Step 1: Understand the Relationship - In an Arithmetic Progression (A.P.), the sum of the first \(n\) terms is \(S_n\). To find a specific term \(a_n\), we use the formula: \(\)a_n = S_n - S_{n-1\(\)
This is because \(S_n\) includes all terms from \(1\) to \(n\), and \(S_{n-1}\) includes terms from \(1\) to \(n-1\). Subtracting them leaves only the \(n^{th}\) term.

Step 2: Calculate \(S_{10}\) -

Substitute \(n = 10\) into the given equation \(S_n = n^2 + 4n\): \(\)S_{10 = (10)^2 + 4(10)\(\) \(\)S_{10 = 100 + 40 = 140\(\)

Step 3: Calculate \(S_{9}\) -

Substitute \(n = 9\) into the same equation: \(\)S_9 = (9)^2 + 4(9)\(\) \(\)S_9 = 81 + 36 = 117\(\)

Step 4: Find the 10th term (\(a_{10}\)) -

Apply the values to the formula from Step 1: \(\)a_{10 = S_{10 - S_9\(\) \(\)a_{10 = 140 - 117 = 23\(\) Quick Tip: For any \(S_n = An^2 + Bn\), the common difference \(d\) is always \(2A\). Here \(A=1\), so \(d=2\). The first term \(a = S_1 = 5\). Using \(a_{10} = a + 9d\), we get \(5 + 9(2) = 23\).

Step 1: Write the given equation -
\(\)\tan\theta + \cot\theta = 4\(\)

Step 2: Square both sides -

To find the terms with squares, we must square the entire equation: \(\)(\tan\theta + \cot\theta)^2 = (4)^2\(\)

Step 3: Expand using \((a+b)^2 = a^2 + b^2 + 2ab\) -
\(\)\tan^2\theta + \cot^2\theta + 2(\tan\theta)(\cot\theta) = 16\(\)

Step 4: Apply Trigonometric Identity -

We know that \(\cot\theta = \frac{1}{\tan\theta}\), so \(\tan\theta \cdot \cot\theta = 1\). Substitute this into the equation: \(\)\tan^2\theta + \cot^2\theta + 2(1) = 16\(\)

Step 5: Solve for \(\tan^2\theta + \cot^2\theta\) -
\(\)\tan^2\theta + \cot^2\theta = 16 - 2 = 14\(\) Quick Tip: This is a common algebraic pattern: if \(x + \frac{1}{x} = k\), then \(x^2 + \frac{1}{x^2} = k^2 - 2\). In trigonometry, this applies to pairs like (\(\tan, \cot\)), (\(\sin, \csc\)), and (\(\cos, \sec\)).

Step 1: Analyze the first blank - The sentence describes an opinion piece. Forthright means direct and outspoken, which fits a critique. Illicit means illegal, which is contextually incorrect.

Step 2: Analyze the second blank - The goal is to produce a reaction. Elicit (verb) means to evoke or draw out. Illicit is an adjective meaning illegal.

Step 3: Combine and verify - "The forthright piece... aiming to elicit a strong reaction." This is the only logically sound combination. Quick Tip: To remember the difference: \textbf{E}licit starts with 'E' like 'Evoke'. \textbf{I}llicit starts with 'I' like 'Illegal'.

Step 1: Identify the Word being Modified - In the phrase "resounding success", success is a Noun.

Step 2: Apply Grammar Rules - A word that describes or modifies a noun is an Adjective. 'Resounding' here describes the nature of the success.

Step 3: Rule out other options - It is not a Noun or Gerund as it doesn't function as a subject/object. It is not an Adverb because it doesn't modify a verb or adjective. Quick Tip: Even though it ends in '-ing', it acts as a descriptive word here. If it were a Noun, it would be a Gerund; if it modified a Verb, it would be an Adverb.

Step 1: Evaluate Idiomatic Usage - In English, "fixed phrases" are often used. Talking point is a standard idiom meaning a topic for discussion.

Step 2: Evaluate Context - A board meeting involves discussing specific issues. The company's performance is the subject being discussed.

Step 3: Rule out other options - "Point of talk" and "talk point" are not standard idioms. Breaking point refers to a state of collapse, which doesn't fit the context of a "meeting topic." Quick Tip: \textbf{Talking point} = Topic of discussion. \textbf{Breaking point} = The moment a situation becomes unbearable.

Step 1: Analyze the phrase - The sentence says the presentation "was a train wreck."

Step 2: Define Metaphor - A metaphor is a figure of speech that describes an object or action in a way that isn't literally true, but helps explain an idea or make a comparison without using "like" or "as."

Step 3: Verify the comparison - Here, the presentation is directly equated to a "train wreck" to emphasize how disastrous it was. Since it doesn't use "like a train wreck" (which would be a simile), it is a metaphor. Quick Tip: \textbf{Metaphor:} A is B ("Life is a highway"). \textbf{Simile:} A is like B ("Life is like a highway").

Step 1: Define the target word - Perspicacious means having a ready insight into and understanding of things; mentally sharp or keen.

Step 2: Evaluate options -

- Astute means having or showing an ability to accurately assess situations or people and turn this to one's advantage. This is a direct synonym.

- Superficial means shallow or on the surface (Antonym).

- Ambiguous means unclear.

- Obstinate means stubborn.

Step 3: Context Check - An "astute" analysis would certainly help a CEO pivot strategy effectively. Quick Tip: Perspicacious, Astute, Shrewd, and Sagacious are all synonyms for being "mentally sharp."

Step 1: Identify Initial and Final Values -

Initial Production (2018) = 100

Final Production (2021) = 147

Step 2: Apply the Percentage Growth Formula -
\(\)% Growth = \left( \frac{\text{Final Value - \text{Initial Value{\text{Initial Value \right) \times 100\(\)

Step 3: Calculate the difference -
\(147 - 100 = 47\)

Step 4: Final Calculation -
\(\text{% Growth = \frac{47}{100} \times 100 = 47%\) Quick Tip: Whenever the base (initial) value is 100, the numerical increase is equal to the percentage increase.

Step 1: Formula - To find the \(n^{th}\) term from the sum \(S_n\):
\(a_n = S_n - S_{n-1}\)

Step 2: Calculate \(S_{10}\) -
\(S_{10} = (10)^2 + 4(10) = 100 + 40 = 140\)

Step 3: Calculate \(S_9\) -
\(S_9 = (9)^2 + 4(9) = 81 + 36 = 117\)

Step 4: Find \(a_{10}\) -
\(a_{10} = 140 - 117 = 23\) Quick Tip: If \(S_n\) is a quadratic expression \(An^2+Bn\), the \(n^{th}\) term \(a_n\) will always be a linear expression.

Step 1: Identify Selection Requirements - We need to select 2 men out of 5 and 1 woman out of 4.

Step 2: Calculate combinations for Men -
\(^{5}C_{2} = \frac{5 \times 4}{2 \times 1} = 10\)

Step 3: Calculate combinations for Women -
\(^{4}C_{1} = 4\)

Step 4: Use Fundamental Principle of Counting - Since we need both men AND women, multiply the results:

Total ways = \(10 \times 4 = 40\) Quick Tip: Remember: \textbf{"AND" means Multiply} and \textbf{"OR" means Add} in Permutations and Combinations.

Step 1: Understand the conditions - We need a number \(N\) such that:
- \(N = 7k + 3\) (where \(k\) is an integer)

- \(N = 5m + 2\) (where \(m\) is an integer)

Step 2: Test the options -

- Option (A) 22: \(22 \div 7\) gives remainder 1. (Incorrect)

- Option (B) 38: \(38 \div 7\) gives remainder 3 (Correct for first part), but \(38 \div 5\) gives remainder 3. (Incorrect for second part)

- Option (C) 17: \(17 \div 7\) gives remainder 3 (\(7 \times 2 = 14\); \(17 - 14 = 3\)).
\(17 \div 5\) gives remainder 2 (\(5 \times 3 = 15\); \(17 - 15 = 2\)). (Correct)

Step 3: Verification - Since 17 satisfies both conditions, it is the correct answer. Quick Tip: For these types of questions, the "Option Substitution Method" is much faster than solving linear congruences.

Step 1: Factorize the number - Write 75 in terms of 5 and 3. \(\)75 = 25 \times 3 = 5^2 \times 3\(\)

Step 2: Apply Logarithm Properties - Use the product rule \(\log(MN) = \log M + \log N\): \(\)\log_{10(75) = \log_{10(5^2 \times 3)\(\) \(\)\log_{10(75) = \log_{10(5^2) + \log_{10(3)\(\)

Step 3: Apply the Power Rule - Use \(\log(M^n) = n\log M\): \(\)\log_{10(75) = 2\log_{10(5) + \log_{10(3)\(\)

Step 4: Substitute the given values - \(\)\log_{10(75) = 2(a) + b = 2a + b\(\) Quick Tip: Prime factorization is the first step for almost all "express log in terms of" problems.

Step 1: Solve the first term - Find the power to which 5 must be raised to get 125. \(\)125 = 5^3 \implies \log_5(5^3) = 3\(\)

Step 2: Solve the second term - Find the power to which 2 must be raised to get \(1/16\). \(\)\frac{1{16 = \frac{1{2^4 = 2^{-4 \implies \log_2(2^{-4) = -4\(\)

Step 3: Combine the results - \(\)3 + (-4) = -1\(\) Quick Tip: Remember: \(\log_b(1/x) = -\log_b(x)\). So \(\log_2(1/16) = -\log_2(16) = -4\).

Step 1: Define x-intercept - The x-intercept is the point where \(y = 0\).

Step 2: Set the equation to zero - \(\)x^2 - 4x + 3 = 0\(\)

Step 3: Factorize the quadratic equation - We need two numbers that multiply to 3 and add to -4. Those numbers are -1 and -3. \(\)(x-1)(x-3) = 0\(\)

Step 4: Solve for x - \(x - 1 = 0 \implies x = 1\)
\(x - 3 = 0 \implies x = 3\)

Step 5: Write as coordinates - The points are \((1,0)\) and \((3,0)\). Quick Tip: x-intercepts always have \(y=0\). y-intercepts always have \(x=0\).

Step 1: Identify given data in terms of failure -
- Failed in QA, \(n(Q) = 55%\)

- Failed in VA, \(n(V) = 45%\)

- Passed in both subjects = 25%. This means 25% of students are outside the "failure" circles.

Step 2: Calculate Total Failures -
Total % students = 100%.

Students who failed in at least one subject \(n(Q \cup V) = 100% - 25% = 75%\).

Step 3: Apply Set Theory Formula - \(\)n(Q \cup V) = n(Q) + n(V) - n(Q \cap V)\(\) \(\)75 = 55 + 45 - n(Q \cap V)\(\)

Step 4: Solve for both failed \(n(Q \cap V)\) - \(\)75 = 100 - n(Q \cap V)\(\) \(\)n(Q \cap V) = 100 - 75 = 25%\(\) Quick Tip: The students who "Passed in Both" are \((100 - Total Failures)\). Always bring all data to one category (either "failed" or "passed") before using the formula.

Step 1: Calculate Marked Price (MP) -

Markup is 25% on Cost Price (CP).
\(Markup Amount = 25% of 800 = \frac{25}{100} \times 800 = 200\).
\(Marked Price (MP) = CP + Markup = 800 + 200 = 1000\).

Step 2: Calculate Discount -

Discount is 20% on Marked Price.
\(Discount Amount = 20% of 1000 = \frac{20}{100} \times 1000 = 200\).

Step 3: Calculate Selling Price (SP) -
\(Selling Price (SP) = MP - Discount = 1000 - 200 = 800\). Quick Tip: When a value is increased by \(x%\) and then decreased by \(x%\), the overall change is always a loss of \(x^2/100%\). However, here the percentages are different. Use the successive change formula: \(a + b + \frac{ab}{100}\).
\(25 - 20 + \frac{25 \times (-20)}{100} = 5 - 5 = 0%\). Net change is 0%, so SP = CP.

Step 1: Use the \(n^{th}\) term formula \(a_n = ar^{n-1}\) -
\(a_3 = ar^2 = 12\) \dots (Eq 1)
\(a_6 = ar^5 = 96\) \dots (Eq 2)

Step 2: Find the common ratio (r) -

Divide Eq 2 by Eq 1: \(\frac{ar^5}{ar^2} = \frac{96}{12} \implies r^3 = 8 \implies r = 2\).

Step 3: Find the first term (a) -

Substitute \(r=2\) in Eq 1: \(a(2)^2 = 12 \implies 4a = 12 \implies a = 3\).

Step 4: Find the sum of first 5 terms (\(S_5\)) -

Formula: \(S_n = \frac{a(r^n - 1)}{r - 1}\)
\(S_5 = \frac{3(2^5 - 1)}{2 - 1} = \frac{3(32 - 1)}{1} = 3 \times 31 = 93\). Quick Tip: In a GP, if you know any two terms \(a_p\) and \(a_q\), the ratio is \(r^{(p-q)} = \frac{a_p}{a_q}\). Here \(r^{(6-3)} = 96/12 = 8\), so \(r=2\) instantly.

Step 1: Find the Rate of Interest (R) -

In CI, the amount at the end of year 2 becomes the principal for year 3.
\(Interest for 1 year = 9261 - 8820 = 441\).
\(R = \frac{441}{8820} \times 100 = 5%\).

Step 2: Find the Principal (P) -
\(A = P(1 + R/100)^n\)
\(8820 = P(1 + 5/100)^2 = P(21/20)^2\)
\(8820 = P \times \frac{441}{400}\)
\(P = \frac{8820 \times 400}{441} = 20 \times 400 = 8000\).

Wait, let's re-verify the calculation: \(8820/441 = 20\). \(20 \times 400 = 8000\). Correct calculation gives 8000.

Note: Based on calculations, (D) is the result, but let's double check common exam values. Often these figures lead to 8000. Quick Tip: Rate is simply the interest earned in that one year divided by the previous year's amount. \(R = \frac{A_3 - A_2{A_2} \times 100\).

Step 1: Use the Gap Method - First, seat the 4 women.

Ways to arrange 4 women = \(4! = 24\).

Step 2: Identify the gaps - When 4 women are seated (_W_W_W_W_), there are 5 possible gaps created where men can sit.

Step 3: Arrange the men in gaps -

Select 2 gaps out of 5 and arrange 2 men = \(^5P_2 = 5 \times 4 = 20\).

Step 4: Total arrangement -

Total ways = (Arrangement of women) \(\times\) (Arrangement of men in gaps)

Total ways = \(24 \times 20 = 480\). Quick Tip: Whenever the condition is "No two X sit together", always arrange the "Y" group first and place "X" in the gaps.

Step 1: Observe the pattern - The numbers are increasing rapidly, suggesting a combination of multiplication and addition.

Step 2: Test the logic -
\(5 \times 1 + 1 = 6\)
\(6 \times 2 + 2 = 14\)
\(14 \times 3 + 3 = 45\)
\(45 \times 4 + 4 = 184\)

Step 3: Find the next term -
\(184 \times 5 + 5 = 920 + 5 = 925\). Quick Tip: In series where numbers grow slowly then very fast, look for the pattern: \(Previous Term \times n + n\) or \(Previous Term \times n + (n^2)\).

Step 1: Identify the relationship - The relationship is "Artist : Primary Tool".

Step 2: Apply to Sculptor - A Sculptor uses a Chisel as their primary tool to create art.

Step 3: Apply to Author - An Author uses a Pen (or keyboard/typewriter) as their primary tool to create their work.

Step 4: Rule out other options - Reader is the consumer, Paper is the medium, and Book is the final product. Quick Tip: In analogy questions, always define the relationship in a short sentence: "A [Sculptor] uses a [Chisel] to work." Then apply it to the second pair.

Step 1: Identify hours (H) and minutes (M) - \(H = 3, M = 40\).

Step 2: Use the standard formula -
\(\)Angle = |30H - \frac{11{2M|\(\)

Step 3: Substitute the values -
\(\text{Angle = |30(3) - \frac{11}{2}(40)|\)
\(Angle = |90 - 11(20)|\)
\(Angle = |90 - 220|\)

Step 4: Calculate the absolute difference -
\(Angle = |-130| = 130^\circ\). Quick Tip: Each minute space is \(6^\circ\) and each hour space is \(30^\circ\). If the result is greater than \(180^\circ\), subtract it from \(360^\circ\) to find the reflex or internal angle as required.

Step 1: Analyze Statement I - Increasing import duty is an action/decision taken by the government, likely to reduce imports or increase revenue. It is an effect of an economic policy or fiscal deficit.

Step 2: Analyze Statement II - Locally made electronics prices remaining unchanged is a market condition. It might be due to stable local raw material costs or competitive strategy.

Step 3: Determine the link - Statement I affects imported goods. It does not directly cause local prices to stay the same (in fact, it often allows them to rise). Therefore, they are not cause-and-effect.

Step 4: Conclusion - Both are results/effects of different economic factors. Quick Tip: To check for cause-effect, use "Because [I], therefore [II]". If it doesn't sound logical, they are either independent causes or effects of different causes.

Step 1: Identify the relationship - The relationship is "Place : Items stored there".

Step 2: Verify the pair - An Armory is a place where Weapons are stored.

Step 3: Evaluate options -

- Hospital is for treating patients, not just storage.

- Bank deals with money/loans, but it's not a storage room for "loans".

- Aviary is for birds, not fishes.

- Warehouse is specifically a place where Merchandise (goods) is stored. (Correct) Quick Tip: Be careful with biological terms: Aviary = Birds, Aquarium = Fish, Apiary = Bees.

Step 1: Use Fixed positions - E is at the left end (Pos 1).

Current: E _ _ _ _ _ _ _

Step 2: Apply condition IV - B is immediate right of E.

Current: E B _ _ _ _ _ _

Step 3: Apply condition V - H, B, G sit together with H in middle. Since B is at Pos 2, H must be at Pos 3 and G at Pos 4.

Current: E B H G _ _ _ _

Step 4: Combine conditions I, III, and VI -

- III says C is 3rd to the left of D (\(C\) _ _ \(D\)).

- I says D is right of A (\(A\) \(D\)).

- VI says F is left of A (\(F\) \(A\)).

Combining these: C _ F A D.

Step 5: Fit this into the remaining 4 slots -

Slots 5, 6, 7, 8 are left. If C is at Pos 5: E B H G C F A D.

Step 6: Check all conditions - D is right of A (Yes), E is left end (Yes), C is 3rd left of D (Yes: slots 8-5=3), B is right of E (Yes), H is between B and G (Yes), F is left of A (Yes). Quick Tip: In linear seating, always start with the "ends" or "fixed positions" (like E in this case) and build the chain from there.

Step 1: Compare the second and third sentences -

- "Exam is tough" \(\rightarrow\) ga ni sa

- "study tough subject" \(\rightarrow\) lo sa ma

Step 2: Identify the common word - The word common to both sentences is "tough".

Step 3: Identify the common code - The code common to both sentences is "sa".

Step 4: Conclusion - Therefore, the code for "tough" is "sa". Quick Tip: In substitution coding, look for the word that repeats in different sentences. Its corresponding code will also repeat.

Step 1: Analyze the first letters - A, D, G, J...

A (+3) \(\rightarrow\) D; D (+3) \(\rightarrow\) G; G (+3) \(\rightarrow\) J.

The next letter will be J + 3 = M.

Step 2: Analyze the numbers - 1, 4, 9, 16...

These are squares of consecutive numbers: \(1^2, 2^2, 3^2, 4^2\).

The next number will be \(5^2 = \textbf{25}\).

Step 3: Analyze the last letters - C, F, I, L...

C (+3) \(\rightarrow\) F; F (+3) \(\rightarrow\) I; I (+3) \(\rightarrow\) L.

The next letter will be L + 3 = O.

Step 4: Combine - The term is M25O. Quick Tip: Always break mixed series into three independent parts (First letter, Number, Last letter) to find the pattern easily.

Step 1: Calculate the remaining days in July - \(31 - 21 = 10\) days.

Step 2: Add days for the following months -

August: 31 days

September: 30 days

October: 31 days

November: 30 days

December (up to 25th): 25 days

Step 3: Find the total days - \(10 + 31 + 30 + 31 + 30 + 25 = 157\) days.

Step 4: Calculate odd days - \(157 \div 7 = 22\) weeks and 3 odd days.

Step 5: Find the day - Wednesday + 3 days = Saturday. Quick Tip: To save time, use odd days for each month: July(3), Aug(3), Sept(2), Oct(3), Nov(2). Here: \(10/7 \rightarrow 3\) (July) + 3 + 2 + 3 + 2 + \(25/7 \rightarrow 4\) (Dec) = \(17\) total odd days. \(17/7 \rightarrow \textbf{3}\) final odd days.

Step 1: Analyze "All oceans are seas and All seas are oceans" - This means the set of Oceans and Seas is identical (the same circle).

Step 2: Analyze "Only a few rivers are oceans" - This means:

1. Some rivers ARE oceans.

2. Some rivers ARE NOT oceans.

Step 3: Evaluate Conclusion I - Since some rivers are already oceans (and thus seas), "Some rivers are seas" is a definite fact, not a possibility. In Syllogism, a definite fact is not considered a possibility.

Step 4: Evaluate Conclusion II - "Only a few" implies that "All rivers can never be oceans/seas." Therefore, this is not possible. Quick Tip: "Only a few A are B" = "Some A are B" + "Some A are NOT B". It's a double-edged sword that restricts "All" possibilities.

Step 1: Create a table based on fixed data (N is 2nd with F-16) -

Rank 1: \dots

Rank 2: N (F-16)

Rank 3: \dots

Rank 4: \dots

Step 2: Apply condition (1) - O is 2 ranks below L. The only possible spots for L and O are Rank 1 and Rank 3.

Rank 1: L

Rank 2: N (F-16)

Rank 3: O

Rank 4: M (Remaining pilot)

Step 3: Apply condition (2) - F-35 pilot finished immediately before M. Since M is 4th, the F-35 pilot must be 3rd (O).

Step 4: Apply condition (4) - L did not fly SU-57. Since O has F-35 and N has F-16, SU-57 must go to the only other person remaining, which is M.

Step 5: Final Placement -

Rank 1: L (F-22)

Rank 2: N (F-16)

Rank 3: O (F-35)

Rank 4: M (SU-57)

Result: L flew the F-22. Quick Tip: In grid puzzles, always fill in the definite ranks first. Use the "gap" information (like O being 2 ranks below L) to narrow down the remaining positions.

Step 1: Find the difference between consecutive terms -
\(8 - 7 = 1\)
\(12 - 8 = 4\)
\(21 - 12 = 9\)
\(37 - 21 = 16\)

Step 2: Observe the pattern of differences -

The differences are \(1, 4, 9, 16\), which are perfect squares: \(1^2, 2^2, 3^2, 4^2\).

Step 3: Find the next difference -

The next square is \(5^2 = 25\).

Step 4: Calculate the next term -
\(37 + 25 = 62\). Quick Tip: Whenever the numbers in a series increase slowly, always check the "difference of differences" or see if the first differences follow a square/cube pattern.

Step 1: Map the generations -

- Generation 1: A (Father of B)

- Generation 2: B (Child of A) and D (Brother of B)

- Generation 3: C (Daughter of B)

- Generation 4: E (Son of C)

Step 2: Analyze the relationship -

- B is E's Grandparent (since B is the parent of C, and C is the parent of E).

- A is the parent of B.

Step 3: Determine the final link -

A is the parent of E's grandparent. Therefore, A is the Great Grandfather of E. Quick Tip: In Blood Relation problems, use '+' for male, '-' for female, and vertical lines for generation gaps to avoid confusion.

Step 1: Draw the Venn Diagram -

- Draw a circle for "Pens" entirely inside a circle for "Pencils".

- Draw the circle for "Markers" completely separate from the "Pencils" circle.

Step 2: Evaluate Conclusion (1) - "Some pens are markers." Since all Pens are inside Pencils, and no Pencil can touch a Marker, no Pen can ever be a Marker. (Does not follow)

Step 3: Evaluate Conclusion (2) - "Some pens are pencils." Since ALL pens are pencils, it is a logical certainty that "some" (at least a part) of them are pencils. (Follows) Quick Tip: In Syllogism, if "All A are B", then "Some A are B" is always considered true.

Step 1: Square the given equation -
\((\tan\theta + \cot\theta)^2 = 4^2\)
\(\tan^2\theta + \cot^2\theta + 2\tan\theta\cot\theta = 16\)

Step 2: Use the identity \(\tan\theta \cdot \cot\theta = 1\) -
\(\tan^2\theta + \cot^2\theta + 2(1) = 16\)

Step 3: Solve -
\(\tan^2\theta + \cot^2\theta = 16 - 2 = 14\) Quick Tip: For any reciprocal pair where \(x + 1/x = a\), then \(x^2 + 1/x^2 = a^2 - 2\).

Step 1: Find the smaller angle using the formula -
\(H = 10, M = 25\).
\(Angle = |30H - \frac{11}{2}M|\)
\(Angle = |30(10) - \frac{11}{2}(25)|\)
\(Angle = |300 - 137.5| = 162.5^\circ\).

Step 2: Calculate the Reflex angle -

Reflex angle is the larger angle, found by subtracting the smaller angle from 360°.
\(Reflex Angle = 360^\circ - 162.5^\circ = 197.5^\circ\). Quick Tip: A reflex angle is always greater than 180° and less than 360°. If your initial calculation gives a result \(> 180^\circ\), that is already the reflex angle.

Step 1: Calculate the semi-perimeter (s) -
\(s = \frac{a + b + c}{2} = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21\)

Step 2: Use Heron's Formula -
\(Area = \sqrt{s(s-a)(s-b)(s-c)}\)

Step 3: Substitute the values -
\(Area = \sqrt{21(21-13)(21-14)(21-15)}\)
\(Area = \sqrt{21 \times 8 \times 7 \times 6}\)

Step 4: Factorize for easier calculation -
\(Area = \sqrt{(7 \times 3) \times (2 \times 2 \times 2) \times 7 \times (2 \times 3)}\)
\(Area = \sqrt{7^2 \times 3^2 \times 2^4} = 7 \times 3 \times 2^2 = 7 \times 3 \times 4 = 84\) Quick Tip: The (13, 14, 15) triangle is a very common exam triangle. It's essentially two right triangles (9-12-15 and 5-12-13) joined at their common side of 12.

Step 1: Apply Logarithm Properties - Using \(\log A + \log B = \log(AB)\):
\(\log_2(x(x-2)) = 3\)

Step 2: Convert to Exponential form -
\(x(x-2) = 2^3\)
\(x^2 - 2x = 8\)

Step 3: Solve the Quadratic Equation -
\(x^2 - 2x - 8 = 0\)
\((x-4)(x+2) = 0\)

Step 4: Check for validity -
\(x = 4\) or \(x = -2\).

Since we cannot take the log of a negative number, \(x = -2\) is invalid.
\(\therefore x = 4\). Quick Tip: Always test options in log equations! For \(x=4\): \(\log_2(4) + \log_2(2) = 2 + 1 = 3\). This is much faster.

Step 1: Use the standard formula for 2 years -
\(Difference (D) = \frac{Pr^2}{100^2}\)

Step 2: Substitute given values -
\(150 = \frac{P \times (10)^2}{10000}\)

Step 3: Solve for P -
\(150 = \frac{P \times 100}{10000}\)
\(150 = \frac{P}{100}\)
\(P = 150 \times 100 = 15,000\). Quick Tip: For 2 years, the difference is simply "Interest on first year's interest." Since 10% of 10% is 1%, 1% of Principal = 150, so Principal = 15,000.

Step 1: Find Total Work (LCM of 12, 18) -

Total Work = 36 units.

Step 2: Find Efficiencies -

Efficiency of A = \(36 / 12 = 3\) units/day.

Efficiency of B = \(36 / 18 = 2\) units/day.

Step 3: Work done in first 2 days -

Together they do \((3+2) = 5\) units/day.

In 2 days: \(5 \times 2 = 10\) units.

Step 4: Remaining work -
\(36 - 10 = 26\) units.

Step 5: Time taken by A to finish -
\(26 / 3 = 8.66\) days.

Step 6: Total time for completion -
\(2 days (initial) + 8.66 days = 10.66\) days. Quick Tip: Always clarify if the question asks for "total time" or "remaining time." Here, "it will take for the work to get completed" implies the total duration.

Step 1: Convert speed to m/s -
\(72 \times \frac{5}{18} = 20\) m/s.

Step 2: Find Train Length (L) using station data -
\(Distance = Speed \times Time\)
\(L + 160 = 20 \times 18\)
\(L + 160 = 360 \implies L = 200\) meters.

Step 3: Find time to cross a man -

When crossing a man, distance = length of train.
\(Time = \frac{Distance}{Speed} = \frac{200}{20} = 10\) seconds. Quick Tip: To cross a platform, distance = (Train + Platform). To cross a man/pole, distance = (Train) only.

Step 1: Calculate Volume of Cylinder -
\(V = \pi r^2 h\)
\(V = \frac{22}{7} \times 7 \times 7 \times 1\)
\(V = 22 \times 7 = 154\) m³.

Step 2: Convert to Liters -
\(1 m^3 = 1000 liters\)
\(154 \times 1000 = 1,54,000 liters\). Quick Tip: Common conversion: \(1 m^3 = 1000\) liters, and \(1000 cm^3 = 1\) liter.