.

Place them in seats 3, 4, 5: P(3), R(4), A(5).
Suresh faces Raghubir: S(1).
Yogendra is to the left of Pramod(3): Y(2).
Ajay(5) is to the left of Dhiraj: D(6). (This must be a typo in the question, probably "Ajay is to the right of Dhiraj"). Assuming the intended logic is just to fill the spots, the final person is Dhiraj. Let's place him at 6. A(5), D(6).
Let's re-read "Ajay is to the left of Dhiraj." With A(5) and D(6), this works if "left" is clockwise.

Initial position (clockwise): S(1), Y(2), P(3), R(4), A(5), D(6).
This arrangement satisfies all conditions.

Step 2: Perform the seat exchanges.

Dhiraj(6) exchanges with Yogendra(2). New positions: Y is at 6, D is at 2.
Pramod(3) exchanges with Raghubir(4). New positions: R is at 3, P is at 4.
Suresh(1) and Ajay(5) do not move.


Step 3: Determine the final arrangement.
The new order (clockwise) is: S(1), D(2), R(3), P(4), A(5), Y(6).

Step 4: Answer the question.
"who will be sitting to the left of Dhiraj?"
Dhiraj is at seat 2. The person to his left (clockwise, seat 3) is Raghubir. \[ \boxed{(2) Raghubir} \] Quick Tip: For seating arrangement puzzles, draw a diagram (like a circle with numbered seats). Place individuals based on the most restrictive clues first (like "facing each other"). Then, fill in the rest based on relative positions.

We need to count the number of 'Two' orders (add an orange) and the number of 'Four' orders (remove an orange).

Count of 'Two' orders (Orange IN):
Let's go through the sequence: 1 2 3 3 2 1 4 2 3 1 4 2 2 3 3 1 4 1 1 3 2 3 4.
The number '2' appears 6 times. So, 6 oranges were put into the basket.
Count of 'Four' orders (Orange OUT):
Let's go through the sequence: 1 2 3 3 2 1 4 2 3 1 4 2 2 3 3 1 4 1 1 3 2 3 4.
The number '4' appears 4 times. So, 4 oranges were taken out of the basket.

Final count of oranges: \[ Oranges = (Oranges IN) - (Oranges OUT) = 6 - 4 = 2 \]
There were 2 oranges in the basket at the end. \[ \boxed{(4) 2} \] Quick Tip: For sequence-based counting problems, it's more efficient to tally the occurrences of each command type first, rather than keeping a running count through the sequence.

We need to find the final count for each type of fruit and then add them together.
From the previous question, we already have the counts of the orders:

Order 'One' (Mango IN): 6 times
Order 'Two' (Orange IN): 6 times
Order 'Three' (Apple IN): 7 times
Order 'Four' (Mango OUT, Orange OUT): 4 times

Now, let's calculate the final count for each fruit:

Mangoes: 6 IN - 4 OUT = 2 mangoes.
Oranges: 6 IN - 4 OUT = 2 oranges.
Apples: 7 IN - 0 OUT = 7 apples.

Total Fruits: \[ Total = Mangoes + Oranges + Apples = 2 + 2 + 7 = 11 \]
There will be a total of 11 fruits in the basket. \[ \boxed{(2) 11} \] Quick Tip: Break down the problem by the type of item. Calculate the final count for each item individually before summing them up for the total.

The problem asks for the number of arrangements of 4 distinct letters chosen from the set of 11 symmetric letters. This is a permutation problem.

Number of available symmetric letters = 11.
Length of the password = 4.
Repetition is not allowed.

The number of ways to arrange 4 items chosen from 11 is given by the permutation formula \(P(n, k) = \frac{n!}{(n-k)!}\). \[ P(11, 4) = \frac{11!}{(11-4)!} = \frac{11!}{7!} = 11 \times 10 \times 9 \times 8 \] \[ 11 \times 10 \times 9 \times 8 = 110 \times 72 = 7920 \]
There are 7,920 possible passwords. \[ \boxed{(1) 7,920} \] Quick Tip: When order matters and repetition is not allowed, use the permutation formula or the multiplication principle (11 choices for the first position, 10 for the second, etc.).

This is a problem that is best solved using the complementary counting principle.
Total Passwords = (Passwords with at least one symmetric letter) + (Passwords with NO symmetric letters).
Therefore, (Passwords with at least one symmetric letter) = Total Passwords - (Passwords with NO symmetric letters).

Step 1: Calculate the total number of 3-letter passwords with no repetition.

Total letters in the alphabet = 26.
Number of ways to form a 3-letter password = \(P(26, 3) = 26 \times 25 \times 24 = 15,600\).


Step 2: Calculate the number of passwords with NO symmetric letters.
This means we can only use the asymmetric letters.

Number of symmetric letters = 11.
Number of asymmetric letters = \(26 - 11 = 15\).
Number of ways to form a 3-letter password using only asymmetric letters = \(P(15, 3) = 15 \times 14 \times 13 = 2730\).


Step 3: Calculate the number of passwords with AT LEAST ONE symmetric letter. \[ At least one symmetric = Total - None symmetric \] \[ = 15,600 - 2,730 = 12,870 \]
(Note: The provided answer key `(2) 2,730` gives the count for passwords with *no* symmetric letters, which is the opposite of what the question asks for). \[ \boxed{(3) 12,870} \] Quick Tip: For counting problems with the phrase "at least one," it is almost always easier to calculate the total number of possibilities and subtract the number of possibilities for the complementary case ("none").

Let the length of the tunnel AB be \(L\). The cat is at a point C, where the distance \(AC = \frac{3}{8}L\). The distance to the other end is \(CB = L - \frac{3}{8}L = \frac{5}{8}L\).
Let the train be at a distance \(D\) from the entrance A when it whistles. Let the speed of the train be \(v_t\) and the speed of the cat be \(v_c\).

Case 1: Cat runs to entrance A.
The train and cat arrive at point A at the same time. The time taken is equal for both. \[ Time = \frac{Distance}{Speed} \] \[ t = \frac{D}{v_t} = \frac{AC}{v_c} = \frac{3L/8}{v_c} \quad (Equation 1) \]

Case 2: Cat runs to exit B.
The train and cat arrive at point B at the same time. The time taken is equal for both.
The distance the train travels is \(D+L\). The distance the cat travels is \(CB = \frac{5}{8}L\). \[ t' = \frac{D+L}{v_t} = \frac{CB}{v_c} = \frac{5L/8}{v_c} \quad (Equation 2) \]

Solving for the speed ratio:
From Equation 1, we can express the distance \(D\) in terms of the other variables: \[ D = v_t \times \frac{3L}{8v_c} \]
Now substitute this expression for \(D\) into Equation 2: \[ \frac{(v_t \times \frac{3L}{8v_c}) + L}{v_t} = \frac{5L/8}{v_c} \]
Divide the entire equation by \(L\) (since \(L \neq 0\)): \[ \frac{(v_t \times \frac{3}{8v_c}) + 1}{v_t} = \frac{5/8}{v_c} \] \[ \frac{3}{8v_c} + \frac{1}{v_t} = \frac{5}{8v_c} \]
Rearrange the terms to solve for the ratio \(\frac{v_t}{v_c}\): \[ \frac{1}{v_t} = \frac{5}{8v_c} - \frac{3}{8v_c} = \frac{2}{8v_c} = \frac{1}{4v_c} \] \[ 4v_c = v_t \] \[ \frac{v_t}{v_c} = 4 \]
The ratio of the speed of the train to the speed of the cat is 4:1. \[ \boxed{(2) 4 : 1} \] Quick Tip: In relative speed problems where time is the common factor, set up equations of the form \(D_1/S_1 = D_2/S_2\). Creating a system of two such equations often allows you to eliminate unknown distances and solve for the ratio of speeds.

Let the lengths of the three pieces be \(L\) (longest), \(M\) (middle-sized), and \(S\) (shortest).

Step 1: Translate the relationships into equations.

\(L + M + S = 40\) (Total length)
\(L = 3M\) (Longest and middle)
\(S = L - 23\) (Shortest and longest)


Step 2: Solve the system of equations.
It's easiest to express all lengths in terms of one variable. Let's use \(L\).

From (2), \(M = \frac{L}{3}\).
From (3), \(S = L - 23\).

Now substitute these into the first equation: \[ L + \left(\frac{L}{3}\right) + (L - 23) = 40 \]
Combine the terms with \(L\): \[ 2L + \frac{L}{3} - 23 = 40 \] \[ \frac{6L + L}{3} = 40 + 23 \] \[ \frac{7L}{3} = 63 \] \[ 7L = 189 \] \[ L = \frac{189}{7} = 27 \]
So, the longest piece is 27 cm.

Step 3: Find the length of the shortest piece.
The question asks for the length of the shortest piece, \(S\). \[ S = L - 23 = 27 - 23 = 4 cm \]
(For completeness, the middle piece is \(M = L/3 = 27/3 = 9\) cm. Check: \(27+9+4=40\). Correct.)
(Note: The provided answer key `(4) 9` gives the length of the middle piece, not the shortest). \[ \boxed{(3) 4} \] Quick Tip: In word problems with multiple relationships, choose one variable to be your "base" and express all other variables in terms of that base variable. This simplifies the final equation.

The key to this problem is that the payment should be proportional to the amount of bread each person *gave up* for the third traveller, not based on how much they had initially.

Step 1: Determine each person's fair share of bread.

There are a total of \(5 + 3 = 8\) loaves of bread.
These are shared equally among 3 travellers.
Each person's fair share is \(\frac{8}{3}\) loaves.


Step 2: Calculate how much bread each of the first two travellers contributed to the third.

First Traveller (T1):
- Had 5 loaves.
- Ate his share of \(\frac{8}{3}\) loaves.
- Amount he contributed to the third traveller = \(5 - \frac{8}{3} = \frac{15}{3} - \frac{8}{3} = \frac{7}{3}\) loaves.
Second Traveller (T2):
- Had 3 loaves.
- Ate his share of \(\frac{8}{3}\) loaves.
- Amount he contributed to the third traveller = \(3 - \frac{8}{3} = \frac{9}{3} - \frac{8}{3} = \frac{1}{3}\) loaves.


Step 3: Distribute the 8 coins based on the contribution ratio.

The third traveller (T3) ate \(\frac{8}{3}\) loaves and paid 8 coins for it.
This total contribution of \(\frac{8}{3}\) loaves was provided by T1 and T2.
The ratio of their contributions is T1 : T2 = \(\frac{7}{3} : \frac{1}{3}\), which simplifies to 7 : 1.
The 8 coins should be split in this 7:1 ratio.
First traveller's share of coins = \(\frac{7}{7+1} \times 8 = \frac{7}{8} \times 8 = 7\) coins.
Second traveller's share of coins = \(\frac{1}{7+1} \times 8 = \frac{1}{8} \times 8 = 1\) coin.

The first traveller should get 7 coins. The second traveller's suggestion to split it 5:3 was incorrect because it was based on initial possession, not contribution. \[ \boxed{(2) 7} \] Quick Tip: In fair division problems, payment should always be based on the net amount of goods a person provides to others, not on their initial holdings. Calculate each person's fair share first, then find out who was a net giver and by how much.

Let's place the triangle on a coordinate plane to find the coordinates of the incenters P and Q.

Let the vertex with the right angle, C, be at the origin (0, 0).
Let A be on the y-axis and B be on the x-axis. So, A = (0, 15) and B = (20, 0).
The hypotenuse is the line segment AB. The line equation is \(\frac{x}{20}+\frac{y}{15}=1 \implies 3x+4y=60\).
The altitude CD is a line from C(0,0) perpendicular to AB. Its equation is \(y = \frac{4}{3}x\).
The point D is the intersection of these two lines. \(3x+4(\frac{4}{3}x)=60 \implies 3x+\frac{16x}{3}=60 \implies \frac{25x}{3}=60 \implies x=\frac{180}{25}=7.2\). Then \(y=\frac{4}{3}(7.2)=9.6\). So, D = (7.2, 9.6).

The vertices of the two smaller right-angled triangles are:

\(\triangle ACD\): Vertices are A(0,15), C(0,0), D(7.2, 9.6). This is not a right-angled triangle with legs on the axes. This method is too complex.


Let's use a simpler geometric approach.

Find side lengths of smaller triangles: \(\triangle ACD\) and \(\triangle BCD\) are similar to \(\triangle ABC\).
- Area of \(\triangle ABC = \frac{1}{2} \times 15 \times 20 = 150\).
- Also, Area = \(\frac{1}{2} \times AB \times CD = \frac{1}{2} \times 25 \times CD\).
- So, \(CD = \frac{2 \times 150}{25} = 12\).
- In right \(\triangle ACD\): \(AD^2 = AC^2 - CD^2 = 15^2 - 12^2 = 225 - 144 = 81 \implies AD=9\).
- In right \(\triangle BCD\): \(BD^2 = BC^2 - CD^2 = 20^2 - 12^2 = 400 - 144 = 256 \implies BD=16\). (Check: AD+BD = 9+16=25=AB. Correct).
Find inradii of smaller triangles: The inradius \(r\) of a right-angled triangle with legs \(p,q\) and hypotenuse \(h\) is \(r = \frac{p+q-h}{2}\).
- For \(\triangle ACD\) (sides 9, 12, 15): \(r_P = \frac{9+12-15}{2} = \frac{6}{2} = 3\).
- For \(\triangle BCD\) (sides 12, 16, 20): \(r_Q = \frac{12+16-20}{2} = \frac{8}{2} = 4\).
Find coordinates of incenters P and Q: Let's place D at the origin (0,0).
- Place C on the negative x-axis, so C = (-12, 0).
- Place A on the positive y-axis, so A = (0, 9).
- Place B on the negative y-axis, so B = (0, -16).
- The incenter of a right triangle with vertices at (0,0), (p,0), (0,q) is at \((r,r)\) in the quadrant of the triangle.
- For \(\triangle ACD\) (legs DC=12, DA=9): The incenter P is at \((-r_P, r_P) = (-3, 3)\).
- For \(\triangle BCD\) (legs DC=12, DB=16): The incenter Q is at \((-r_Q, -r_Q) = (-4, -4)\).
Calculate distance PQ: Use the distance formula.
\[ PQ^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 \]
\[ PQ^2 = (-4 - (-3))^2 + (-4 - 3)^2 = (-1)^2 + (-7)^2 = 1 + 49 = 50 \]
\[ PQ = \sqrt{50} \]
\[ \boxed{(2) \sqrt{50}} \] Quick Tip: To find the distance between two points in a complex geometric figure, it's often easiest to place the figure on a coordinate plane. Choose a convenient point as the origin (like the vertex of a right angle) and calculate the coordinates of the points of interest.

The equation is \(u^m + v^m = w^m\), where \(u,v,w,m\) are natural numbers (positive integers).

Fermat's Last Theorem states that there are no natural number solutions for this equation when the exponent \(m\) is greater than 2 (\(m \)>\( 2\)).
Case m = 1: The equation is \(u+v=w\). This has infinite solutions. For example, \(u=3, v=4, w=7\). Here, \(m=1\). Let's check the options for this case.
- (1) \(m \ge \min(3,4,7) \implies 1 \ge 3\) (False).
- (2) \(m \)>\( \max(3,4,7) \implies 1 \)>\( 7\) (False).
- (3) \(m \)<\( \min(3,4,7) \implies 1 \)<\( 3\) (True for this example).
Case m = 2: The equation is \(u^2+v^2=w^2\). This is the Pythagorean theorem, which has infinite integer solutions (Pythagorean triples). For example, \(u=3, v=4, w=5\). Here, \(m=2\).
- (1) \(m \ge \min(3,4,5) \implies 2 \ge 3\) (False).
- (2) \(m \)>\( \max(3,4,5) \implies 2 \)>\( 5\) (False).
- (3) \(m \)<\( \min(3,4,5) \implies 2 \)<\( 3\) (True for this example).

We have found that for \(m=1\) and \(m=2\), option (3) holds true for our examples. But is it *necessarily* true? Let's try another example for \(m=1\). Let \(u=1, v=1, w=2\). Here \(m=1, \min(u,v,w)=1\). The condition \(m \)<\( \min(u,v,w)\) becomes \(1 \)<\( 1\), which is false.
Since none of the statements are true for *all* possible solutions, none of them is a necessary truth. \[ \boxed{(4) None of these} \] Quick Tip: When testing a statement's necessary truth in number theory, always check for edge cases or simple counterexamples. The existence of just one counterexample is enough to prove a statement is not universally true.

This problem can be solved by choosing the white square first and then counting the available black squares.

Step 1: Choose a white square.

A standard chessboard has 64 squares.
Half of them are white, and half are black.
Number of ways to choose a white square = 32.


Step 2: Choose a black square not in the same row or column.

Let's say we have chosen a white square. This square occupies one row and one column.
The total number of black squares on the board is 32.
We must exclude the black squares that are in the same row or same column as the chosen white square.
In any given row on a chessboard, there are 4 white and 4 black squares. So, there are 4 black squares in the same row.
In any given column, there are 4 white and 4 black squares. So, there are 4 black squares in the same column.
The set of squares in a row and the set of squares in a column are disjoint except for the square at their intersection. Since we chose a white square, the black squares in its row and the black squares in its column are completely distinct sets.
Number of forbidden black squares = (Black squares in the same row) + (Black squares in the same column) = \(4 + 4 = 8\).
Number of available black squares for the second choice = Total black squares - Forbidden black squares = \(32 - 8 = 24\).


Step 3: Calculate the total number of ways.

Total ways = (Ways to choose a white square) \(\times\) (Ways to choose a valid black square).
Total ways = \(32 \times 24\).
\(32 \times 24 = 32 \times (20 + 4) = 640 + 128 = 768\).

There are 768 ways to make such a choice. \[ \boxed{(4) 768} \] Quick Tip: Use the multiplication principle for sequential choices. For the second choice, calculate the total possibilities and subtract the ones that are disallowed by the first choice.

We use the algebraic identity \(a^k - b^k = (a-b)(a^{k-1} + a^{k-2}b + \dots + b^{k-1})\), which implies that \(a^k - b^k\) is always divisible by \(a-b\).

Step 1: Use the difference of squares/cubes.
Let the expression be \(E = 7^{6n} - 6^{6n}\). We can write this as \((7^6)^n - (6^6)^n\).
This is divisible by \(7^6 - 6^6\).
Let's factor \(7^6 - 6^6\):

As a difference of squares: \((7^3)^2 - (6^3)^2 = (7^3 - 6^3)(7^3 + 6^3)\).
- \(7^3 = 343\).
- \(6^3 = 216\).
- \(7^3 - 6^3 = 343 - 216 = 127\).
- \(7^3 + 6^3 = 343 + 216 = 559\).
So, \(7^6 - 6^6 = 127 \times 559\).

Since \(E\) is divisible by \(7^6-6^6\), it must be divisible by its factors, 127 and 559.
This confirms that the expression is divisible by the numbers in options (2) and (3).

Step 2: Check divisibility by 13.
We need to check if either 127 or 559 is divisible by 13.

\(127 \div 13 = 9\) with a remainder of \(10\). So 127 is not divisible by 13.
\(559 \div 13\). Let's do the division: \(559 = 13 \times 40 + 39 = 13 \times 40 + 13 \times 3 = 13 \times 43\).
Since \(559 = 13 \times 43\), 559 is divisible by 13.

Because the expression is divisible by 559, and 559 is divisible by 13, the entire expression must be divisible by 13.

Conclusion:
The expression \(7^{6n} - 6^{6n}\) is divisible by 13, 127, and 559. Therefore, it is divisible by all of them. \[ \boxed{(4) All of these} \] Quick Tip: For expressions of the form \(a^k - b^k\), immediately use factorization identities. The difference of squares (\(x^2-y^2=(x-y)(x+y)\)) and difference of cubes (\(x^3-y^3=(x-y)(...)\)) are the most useful.

We are given \(pqr=1\). We will manipulate the terms in the expression to find a common denominator. The key is to use the condition \(pqr=1\) to substitute for the inverse terms.

\(q^{-1} = 1/q\). From \(pqr=1\), we have \(pr=1/q\). So, \(q^{-1} = pr\).
\(r^{-1} = 1/r = pq\).
\(p^{-1} = 1/p = qr\).

Let's substitute these into the denominators of the second and third terms.

Term 1: Stays as it is. \[ T_1 = \frac{1}{1 + p + pr} \]
(Note: I substituted \(q^{-1}=pr\) to make it consistent with the other terms).

Term 2: \[ T_2 = \frac{1}{1 + q + r^{-1}} = \frac{1}{1 + q + pq} \]
Multiply numerator and denominator by \(pr\). Since \(pqr=1\), \(pr=1/q\). Let's multiply by \(r^{-1}=pq\). No. Let's multiply by \(p\).
This is not working. Let's try to make the denominators the same.
Let's make them all look like the first denominator, \(1+p+pr\).

Term 2: \[ T_2 = \frac{1}{1 + q + pq} \]
We have \(q=1/pr\).
This is getting messy. Let's restart with a different substitution.

Let's work with the given terms: \(q^{-1}, r^{-1}, p^{-1}\).
Term 1: \[ T_1 = \frac{1}{1 + p + q^{-1}} \]
Term 2: \[ T_2 = \frac{1}{1 + q + r^{-1}} \]
Multiply numerator and denominator by \(q^{-1}\): \[ T_2 = \frac{q^{-1}}{q^{-1}(1 + q + r^{-1})} = \frac{q^{-1}}{q^{-1} + 1 + q^{-1}r^{-1}} = \frac{q^{-1}}{1 + q^{-1} + (qr)^{-1}} \]
Since \(pqr=1\), \((qr)^{-1} = p\).
So, \(T_2 = \frac{q^{-1}}{1 + q^{-1} + p}\). This has the same denominator as \(T_1\).

Term 3: \[ T_3 = \frac{1}{1 + r + p^{-1}} \]
Multiply numerator and denominator by \(p\): \[ T_3 = \frac{p}{p(1+r+p^{-1})} = \frac{p}{p+pr+1} \]
We know \(q^{-1} = pr\). So the denominator is \(p + q^{-1} + 1\). This also has the same denominator.

Summing the terms: \[ E = T_1 + T_2 + T_3 = \frac{1}{1 + p + q^{-1}} + \frac{q^{-1}}{1 + q^{-1} + p} + \frac{p}{p+q^{-1}+1} \]
Since all terms have the same denominator, we add the numerators: \[ E = \frac{1 + q^{-1} + p}{1 + p + q^{-1}} = 1 \] \[ \boxed{(3) 1} \] Quick Tip: For symmetric expressions with a constraint like \(pqr=1\), the answer is often a simple constant like 0, 1, or -1. The trick is usually to manipulate the fractions by multiplying the numerator and denominator by one of the variables to transform the denominators into a common form.

Step 1: Calculate the total work required.

The total work is measured in "technician-hours".
Total Work = 6 technicians \(\times\) 10 hours = 60 technician-hours.


Step 2: Calculate the work done before new technicians are added.

The initial 6 technicians work from 11 am to 5 pm.
Duration = 6 hours.
Work Done = 6 technicians \(\times\) 6 hours = 36 technician-hours.


Step 3: Calculate the remaining work.

Remaining Work = Total Work - Work Done = \(60 - 36 = 24\) technician-hours.


Step 4: Track the work done hour by hour after 5 pm.

From 5 pm to 6 pm: One technician is added. Total technicians = \(6+1=7\).
- Work done in this hour = 7 technicians \(\times\) 1 hour = 7 technician-hours.
- Work still remaining = \(24 - 7 = 17\) technician-hours.
From 6 pm to 7 pm: Another technician is added. Total technicians = \(7+1=8\).
- Work done in this hour = 8 technicians \(\times\) 1 hour = 8 technician-hours.
- Work still remaining = \(17 - 8 = 9\) technician-hours.
After 7 pm: Another technician is added. Total technicians = \(8+1=9\).
- The rate of work is now 9 technician-hours per hour.
- Time required to complete the remaining 9 technician-hours = \(\frac{Work Remaining}{Rate} = \frac{9 tech-hours}{9 techs} = 1\) hour.


Step 5: Determine the completion time.
The work will be completed 1 hour after 7 pm, which is 8:00 pm.
(Note: The provided answer key `(3) 7:20 pm` is incorrect). \[ \boxed{(4) 8:00 pm} \] Quick Tip: In work problems where the rate of work changes over time, calculate the work done in each interval where the rate is constant. Subtract this from the remaining work and proceed to the next interval.

Let the number of samosas be \(x\). We want to maximize the revenue, \(R\).

Step 1: Define the price and revenue functions for \(x \)>\( 200\).

Let \(k\) be the number of additional lots of 20 samosas.
Then the number of samosas is \(x = 200 + 20k\).
The price reduction is 10 paise (or Rs. 0.10) for each lot \(k\).
The price per samosa, \(P(k)\), is \(2 - 0.10k\).
The total revenue, \(R\), is the number of samosas times the price per samosa.
\[ R(k) = x \times P(k) = (200 + 20k)(2 - 0.10k) \]


Step 2: Maximize the revenue function.

Expand the expression for revenue:
\[ R(k) = 200(2) + 200(-0.10k) + 20k(2) + 20k(-0.10k) \]
\[ R(k) = 400 - 20k + 40k - 2k^2 \]
\[ R(k) = -2k^2 + 20k + 400 \]
This is a quadratic function of \(k\) representing a downward-opening parabola. Its maximum value occurs at the vertex.
The vertex of a parabola \(ax^2+bx+c\) is at \(x = -b/(2a)\).
Here, the vertex is at \(k = \frac{-20}{2(-2)} = \frac{-20}{-4} = 5\).

The revenue is maximized when \(k=5\).

Step 3: Find the corresponding box size.

The size of the box is \(x = 200 + 20k\).
Substitute \(k=5\):
\[ x = 200 + 20(5) = 200 + 100 = 300 \]

The maximum revenue is obtained with a box size of 300 samosas. \[ \boxed{(2) 300} \] Quick Tip: When revenue or profit is described by a quadratic function (\(ax^2+bx+c\)), you can find the maximum (if \(a\)<\(0\)) or minimum (if \(a\)>\(0\)) by finding the vertex at \(x=-b/(2a)\), without using calculus.

Let the rate of the large pump be \(R_L\) (in tanks per hour).
The time it takes the large pump alone is \(T_L = \frac{1}{R_L}\).

Step 1: Find the rate of the small pumps.

The rate of one small pump is \(R_S = \frac{2}{3}R_L\).


Step 2: Find the combined rate of all four pumps.

There is one large pump and three small pumps.
Combined Rate, \(R_{total} = R_L + 3 \times R_S\).
Substitute the expression for \(R_S\):
\[ R_{total} = R_L + 3 \times \left(\frac{2}{3}R_L\right) = R_L + 2R_L = 3R_L \]


Step 3: Find the time taken by all four pumps.

The time taken is the reciprocal of the rate.
\(T_{total} = \frac{1}{R_{total}} = \frac{1}{3R_L}\).


Step 4: Find the required fraction.
The question asks for the fraction of the time, which is the ratio \(\frac{T_{total}}{T_L}\). \[ Fraction = \frac{T_{total}}{T_L} = \frac{1/(3R_L)}{1/R_L} = \frac{1}{3} \]
The four pumps together take \(\frac{1}{3}\) of the time that the large pump would have taken alone.
(Note: The provided answer key `(3)` is incorrect). \[ \boxed{(2) \frac{1}{3}} \] Quick Tip: In work-rate problems, time is inversely proportional to the rate of work (\(T=1/R\)). To find the fraction of time, find the ratio of the rates and take its reciprocal. If the combined rate is \(k\) times the single rate, the combined time will be \(1/k\) times the single time.

Let's assign a variable to the lengths to make calculations easier.
Let \(KL = x\). Based on the given equalities:

\(4KL = 4x\).
\(FK = 4x\).
\(2HK = 4x \implies HK = 2x\).
\(MN = 4x\).
\(2LM = 4x \implies LM = 2x\).

From the diagram, FKLG seems to be a composite shape. Let's analyze triangle FOG. O is on the line DG. The line DG is parallel to the base BM. FI is an altitude from F to DG.

FI = KL = x.
IG = KL. This is not necessarily true. IG = KL - HG. We don't have enough info.

Let's try another approach. Consider the coordinates of the points.
Let L = (0,0). Then M = (2x, 0), G = (0, GL), K = (-x, 0), F = (-x, FK) = (-x, 4x).
Since \(\angle GLM = 90^\circ\), GL is on the y-axis and LM is on the x-axis.
Coordinates: L(0,0), M(2x, 0). From \(\angle LMN = 90^\circ\), N is at (2x, MN) = (2x, 4x).
G is on the y-axis. We don't know the length GL.

Let's assume the question implies O is the midpoint of DG. We cannot solve this without more information or assumptions about the relationships between the points. The lines AB, CD, EH, etc. being perpendicular to BM implies they are all vertical, and BM is horizontal. The statement "all other angle relations and lengths are symmetric" is vague.

Let's assume O is on line DG and FI is perpendicular to DG, where I is on DG. Then in right triangle FIG, we need FI and IG to find the angle. We don't have enough information. The problem is ill-defined. \[ \boxed{(4) None of these (Unsolvable)} \] Quick Tip: In geometry problems, if the given information is insufficient to determine lengths and angles required to solve the problem, and there are no obvious geometric theorems that apply, the problem may be unsolvable as stated.

The figure does not define quadrilaterals ABCD and DEFG. Point O is shown, but its relation to the other points is unclear. The segments AB, BC, CD are parts of the overall shape, but do not form a closed quadrilateral ABCD. Similarly for DEFG.
The question is unanswerable because the shapes it refers to are not defined in the diagram. \[ \boxed{(4) None of these (Unsolvable)} \] Quick Tip: Always ensure that the geometric figures mentioned in a question are clearly and unambiguously defined in the provided diagram or text before attempting to calculate properties like area.

We need to count all the numbers that can be formed using the digits {0, 7, 8 that are between 1 and 999,999 inclusive. This means we need to count all the 1-digit, 2-digit, 3-digit, 4-digit, 5-digit, and 6-digit numbers that can be formed.


1-digit numbers: The number must be greater than 0. The possible digits are {7, 8.
Number of 1-digit numbers = 2.
2-digit numbers: The first digit cannot be 0. So there are 2 choices (7 or 8). The second digit can be any of the 3 digits.
Number of 2-digit numbers = \(2 \times 3 = 6\).
3-digit numbers: The first digit has 2 choices. The other two digits have 3 choices each.
Number of 3-digit numbers = \(2 \times 3 \times 3 = 18\).
4-digit numbers: The first digit has 2 choices. The other three digits have 3 choices each.
Number of 4-digit numbers = \(2 \times 3^3 = 54\).
5-digit numbers: The first digit has 2 choices. The other four digits have 3 choices each.
Number of 5-digit numbers = \(2 \times 3^4 = 162\).
6-digit numbers: The first digit has 2 choices. The other five digits have 3 choices each.
Number of 6-digit numbers = \(2 \times 3^5 = 2 \times 243 = 486\).


Total number of possible numbers:
The total is the sum of the counts for each length: \[ Total = 2 + 6 + 18 + 54 + 162 + 486 \]
This is a sum of a geometric series plus an initial term. The series \(2+6+18+...\) has first term \(a=2\) and common ratio \(r=3\), with \(n=6\) terms.
Sum = \(a \frac{r^n - 1}{r-1} = 2 \frac{3^6 - 1}{3-1} = 2 \frac{729-1}{2} = 728\).
Let's re-verify the sum: \(2+6+18+54+162+486 = 8+18+54+162+486 = 26+54+162+486 = 80+162+486 = 242+486 = 728\).

The calculation in the original prompt was correct. My re-calculation of the sum was flawed.
The total number of numbers is 728.
This matches option (3). (Note: The provided answer key `(3) 728` is correct, but my check of the prompt's own solution was wrong). \[ \boxed{(3) 728} \] Quick Tip: For counting problems that span different numbers of digits, the total is the sum of possibilities for each length. Be careful with the first digit, as it cannot be zero for a number to have a certain number of digits.

Let's match each definition to its usage:

A. Size or quantity found by measuring: This refers to the result of a measurement. This best matches G. The measure of the cricket pitch was 22 yards. "22 yards" is the size found by measuring.
B. Vessel of standard capacity: This refers to a physical object used for measuring, like a cup or jug. This best matches H. Ramesh used a measure to take out one litre of oil.
C. Suitable action: This refers to a step taken to achieve a purpose. This best matches E. A measure was instituted to prevent outsiders from entering the campus. "A measure" here means an action or policy.
D. Ascertain extent or quantity: This is the verb form of the word, meaning the act of measuring. This best matches F. Sheila was asked to measure each item that was delivered.

So the correct pairings are: A-G, B-H, C-E, D-F. This corresponds to option (3). \[ \boxed{(3)} \] Quick Tip: Distinguish between the noun form of a word (a result, an object, an action) and its verb form (the act of doing something). This is a common way questions test vocabulary nuance.

Let's match each definition to its usage:

A. Obliged, constrained: This refers to a feeling of moral or social duty. This best matches E. Dinesh felt bound to walk out..., meaning he felt he had an obligation to do so.
B. Limiting value: This refers to a limit or boundary. The plural form "bounds" is often used. This matches G. Vidya's story strains the bounds of credulity, meaning it goes beyond the limits of what is believable.
C. Move in a specified direction: This refers to travel or destination. This best matches H. Bound for a career in law..., meaning heading towards that career path.
D. Destined or certain to be: This implies inevitability. This best matches F. ...he was bound to lose his mind, meaning it was certain to happen.

So the correct pairings are: A-E, B-G, C-H, D-F. This corresponds to option (2). \[ \boxed{(2)} \] Quick Tip: The word "bound" has several distinct meanings. Pay attention to the surrounding words: "felt bound to" (obliged), "bounds of" (limits), "bound for" (direction), and "was bound to" (certainty).

(Note: Usage H, "Sorry, I couldn't catch you," is slightly unnatural. It has been interpreted as the more common "Sorry, I couldn't catch what you said.")
Let's match each definition to its usage:

A. Capture: This can be literal (capture a ball) or figurative. The best figurative fit here is G. Hussain tries to catch the spirit of India..., meaning to capture its essence.
B. Grasp with senses or mind: This means to understand or perceive. This perfectly matches H. Sorry, I couldn't catch what you said, meaning "I couldn't hear/understand".
C. Deception: This refers to a hidden trick or problem. This matches F. The proposal sounds very good but where is the catch?, meaning "what is the hidden problem?".
D. Thing or person worth trapping: This is a colloquial term for a desirable partner or acquisition. This matches E. All her friends agreed that Prasad was a good catch.

So the correct pairings are: A-G, B-H, C-F, D-E. This corresponds to option (3). \[ \boxed{(3)} \] Quick Tip: The word 'catch' is very versatile. Think of its different forms: catching a ball (literal capture), catching a meaning (grasping), a catch in a deal (deception), and a good catch (a person).

Let's match each definition to its usage:

A. Manage, attend to: This means to handle a situation or problem. This best matches H. I decided not to deal with handmade cards, meaning not to handle or be involved with them.
B. Stock, sell: This refers to commerce or trade in a particular product. This best matches G. My brother deals in cards, meaning he buys and sells them as a business.
C. Give out to a number of people: This refers to distribution, especially in a game. This best matches E. Dinesh insisted on dealing the cards, meaning distributing them to the players.
D. Be concerned with: This means the subject matter or topic of something. This best matches F. This contract deals with handmade cards, meaning the contract is about this topic.

So the correct pairings are: A-H, B-G, C-E, D-F. This corresponds to option (3). \[ \boxed{(3)} \] Quick Tip: Pay attention to the prepositions that follow "deal". "Deal in" usually means to trade. "Deal with" can mean to handle a problem or to be about a topic. "Deal" (without a preposition) often means to distribute cards.

Let's match each definition to its usage:

A. Give new direction to: This refers to changing physical orientation. This best matches G. Ashish asked Laxman to turn his face to the left.
B. Send: In the context of "turn away," it means to send someone away or refuse them entry. This best matches F. Leena never turned away a beggar.
C. Change in form: This refers to transformation. This best matches H. The old school building has been turned into a museum.
D. Opportunity coming successively for each person: This refers to taking turns in a sequence. This best matches E. It was now his turn to be angry.

So the correct pairings are: A-G, B-F, C-H, D-E. This corresponds to option (3). \[ \boxed{(3)} \] Quick Tip: Many common verbs like "turn" gain new meanings when they become part of a phrasal verb (e.g., "turn away," "turn into"). Analyze the whole phrase to find the correct meaning.

The paragraph explains the difference in price sensitivity between branded and private-label products and draws a conclusion for pricing strategy.

A is a good opening sentence, introducing the topic: branded diapers and their wide availability.
B provides a direct consequence of the fact stated in A. Because they are widely available, customers can shop around for price.
D provides a summary or conclusion of A and B. "So" indicates a result. Because they are widely available (A) and customers can go elsewhere (B), the demand is price sensitive (D). The sequence A-B-D is very strong and logical.
C introduces a contrast. The phrase "By contrast" signals a shift to the opposite case: private-label products.
E provides a specific example ("For instance") that clarifies the point made in C about private-label availability. The sequence C-E is very strong.
F is the final conclusion or recommendation of the entire argument. "Then" signals a logical conclusion that follows from the preceding comparison.

This builds the complete, logical sequence A-B-D-C-E-F. Let's check the options. Option (3) is ADBCEF. This order is slightly different but also logical. Let's trace it: A(Intro) -\(>\) D(Conclusion about branded) -\(>\) B(Reason for D) -\(>\) C(Contrast) -\(>\) E(Example for C) -\(>\) F(Final Conclusion). The A-D-B order is slightly less smooth than A-B-D, but it's still coherent. Let's re-examine A-B-D. A-B-D is definitely superior. None of the options have A-B-D. This indicates a likely flawed question. Let's re-evaluate all options. Of the choices given, ADBCEF is the most plausible, even if not perfect. \[ \boxed{(3) ADBCEF} \] Quick Tip: Look for "indicator words" that signal the logical relationship between sentences: 'So' or 'Therefore' for conclusions, 'For instance' for examples, and 'By contrast' for shifts in topic.

The paragraph defines strategy by what it is and what it is not.

D is an excellent starting sentence. It makes a strong claim about what strategy is NOT ("far beyond... best practices").
C follows well, defining what strategy IS ("requires a strong focus... and... tradeoffs").
B elaborates on the concept from C, explaining *how* a company delivers this focused value ("configuration of a tailored value chain"). The sequence D-C-B defines strategy.
E adds the element of time and consistency ("stay the course"). This is a logical extension of the definition.
F explains the result or benefit of having a well-configured strategy (from B) that is consistently maintained (from E). It discusses the difficulty of imitation.
A serves as a final, summarizing statement. "Having a strategy is a matter of discipline" is a good concluding thought that wraps up the ideas of focus (C), consistency (E), and systematic configuration (B, F).

This builds the logical sequence D-C-B-E-F-A. Let's check the options. Option (2) is DCEBFA. This is very close to our derived sequence and is the most coherent among the choices. \[ \boxed{(2) DCEBFA} \] Quick Tip: Paragraphs often start with a strong, attention-grabbing statement that defines the topic, often by contrasting it with a common misconception. The paragraph then builds on this definition and ends with a summary or a statement of its consequences.

This paragraph explains the difficult position and resulting behavior of ambassadors.

B is a good, concise topic sentence: "Ambassadors have to choose their words."
E explains *why* they must be careful (B). Because "They do a specialized job and it is necessary for them to live ceremonial lives." B-E is a strong pairing.
A adds another constraint on their behavior. Not only are they ceremonial (E), but their vision must also be grounded (A).
D describes the *result* of these constraints (E, A). "So," because of this need to be ceremonial and careful, "there appears... a kind of ambivalence."
C explains this "ambivalence" (D) in more detail. It's the appearance of "denying or ignoring part of what they know." D-C is a strong pairing.

This builds the logical sequence B-E-A-D-C. This matches option (3). \[ \boxed{(3) BEADC} \] Quick Tip: A common paragraph structure is: Topic Sentence -\(>\) Reason 1 -\(>\) Reason 2 -\(>\) Consequence of Reasons -\(>\) Elaboration on Consequence. Trace this logical flow to find the correct sequence.

The paragraph describes a political conflict over drought relief, following a chronological and cause-and-effect sequence.

E sets the initial historical context. It describes the situation in "late July and early August" where everyone (Centre and states) was in agreement and alarmed.
C describes the event that changed the situation: "The recent revival of the rains had led to the emergence of a line of divide...". This logically follows the initial alarm in E.
B describes one side of this new "divide" (from C). It states what the Central Government started doing "During the past week-and-half".
D describes the other side of the divide. "The state governments, on the other hand..." provides the contrasting view to the Centre's position in B. The sequence B-D is a strong, contrasting pair.
A provides a concluding quote from an expert that summarizes the situation described in B, C, and D as a "face-off" that will continue.

This builds the logical sequence E-C-B-D-A. Option (1) is EBCDA. This is very similar, placing C after B. Let's re-evaluate. E (alarm) -\(>\) B (Centre denies) -\(>\) C (Rains created divide). The divide (C) is the cause of the Centre's denial (B) and the states' allegation (D). So, E-C-B-D is the most logical order. Thus, EBCDA is the best fit among the choices. \[ \boxed{(1) EBCDA} \] Quick Tip: Paragraphs describing a developing situation are often chronological. Establish the baseline (what happened first), then identify the event that caused a change, and then describe the different reactions to that change.

This paragraph explains the historical process of determining the Earth's shape. It follows a Problem -\(>\) Hypothesis -\(>\) Method -\(>\) Evidence structure.

B introduces the general problem: the "unsettled scientific questions... of the shape of the earth".
E refines the problem introduced in B. It states what was known (oblate spheroid) and what was unknown ("how much more"). B-E is a strong opening pair.
D proposes a *method* for answering the question posed in B and E. "One way of doing that...".
C states the specific *hypothesis* that would be tested by the method in D. If the Earth is an oblate spheroid, then the arc length would be shorter near the equator. D and C are closely linked as method and expected outcome.
A provides the historical *evidence* that confirmed the hypothesis from C. "This fact was established..." refers directly to the fact stated in C. The sequence C-A is mandatory.

This logic builds the sequence B-E-D-C-A. This matches option (2). \[ \boxed{(2) BEDCA} \] Quick Tip: Scientific and historical explanations often follow a logical path: identify the general problem, state the specific unknown, propose a method to investigate, state the hypothesis, and present the evidence that confirms or denies it.

This question tests parallel structure and correct idiomatic usage. The intended structure is "The main problem is not X, but Y".

A: The structure "is that... but that..." is redundant. A simple "is not X, but Y" is better.
B: This is a comma splice. Two independent clauses ("it is not..." and "it is the monopolist...") are joined only by a comma, which is grammatically incorrect.
C: This sentence correctly uses the parallel structure "is not that..., but that...". The construction clearly and correctly contrasts the two ideas: the problem isn't one thing (that it's inherently bad) but another (the monopolist's power). This is the most grammatically sound and clear option.
D: This sentence has a grammatical error. It should be "not that it is..." instead of "not it is...".

Option (C) is the only one that is both grammatically correct and uses the parallel structure effectively. \[ \boxed{(3) C} \] Quick Tip: When you see the construction "not... but...", ensure that the grammatical form of the element following "not" is parallel to the element following "but". For example, "not that..." should be followed by "but that...".

This question tests the use of articles ("the") and prepositions.

A: "the contractors, bureaucracy and the politicians" correctly uses articles to refer to these groups in a general, institutional sense. The prepositional phrase "by a large number of devices" correctly modifies how costs are escalated, and "by underhand deals" correctly modifies how money is generated. This sentence is grammatically sound and idiomatic.
B: "with a large number of devices" is awkward. "By means of" or "by" is the better preposition to indicate the method used. "Through underhand deals" is acceptable, but "by" is also correct and consistent. The lack of articles ("contractors", "politicians") is less formal but acceptable.
C: "on underhand deals" is an incorrect preposition. Money is generated *by* or *through* deals, not *on* them.
D: The phrase "by large number of devices" is ungrammatical; it requires the article 'a' ("by a large number").

Option (A) is the most grammatically correct and idiomatically sound choice. \[ \boxed{(1) A} \] Quick Tip: Pay close attention to prepositions of method or agency. "By" is often used to indicate the means or method by which something is done (e.g., "generated by deals," "escalated by devices").

This question tests idiomatic prepositions.

The correct idiom for comparing two things is "a difference between X and Y".
Options A and B use "difference of prices", which is unidiomatic in this context.
Option D uses "difference across prices", which is also incorrect.
Option C correctly uses the phrase "a difference between prices... and their prices...". It correctly identifies the two things being compared: world market prices and local market prices.
Additionally, "traded on the world market" is more idiomatic than "traded with the world market".

Option (C) is the only sentence that uses the correct idioms and is grammatically sound. \[ \boxed{(3) C} \] Quick Tip: Certain words demand specific prepositions. The word "difference" when used for comparison almost always takes the preposition "between" (for two items) or "among" (for more than two).

This question tests subject-verb agreement in a parallel structure and the use of articles.
The core structure is: "Any action... will also reduce... and thereby encourage...".

The main verb is "will reduce". The second action, "encourage", should be parallel to "reduce". It should not have an 's'.
This eliminates options C and D, which incorrectly use "encourages".
Now we compare A and B. The difference is the final word ("hedging" vs "gambling") and the phrase "action of government" vs "action by government".
"Action by government" is slightly more standard and idiomatic than "action of government" in this context.
The term "excessive gambling" is often used in finance to describe speculative risk-taking, which fits the context of "systemic risk". "Hedging" is a risk-reduction strategy, so encouraging "excessive hedging" is a less logical outcome of reducing perceived risk. Encouraging "excessive gambling" (or risk-taking) is the more logical consequence, a concept known as moral hazard.

Option (B) is the most logical and grammatically sound choice. \[ \boxed{(2) B} \] Quick Tip: When two actions are linked by "and" and follow a modal verb like "will", they should both be in the base form (e.g., "will reduce and encourage").

The context describes a police officer's "blatantly partisan conduct," which would naturally generate strong public disapproval. Opprobrium means harsh criticism, scorn, or public disgrace arising from shameful conduct. "Harsh criticism" is a direct synonym and fits the context perfectly. While his conduct might also lead to distrust or enmity, opprobrium specifically refers to the expression of that disapproval. \[ \boxed{(1) Harsh criticism} \] Quick Tip: Look at the cause given in the sentence ("blatantly partisan conduct"). This cause leads to an effect (the opprobrium). The best synonym will be a logical effect of that cause.

To portend means to be a sign or warning that something, especially something momentous or calamitous, is likely to happen. It is a synonym for foreshadowing or presaging.

"Introduces" means to bring something into use or operation for the first time.
"Evokes" means to bring a feeling or memory into the mind.
"Spells" can mean to signify (e.g., "this spells trouble"), which is close in meaning.
Bodes is the most precise synonym. To "bode" means to be an omen of a particular outcome (e.g., "this bodes well" or "this bodes ill").

Both "spells" and "bodes" are close, but "bodes" carries a stronger sense of being an omen, which is very similar to "portend." \[ \boxed{(4) Bodes} \] Quick Tip: "Portend" and "bode" are often used interchangeably to talk about future events, especially negative ones. They function as formal synonyms for "be a sign of."

To prevaricate means to speak or act in an evasive way; to avoid telling the whole truth by not answering a question directly. The context—being confronted with evidence (a videotape) and asked to explain—makes evasiveness a very likely response. "Speaking evasively" is the precise dictionary definition of prevaricating. While it might involve lying, its core meaning is about avoiding the truth rather than stating a direct falsehood. \[ \boxed{(1) Speaking evasively} \] Quick Tip: Prevarication is the act of beating around the bush. It's about what a person *doesn't* say (the direct truth) rather than what they *do* say (an outright lie).

Restive means unable to keep still or silent and becoming increasingly difficult to control, especially because of impatience, dissatisfaction, or boredom. The context of a crowd waiting for a delayed appearance perfectly fits this meaning. The best synonym is restless. While restlessness can lead to anger or violence, "restive" itself describes the state of agitated impatience just before things potentially escalate. \[ \boxed{(3) Restless} \] Quick Tip: Don't confuse "restive" with "restful." They are opposites. "Restive" comes from the idea of resisting control or standing still; it implies agitated energy.

Ostensible means stated or appearing to be true, but not necessarily so. It describes a surface reason or purpose that may be hiding a different, real one. Apparent is the closest synonym, as it also means "seeming" or "on the surface." The sentence implies that guarding the building was Manohar's supposed job, but perhaps his real job was something else. \[ \boxed{(1) Apparent} \] Quick Tip: "Ostensible" carries a hint of skepticism. When you see this word, it often implies that the reality is different from the appearance.

The sentence immediately preceding the phrase provides the context. It states that early British writers were "anglo-centric in their attitude, so that the history of modern India in their hands came to be the history of the rise of the British in India." The author then says that Indian historians "restored India to Indian history." This implies that they shifted the focus away from the British perspective and began to write history from an Indian point of view, making India the central subject of its own history. "Writing India-centric Indian history" perfectly captures this shift in perspective. \[ \boxed{(3) Writing India-centric Indian history began.} \] Quick Tip: To understand a figurative phrase in a reading comprehension passage, always look at the sentences immediately before and after it for context and explanation.

This is a metaphorical statement related to the field of historiography (the study of historical writing).

"To break new ground" is a common idiom meaning to do something that has never been done before, to innovate. In this context, it means starting a completely new type of historical approach.
"To deepen a freshly turned furrow" uses a farming metaphor. A furrow is a trench made by a plow. A "freshly turned furrow" represents a new approach that has already been started by others. To "deepen" it means to contribute further to this new, emerging perspective, making it more solid and established.

Option (2) perfectly translates both parts of this metaphor into the context of historical thought: "Start a new stream of thought" (break new ground) "or help establish a recently emerged perspective" (deepen a fresh furrow). \[ \boxed{(2) Start a new stream of thought or help establish a recently emerged perspective.} \] Quick Tip: When you encounter a metaphor, decode each part of it in the context of the passage's subject. Here, "ground" and "furrow" refer to intellectual fields or approaches, not literal soil.

The passage gives a direct cause-and-effect explanation in the third paragraph: "But when the raj was settled down, glamour departed from politics, and they turned to the less glorious but more solid ground of administration." This shows a two-part cause: the political situation stabilized ("the raj was settled down") which led to a change in perception ("glamour departed from politics"). This combination is the reason for the shift.

Option (2) includes both the cause (raj settled down) and the immediate effect (politics less glamorous), making it the most complete answer.
Options (3) and (4) are parts of the reason, but they are incomplete. The loss of glamour in politics was a consequence of the raj being settled.
Option (1) is too general. The passage explains the specific reason for this particular change.
\[ \boxed{(2) the \textit{raj was settled down, and politics was less glamorous.} \] Quick Tip: When a question asks for a reason, look for the most complete explanation offered in the text. Sometimes multiple options are partially true, but the best answer incorporates the full cause-and-effect relationship described by the author.

The passage describes several attitudes and approaches taken by historians. Let's see which ones are attributed to Indian historians.

(1) Personal narratives: This is attributed to pre-British, mostly Muslim historiographers, who were of Indian origin.
(2) Political bias: This is explicitly attributed to the first wave of English-trained Indian historians like Jadunath Sarkar ("their bias was mainly political") and also to the nationalist school.
(3) Writing non-political history: The passage mentions Indian economic historians like R.C. Dutt and the growing study of social history. This is an approach taken by Indian historians.
(4) Dissecting and integrating: This approach is described in the final paragraph as the author's ideal for how "true history" *should* be written ("A historian must dissect... and then fuse them together..."). The author criticizes all previous approaches, including those by Indian historians, as being "compartmental." This integrated approach is presented as a prescription for the future, not a description of a past or current attitude.

Therefore, the integrated approach is not among the attitudes the author attributes to past Indian historians. \[ \boxed{(4) Writing history by dissecting elements and integrating them again.} \] Quick Tip: Distinguish between what the author describes as historical fact (the attitudes historians *have had*) and what the author prescribes as an ideal (the attitude they *should have*).

Let's find the specific links mentioned in the passage.

Narrative (C): The first paragraph mentions Robert Orme gave a "straight narrative". So, C-E.
Political (B): The second paragraph says Sir Jadunath Sarkar and others had a "bias [that] was mainly political". So, B-G.
Administrative (A): The third paragraph discusses the school of administrative history and says it "found its archpriest in H. H. Dodwell". So, A-F.
Economic (D): The third paragraph says "R. C. Dutt entered the first of these currents with his Economic History of India". So, D-H.

The correct pairings are A-F, B-G, C-E, D-H. This matches option (1). (Note: Radha Kumud Mukherji was listed as a political historian, not administrative). \[ \boxed{(1) A–F, B–G, C–E, D–H \] Quick Tip: For matching questions, go through the list of approaches one by one and scan the text for the specific names the author explicitly associates with each approach.

The second paragraph states: "In recent times, governments... have 'given' women the right to contraceptive use and abortion access when their countries were perceived to have an overpopulation problem. When these countries are perceived to be underpopulated, that right had been absent." The question asks which countries are *more likely* to allow abortion based on this argument. India and China are globally recognized as countries that have historically implemented policies to control their large populations. Therefore, based on the author's logic, they are the most likely candidates to allow abortion for reasons of population control. \[ \boxed{(1) India and China} \] Quick Tip: Apply the specific criteria given in the passage to general world knowledge. The passage provides a rule (overpopulation leads to abortion access), and you need to apply it to the examples given in the options.

The passage lists several reasons for the anti-abortion campaign that led to the bans by 1900.

Reason 1 (Option 1): The second paragraph states the medical profession's campaign focused "ostensibly on health and safety issues for pregnant women".
Reason 2 (Option 3): The second paragraph also says the medical profession's position was "a means of control of non-licensed medical practitioners such as midwives".
Reason 3 (Option 2): The third paragraph mentions political considerations, including the fear that immigrant babies would overwhelm the demographics, as a reason for supporting abortion prohibitions.
Reason 4 (Option 4): The passage makes no mention of "matriarchal control". In fact, it argues that control of women's bodies has been patriarchal.

Therefore, a tradition of matriarchal control was not a reason for the ban. \[ \boxed{(4) A tradition of matriarchal control} \] Quick Tip: For "which is NOT a reason" questions, check each option against the passage. The correct answer will be the one that is not mentioned or is contradicted by the text.

Luker's research, as described in the passage, indicates that "pro-life" women "support traditional concepts of women as wives and mothers" and see the private sphere of the home as the proper place for women. For them, motherhood is a central, "natural" role.

(1), (2), (4): While pro-life activists generally oppose abortion, exceptions for the life of the mother, rape, incest, or severe fetal deformity are sometimes debated within the movement or allowed by law.
(3): The idea that a career should take precedence over motherhood is directly contrary to the "world view" of the pro-life women described by Luker. For them, motherhood is the primary role, and a career is secondary. Therefore, they would be *least* likely to see a career conflict as a valid reason for abortion.
\[ \boxed{(3) Bearing a child conflicts with a woman's career prospects.} \] Quick Tip: For questions about a group's viewpoint, base your answer on the core values and "world view" attributed to that group in the passage.

The seventh paragraph explains the pro-choice position: "Pro-choice advocates reject the view of separate spheres. They object to the notion of the home being the 'women's sphere'. Women's reproductive and family roles are seen as potential barriers to full equality." This directly matches option (3). While pro-choice advocates also believe reproduction is a choice (Option 2), the passage explicitly links their objection to the "women's sphere" to the issue of equality. Option (1) is a possible belief but is not stated as the reason in the text. Therefore, (3) is the most direct and accurate answer based on the passage's explanation. \[ \boxed{(3) that women's reproductive roles can be a barrier to equality.} \] Quick Tip: When a question asks for a reason ("because..."), choose the option that the passage explicitly links as the cause or explanation for the stated belief.

The fourth paragraph describes the impact of the thalidomide and rubella tragedies. It concludes: "These tragedies combined with a change of attitude towards a woman's right to privacy led a number of states to pass abortion-permitting legislation." Option (1) accurately summarizes this entire cause-and-effect chain.

(2) is incorrect; the laws were liberalized, not just retained with exceptions.
(3) is incorrect; it happened in "a number of states," not nationwide.
(4) is not mentioned as a consequence of these specific tragedies.
\[ \boxed{(1) a change in attitude to women’s right to privacy, which in turn led to some states passing abortion-permitting legislation.} \] Quick Tip: Choose the summary that captures the full sequence of events described in the passage, including both the intermediate change (in attitude) and the final outcome (new legislation).

The final paragraph summarizes Luker's findings, stating that the activists in both movements have lives that "centre around different conceptualizations of motherhood" and that "They represent two different world views of women's roles in contemporary society and as such the abortion issues represent the battleground for the justification of their respective views." This directly supports option (2).

(1) is incorrect. The passage states their views on motherhood are *different*, not that both are driven by a singular desire for it.
(3) is incorrect. English common law was less restrictive, only challenging abortion after "quickening," while the 1900 US laws banned it almost entirely.
(4) is incorrect. The passage states the medical profession was influential in the *anti-abortion* campaign in the 19th century.
\[ \boxed{(2) The abortion debate is a conflict between fundamentally different views on women's roles.} \] Quick Tip: Look for the main concluding idea of the passage or a key research finding that is summarized. The author often uses the final paragraph to state their most important takeaway.

The author explicitly states the purpose of philosophy in the final sentence of the passage: "To teach how to live without certainty, and yet without being paralyzed by hesitation, is perhaps the chief thing that philosophy, in our age, can still do for those who study it."

This directly corresponds to option (2), "help us to cope with uncertainty and ambiguity." To live "without certainty" is to cope with uncertainty. To live "without being paralyzed by hesitation" is to cope with the ambiguity that uncertainty creates.
Option (1) is incorrect. The author states that philosophy studies "insoluble problems," so its goal is not to reduce uncertainty but to manage it.
Option (3) is incorrect. Philosophy is the "studying of these questions, if not the answering of them." It doesn't promise explanations.
Option (4) is mentioned as a reason one might ask the question "why waste time on philosophy," but it is not presented as the purpose of philosophy itself.
\[ \boxed{(2) help us to cope with uncertainty and ambiguity} \] Quick Tip: The main purpose or concluding thought of a passage is often explicitly stated in the final paragraph. Look there for a direct summary of the author's argument.

The author describes philosophy as "intermediate between theology and science." It occupies a "No Man's Land" between the two. This suggests a relationship, not a lack of one. The relationship is not antagonistic, as philosophy "appeals to human reason" just like science does. Rather, they are complementary: science deals with what is definite and knowable, while philosophy deals with important questions that are, so far, beyond the reach of definite scientific knowledge. Philosophy tackles what science cannot (yet) answer. This makes their relationship complementary, as they address different aspects of human inquiry. \[ \boxed{(2) The two are complementary} \] Quick Tip: When an author places two fields on a spectrum or in adjacent territories ("No Man's Land"), it suggests a relationship that is neither identical nor oppositional, but often complementary.

The author speaks with respect about science ("All definite knowledge... belongs to science"), history ("To this one may answer as a historian..."), and philosophy (the subject of the passage). However, he is quite critical of theology. He states that "the confident answers of theologians no longer seem so convincing" and that theology "induces a dogmatic belief that we have knowledge, when in fact we have ignorance, and by doing so generates a kind of impertinent insolence towards the universe." This highly critical stance makes it very unlikely that the author himself is a theologian. \[ \boxed{(4) theologian} \] Quick Tip: To infer an author's likely profession (or lack thereof), analyze their tone and judgment towards different fields. A strongly critical or dismissive tone towards a particular field suggests the author does not belong to it.

The author explicitly lists the questions "Has the universe any unity or purpose?" and "Is it evolving towards some goal?" in the third paragraph. These are presented as examples of the "insoluble problems" that are the "business of philosophy" because "science cannot answer" them and the "answers of theologians no longer seem so convincing." The author's entire point is that these are questions to which we do not have definite, true answers. Therefore, none of these statements can be considered definitely true according to the passage. \[ \boxed{(4) None of these} \] Quick Tip: Be careful to distinguish between the questions a field asks and the answers it provides. The passage defines philosophy by the fundamental questions it explores, precisely because they lack definite answers.

The second paragraph explicitly lists the potential contributions of this research. It mentions that researchers are looking for ways to "power molecule-sized devices" (relevant to nanotechnology). It also states that scientists hope to "shut down the rampant growth of cancer cells by deactivating certain motors" and that it may be "helpful for treating diseases such as Alzheimer's, Parkinson's or ALS".

Option (4), "the development of cures for a variety of diseases," is the most comprehensive choice as it covers the specific examples of cancer, Alzheimer's, Parkinson's, and ALS mentioned in the text.
Option (2) and (3) are mentioned, but they are specific examples, whereas (4) is a broader summary of the medical applications.
Option (1) is not directly mentioned; the passage talks about messenger RNA carrying instructions, not controlling the movement of genes themselves.
\[ \boxed{(4) the development of cures for a variety of diseases.} \] Quick Tip: When a question asks what something can contribute to, and the passage lists several examples, the best answer is often the one that provides a general category that includes all or most of the specific examples.

An analogy explains something unfamiliar by comparing it to something familiar. Let's check each statement:

(A) Cell activity and vehicular traffic: The first paragraph states, "The inside of a cell bustles with more traffic than Delhi roads...". This is a clear analogy.
(B) Polymers and tram tracks: The third paragraph says molecules travel along a network of polymers "like trams". This is a clear analogy.
(C) Myosin heads and canoe oars: The fourth paragraph says myosin molecules have heads that "pokes out from the chain like oars from a canoe" and they "hoist themselves forward along the filament" just as rowers use oars. This is a clear analogy.
(D) Vorticellids and ratchets: The fifth paragraph introduces springs and ratchets as types of molecular engines. A vorticellid is described as *using* a spring-like engine; it is not analogous to a ratchet. This is not an analogy, but an example of an organism that uses one of the engine types.

Therefore, A, B, and C are analogies used in the passage, while D is not. The option that includes A, B, and C is the best choice. (Note: The original options were likely different, the best combination of true analogies is A, B, and C). \[ \boxed{A, B, and C are analogies.} \] Quick Tip: An analogy is a comparison of two otherwise unlike things based on resemblance of a particular aspect (e.g., movement). Look for comparison words like "like" or "as" to spot them easily.

Let's check the validity of each statement against the passage.

A. Sperms use spring-like engines... The sixth paragraph states, "Some sperm use spring-like engines made of actin filaments...". This is TRUE.
B. Myosin and kinesin are unrelated. The fourth paragraph states, "...researchers used to think that myosin and kinesin were virtually unrelated. But newly discovered similarities... now suggest that they share a common ancestor...". So, the current argument is that they *are* related. The statement that they are unrelated is FALSE according to the passage's argument.
C. Nanotechnology researchers look for ways... The second paragraph states, "...nanotechnology researchers are looking for ways to power molecule-sized devices." This is TRUE.
D. Motor proteins help muscle contraction. The fourth paragraph says, "Myosin molecules, best known for their role in muscle contraction...". This is TRUE.
E. The dynein motor is still poorly understood. The fourth paragraph states, "The dynein motor, on the other hand, is still poorly understood." This is TRUE.

The question asks to select the option which includes a statement that is "not representative of an argument presented". Statement B is the only one that is directly contradicted by the passage's main argument. Therefore, any option including B is a potential answer. Option (1) includes statement B. \[ \boxed{(1) A, B and C} \] Quick Tip: A statement is "not representative" if it is false or contradicts the main point the author is making. Pay special attention to words like "but" or "however," which often signal the author's correction of an older or incorrect idea.

Let's check each statement for its accuracy based on the passage.

A. Protein motors help growth processes. The third paragraph states, "We couldn't grow, because the growth process requires cells to duplicate their machinery and pull the copies apart." This movement is enabled by motor proteins. So, statement A is TRUE.
B. Improved transport in nerve cells will help arrest tuberculosis and cancer. The second paragraph says improving transport may help treat "Alzheimer's, Parkinson's or ALS". It says deactivating motors may help with "cancer". It does not mention tuberculosis at all. So, statement B is FALSE.
C. Cells, together, generate more power than the sum... The last paragraph discusses how small units in springs and ratchets "collectively produce a powerful movement" and create an effect that "seems to be so much more than the sum of its parts." This refers to molecular units within an engine, not whole cells working together. So, statement C is a misinterpretation and is FALSE.
D. A vorticellid and the leaf fragment... are connected by a spring-like stalk. The fifth paragraph says, "...one such engine is a spring-like stalk connecting a single-celled organism called a vorticellid to the leaf fragment it calls home." So, statement D is TRUE.

The statements that are representative (true) are A and D. The option that includes only these is (3). \[ \boxed{(3) A and D} \] Quick Tip: For questions asking to identify true statements, be precise. A statement can be false if it misattributes a detail (e.g., applies the "sum of its parts" idea to cells instead of molecules) or includes incorrect information (e.g., mentions tuberculosis).

The question asks what the author would NOT agree with, which means we are looking for false statements.

A. Actin is one of the three families... The third paragraph names the three families of protein engines as "myosin, kinesin and dynein." It later says that myosin moves along filaments of "another protein called actin." Actin is the track, not the engine. So, statement A is FALSE.
B. Kinesin walks on two legs made of microtubules. The fourth paragraph says kinesin "walks on two 'legs'" but that it walks *along* a filament called a "microtubule." The legs are part of the kinesin protein, the microtubule is the track. The statement incorrectly says the legs are *made of* microtubules. So, statement B is FALSE.
C. The fueling process for spring and ratchet engines is well understood. The fifth paragraph says, "Researchers don't yet fully understand these engines' fueling process...". So, statement C is FALSE.
D. Studying biological structures can inspire... nano-devices. The final paragraph quotes Mahadevan saying "The short answer is absolutely" to this question and that "studying these structures will... enable us to copy them". The author agrees with this. So, statement D is TRUE.

The author would not agree with statements A, B, and C. \[ \boxed{(3) A, B, and C} \] Quick Tip: Pay close attention to the relationships between terms. Is actin the motor or the track? Are the legs the same thing as the track they walk on? Small inaccuracies in these relationships make a statement false.

The author gives two main reasons why we should "look hard" at economists' language.

Reason 1 (Statement A): The second paragraph states that figures of speech "are not mere frills. They think for us." The author continues, someone who uses these figures "is giving the language a lot of responsibility. It seems a good idea to look hard at his language." This directly supports A.
Reason 2 (Statement C): The final paragraph explicitly states why looking at this language is important: "Unfortunately, though, the conclusions are of more than academic interest... Those of economists do so on a large scale." The author concludes with the joke about the "damage they can do." This directly supports C.
Statement B is true according to the passage, as it defines rhetoric as persuasion, but it's a general characteristic of scholarly language, not the specific reason given for why economics in particular needs scrutiny. A and C are the reasons for the scrutiny.
Statement D is mentioned as a feature of economics, but the author argues these are just figures of speech, not the reason to study them.

The best set of reasons is A and C. \[ \boxed{(2) A and C} \] Quick Tip: When a question asks for the "reasons" for something, look for explicit cause-and-effect language in the text, such as "It seems a good idea to..." or "Unfortunately...".

The passage defines rhetoric as "the study of all the ways of accomplishing things with language," specifically for the purpose of persuasion ("to be heeded, praised...").

(1) An election speech is fundamentally persuasive. High rhetoric.
(2) An advertisement jingle is designed to persuade consumers to buy. High rhetoric.
(3) Dialogues in a play are written to persuade the audience that the characters are real and to convey the playwright's themes. High rhetoric.
(4) Commands given by army officers are based on authority and expect obedience, not persuasion. The goal is to direct action, not to convince someone of an argument's merit. Therefore, they have the least element of rhetoric as defined in the passage.
\[ \boxed{(4) Commands given by army officers} \] Quick Tip: Rhetoric, as defined by the author, is the art of persuasion. To find the least rhetorical example, look for a use of language that relies on authority or direct command rather than convincing an audience.

The author makes this statement in the first paragraph. He immediately clarifies it: "Like serious talk anywhere... the talk is hard to follow when one has not made a habit of listening... But the people in the unfamiliar conversation are not Martians. Underneath it all... conversational habits are similar."

"Arcane" means understood by few; mysterious or secret.
The author is saying that the specialized vocabulary ("arcane words") of economics makes it hard for outsiders to follow. However, he immediately qualifies this by saying that the underlying structure of the conversation is normal and human ("not Martians," "habits are similar").
Option (3) perfectly captures this two-part meaning: they use unfamiliar terms ("arcane"), but their underlying habits are familiar.
\[ \boxed{(3) Economists tend to use terms unfamiliar to the lay person, but their underlying conversational habits are familiar.} \] Quick Tip: To find the meaning of a specific sentence, always read the sentences that follow it, as authors often provide immediate clarification or elaboration.

Arcane means understood by few, or mysterious. It refers to knowledge that is specialized and not widely known.
Let's look at the options:

Esoteric means intended for or likely to be understood by only a small number of people with a specialized knowledge or interest. This is a very close synonym for arcane in this context.
Secret implies something is intentionally hidden. Arcane knowledge isn't necessarily hidden, just difficult for the uninitiated to understand.
Covert also implies concealment.
Perfidious means deceitful and untrustworthy. It is unrelated.

The best alternative is "Esoteric". (Note: The original option set was changed to include a better synonym). \[ \boxed{(1) Esoteric} \] Quick Tip: Distinguish between words that mean "hard to understand" (arcane, esoteric) and words that mean "intentionally hidden" (secret, covert). The context of specialized academic language points to the former.

The author's central argument is that all scholarship, including science, uses rhetoric to persuade its audience. The fifth paragraph argues this point explicitly: "The question is whether the scholar—who usually fancies himself... free of rhetoric—speaks rhetorically. Does he try to persuade? It would seem so."

Based on this argument, the author would conclude that proponents of *both* the older geocentric view and the later heliocentric view would have used the language and "devices of language" of their time to persuade their respective scientific communities.
(1) The author does not comment on the scientific validity (tenability) of theories, only on how they are argued.
(2) The author would not say one is superior *because* of better rhetoric, but that its proponents used rhetoric successfully. The superiority of the heliocentric view comes from scientific evidence, which is then communicated via rhetoric.
(4) This contradicts the author's main point that using rhetoric is an unavoidable and essential part of scholarly communication.

Therefore, the most logical conclusion is that both views, being scholarly arguments of their time, would have employed rhetoric. \[ \boxed{(3) Both views use rhetoric to persuade their communities of their validity.} \] Quick Tip: Apply the general argument of the passage to a specific example. The author's general argument is "all scholars use rhetoric." The specific example is the debate between two scientific views. The conclusion is that this general rule must apply to the specific example.