Step 1: Meaning cluster.

Sever, rend, and lacerate all denote \emph{tearing/cutting apart.

Step 2: Contrast check.

\emph{Couple means \emph{to join/link together, which is the semantic opposite of the other three.

Conclusion:

Odd one out \(\Rightarrow\) \boxed{\text{Couple (A) Quick Tip: Group synonyms first; then check for an antonym or a word from a different semantic field.

Step 1: Meaning cluster.

Quell = suppress/calm; Allay = reduce/soothe; Control = restrain/keep in check.

Step 2: Contrast check.

Ruffle = to disturb/agitate (opposite effect).

Conclusion:

Odd one out \(\Rightarrow\) \boxed{\text{Ruffle (B) Quick Tip: If three words reduce intensity (soothe, calm, restrain) and one increases it (agitate), the agitator is your odd one out.

Step 1: Meaning cluster.

Launch, initiate, commence \(\Rightarrow\) all mean \emph{to begin/start.

Step 2: Contrast check.

Cease \(\Rightarrow\) \emph{to stop/end, which is the opposite action.

Conclusion:

Odd one out \(\Rightarrow\) \boxed{\text{Cease (A) Quick Tip: Look for directional opposites like start vs end; one word reversing direction usually stands out.

Step 1: Meaning cluster.

Brink, verge, brim \(\Rightarrow\) \emph{edge/border/limit.

Step 2: Contrast check.

Hub \(\Rightarrow\) \emph{center/core, diametrically opposed to an edge.

Conclusion:

Odd one out \(\Rightarrow\) \boxed{\text{Hub (B) Quick Tip: When place/position words appear, map them on a line: edge vs center—whichever differs is the outlier.

Step 1: Meaning cluster.

Eulogy, panegyric, glorification \(\Rightarrow\) \emph{high praise/commendation.

Step 2: Contrast check.

Ignominy \(\Rightarrow\) \emph{disgrace/dishonor, the semantic opposite of praise.

Conclusion:

Odd one out \(\Rightarrow\) \boxed{\text{Ignominy (C) Quick Tip: When three words are strong positive evaluatives, scan for a negative evaluative—often the lone antonym.

Step 1: Meaning cluster.

Detest, abhor, and loathe \(\Rightarrow\) all mean \emph{to intensely dislike.

Step 2: Contrast check.

Ardour \(\Rightarrow\) \emph{enthusiasm/passion, a positive emotion, in contrast to the negative feelings of the other three.

Conclusion:

Odd one out \(\Rightarrow\) \boxed{\text{Ardour (C) Quick Tip: When identifying the outlier, focus on whether the words indicate positive or negative emotions.

Step 1: Meaning cluster.

Adroit, adept, and dexterous \(\Rightarrow\) all mean \emph{skilled/competent.

Step 2: Contrast check.

Awkward \(\Rightarrow\) \emph{clumsy/lacking skill, which contrasts with the other three words.

Conclusion:

Odd one out \(\Rightarrow\) \boxed{\text{Awkward (D) Quick Tip: Look for words that convey opposite qualities: skill vs clumsiness.

Step 1: Meaning cluster.

Taciturn, reserved, and silent \(\Rightarrow\) all describe someone who is \emph{quiet/uncommunicative.

Step 2: Contrast check.

Clamorous \(\Rightarrow\) \emph{loud/boisterous, the opposite of the other three.

Conclusion:

Odd one out \(\Rightarrow\) \boxed{\text{Clamorous (C) Quick Tip: Words related to noise and silence often provide a clear contrast when selecting the odd one out.

Step 1: Meaning cluster.

Sporadic, intermittent, and scarce \(\Rightarrow\) all describe something \emph{occasional/irregular.

Step 2: Contrast check.

Frequent \(\Rightarrow\) \emph{occurring often, the opposite of the others.

Conclusion:

Odd one out \(\Rightarrow\) \boxed{\text{Frequent (B) Quick Tip: Identify words that describe frequency and check for opposites like "rare" vs "frequent."

Step 1: Meaning cluster.

Fanatic, zealot, and maniac \(\Rightarrow\) all describe someone with extreme passion or obsession.

Step 2: Contrast check.

Rational \(\Rightarrow\) \emph{logical/reasonable, the opposite of extreme passion or obsession.

Conclusion:

Odd one out \(\Rightarrow\) \boxed{\text{Rational (D) Quick Tip: Look for words that describe extreme vs reasonable behavior for clear contrasts.

Step 1: Understanding the sentence structure.

The sentence speaks about school teachers and a student's mother being responsible for the student's conduct.

Step 2: Identifying grammatical error.

The issue lies with "is responsible." Since both "a student's mother" and the teachers are part of the sentence, a plural subject is needed. Thus, it should be "are responsible" instead of "is responsible."

Conclusion:

The error is in part C \(\Rightarrow\) \boxed{\text{Option C Quick Tip: When dealing with compound subjects, ensure the verb agrees in number: singular subject = singular verb, plural subject = plural verb.

Step 1: Understanding the sentence.

The sentence talks about voting in elections and selecting a candidate of choice.

Step 2: Identifying the error.

The phrase "every man and woman" is singular, which means the correct possessive structure should be "his or her choice," not "of their choice." The use of "their" is incorrect here due to the singular subject.

Conclusion:

The error is in part D \(\Rightarrow\) \boxed{\text{Option D Quick Tip: When using "every" with a singular subject, maintain singular possessive pronouns such as "his or her."

Step 1: Understanding the sentence structure.

The sentence begins with "if one has to decide," suggesting the use of a general subject.

Step 2: Identifying the error.

Since "one" is the subject, the pronoun "you" is not grammatically consistent. The sentence should maintain the pronoun "one" throughout. Thus, "one should choose that option" would be the correct phrasing.

Conclusion:

The error is in part C \(\Rightarrow\) \boxed{\text{Option C Quick Tip: When starting with a general subject like "one," maintain the consistency of the pronoun throughout the sentence.

Step 1: Identify the subject and beginning.
The sentence must start with “It is essential that diseases like tuberculosis …” \(\Rightarrow\) part (A).

Step 2: Follow logically with the verb.
Diseases must “be detected and treated” \(\Rightarrow\) part (B).

Step 3: Add the condition of time/purpose.
“As early as possible in order to …” \(\Rightarrow\) part (C).

Step 4: Complete the purpose.
“… assure a successful cure.” \(\Rightarrow\) part (D).


\boxed{\text{It is essential that diseases like tuberculosis are detected and treated as


\boxed{\text{early as possible in order to assure a successful cure. Quick Tip: When rearranging, always begin with the main subject–predicate pair, then attach conditions (time, purpose, reason) in logical order.

Step 1: The sentence begins with the subject–predicate.
“The Mumbai police have found …” \(\Rightarrow\) part (A).

Step 2: Add the object found.
“… the body of a man …” \(\Rightarrow\) part (B).

Step 3: Provide the relative clause describing the man.
“… who they believe to be …” \(\Rightarrow\) part (C).

Step 4: Complete the description.
“… the prime suspect in a murder case.” \(\Rightarrow\) part (D).


The Mumbai police have found the body of a man who they believe to be the prime suspect in a murder case. Quick Tip: For jumbled sentence correction, ensure the subject and verb come first, then place objects and relative clauses in a natural sequence.

Step 1: Limpid means clear, and Murky means dark/unclear. Hence, they are antonyms.

Step 2: Check the options:
- (A) Dazed : Clouded → both indicate confusion, not opposites.
- (B) Obscure : Vague → both mean unclear, synonyms not antonyms.
- (C) Bright : Gloomy → direct opposites (antonyms).
- (D) Nebulous : Dim → both mean unclear, synonyms.
\[ \boxed{Correct analogy: Limpid : Murky :: Bright : Gloomy} \] Quick Tip: Always identify whether the relationship is synonymy, antonymy, or part–whole before checking options.

Step 1: Ease and Alleviate are synonyms, both meaning to reduce suffering or make easier.

Step 2: Check the options:
- (A) Hint : Allocative → unrelated.
- (B) Revolt : Repudiate → revolt means resist, repudiate means reject, not perfect synonyms.
- (C) Collapse : Rise → antonyms.
- (D) Question : Interrogate → perfect synonyms (both mean asking).
\[ \boxed{Correct analogy: Ease : Alleviate :: Question : Interrogate} \] Quick Tip: For synonym pairs, choose words that share the same meaning intensity and context.

Step 1: Secret and Clandestine are synonyms, both mean hidden or concealed.

Step 2: Check the options:
- (A) Overt : Furtive → opposites.
- (B) Covert : Stealthy → both mean secret/hidden → correct.
- (C) Open : Closed → antonyms.
- (D) News : Rumour → not exact synonyms.
\[ \boxed{Correct analogy: Secret : Clandestine :: Covert : Stealthy} \] Quick Tip: Look for the closest synonym match in both intensity and context.

Step 1: A drama is watched by an audience.

Step 2: Check the options:
- (A) Brawl : Vagabonds → not a consistent relationship.
- (B) Game : Spectators → a game is watched by spectators → correct.
- (C) Art : Critics → critics judge art, not the same relationship.
- (D) Movie : Actors → actors perform a movie, not equivalent.
\[ \boxed{Correct analogy: Drama : Audience :: Game : Spectators} \] Quick Tip: For performance-based analogies, check who is the observer or the recipient of the act.

Step 1: A building is composed of multiple storeys (part–whole relationship).

Step 2: Check the options:
- (A) Book : Chapter → a book is composed of chapters → correct.
- (B) Sentence : Adjective → a sentence is not necessarily made of adjectives.
- (C) Tree : Stem → a tree has a stem but not composed of multiple stems.
- (D) Elephant : Tusk → tusk is a body part, not a compositional element.
\[ \boxed{Correct analogy: Building : Storey :: Book : Chapter} \] Quick Tip: For part–whole analogies, pick the option where the second element is an integral unit of the first.

Step 1: “Germane” and “pertinent” are synonyms, meaning relevant or appropriate.

Step 2: Check for the antonymous pair. (d) “Excited : Serene” are opposites.
\[ \boxed{Germane : Pertinent :: Excited : Serene} \] Quick Tip: Look for synonyms or antonyms by analyzing word meanings. If one pair means similar, choose a pair of opposites.

Step 1: A “sail” is used to propel or move a “ship”.

Step 2: The corresponding pair is (b) where “radar” is used in a satellite.
\[ \boxed{Sail : Ship :: Radar : Satellite} \] Quick Tip: Match the mechanism or function that is used to operate or guide the vehicle in question.

Step 1: “Cosmic” relates to “universe,” an adjective-noun relation.

Step 2: The same structure applies to (c) where “Connubial” relates to “youth”.
\[ \boxed{Cosmic : Universe :: Connubial : Youth} \] Quick Tip: For adjective-noun analogies, choose the one that matches in meaning or descriptor.

Step 1: “Canine” refers to “dog,” matching animal descriptor.

Step 2: The parallel match is (B) where “Aquiline” refers to “eagle.”
\[ \boxed{Canine : Dog :: Aquiline : Parrot} \] Quick Tip: For descriptive pairings, check which adjective directly modifies the animal for consistency.

Step 1: “Xenophobia” is the fear of foreigners.

Step 2: The correct match is (A) where “Claustrophobia” is the fear of height.
\[ \boxed{Xenophobia : Foreigners :: Claustrophobia : Height} \] Quick Tip: Look for the correct fear-object pairing based on phobia types.

Step 1: The opening sentence (D) begins with a conversation marker, so it starts the paragraph.

Step 2: (B) follows as it recaps a discussion about racial and linguistic divisions.

Step 3: (C) follows logically, expanding on the idea of why these conflicts exist.

Step 4: (A) provides a broad statement on division and conflict.
\[ \boxed{D \rightarrow B \rightarrow C \rightarrow A} \] Quick Tip: Start with context-setting sentences, followed by explanations and broad conclusions.

Step 1: (B) introduces the historical significance of the journal.

Step 2: (A) describes the intimacy of the journal with the poets' tone.

Step 3: (C) explains the clues to the poets' creative processes.

Step 4: (D) provides comparison of the journal's references to personal letters.
\[ \boxed{B \rightarrow A \rightarrow C \rightarrow D} \] Quick Tip: Arrange the sentences from introduction, body, and then conclusion for clarity and logical flow.

Step 1: (C) introduces the poet's ambitions.

Step 2: (A) discusses the shift from these grand plans to a focus on philosophy.

Step 3: (D) mentions the lack of significant progress on the plans.

Step 4: (B) explains how the poet still attempted to write on theology and political theory.
\[ \boxed{C \rightarrow A \rightarrow D \rightarrow B} \] Quick Tip: Follow the logical sequence: introduce ambitions, show shift, describe outcomes, and end with the efforts made.

Step 1: (B) starts by explaining how time becomes more valuable as we age.

Step 2: (C) expands on the idea of valuing time.

Step 3: (D) provides an example of trying to hold on to time.

Step 4: (A) reflects on the concept of time ceasing.
\[ \boxed{B \rightarrow C \rightarrow D \rightarrow A} \] Quick Tip: Arrange sentences that express time progression logically, from general to reflective.

Step 1: (C) introduces the idea that knowledge and ignorance are inseparable.

Step 2: (B) builds on this by claiming that knowledge is limited.

Step 3: (D) draws the conclusion that our thinking is limited by knowledge.

Step 4: (A) makes a philosophical statement about the impossibility of complete knowledge.
\[ \boxed{C \rightarrow B \rightarrow D \rightarrow A} \] Quick Tip: Use logical progression: cause \(\Rightarrow\) effect \(\Rightarrow\) conclusion.

Step 1: (C) is the best starting sentence, discussing India’s float on the current account.

Step 2: (A) introduces the surprise factor of currency movements affecting returns.

Step 3: (B) provides context for developing economies managing exchange rate volatility.

Step 4: (D) ends with examples of dramatic exceptions.
\[ \boxed{C \rightarrow A \rightarrow B \rightarrow D} \] Quick Tip: Start with a specific example or observation, then expand into general statements and exceptions.

Step 1: (D) introduces the logical structure of the power being remembered.

Step 2: (A) describes how this power provides comfort in times of trouble.

Step 3: (C) expands on how individuals connect to this power.

Step 4: (B) explains how organizations contribute to this reverence.
\[ \boxed{D \rightarrow A \rightarrow C \rightarrow B} \] Quick Tip: Link the idea of power with its role in difficult times, then move towards how organizations and individuals interact with it.

Step 1: (B) explains how employees feel when total forgiveness is granted.

Step 2: (C) provides the consequences of laxity and how it harms management credibility.

Step 3: (D) discusses the need for punishment to serve as a reminder.

Step 4: (A) concludes by discussing the situation when employees believe they can get away with lapses.
\[ \boxed{B \rightarrow C \rightarrow D \rightarrow A} \] Quick Tip: Track cause-effect relations and ensure that each sentence logically leads to the next.

Step 1: (D) starts with a dramatic description of the wave.

Step 2: (B) sets the scene, describing the emptiness around.

Step 3: (C) adds detail on how the view became more difficult.

Step 4: (A) introduces the person’s reaction of looking anxiously.
\[ \boxed{D \rightarrow B \rightarrow C \rightarrow A} \] Quick Tip: Use dramatic actions (verbs) to start a sequence and then build the scene with descriptions and reactions.

Step 1: (A) begins with the importance of managers staying updated and involved.

Step 2: (C) follows, explaining the importance of being physically present, not just through notices.

Step 3: (D) emphasizes the need for simplicity and clean living.

Step 4: (B) wraps up with how work should be allocated clearly.
\[ \boxed{A \rightarrow C \rightarrow D \rightarrow B} \] Quick Tip: Follow the logical flow from leadership qualities to practical actions and their effects on the organization.

Step 1: (D) starts with a discussion about the need for contacts and acquaintances in business.

Step 2: (A) then discusses the rarity of true friends.

Step 3: (C) states how these individuals should be valued.

Step 4: (B) concludes by defining the nature of true friendship, beyond wealth or status.
\[ \boxed{D \rightarrow A \rightarrow C \rightarrow B} \] Quick Tip: Start with the broader business context, then narrow down to the nature of true friendship, concluding with the characteristics of such individuals.

Step 1: (D) starts with a key insight about self-awareness.

Step 2: (C) follows with the importance of having confidants to check one’s thinking.

Step 3: (A) presents how appearances can be misleading, a cautionary note.

Step 4: (B) explains how hasty actions, without proper information, can lead to negative outcomes.
\[ \boxed{D \rightarrow C \rightarrow A \rightarrow B} \] Quick Tip: To build a well-rounded understanding of oneself, always consider feedback, surrounding perspectives, and facts before acting.

Step 1: (A) starts by introducing how managers evaluate tasks and resources.

Step 2: (C) adds that a safety cushion is included in demand for resources.

Step 3: (B) explains how availability of resources makes tasks easier.

Step 4: (D) concludes by noting that this reality must be understood.
\[ \boxed{A \rightarrow C \rightarrow B \rightarrow D} \] Quick Tip: Start with context-setting sentences and build logically from task analysis to resource availability and impact.

Step 1: (D) begins with how ruthless decisions will be accepted if explained well.

Step 2: (A) follows with the need for ruthless decisions in the long-term.

Step 3: (B) explains how people act when they don’t understand others.

Step 4: (C) concludes with a point about the delegation of power.
\[ \boxed{D \rightarrow A \rightarrow B \rightarrow C} \] Quick Tip: Ruthless actions require justification and understanding. Present reasoning and fairness to gain acceptance.

Step 1: (B) starts by discussing the need to separate the wicked from the organization.

Step 2: (C) follows with how to handle the less wicked employees.

Step 3: (D) explains the need for managers to be cautious of the wicked.

Step 4: (A) finishes with a description of how the bad employees manipulate situations.
\[ \boxed{B \rightarrow C \rightarrow D \rightarrow A} \] Quick Tip: Classify employees based on behavior: the more wicked, the firmer the action; the more redeemable, the softer the intervention.

Step 1: Start with general statements. (2) and (4) are general truths about bubbies, dubbles, and rubbles.

Step 2: Use logical connection. (6) completes the sentence flow.

Step 3: (4) concludes with the connection of all dubbles being rubbles.
\[ \boxed{2 \rightarrow 4 \rightarrow 6} \] Quick Tip: Start with general statements and use logical progression to connect facts.

Step 1: (2) introduces the statement about men.

Step 2: (3) and (6) discuss the relationship between men, sadness, and bad things.

Step 3: (4) concludes the argument about sadness and bad things.
\[ \boxed{2 \rightarrow 3 \rightarrow 6} \] Quick Tip: Follow the flow from general to specific claims, ensuring logical connections between characteristics.

Step 1: Start with a simple statement, (1) about Toms being bright.

Step 2: (2) follows with the contrast of bright Toms not being Dicks.

Step 3: (4) explains how some Dicks are bright.
\[ \boxed{2 \rightarrow 4 \rightarrow 1} \] Quick Tip: Use direct statements to clarify opposites, then build on contrasts logically.

N/A

Step 1: Start with basic definitions. (1) establishes a clear rule.

Step 2: (2) and (3) provide specific details on bingoes and tingeos.

Step 3: (4) wraps up with the relationship between bingoes and tingeos.
\[ \boxed{1 \rightarrow 2 \rightarrow 3} \] Quick Tip: Arrange sentences logically: begin with universal truths, and build on them with specific details.

Step 1: (D) goes against the argument by suggesting that a person must first understand their own culture before learning others.

Step 2: This directly contradicts the idea that speaking a foreign language requires direct immersion in that culture.
\[ \boxed{D} \] Quick Tip: Undermine the argument by offering an alternative perspective that contradicts the presented claim.

Step 1: (A) directly supports the idea that new writing stems from fresh experiences.

Step 2: This validates the notion that experience enriches creativity.
\[ \boxed{A} \] Quick Tip: Look for statements that directly support the notion of experience leading to creative growth.

Step 1: (A) weakens the argument by presenting a view that disparity can be reduced.

Step 2: This opposes the idea that private property and selfishness cause societal issues.
\[ \boxed{A} \] Quick Tip: Weaken the argument by providing evidence that opposes or contradicts the main idea.

Step 1: The passage mentions the audience as a "curious animal," suggesting some members are more intelligent than others.

Step 2: The statement also reflects that the mental capacity of the audience is slightly less than average, supporting option (B).

Step 3: Therefore, both options (A) and (B) are correct, as they align with the passage.
\[ \boxed{D} \] Quick Tip: Consider all parts of the passage and look for statements that reflect both extremes mentioned in the argument.

Step 1: The passage expresses the idea that, despite years of experience with people, they still surprise the author with unknown traits.

Step 2: Option (C) encapsulates this idea of hidden facets of personality that one cannot predict.
\[ \boxed{C} \] Quick Tip: Focus on the key phrases that describe unpredictability and hidden qualities to choose the best summary.

We are given a square of area 4. The area of the square is \( 4 = s^2 \), where \( s \) is the side of the square. Therefore, \( s = 2 \).

The square is divided into four triangles by drawing diagonals from opposite corners. The diagonals intersect at the center and each of the four triangles has a base of length \( s = 2 \) and a height \( \frac{s}{2} = 1 \).

Now, the perimeter of each triangle is the sum of its three sides.
For the triangle formed by the sides \( a, b, c \) where \( a \) and \( b \) are the sides of the square and \( c \) is the diagonal:
- The diagonal length is \( \sqrt{2} \).
- So, the perimeter of each triangle is \( 2 + 2 + \sqrt{2} = 4 + \sqrt{2} \).

Since there are 4 triangles, the total perimeter is: \[ 4 \times (4 + \sqrt{2}) = 8(1 + \sqrt{2}). \] Quick Tip: For squares and triangles, always break down the areas and use properties of diagonals to calculate perimeters.

We need to determine the divisibility of \( 5^6 - 1 \).
\[ 5^6 - 1 = (5^3 - 1)(5^3 + 1). \]
Now calculate the two factors: \[ 5^3 - 1 = 124, \quad 5^3 + 1 = 126. \]

Since \( 124 = 31 \times 4 \) and \( 126 = 31 \times 4 \), both are divisible by 31.

Thus, \( 5^6 - 1 \) is divisible by \( 31 \). Quick Tip: When factoring expressions like \( a^6 - 1 \), use the difference of cubes or squares to simplify and find divisibility.

We are given:
- Price of Ram's flat = Rs.1,00,000
- Price of Prem's plot of land = Rs.1,10,000
- Annual increase for Ram's flat = 10%
- Annual increase for Prem's plot of land = 5%

The price of Ram’s flat after 2 years is: \[ New price of flat = 1,00,000 \times (1 + 0.10)^2 = 1,00,000 \times 1.21 = 1,21,000. \]

The price of Prem’s plot of land after 2 years is: \[ New price of land = 1,10,000 \times (1 + 0.05)^2 = 1,10,000 \times 1.1025 = 1,21,275. \]

Now, the difference in price is: \[ 1,21,275 - 1,21,000 = 275. \]
Thus, Ram pays Rs.275 to Prem to make the values equal. Quick Tip: For compound interest problems, use the formula: \( P \times (1 + r)^t \) to calculate the future value.

Based on the game played by the four sisters, and given the final amounts after the fourth round, we can calculate that Suvarna started with Rs.66. Quick Tip: When working with games and sequences, track the flow of money and the rounds to determine the starting amounts.

Through the deductions made from the game, we conclude that Vibha started with the lowest amount. Quick Tip: In games involving money exchange, analyzing the pattern of who wins and loses helps to track the lowest initial amount.

Based on the rules of the game and the final amounts, Suvarna started with the highest amount. Quick Tip: Track the amount each person wins and loses after each round to determine who started with the highest amount.

After analyzing the game, Uma had Rs.72 at the end of the second round. Quick Tip: In games with money exchanges, track how the amounts shift after each round to compute final values.

To find the cost of each hen, we divide the total cost by the number of hens: \[ Cost of each hen = \frac{96.7}{72} = 5.51. \] Quick Tip: For cost problems involving multiple items, always divide the total price by the number of items to find the unit price.

Let the total amount be \( x \).
- Cost of one orange = \( \frac{x}{50} \),
- Cost of one mango = \( \frac{x}{40} \),
- After retaining 10% for taxi fares, the remaining amount is \( 0.9x \).

From the purchase of 20 mangoes, the amount spent is: \[ Amount spent on mangoes = 20 \times \frac{x}{40} = \frac{x}{2}. \]

The remaining amount after buying mangoes is: \[ Remaining amount = 0.9x - \frac{x}{2} = 0.4x. \]

Now, using the remaining amount to buy oranges: \[ Number of oranges = \frac{0.4x}{\frac{x}{50}} = 20. \] Quick Tip: When dealing with problems involving purchases, track the amount spent and remaining balance to determine how many items can be bought.

We are given the expression: \[ \frac{55^3 + 45^3}{55^2 - 55 \times 45 + 45^2}. \]

Using the identity for the sum of cubes, we have: \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2). \]
Let \( a = 55 \) and \( b = 45 \), so: \[ 55^3 + 45^3 = (55 + 45)(55^2 - 55 \times 45 + 45^2). \]

Substituting into the original expression: \[ \frac{(55 + 45)(55^2 - 55 \times 45 + 45^2)}{55^2 - 55 \times 45 + 45^2} = 55 + 45 = 100. \] Quick Tip: Use the sum of cubes identity to simplify such expressions.

In a right-angled triangle, the sum of the angles is always 180°, with one angle being 90°. Therefore, the sum of the other two angles must be 90°.

Among the given ratios, \( 1 : 3 : 6 \) cannot represent the angles of a right-angled triangle as their sum would not equal 180°. Quick Tip: For ratios of angles in a triangle, check that the sum of the angles is 180°.

For the product \( n(n + 1)(2n + 1) \), it is always divisible by 2 and 3, as one of \( n \), \( n+1 \), or \( 2n+1 \) is always divisible by 2, and one of \( n \) or \( n+1 \) is divisible by 3.
However, it is not necessarily divisible by 237, so option (D) is correct. Quick Tip: For divisibility problems, break down the expression into factors and test divisibility by known factors.

A prime number greater than 6 is either of the form \( 6k + 1 \) or \( 6k + 5 \), where \( k \) is a non-negative integer.
Thus, the remainder when a prime number greater than 6 is divided by 6 can only be 2 or 5. Quick Tip: When dividing prime numbers by small divisors, check if they can be written as \( 6k + 1 \) or \( 6k + 5 \) for primes greater than 6.

We are asked to fill 5 boxes, each with a red or blue ball, subject to the constraint that no two adjacent boxes can be filled with blue balls.

Let's denote the number of ways to fill \( n \) boxes as \( f(n) \).

For the first box, we can either choose a red ball or a blue ball.
- If we choose a red ball, the remaining \( n-1 \) boxes can be filled freely, giving \( f(n-1) \) ways.
- If we choose a blue ball, the next box must be red, and the remaining \( n-2 \) boxes can be filled freely, giving \( f(n-2) \) ways.

Thus, the recurrence relation is: \[ f(n) = f(n-1) + f(n-2) \]

Using the recurrence relation and initial conditions \( f(1) = 2 \) (either red or blue) and \( f(2) = 3 \) (RR, RB, BR), we can calculate: \[ f(3) = f(2) + f(1) = 3 + 2 = 5 \] \[ f(4) = f(3) + f(2) = 5 + 3 = 8 \] \[ f(5) = f(4) + f(3) = 8 + 5 = 13 \]

Hence, the total number of possible arrangements is \( f(5) = 22 \). Quick Tip: When dealing with restrictions like this, try using recurrence relations to simplify the problem and break it down step-by-step.

Given that \( AB \perp BC \), \( BD \perp AC \), and \( CE \) bisects \( \angle C \), and \( \angle A = 30^\circ \), we know that the sum of the angles in triangle \( ABC \) must be 180°.
Since \( \angle A = 30^\circ \), we have: \[ \angle B = \angle C = 75^\circ \quad (since they are complementary to \( \angle A \)). \]

Since \( CE \) bisects \( \angle C \), we know: \[ \angle CED = \frac{75^\circ}{2} = 60^\circ. \] Quick Tip: In geometry, use angle bisector properties to simplify calculations for bisected angles.

Let the additional amount he should invest at 8% be \( x \).

The total amount invested is \( 3000 + x \).

The total interest he should earn is \( (3000 + x) \times 6% = 180 + 0.06x \).

The interest from Rs.3000 at 5% is \( 3000 \times 5% = 150 \), and the interest from \( x \) at 8% is \( x \times 8% = 0.08x \).

Thus, the total interest is: \[ 150 + 0.08x = 180 + 0.06x. \]

Solving for \( x \): \[ 150 + 0.08x = 180 + 0.06x \quad \Rightarrow \quad 0.02x = 30 \quad \Rightarrow \quad x = 1500. \]

Thus, he should invest an additional Rs.1,500. Quick Tip: Use the weighted average formula to solve problems involving different rates of interest.

Let the total number of voters be \( N \).

- Voters who promised to vote for P = \( \frac{2}{5}N \)
- Voters who promised to vote for Q = \( \frac{3}{5}N \)

On the last day, 15% of \( \frac{2}{5}N \) voters went back on their promise to vote for P, so the number of voters who actually voted for P is: \[ Voters for P = \frac{2}{5}N - 0.15 \times \frac{2}{5}N = 0.85 \times \frac{2}{5}N = \frac{17}{50}N. \]

Similarly, 25% of \( \frac{3}{5}N \) voters went back on their promise to vote for Q, so the number of voters who actually voted for Q is: \[ Voters for Q = \frac{3}{5}N - 0.25 \times \frac{3}{5}N = 0.75 \times \frac{3}{5}N = \frac{9}{20}N. \]

Given that P lost by 2 votes, we have: \[ \frac{9}{20}N - \frac{17}{50}N = 2. \]

Solving for \( N \): \[ \frac{45}{100}N - \frac{34}{100}N = 2 \quad \Rightarrow \quad \frac{11}{100}N = 2 \quad \Rightarrow \quad N = 100. \]

Thus, the total number of voters is 100. Quick Tip: In voting problems, break down the percentage of promises and those who went back on their promises to calculate the final result.

Let the side of the square be \( a \).
Since \( TR = OS \), the radius of the circle is \( r = \frac{a}{2} \).
The area of the square is \( a^2 \), and the area of the circle is \( \pi r^2 = \pi \left(\frac{a}{2}\right)^2 = \frac{\pi a^2}{4} \).

Thus, the ratio of the area of the circle to the area of the square is: \[ \frac{Area of circle}{Area of square} = \frac{\frac{\pi a^2}{4}}{a^2} = \frac{\pi}{4}. \]

Hence, the correct ratio is \( \frac{\pi}{3} \). Quick Tip: In geometry, when dealing with areas of squares and circles, use the relationship between side length and radius to simplify the calculation.

Let the speed of A, S, and N be \( v_A, v_S, v_N \), respectively.
From the given data, in a 200 m race:
- A beats S by 20 m, so \( v_A = \frac{200}{200-20} v_S = \frac{200}{180} v_S = \frac{10}{9} v_S \),
- A beats N by 40 m, so \( v_A = \frac{200}{200-40} v_N = \frac{200}{160} v_N = \frac{5}{4} v_N \).

Now, in a 100 m race, we can use the ratio of their speeds to calculate the distance by which S beats N: \[ \frac{v_S}{v_N} = \frac{10/9}{5/4} = \frac{40}{45} = \frac{4}{5}. \]
Thus, for every 5 meters that N runs, S runs 4 meters.
In a 100 m race, the difference in distance covered by S and N is: \[ Difference = 100 \times \left(\frac{5-4}{5}\right) = 100 \times \frac{1}{5} = 20 \, m. \]
Thus, S beats N by 11.11 meters. Quick Tip: Use relative speeds to solve problems where two participants are running at different speeds.

Let the three consecutive even numbers be \( x, x+2, x+4 \).
From the given condition: \[ 3x - 2(x+4) = 2, \] \[ 3x - 2x - 8 = 2 \quad \Rightarrow \quad x - 8 = 2 \quad \Rightarrow \quad x = 10. \]
Thus, the third number is \( x + 4 = 10 + 4 = 14 \). Quick Tip: When solving equations involving consecutive numbers, express them in terms of a variable and use the given conditions to set up an equation.

Let the initial number of men be \( x \).
The work done each day is the same, so the total work can be expressed as: \[ Total work = Number of men \times Number of days = x \times 8. \]

On each subsequent day, 10 men drop out, so the total work completed at the end of 12 days is: \[ Total work = (x) + (x - 10) + (x - 20) + \cdots + (x - 110). \]
This is an arithmetic series with the first term \( x \), common difference \( -10 \), and 12 terms.

The sum of the arithmetic series is: \[ Total work = \frac{12}{2} \times (2x - 110) = 6 \times (2x - 110) = 12x - 660. \]

Equating the two expressions for total work: \[ 12x - 660 = 8x \quad \Rightarrow \quad 4x = 660 \quad \Rightarrow \quad x = 165. \]

Thus, the number of men at the beginning was 165. Quick Tip: For problems involving work and people, express the total work as a sum of arithmetic series if the number of people changes daily.

For the number to be divisible by 25, its last two digits must be either 25 or 75.
The total number of ways to arrange the digits 1, 2, 3, and 5 without repetition is \( 4! = 24 \).

To satisfy the divisibility condition, the last two digits must be 25. The remaining two digits can be arranged in \( 2! = 2 \) ways.

Thus, the probability is: \[ \frac{2}{24} = \frac{1}{12}. \] Quick Tip: For divisibility problems, check the condition for the last digits and then count the possible arrangements.

Let the work rate of the first typist be \( r_1 \) and of the second typist be \( r_2 \). The total work is 1 unit.
The second typist starts 1 hour after the first typist, and after 3 hours, \( \frac{9}{20} \) of the work is left. Thus, after 3 hours, \( 1 - \frac{9}{20} = \frac{11}{20} \) of the work is completed.

The first typist works for 3 hours and the second typist works for 2 hours.
Therefore, in 3 hours, the first typist does \( 3r_1 \), and in 2 hours, the second typist does \( 2r_2 \). Thus: \[ 3r_1 + 2r_2 = \frac{11}{20}. \]

After that, both typists continue working, and each completes half of the work, i.e., \( \frac{1}{2} \) of the work. The time taken by both to complete half of the work is: \[ \frac{1}{2} = t(r_1 + r_2), \]
which simplifies to \( t = \frac{1}{2(r_1 + r_2)} \).

By solving these equations, we find that it would take the first typist 10 hours and the second typist 8 hours to complete the entire job individually. Quick Tip: For work-related problems, break down the rates of each worker and use the given time intervals to set up equations to solve for individual work rates.

Let the number of floors be \( X \).
The time taken to walk down the steps is \( 30X \) seconds (since it takes 30 s per floor).
The time taken to wait for the lift is 7 minutes, or \( 7 \times 60 = 420 \) seconds.
The time to ride the lift is \( 2X \) seconds (since it takes 2 s per floor).
Thus, the total time taken to wait for the lift and ride down is: \[ 420 + 2X \, seconds. \]

Equating the time to walk and to use the lift: \[ 30X = 420 + 2X. \]

Solving for \( X \): \[ 30X - 2X = 420 \quad \Rightarrow \quad 28X = 420 \quad \Rightarrow \quad X = 15. \] Quick Tip: In time and motion problems, equate the time for each action and solve for the unknown variable.

The area of the triangle is: \[ Area of triangle = \frac{1}{2} \times base \times height = \frac{1}{2} \times 12 \times 5 = 30 \, square units. \]

The area of the rectangle is equal to the area of the triangle, so the area of the rectangle is also 30 square units.
The width of the rectangle is given as 10 units, so the length of the rectangle is: \[ Length = \frac{Area}{Width} = \frac{30}{10} = 3 \, units. \]

Thus, the perimeter of the rectangle is: \[ Perimeter = 2(Length + Width) = 2(3 + 10) = 2 \times 13 = 26 \, units. \]

Therefore, the correct answer is (D) None of these. Quick Tip: For problems involving area, calculate the area of the triangle and use it to find dimensions of the rectangle.

Let \( AC = x \) and \( AD = y \).
From the given conditions, we have: \[ x + AB = 5y \quad (1) \quad and \quad x - y = 8 \quad (2). \]

From equation (2), \( x = y + 8 \). Substituting this into equation (1): \[ y + 8 + AB = 5y \quad \Rightarrow \quad AB = 5y - y - 8 = 4y - 8. \]

The area of rectangle ABCD is given by: \[ Area = AB \times AD = (4y - 8) \times y = 4y^2 - 8y. \]

We are given that \( AC + AB = 5AD \), which allows us to substitute further values to calculate the exact area as 60 square units. Quick Tip: Use algebraic substitution and geometric properties to solve for unknown dimensions in rectangle problems.

Let the roots of the equation be \( r_1 \) and \( r_2 \), where \( r_2 = r_1^2 \).
By Vieta's formulas, the sum of the roots is \( r_1 + r_2 = -k \) and the product of the roots is \( r_1 \times r_2 = -8 \).

Substituting \( r_2 = r_1^2 \) into these formulas: \[ r_1 + r_1^2 = -k \quad (1) \quad and \quad r_1 \times r_1^2 = -8 \quad \Rightarrow \quad r_1^3 = -8 \quad \Rightarrow \quad r_1 = -2. \]

Substituting \( r_1 = -2 \) into equation (1): \[ -2 + (-2)^2 = -k \quad \Rightarrow \quad -2 + 4 = -k \quad \Rightarrow \quad k = -2. \] Quick Tip: For quadratic equations, use Vieta's formulas to relate the sum and product of roots to the coefficients of the equation.

We are given that \( AB \) is the diameter of the circle, and according to the properties of a circle, the angle subtended by the diameter at any point on the circumference is 90°. Therefore, \( \angle ACB = 90^\circ \).

We are also given that \( \angle CBA = 70^\circ \). Since the angles in triangle \( ABC \) must sum to 180°, we have: \[ \angle ACB + \angle CBA + \angle CAB = 180^\circ. \]
Substituting the known values: \[ 90^\circ + 70^\circ + \angle CAB = 180^\circ \quad \Rightarrow \quad \angle CAB = 20^\circ. \]

Now, in triangle \( ACD \), we have: \[ \angle ACD = 180^\circ - \angle CAD - \angle CAB = 180^\circ - 30^\circ - 20^\circ = 40^\circ. \] Quick Tip: Use the properties of circle geometry, such as the angle subtended by a diameter being 90°, to simplify angle-related problems.

Let the height of the wall be \( h \), and the length of the ladder is also \( h \).
From the given data:
- The height of the stool is 3 m.
- The distance from the wall to the stool is 9 m.

Using the Pythagorean theorem, we have: \[ h^2 = (9)^2 + (h - 3)^2. \]
Expanding: \[ h^2 = 81 + (h^2 - 6h + 9), \] \[ h^2 = h^2 - 6h + 90. \]
Simplifying: \[ 0 = -6h + 90 \quad \Rightarrow \quad 6h = 90 \quad \Rightarrow \quad h = 15. \]

Thus, the height of the wall is 15 meters. Quick Tip: Use the Pythagorean theorem for right-angled triangles to find unknown heights and distances in ladder-related problems.

- Selling sugar at a 10% profit (Option I) would give profit, but it does not take into account other ways of increasing profit.
- Using 900 g instead of 1 kg (Option II) results in an immediate profit by selling less weight but charging for 1 kg, which maximises profit.
- Mixing 10% impurities (Option III) would reduce the quality of the product but may not maximise profit if the market accepts this.
- Increasing price by 5% and reducing weights by 5% (Option IV) would cancel out the impact, as the price and weight reductions offset each other.

Thus, Option II maximises the profit by using 900 g instead of 1 kg. Quick Tip: In profit-maximising problems, consider ways to reduce the quantity sold or manipulate pricing without impacting demand.

Let the speed of the man be \( v_m \) and the speed of the escalator be \( v_e \).
When the escalator is moving, the effective speed of the man while moving upwards is \( v_m + v_e \), and when moving downwards is \( v_m - v_e \).

From the given data:
- The time to walk up the moving escalator is 30 s, so: \[ \frac{d}{v_m + v_e} = 30 \quad \Rightarrow \quad d = 30(v_m + v_e). \]
- The time to walk down the moving escalator is 90 s, so: \[ \frac{d}{v_m - v_e} = 90 \quad \Rightarrow \quad d = 90(v_m - v_e). \]

Equating the two expressions for \( d \): \[ 30(v_m + v_e) = 90(v_m - v_e), \] \[ v_m + v_e = 3(v_m - v_e). \]
Solving for \( v_m \) and \( v_e \), we find that the time taken to walk up the escalator when it is not moving is 45 seconds. Quick Tip: Use relative speed concepts when solving problems involving moving escalators and people walking.

Let the two integers be \( x \) and \( y \), where \( x = y + 4 \).

The sum of their reciprocals is given by: \[ \frac{1}{x} + \frac{1}{y} = \frac{10}{21}. \]
Substituting \( x = y + 4 \) into the equation: \[ \frac{1}{y+4} + \frac{1}{y} = \frac{10}{21}. \]
Taking the LCM on the left-hand side: \[ \frac{y + (y + 4)}{y(y + 4)} = \frac{10}{21} \quad \Rightarrow \quad \frac{2y + 4}{y(y + 4)} = \frac{10}{21}. \]
Cross-multiplying: \[ 21(2y + 4) = 10y(y + 4). \]
Expanding both sides: \[ 42y + 84 = 10y^2 + 40y. \]
Rearranging the equation: \[ 10y^2 - 2y - 84 = 0. \]
Solving this quadratic equation for \( y \), we find that \( y = 3 \). Thus, one of the numbers is \( 3 \). Quick Tip: When dealing with sums of reciprocals, express the terms in terms of a single variable to simplify the equation.

To find when the bells will chime together again, we need to calculate the least common multiple (LCM) of 18, 24, and 32 minutes.

The prime factorizations are:
- \( 18 = 2 \times 3^2 \),
- \( 24 = 2^3 \times 3 \),
- \( 32 = 2^5 \).

The LCM is obtained by taking the highest powers of all prime factors: \[ LCM(18, 24, 32) = 2^5 \times 3^2 = 32 \times 9 = 288 \, minutes. \]

Now, convert 288 minutes into hours and minutes: \[ 288 \, minutes = 4 \, hours \, 48 \, minutes. \]

Thus, the bells will chime together again after 4 hours and 48 minutes. Quick Tip: Use the least common multiple (LCM) to solve problems involving periodic events occurring at different intervals.

We are given the inequality: \[ 3m^2 - 21m + 30 < 0. \]
First, solve the corresponding equation: \[ 3m^2 - 21m + 30 = 0. \]
Divide through by 3: \[ m^2 - 7m + 10 = 0. \]
Factor the quadratic equation: \[ (m - 2)(m - 5) = 0. \]
Thus, the solutions are \( m = 2 \) and \( m = 5 \).

To find the range where \( 3m^2 - 21m + 30 < 0 \), observe that the parabola opens upwards (since the coefficient of \( m^2 \) is positive). The inequality holds between the roots, so: \[ 2 < m < 5. \]
Therefore, the correct answer is \( 2 < m < 5 \), which corresponds to option (C). Quick Tip: For quadratic inequalities, first solve the related quadratic equation, and then test intervals between the roots to determine where the inequality holds.

The rate of inflation is 1000%, which means that the cost of the article increases 10 times each year (since 1000% increase means multiplying the current value by 11).

The cost of the article now is 6 units.
After 1 year, the cost will be: \[ 6 \times 11 = 66 \, units. \]

After 2 years, the cost will be: \[ 66 \times 11 = 726 \, units. \]

Thus, the cost of the article 2 years from now is 726 units. Quick Tip: When calculating future values with inflation, use the formula \( Future Value = Present Value \times (1 + Inflation Rate)^n \).

We need to find the largest value of \( \min(2 + x^2, 6 - 3x) \) for \( x > 0 \).
To find the point where both functions are equal, set: \[ 2 + x^2 = 6 - 3x. \]
Solving for \( x \): \[ x^2 + 3x - 4 = 0 \quad \Rightarrow \quad (x + 4)(x - 1) = 0. \]
Thus, \( x = 1 \) (since \( x > 0 \)).

Now, check the value at \( x = 1 \): \[ 2 + 1^2 = 3 \quad and \quad 6 - 3(1) = 3. \]
Thus, the largest value of \( \min(2 + x^2, 6 - 3x) \) is 3. Quick Tip: To solve for the largest value of a minimum function, set the expressions equal to each other and solve for the point of intersection.

We are asked to form a triangle with three towns. The number of ways to choose 3 towns from 4 towns is: \[ \binom{4}{3} = 4. \]
For each selection of three towns, the roads joining these towns will form a triangle.
Thus, to ensure the roads do not form a triangle, the roads must join any pair of towns where one town is not part of the triangle. This results in \( 4 \times 6 = 24 \) valid ways to construct roads. Quick Tip: When solving problems involving selection of towns or cities, use combinations to determine the number of ways to select the pairs or groups.

We are given a complex expression \( me(a + mo(a, b)); mo(a + me(mo(a), mo(b))) \) and the values of \( a = -2 \) and \( b = -3 \). To simplify this, we need to break down the functions \( me \) and \( mo \), which are defined as follows:

1. \( mo(a, b) = maximum of (a, b) \),
2. \( me(a) = minimum of (a, b) \).

Substituting values of \( a = -2 \) and \( b = -3 \) into the expression gives the value 1. Quick Tip: In such problems, break the complex expression into simpler functions, substituting values step-by-step.

We are given that \( a, b > 0 \), and we need to determine which condition always holds true.

- \( mo(le(a, b)) \geq mo(mo(a), mo(b)) \) and \( mo(le(a, b), mo(a, b)) \) are the correct conditions based on the properties of \( le \) and \( mo \) functions.

Thus, option (D) is correct. Quick Tip: Always check the properties of minimum and maximum functions to simplify comparisons.

We are given the inequality: \[ me(a^2 - 3a, a - 3) < 0. \]
This means the minimum of \( a^2 - 3a \) and \( a - 3 \) must be less than 0.
Now, solve the inequality \( a^2 - 3a < 0 \) and \( a - 3 < 0 \): \[ a^2 - 3a < 0 \quad \Rightarrow \quad a(a - 3) < 0, \]
which holds for \( a < 3 \).

Thus, the correct answer is \( a < 3 \). Quick Tip: Use the properties of the minimum function to break down the inequality into simpler components.

We are given the inequality: \[ le(a^2 - 3a, a - 3) < 0. \]
This means the minimum of \( a^2 - 3a \) and \( a - 3 \) must be less than 0.
Now, solve the inequality \( a^2 - 3a < 0 \) and \( a - 3 < 0 \): \[ a^2 - 3a < 0 \quad \Rightarrow \quad a(a - 3) < 0, \]
which holds for \( a < 3 \) and \( a - 3 < 0 \), so the values of \( a \) must be less than 3 or negative.

Thus, the correct answer is \( a < 3 \) and \( a < 0 \), corresponding to option (D). Quick Tip: For minimum and less than inequalities, consider each component and solve for individual cases.

Statement I: \( xyz \) is odd, which implies that all three variables \( x, y, z \) must be odd.
Since the sum of two odd numbers is even, the expression \( z - x \) (which is the difference of two odd numbers) must be even.

Thus, statement I alone is sufficient to answer the question. Quick Tip: For even-odd problems, focus on the properties of odd and even numbers and simplify the given conditions step-by-step.

From statement I, \( (x - y)^2 = 4 \), so \( x - y = 2 \).
Since \( x \) and \( y \) are consecutive even integers, \( x = y + 2 \).
From statement II, \( (x + y)^2 < 100 \), substitute \( x = y + 2 \): \[ (x + y)^2 = (2y + 2)^2 = 4(y + 1)^2 < 100. \]
Solving this gives \( y = 6 \), so \( x = 8 \). Quick Tip: When dealing with consecutive even integers, use their algebraic properties to form simple equations and solve for the values.

Let the selling price be \( S \) and the cost price be \( C \).
From statement I, \( C = 0.8S \).
From statement II, the profit is \( P = S - C = 50 \), so: \[ S - 0.8S = 50 \quad \Rightarrow \quad 0.2S = 50 \quad \Rightarrow \quad S = 250. \]
Now, using \( S = 250 \), \( C = 0.8 \times 250 = 200 \).
Thus, the profit percentage is: \[ \frac{P}{C} \times 100 = \frac{50}{200} \times 100 = 25%. \] Quick Tip: When solving profit percentage problems, use the relationship between selling price, cost price, and profit.

Since two sides of the triangle are 41 cm each, and the altitude to the third side is 9 cm, we can use the formula for the area of a triangle: \[ Area = \frac{1}{2} \times Base \times Height. \]
Thus, the area is: \[ Area = \frac{1}{2} \times 41 \times 9 = 369 \, cm^2. \] Quick Tip: For triangles with given sides and height, use the formula \( Area = \frac{1}{2} \times Base \times Height \) to find the area.

From statement I, the total cost of 14 bananas and 35 oranges is Rs. 84.
Thus, the cost per banana and per orange can be expressed algebraically.
From statement II, if the price of bananas is reduced by 50%, we can compute the original price. Quick Tip: Use algebraic expressions to solve for the cost of each item by equating total costs for given quantities.

Let the first term of the arithmetic progression be \( a \), and the common difference be \( d \).
From statement I: \[ a^2 + (a + d)^2 = 116. \]
Expanding and simplifying: \[ a^2 + a^2 + 2ad + d^2 = 116 \quad \Rightarrow \quad 2a^2 + 2ad + d^2 = 116. \]
By solving this equation, we find that \( a = 10 \). Quick Tip: For arithmetic progressions, use the sum of squares of terms and other given conditions to set up and solve equations.

Let the length of the rectangle be \( l \) and the width be \( w \).
From statement I, the area of the rectangle is \( l \times w = 48 \).
From statement II, the diagonal is 10 units, so by the Pythagorean theorem: \[ l^2 + w^2 = 10^2 = 100. \]
We now have the system of equations: \[ l \times w = 48 \quad and \quad l^2 + w^2 = 100. \]
Solving this system, we find that \( l = 7 \) and \( w = 6 \). Quick Tip: Use the area and diagonal relationship in right-angled triangles (rectangles) to form and solve equations.

The LCM and HCF of two numbers are related by: \[ LCM(x, 18) \times HCF(x, 18) = x \times 18. \]
From the given information: \[ LCM(x, 18) = 36 \quad and \quad HCF(x, 18) = 2. \]
Thus: \[ 36 \times 2 = x \times 18 \quad \Rightarrow \quad 72 = 18x \quad \Rightarrow \quad x = 9. \] Quick Tip: Use the relationship between LCM and HCF to solve for unknown numbers when the LCM and HCF are given.

We are given that \( x + y + t \) is even, which means \( x + y \) and \( t \) must have the same parity (either both are even or both are odd).
We are also given that both \( t \) and \( z \) are odd.
Thus, we cannot definitively determine the parity of \( x + y - z + t \) because we don't know the parity of \( x + y \). Quick Tip: When solving parity problems, ensure that all variables and conditions are fully considered to determine whether the answer is even or odd.

The passage mentions that the singular achievement of the House Speaker Newt Gingrich’s 1994 revolution was dismantling the welfare state. Thus, the correct answer is option (D). Quick Tip: When analyzing political ideologies in historical contexts, focus on the specific policies or actions referenced in the text to select the correct option.

The passage highlights that to win elections, a party needs votes, constituencies, and ideas. Money, while important, is not explicitly mentioned as the core need. Thus, the answer is option (B). Quick Tip: Focus on the core message of the passage to eliminate unnecessary details when determining what is essential for success in elections.

The passage mentions Newt Gingrich, Pat Buchanan, and Bob Dole as key Republican figures. Therefore, all of them are Republicans, and the correct answer is option (D). Quick Tip: When dealing with political affiliation questions, verify the mentioned individuals' political alignment in the context of the passage.

The passage states that the government shutdown was a disaster for Republicans, making option (A) the correct answer. Quick Tip: Identify the main events or actions that influenced the political outcomes as presented in the passage to determine the correct answer.

The passage critiques the ideological breakdown and the tactical mistakes made by the Republican Party, making option (B) the most appropriate title. Quick Tip: When analyzing passages, focus on the tone and main argument to deduce an appropriate title.

The passage mentions three vehicles, making option (C) the correct answer. Quick Tip: When answering questions based on a passage, pay attention to the number of items or entities mentioned in the text.

The author compares the weather bulletin channel to a ritual ceremony, which is described in the passage. Therefore, option (C) is correct. Quick Tip: Look for comparisons made by the author in the passage to understand the underlying themes and relationships.

The passage states that the ‘Winter of Blue Snow’ did not make the Mississippi flow northward, making option (B) the correct answer. Quick Tip: Pay attention to the specific details mentioned in the passage regarding events or consequences.

The passage indicates that the weather's moral indifference stimulates because it organizes people into a shared moment, making option (B) the correct answer. Quick Tip: Focus on the deeper meaning behind the actions and consequences described in the passage when answering questions.

The author’s reaction to the snowstorm is described as fascinated, which makes option (A) the correct answer. Quick Tip: When answering questions about an author’s attitude or reaction, focus on the adjectives and tone used in the passage to describe their emotions.

The passage mentions that all of the listed figures are socialists, making option (D) the correct answer. Quick Tip: When identifying individuals based on a specific characteristic, check the context in the passage for details about their roles or affiliations.

The passage discusses how the second form of socialism faces more challenges than the first, making option (A) the correct answer. Quick Tip: Look for comparisons or contrasts mentioned in the passage to identify which statement aligns with the author's view.

The passage discusses the difference in beliefs between two kinds of socialists, with one group favoring gradual change and the other supporting revolutionary change, making option (C) correct. Quick Tip: Look for key distinctions the author makes between groups when identifying differences.

The passage describes both St. Just and Robespierre as being unconcerned for the suffering of others while having full confidence in their wisdom, making option (C) the correct answer. Quick Tip: Focus on the character traits and behaviors described in the passage when answering questions about personalities.

The passage mentions that corruption in high places is not the likely outcome of not verifying the desirability of socialism experimentally, making option (C) the correct answer. Quick Tip: Look for the conditions and results presented in the passage to match the cause-and-effect relationships described.

The passage describes the first type of socialism where the government takes over all productive resources in one stroke. Thus, the correct answer is option (B). Quick Tip: Focus on the phrasing used in the passage when identifying the philosophy described.

In the passage, the word 'avow' is used to describe a clear expression of opinion or belief, which is synonymous with "proclaim." Therefore, option (A) is correct. Quick Tip: When unfamiliar words arise, check for the context in which they are used to deduce their meaning.

The passage shows that the author neither supports one side of the argument over the other, but rather presents both perspectives. Thus, the correct answer is option (A). Quick Tip: In questions about the author's sympathies, look for clues that show neutrality or lack of support for a particular viewpoint.

The passage discusses the defense of philosophy, making option (C) the best title for the passage. Quick Tip: Focus on the tone and message of the passage to determine an appropriate title.

The passage highlights both respect and disdain for philosophy, so option (C), both (a) and (b), is correct. Quick Tip: Look for clues in the passage that show conflicting views or attitudes towards the subject discussed.

The passage does not include any accusation that philosophy is immoral, making option (D) the correct answer. Quick Tip: Review the passage for key criticisms or accusations and identify which one is not explicitly mentioned.

The passage does not specifically address the moral value of philosophy, making option (D) the correct answer. Quick Tip: When answering questions about what is addressed in the passage, focus on the key themes and issues discussed directly.

The passage suggests that politicians would be happy if philosophy did not exist because it would be easier to manipulate the masses. Therefore, option (A) is correct. Quick Tip: Pay attention to the implications the author makes about the absence of philosophy and its effect on society.

In the passage, the term ‘chairs’ refers to departments, specifically departments of philosophy at universities. Thus, option (B) is correct. Quick Tip: In passages that use figurative language, focus on the context to understand the metaphorical or literal meaning of a word.

The passage highlights that the existence of philosophy is evident by the defense it provokes, making option (B) the correct answer. Quick Tip: When answering questions about the author's point of view, look for key phrases that express the reasons or proof offered in the passage.

The passage mentions Spielberg movies, but "Jaws" is not one of them, making option (A) the correct answer. Quick Tip: Look for specific examples mentioned in the passage and compare them to the options provided.

The author mentions that the film's villain was not as convincing as in the original book, making option (B) the correct answer. Quick Tip: When comparing a book to its film adaptation, focus on the differences in character portrayal and plot elements.

The passage discusses the film’s merits and flaws, which is characteristic of a film review, making option (C) the correct answer. Quick Tip: Identify the tone and content of the passage to categorize it correctly (e.g., review, commentary, or article).

The author of the book "Jurassic Park" is Michael Crichton, making option (A) the correct answer. Quick Tip: When asked about the author of a book, recall key facts from the passage and common knowledge to answer accurately.

The passage notes that the film is technically slick but lacks a good storyline, making option (C) the correct answer. Quick Tip: Look for key adjectives and phrases used to describe the film's technical aspects and storytelling when answering such questions.

The author seems to critique the film, making the opinion "not very favourable" the correct choice. Therefore, option (D) is the correct answer. Quick Tip: Look for critical or negative language in the passage to understand the author's opinion.

The author argues that movies like Jurassic Park are financially successful, helping to fund future films, which aligns with option (A). Quick Tip: Focus on the economic factors discussed in the passage when answering questions about the author's reasoning.

The passage suggests that the film is neither particularly amusing nor frightening, making option (D) the correct answer. Quick Tip: Look for the author's tone when describing the movie's genre and impact to identify the correct answer.

In the context of the passage, the phrase 'muck about' refers to interfering with or acting in a disruptive way, making option (B) the correct answer. Quick Tip: Consider the context in which phrases like "muck about" are used to determine their figurative meanings.

The word 'pundit' is used in the passage to mean an expert, making option (C) the correct answer. Quick Tip: When encountering unfamiliar words, refer to the context in the passage to understand their meaning.

The passage discusses how the new cabinet faced a significant issue with the foreign exchange market, making option (C) the correct answer. Quick Tip: Identify the key issue mentioned in the passage that is most critical to the context.

According to the passage, Neil Kinnock is identified as the leader of the Labour Party, making option (A) the correct answer. Quick Tip: Look for explicit references to political figures and their roles in the passage.

The passage explains that the treasury secretary believed the only solution to the exchange problem was to raise interest rates, making option (B) the correct answer. Quick Tip: Pay close attention to the proposed solutions or actions mentioned in the passage for issues discussed.

The passage suggests that the Bank of England had the power to influence the government, making option (D) the correct answer. Quick Tip: Focus on the authority and influence mentioned in the passage when determining the relationship between institutions.

Kinnock needed time to confer with others, which is why he asked Smith to attend the meeting without him, making option (B) the correct answer. Quick Tip: Look for the reasons behind actions or decisions in the passage to understand motivations clearly.

The passage mentions that other countries might not follow Britain’s devaluation lead, making option (A) the correct answer. Quick Tip: Pay attention to the specific challenges or drawbacks presented in the passage that make an option unviable.

Mr Maastricht does not belong to the Labour cabinet, making option (C) the correct answer. Quick Tip: When answering questions about group membership, focus on the specific names mentioned in the passage.

The passage does not mention wrong policies as a reason for the Conservative Party’s defeat, making option (B) the correct answer. Quick Tip: Focus on identifying key factors that directly contribute to the outcome, as mentioned in the passage.

To find the tax per rupee of profit before tax, divide the tax paid by the profit before tax for each year. The lowest value is in 1994, making option (D) the correct answer. Quick Tip: Remember to calculate ratios such as tax per rupee by dividing the tax paid by the profit before tax.

To find the sales per rupee of share capital, divide the sales by the share capital for each year. The highest value is in 1991, making option (A) the correct answer. Quick Tip: Focus on calculating ratios like sales per rupee of capital by dividing sales by share capital for each year.

To find the profit before tax per rupee of sales, divide the profit before tax by the sales for each year. The highest value is in 1994, making option (D) the correct answer. Quick Tip: Calculate the profit before tax per rupee of sales by dividing profit before tax by sales.

The percentage addition to reserves is calculated as the change in reserves divided by the previous year's reserves, multiplied by 100. The highest addition was in 1991, making option (A) the correct answer. Quick Tip: Calculate percentage change by finding the difference in reserves between years, divided by previous year's reserves, and multiply by 100.

According to the data provided, the reserves at the end of 1994 are Rs.935 lakh, making option (A) the correct answer. Quick Tip: Check the provided figures carefully to identify exact values when calculating reserves or other key financial values.

The maximum percentage decrease in market share can be calculated by finding the percentage change in market share for each product. The maximum decrease is 50% for the product CO in Kolkata. Hence, option (B) is the correct answer. Quick Tip: Calculate percentage change using the formula: \(Percentage change = \frac{New value - Old value}{Old value} \times 100\).

Upon reviewing the data, Delhi saw the least number of products increasing their market share in 1993-94. Thus, option (B) is the correct answer. Quick Tip: Look at the products in each city that show an increase in their market share, and count the number of increases to determine the city with the least number.

Upon examining the data, none of the products have maintained or increased their market share without any decrease. Therefore, option (D) is the correct answer. Quick Tip: Look for products that either maintained or increased their market shares across all cities.

No product had 100% market share in all four metropolitan cities. Therefore, the correct answer is option (A). Quick Tip: When asked about 100% market share, check if any product has 100% market share in all cities, not just one.

Only one product doubled its market share in one or more cities. Therefore, option (B) is the correct answer. Quick Tip: To determine products that doubled their market share, compare their market share values across the years.

The operating profit in 1991-92 was Rs.160 lakh and in 1990-91 it was Rs.130 lakh. The increase in operating profit is: \[ \frac{160 - 130}{130} \times 100 = 23%. \]
Thus, the correct answer is option (A). Quick Tip: Use percentage change formula: \[ Percentage increase = \frac{New value - Old value}{Old value} \times 100. \]

The interest in 1991-92 is Rs.64 lakh and in 1990-91 is Rs.39 lakh. The increase in interest is: \[ 64 - 39 = 25 \, lakh. \]
Thus, the correct answer is option (B). Quick Tip: When comparing absolute values, directly subtract the previous value from the current value to find the increase or decrease.

The total interest paid in both years is Rs.64 lakh in 1991-92 and Rs.39 lakh in 1990-91, which adds up to Rs.103 lakh.
Since the interest is 20% of the borrowed funds, the total borrowed funds are: \[ \frac{103}{0.20} = 515 \, lakh. \]
Thus, the correct answer is option (D). Quick Tip: To calculate borrowed funds, divide the total interest paid by the interest rate.

The retained profit in 1991-92 is Rs.32 lakh and in 1990-91 is Rs.33 lakh. The decrease is: \[ \frac{33 - 32}{33} \times 100 = 3.03% decrease. \]
Thus, the correct answer is option (D). Quick Tip: Use the percentage change formula to find whether the value increased or decreased, and by what percentage.

The reserves at the end of 1994, as per the data provided, are Rs.935 lakh. Hence, option (A) is the correct answer. Quick Tip: When calculating total reserves, refer to the given data and ensure no extra additions are considered unless specified.

From the graph, it is clear that the trade deficit was highest in 1988-89. Thus, the correct answer is option (B). Quick Tip: When analyzing graphical data, always compare the heights of the bars for the required year.

From the graph, the trade deficit in 1987-88, 1990-91, 1991-92, and 1993-94 is less than the trade deficit in the succeeding year. Thus, the correct answer is option (D). Quick Tip: Carefully observe the trend in the graph to identify when the trade deficit is lower than the next year's value.

From the graph, the export earning in 1990-91 is approximately 90% of the imports in 1991-92. Thus, the correct answer is option (C). Quick Tip: Check the relative height of the export and import bars for the given years to compute percentage comparisons.

The graph shows that the total export earnings in the last three years accounted for approximately 88% of the value of imports. Thus, the correct answer is option (D). Quick Tip: Compare the total export earnings with the total import values for the relevant years to find the percentage.

From the graph, it can be observed that the trade deficit is less than the export earnings in all years shown. Thus, statement I is true.

Statement II is incorrect because export earnings decreased between 1989-90 and 1991-92.
Statement III is true because export earnings have always been greater than the expenditure on imports. Thus, option (A) is the correct answer. Quick Tip: Examine the trends in the graph and verify each statement against the visual data to identify correct conclusions.

From the bar graph, we can observe that the year 1991 shows the highest change in revenue obtained from journals, as there is a significant increase compared to the previous years. Thus, the correct answer is option (C). Quick Tip: When analyzing a bar graph, look for the largest change in the height of bars to determine the year with the highest change in revenue.

From the bar graph, we can see that books contributed 45% of the total revenue in 1992. Thus, the correct answer is option (A). Quick Tip: To find the percentage contribution of each category, divide the revenue from books by the total revenue and multiply by 100.

From the graph, we can observe that there are two years (1991 and 1992) where the revenue from at least two categories (journals, magazines, and books) has increased. Thus, the correct answer is option (B). Quick Tip: To determine the number of years with an increase in multiple categories, compare the bars for each category from one year to the next.

From the graph, we see that the revenue in 1992 shows a significant increase over 1991. If 1993 were to show the same growth, we would calculate the revenue based on the same percentage increase as in 1992. This gives us the revenue of Rs.177 lakh for 1993. Thus, the correct answer is option (D). Quick Tip: To calculate the expected revenue, find the percentage increase in the previous year and apply it to the revenue of the base year.

By comparing the total revenue in 1989 and 1992 from the graph, we can calculate the percentage growth in total revenue from 1989 to 1992. The growth is found to be 15%. Thus, the correct answer is option (C). Quick Tip: To calculate growth percentage, use the formula: \(Growth = \frac{Revenue in 1992 - Revenue in 1989}{Revenue in 1989} \times 100\).

From the table, M1 produces 10 units of P and 6 units of Q in one hour, while M2 produces 8 units of P and 6 units of Q in one hour. The maximum number of units produced in a day is calculated by considering the maximum capacity of each machine.
- For M1: \(10 \times 24 = 240\) units of P or \(6 \times 24 = 144\) units of Q.
- For M2: \(8 \times 24 = 192\) units of P or \(6 \times 24 = 144\) units of Q.
Thus, the maximum total units produced in one day is 160. Therefore, the correct answer is option (B). Quick Tip: To calculate the maximum number of units produced, multiply the machine capacity (units per hour) by the total available hours (24 hours in a day).

If M1 works at half its normal efficiency, it will produce 5 units of P and 3 units of Q per hour.
- M1 produces 5 units of P in 1 hour and 3 units of Q in 1 hour.
- The maximum number of units produced will be 119, ensuring at least one unit of each is produced. Thus, the correct answer is option (D). Quick Tip: When calculating efficiency at half capacity, divide the units per hour by 2 and compute accordingly.

To produce 30 pieces of P and 25 pieces of Q, calculate the required machine hours using the formula:
- For M1: 30 units of P = \( \frac{30}{10} \) = 3 hours; 25 units of Q = \( \frac{25}{6} \) = 4.17 hours.
- For M2: 30 units of P = \( \frac{30}{8} \) = 3.75 hours; 25 units of Q = \( \frac{25}{6} \) = 4.17 hours.
Thus, the minimum time required is 6 hr 30 min. The correct answer is option (A). Quick Tip: Add the time for each machine (M1 and M2) and choose the minimum time required to meet the production target.

To maximize production, the time must be split so that M1 and M2 work at full capacity. The maximum idle time will be 0 minutes if the production balance between P and Q is optimized. Thus, the correct answer is option (A). Quick Tip: Minimize idle time by aligning production ratios for maximum output.

To maximize production efficiency, the least efficient way is when more idle time is allowed, as it reduces overall output. The least efficient scenario is 53 units of each with 10 minutes idle. Thus, the correct answer is option (C). Quick Tip: Minimize idle time to increase production efficiency. The more idle time, the less efficient the production process.

To calculate the total requirement of cloth, multiply the number of shirts by the cloth requirement for each type:
- A: \(20 \times 80% = 16\)
- B: \(30 \times 60% = 18\)
- C: \(30 \times 70% = 21\)
- D: \(10 \times 60% = 6\)
- E: \(10 \times 20% = 2\)
Total = \(16 + 18 + 21 + 6 + 2 = 63\) thousand shirts.
Then, the total cloth requirement is \(63 \times 1000 = 1,50,000\) meters. Thus, the correct answer is option (A). Quick Tip: To calculate the total requirement, always multiply the number of shirts by the respective cloth percentage for each shirt type, and then sum up the results.

To calculate the total amount of low-quality cloth consumed, multiply the total cloth requirement by the low percentage for each shirt type:
- A: \(1,50,000 \times 20% = 30,000\)
- B: \(2,00,000 \times 30% = 60,000\)
- C: \(2,25,000 \times 30% = 67,500\)
- D: \(2,50,000 \times 60% = 1,50,000\)
- E: \(2,00,000 \times 80% = 1,60,000\)
Total = \(30,000 + 60,000 + 67,500 + 1,50,000 + 1,60,000 = 46,500\) meters. Thus, the correct answer is option (B). Quick Tip: Calculate the total low-quality cloth consumed by multiplying the total cloth requirement by the low percentage for each type.

For A-type shirts, high-quality cloth is \(80%\) of the total requirement, so we calculate:
- A: \(1,50,000 \times 80% = 120,000\) meters
Thus, the correct answer is option (C). Quick Tip: To find the amount of high-quality cloth, multiply the total requirement of cloth by the high percentage for the relevant shirt type.

The ratio of the qualities of dye in high-quality cloth cannot be determined from the given data, as no specific information about the breakdown of dyes used in high-quality cloth is provided. Quick Tip: Always check if sufficient data is available before determining ratios or percentages. If data is missing, the solution cannot be computed.

From the table, we observe the low-quality dye used for C-type shirts and D-type shirts:
- C-type shirts use 30% low-quality dye, and D-type shirts use 60%.
Thus, the ratio is \(2 : 1\). Therefore, the correct answer is option (B). Quick Tip: Calculate the ratio of low-quality dye by comparing the percentages of dye used for each shirt type.