ATMA 2017 Set A Question Paper with Answer Key PDFs (February 12)

Step 1: The term "Patricide" refers to the act of killing one's father.
The prefix "-cide" means "killing of," and when combined with "pater" (Latin for father), it specifically refers to the killing of one's father.


Step 2: Similarly, the term "Sororicide" refers to the act of killing one's sister, as "soror" is Latin for sister.

Thus, the analogy is based on the relationship between the act of killing and the person killed.


Step 3: Therefore, the correct term related to "Sororicide" is "Sister."
Thus, the correct answer is "Sister." Quick Tip: Remember, the suffix "-cide" refers to the act of killing. Different prefixes specify the relationship or the target of the act (e.g., fratricide for brother, matricide for mother).

Step 1: "Ornithology" is the study of birds. It is a branch of zoology that focuses specifically on birds and their behaviors, anatomy, and classifications.

Step 2: Similarly, "Entomology" is the scientific study of insects. The term is derived from the Greek word "entomos," meaning "insect."

Step 3: Both terms are fields of study, and they are related to animals, but each one focuses on a specific type of organism.

Thus, the correct answer is "Insects." Quick Tip: Branches of zoology often end in "-logy," which indicates the study of a particular group of organisms.

Step 1: A carpenter is a person who specializes in making furniture, such as tables, chairs, cabinets, and other wooden items.

Step 2: A sculptor, on the other hand, is an artist who specializes in creating statues, often by carving or shaping materials like stone, wood, or metal.

Step 3: The analogy is based on the relationship between the profession and the object it produces.

Thus, the correct answer is "Statue." Quick Tip: Focus on the professional's primary product or output when dealing with occupation-to-object analogies.

Step 1: In the sport of shooting, a bullet is the projectile that is shot from a firearm.

Step 2: Similarly, in archery, an arrow is the projectile that is shot from a bow.

Step 3: The analogy focuses on the tool used for shooting and the corresponding projectile.

Thus, the correct answer is "Arrow." Quick Tip: Understanding the context of the action and the tool or object primarily used is crucial in such analogies.

Step 1: A monkey is known for its characteristic sound, which is chatter or a chattering noise.

Step 2: An elephant, on the other hand, is known for its loud, trumpeting sound.

Step 3: The analogy compares the sound made by two animals, with the monkey's chatter being paired with the elephant's trumpet.

Thus, the correct answer is "Trumpet." Quick Tip: When dealing with analogies involving animals and sounds, focus on the distinctive sounds typically associated with each animal.

Step 1: The term "docile" refers to a person who is calm, submissive, and easy to control or manage.

Step 2: A docile person is someone who can be easily directed or handled, not someone who is rebellious or hard to manage.

Step 3: Therefore, the correct answer is "Easily managed." Quick Tip: Understanding the definitions of terms directly helps in selecting the correct answer for adjective-based questions.

Step 1: An "egocentric" person is one who views everything from their own perspective, often considering their own needs and interests above others.

Step 2: While "egoist" and "egotist" can describe people who are self-centered, the term "egocentric" specifically refers to someone who consistently interprets the world in relation to themselves.


Step 3: Therefore, the correct answer is "Egocentric." Quick Tip: Remember, the suffix "-centric" means centered on, which in this case, refers to self-centered.

Step 1: "Fiasco" refers to a situation or event that ends in failure, especially in a ridiculous or disastrous manner.

Step 2: "Chimera" refers to an illusion or something that is hoped for but impossible to achieve.

Step 3: "Anomaly" refers to something that deviates from what is standard or normal, not necessarily a failure.

Step 4: "Blunder" is a mistake, but it does not necessarily mean a complete or spectacular failure like "fiasco."

Thus, the correct answer is "Fiasco." Quick Tip: The term "fiasco" often relates to events or projects that end in a significantly negative or disastrous manner.

Step 1: "Theism" is the belief in the existence of God or gods.

Step 2: "Atheism" is the lack of belief in the existence of God or gods, which directly aligns with the statement "God is nonexistent."

Step 3: "Agnosticism" refers to the belief that the existence of God is unknown or unknowable, not necessarily nonexistent.

Step 4: "Reverence" refers to a deep respect, often towards a divine being, which does not fit the context of the sentence.

Thus, the correct answer is "Atheism." Quick Tip: Remember, atheism explicitly denies the existence of deities, differing from agnosticism, which is about uncertainty or indecision regarding the existence of deities.

Step 1: "Acrimony" refers to bitterness or sharpness, especially in speech or temper. It describes a feeling of hostility or resentment that can be expressed through sharp language.

Step 2: "Apathy" refers to a lack of interest or emotion, which is not related to bitterness or sharpness.

Step 3: "Alacrity" refers to eagerness or readiness, which is also not relevant to bitterness.

Step 4: "Assiduity" refers to diligence or constant effort, which is unrelated to bitterness or sharpness.

Thus, the correct answer is "Acrimony." Quick Tip: The term "acrimony" is useful in describing the tone of discussions where there is noticeable bitterness or harshness.

The passage describes Marie as having a "blithe personality," which is synonymous with being lighthearted. Quick Tip: "Blithe" often means showing a casual and cheerful indifference considered to be callous or improper; in a positive sense, it aligns with lighthearted.

Marie became "disgruntled" when she learned that the university in Warsaw was closed to women, which corresponds to feeling annoyed. Quick Tip: "Disgruntled" generally refers to feeling dissatisfied and annoyed.

Marie defiantly left Poland to study in France, representing a challenge to the authority or norms that restricted her education. Quick Tip: Defying norms or restrictions often signifies a challenge to authority.

Marie recalled their past joy together "despondently," which aligns with doing so dejectedly. Quick Tip: "Despondently" conveys a deep sadness or dejection.

The passage indicates that Marie's feeling of desolation, or wretchedness, began to fade when she was appointed as a physics professor. Quick Tip: "Wretchedness" is a state of extreme unhappiness or misfortune, similar to desolation.

Despite her fatal illness from exposure to radium, the passage notes that Marie Curie was never disillusioned or disappointed about her work. Quick Tip: "Disillusioned" in this context means losing belief or disappointment in something once thought to be good or valuable.

Step 1: The phrase "opposite to" is commonly used when referring to something directly across from something else in a physical or spatial context.

Step 2: "Opposite of" is incorrect in this context, as "opposite of" refers to something completely different in nature, not directionally across.

Step 3: "Opposed to" means to be in opposition to, which does not fit the spatial meaning intended in the sentence.

Step 4: "Opposite with" is not a standard preposition phrase.
Thus, the correct answer is "opposite to."
Quick Tip: "Opposite to" is commonly used to denote location across from something.

Step 1: "Either" is used in negative sentences to agree with a negative statement. In this case, the speaker is agreeing with the idea that they don't need the woolen sweaters.

Step 2: "Anyway" is used to dismiss the importance of something but does not fit the context of agreeing with a negative statement.

Step 3: "Neither" is typically used in a context where two negative choices are being discussed, and "too" is used to mean "also," which doesn't apply here.

Thus, the correct answer is "either." Quick Tip: In negative agreements, use "either" to affirm a preceding negative statement.

Step 1: The correct preposition with "introduced" when referring to a person is "to."

Step 2: "Introduced with" refers to introducing two people together, not one person to another.

Step 3: "Introduced at" is not a common phrase when referring to meeting a person for the first time.

Thus, the correct answer is "introduced to." Quick Tip: "Introduced to" is the standard form when someone is formally presented to another person.

Step 1: The correct structure after "don't hesitate" is "to + verb," as it is an infinitive form.

Step 2: "Ask a question" is incorrect because it lacks the infinitive "to."

Step 3: "To asking a question" is incorrect because "asking" is a gerund, not an infinitive.

Thus, the correct answer is "to ask a question." Quick Tip: When using "hesitate," the structure "hesitate to + verb" is the grammatically correct format.

Step 1: The correct phrase is "called on," which means to visit someone, especially for a specific purpose.

Step 2: "Called to the house of" and "called on the house of" are incorrect, as they imply that the house is the object of the call, not the people.

Step 3: "Called to" is generally used in the context of calling someone to a place, not visiting.

Thus, the correct answer is "called on." Quick Tip: "Called on" means to visit someone or somewhere, typically for a specific purpose, which in this context is to check on the ill sisters.

Step 1: "Enormous" means something extremely large or immense in size.

Step 2: The opposite of "enormous" is "tiny," which refers to something very small.

Step 3: "Soft," "average," and "weak" are unrelated to the concept of size or magnitude, so they cannot be correct.

Thus, the correct answer is "tiny." Quick Tip: When asked for an antonym, look for a word that represents the opposite meaning.

Step 1: "Quiescent" refers to a state of being inactive or quiet, often used in biological or physical contexts.

Step 2: The opposite of "quiescent" would be "active," referring to something that is engaging or functioning.

Step 3: "Dormant" refers to a state of inactivity or rest, but it is not the opposite of quiescent in the same active sense. "Weak" and "unconcerned" do not relate to the opposite of "quiescent."

Thus, the correct answer is "active." Quick Tip: "Quiescent" often appears in contexts involving rest or inactivity, opposite of "active."

Step 1: "Relinquish" means to give up or surrender something.

Step 2: The opposite of "relinquish" is "possess," which means to own or retain something.

Step 3: "Abdicate" and "renounce" are related to giving up power or position, which can be a type of relinquishing, but they are not the direct opposite. "Deny" does not fit in this context.

Thus, the correct answer is "possess." Quick Tip: Understanding the basic meaning of a word can directly help identify its opposite or related terms.

Step 1: "Exodus" refers to a mass departure or migration, typically of people.

Step 2: "Arrival" refers to the coming or reaching of a place, which contrasts with "exodus," meaning mass departure.

Step 3: "Influx" refers to an arrival, but in a more specific sense of a large number of people or things entering. "Return" and "restoration" do not convey the opposite of departure.

Thus, the correct answer is "arrival." Quick Tip: When dealing with terms that seem to have obvious antonyms, it's crucial to consider the context and any provided instructions that might suggest unconventional interpretations.

Step 1: "Commissioned" refers to the act of assigning or authorizing a task, typically the initiation of a project or work.

Step 2: The opposite of commissioning something would be to "revoke" the commission, which means to cancel or withdraw it.

Step 3: "Started" and "finished" relate to the completion or initiation of a task, but do not reflect the cancellation of an order or task. "Terminated" implies the ending of a process but not necessarily revocation of a commission.

Thus, the correct answer is "revoked." Quick Tip: Understanding official or legal terms like "commissioned" and "revoked" is crucial for accurately interpreting contractual or organizational contexts.

Step 1: "Inured" means being accustomed to something, especially something unpleasant, to the point where it no longer has a negative effect.

Step 2: "Hardened" fits with the meaning of becoming less sensitive or more resilient to something over time, which is similar to being inured to an experience.

Step 3: "Softened" means to become gentler or more tender, which is the opposite of being "inured." "Adulterated" refers to something being made impure, and "flattened" refers to making something level or even, neither of which match the meaning of "inured."

Thus, the correct answer is "hardened." Quick Tip: "Inured" is often used in contexts where individuals or groups become accustomed to challenging or harsh conditions.

Step 1: "Mundane" refers to something that is dull, ordinary, or lacking excitement. It often refers to the everyday or worldly aspects of life.

Step 2: "Worldly" is a close synonym to "mundane," as it refers to matters related to the physical world, as opposed to spiritual or extraordinary matters.

Step 3: "Global" refers to something that is worldwide in scope, "futile" refers to something pointless, and "spatial" refers to things relating to space, all of which are unrelated to the meaning of "mundane."

Thus, the correct answer is "worldly." Quick Tip: Remember, "mundane" is not only about being worldly but also implies being unremarkable or lacking excitement.

Step 1: "Patronize" can have two meanings: to treat someone in a condescending manner or to support or sponsor something, such as a business or cause.

Step 2: In the context of the question, "support" aligns with the meaning of "patronize" as in supporting or giving business to.

Step 3: "Oppose" means to act against something, "bless" refers to giving a blessing, and "organize" means to arrange or manage something, none of which match the meaning of "patronize" in the context of support.

Thus, the correct answer is "support." Quick Tip: "Patronize" has dual meanings: it can imply support in a positive sense, or a demeaning attitude when implying superiority.

Step 1: "Proxy" refers to a person or entity authorized to act on behalf of another, often in a specific situation such as voting or making decisions.

Step 2: "Authorized agent" is the closest match because it refers to someone who has been granted authority to act for another person.

Step 3: "Absent" refers to being away or not present, which does not fit the meaning of "proxy." "Substituted" refers to replacing something, but not necessarily with the proper authorization. "Emulated" means to imitate, which also does not fit the meaning of "proxy."

Thus, the correct answer is "authorized agent." Quick Tip: Proxies are commonly used in corporate and legal environments where one individual or group acts on behalf of another in decision-making processes.

The preposition "in" is used to indicate a general period of the day, such as the morning, making it the correct choice for this context. Quick Tip: Remember, "in the morning," "in the afternoon," and "in the evening" are standard phrases to describe activities during these times.

"From" is used to indicate the starting point in a range of time, hence "from nine until four" correctly sets the range of operation hours for the library. Quick Tip: Use "from" to start a time period and "to" or "until" to end it, depending on the context.

"By" is used to indicate a deadline by which something must be completed. Therefore, "by next Monday" specifies that the job must be finished no later than that day. Quick Tip: "By" is typically used to set deadlines for tasks, emphasizing the latest possible time for completion.

Given Radha's perception of Krishna as candid (open and honest) and trustworthy, the opposite quality to consider in his statement would be "insincere," which implies dishonesty or deceit, directly conflicting with her belief in his sincerity. Quick Tip: When evaluating statements based on character perceptions, focus on the attributes directly opposed to the qualities being affirmed.

The phrase "delivered on" is idiomatically correct for fulfilling or meeting an expectation, particularly in reference to promises or plans. "He never delivered on the grandiose promises" means he failed to fulfill the ambitious promises he made. Quick Tip: "Delivered on" is commonly used in the context of meeting or fulfilling obligations or promises, contrasting with simply accomplishing a task.

The term "cogent" refers to an argument or case that is clear, logical, and convincing. In the context of the sentence, describing the reasoning as "cogent" suggests it is so well-presented and convincing that it should be obvious to anyone, making it difficult to understand how it could deceive. Quick Tip: A "cogent" argument is compelling and powerful in its clarity and persuasive force, making it a key term in discussions involving logic and reasoning.

Analyzing the Organization.

The instructions are given in a sequence that follows the logical steps for smoke detector placement and considerations, hence they are organized chronologically by topic, addressing different aspects in a systematic way. Quick Tip: Understanding organizational patterns helps in comprehending the flow of information, especially in instructional or informative texts.

Identifying the Focus.

The main focus of the passage is providing detailed guidelines on the proper installation of smoke detectors in a home, emphasizing placement to avoid dead-air spaces and false alarms. Quick Tip: When identifying the main focus, look for the topic that is most extensively covered or detailed in the text.

Analyzing the Context.

The passage discusses installing detectors away from dead-air spaces, which are areas where stagnant air might prevent smoke from reaching the detector. It specifies that detectors be placed away from the corners where the wall meets the ceiling to avoid these spaces. Quick Tip: Understanding how air moves in a room is crucial for placing smoke detectors effectively to detect smoke from fires.

Interpreting the Information.

The passage emphasizes the importance of smoke detectors in providing early warnings, which significantly increase the chances of surviving a fire by allowing for timely evacuation. Quick Tip: Statistical benefits like those described for smoke detectors highlight the practical importance of safety devices in reducing risks in emergency situations.

Analyzing the Problem.

The passage explicitly warns against installing smoke detectors near windows or exterior doors because drafts in these areas could direct smoke away from the detector, preventing it from detecting smoke effectively. Quick Tip: Placement of smoke detectors should avoid any areas where external air movements could interfere with smoke detection.

Identifying the Firefighter's Role.

The passage discusses firefighters visiting schools specifically to speak about fire safety, including the importance and installation of smoke detectors, indicating their role in educating young students about fire prevention. Quick Tip: Firefighters often engage in community outreach and education as part of their responsibilities to enhance fire safety awareness.

Interpreting Placement Guidelines.

According to the fire safety guidelines mentioned in the passage, smoke detectors should be installed outside every sleeping area. This ensures optimal coverage for detecting smoke that may arise from any part of the home while people are sleeping, thereby enhancing safety for all occupants. Quick Tip: Placing smoke detectors outside all bedrooms ensures that the alarm can be heard more clearly and promptly by everyone in the event of a fire, increasing the chances for a safe evacuation.

Identifying the Profession.

The passage clearly identifies Charles Darwin as a biologist, noted for his contributions to the theory of evolution and natural selection. Quick Tip: Remember, Charles Darwin's profession and contributions are fundamental to understanding his theories and their impact on science and philosophy.

Aligning with Darwin’s Theories.

The statement that all life forms developed slowly over time from lower life forms is a direct reflection of Darwin’s theory of evolution through natural selection, as described in the passage. Quick Tip: Understanding Darwin's theories can help interpret various biological and evolutionary statements accurately in the context of his work.

Identifying the Theory.

Darwin’s description of the survival and reproduction of the fittest individuals is the essence of his theory of natural selection and survival of the fittest, as explicitly mentioned in the passage. Quick Tip: Linking Darwin’s observations about competition, survival, and reproduction to his broader theories is key to understanding his impact on biology and philosophy.

Evaluating the Reception.

The passage specifically notes that many religious opponents condemned \textit{On the Origin of the Species, indicating a significant pushback against the ideas presented in it. Quick Tip: The reception of scientific theories can often be influenced by contemporary social, religious, and cultural contexts, which in Darwin's case led to controversy.

A "connoisseur" is someone who has expert knowledge and keen discrimination, especially in the fine arts or in matters of taste. Quick Tip: "Connoisseur" is often used to describe someone with a deep appreciation and expert judgment in the arts, wine, food, etc.

"Clandestine" activities are done secretly, especially when something illicit or unauthorized is involved. Quick Tip: The term "clandestine" is typically associated with actions that are not only secret but often intended to deceive or evade public notice.

To "grovel" means to act in an obsequious manner in order to obtain someone's forgiveness or favor, often depicted as crawling or lying prostrate. Quick Tip: "Grovel" often conveys a sense of extreme submission or humility, typically in a demeaning or self-deprecating manner.

To "subvert" means to undermine the power and authority of an established system or institution, synonymous with overthrow. Quick Tip: "Subvert" often implies a radical attempt to dismantle or undermine a system or structure, particularly in a secretive or deceitful way.

"Banal" means lacking originality, freshness, or novelty; thus, it is synonymous with being overly common or ordinary. Quick Tip: When considering ideas or expressions in creative works, "banal" is used to criticize them as unoriginal or trite.

The phrase "more preferable" is redundant because "preferable" already suggests a comparison. The correct phrase should be "preferable to." Quick Tip: Avoid redundancy in language. "Preferable" does not require "more" as it already conveys preference.

The error in this sentence lies in the fragment "I enjoyed," which is incomplete as it lacks a direct object. The sentence should be structured to specify what was enjoyed, such as "I enjoyed my stay in Holland." Quick Tip: Always ensure that verbs which require a direct object are complete with one to avoid fragments and incomplete thoughts in sentence structure.

The preposition "than" is incorrect when used with "superior." The correct preposition should be "to," making it "are far superior to those employed in the past." Quick Tip: Remember, "superior" always pairs with "to" in comparisons, not "than."

"Irascible" means easily angered; fitting for describing someone prone to quarrel at slight provocations, matching the senator's described temperament. Quick Tip: "Irascible" is often used to describe a person's quick-tempered nature in literary and formal contexts.

The employees "loathed" Jill, a strong negative reaction, because she "berated" them, meaning she criticized them harshly, which naturally could cause resentment. Quick Tip: "Berate" implies severe or harsh criticism, often leading to strong negative feelings among those on the receiving end.

"Ideological divisions" refers to fundamental disagreements, which "paralyzed" the government, effectively stopping it from functioning or making decisions. Quick Tip: When ideological differences are described, their impact is typically negative, often impeding progress or decision-making processes.

The glue produced by bees is used to "build" their hives, structurally reinforcing and maintaining them, making it essential for hive construction. Quick Tip: The function of bee glue, or propolis, extends beyond simple construction; it also serves as protection against invaders and environmental conditions.

"Macabre" refers to the quality of having a grim or ghastly atmosphere. The use of "macabre" fits the context of describing the details of a murder in a disturbing and horror-like manner. Quick Tip: "Macabre" is particularly used to describe the horrifying aspects of a subject, often associated with death or gruesome events.

Using the distributive property: \[ 303^2 + 208^2 = (300 + 3)^2 + (200 + 8)^2 \] \[ = (300^2 + 2 \times 300 \times 3 + 3^2) + (200^2 + 2 \times 200 \times 8 + 8^2) \] \[ = (90000 + 1800 + 9) + (40000 + 3200 + 64) \] \[ = 91809 + 43264 = 135073 \]
Thus, the correct answer is 135073. Quick Tip: Square numbers directly to solve problems involving sums of squares to avoid errors in multi-step calculations.

Step 1: A number is divisible by 3 if the sum of its digits is divisible by 3.

Step 2: For 541326:
\[ 5 + 4 + 1 + 3 + 2 + 6 = 21 \]
Since 21 is divisible by 3, 541326 is divisible by 3.

Step 3: Check the sum of digits for other numbers:
For 5967013:
\[ 5 + 9 + 6 + 7 + 0 + 1 + 3 = 31 \quad (not divisible by 3) \]
For 5967019: \[ 5 + 9 + 6 + 7 + 0 + 1 + 9 = 37 \quad (not divisible by 3) \]
For 558692: \[ 5 + 5 + 8 + 6 + 9 + 2 = 35 \quad (not divisible by 3) \]

Thus, the correct answer is 541326. Quick Tip: Remember, a number is divisible by 3 if the sum of its digits is divisible by 3.

Step 1: Let the numbers be \( 15x \) and \( 11x \), where \( x \) is the common multiplier.

Step 2: Since the HCF of the numbers is 12, we have:
\[ HCF(15x, 11x) = 12 \]
Step 3: The HCF of 15 and 11 is 1, so:
\[ x = 12 \]

Step 4: The numbers are:
\[ 15x = 15 \times 12 = 180 \] \[ 11x = 11 \times 12 = 132 \]

Thus, the numbers are 180 and 132. Quick Tip: To find actual numbers from a ratio given their HCF, multiply each part of the ratio by the HCF.

Step 1: The given expression is:
\[ \left( 1 - \frac{1}{3} \right) \left( 1 - \frac{1}{4} \right) \left( 1 - \frac{1}{5} \right) \ldots \left( 1 - \frac{1}{100} \right) \] \[ = \frac{2}{3} \times \frac{3}{4} \times \frac{4}{5} \times \ldots \times \frac{99}{100} \]

Step 2: Most of the terms cancel out, leaving:
\[ \frac{2}{100} \]

Step 3: Simplifying the expression:
\[ \frac{2}{100} = \frac{1}{50} \]

Thus, the correct answer is \( \frac{1}{50} \). Quick Tip: For products of sequences, identify any patterns or cancellation opportunities to simplify the calculations.

Step 1: Let the total salary be \( x \).

Step 2: The expenditure on house rent is \( \frac{2}{5}x \), on food is \( \frac{3}{10}x \), and on conveyance is \( \frac{1}{8}x \).

The total expenditure is:
\[ \frac{2}{5}x + \frac{3}{10}x + \frac{1}{8}x \]

Step 3: Find the common denominator for the fractions:
\[ \frac{16}{40}x + \frac{12}{40}x + \frac{5}{40}x = \frac{33}{40}x \]

Step 4: The remaining amount is \( x - \frac{33}{40}x = \frac{7}{40}x \), and this is Rs.1400. \[ \frac{7}{40}x = 1400 \] \[ x = 1400 \times \frac{40}{7} = 8000 \]

Step 5: The expenditure on food is \( \frac{3}{10}x = \frac{3}{10} \times 8000 = 2400 \).

Thus, the correct answer is Rs. 2400. Quick Tip: Always check for simplification and verification by recalculating based on the total to ensure accuracy.

Calculating the total number of oranges.

If 12 unsalable oranges represent three-quarters, then there are 16 bruised oranges. For every bruised orange, there are 40 oranges, so \( 16 \times 40 = 640. \) Quick Tip: Convert conditions into equations or ratios to solve problems involving proportions or rates.

Setting up the equation.

88000 - 1200t = 42000 + 800t leads to 2000t = 46000, so \( t = 23 \) years. Quick Tip: Check calculations for consistency with initial conditions and the rates of change given in problems involving linear changes over time.

Let the cost of a pen be \( p \) and the cost of a pencil be \( q \).
From the given conditions: \[ 2p + 3q = 86 \quad (Equation 1) \] \[ 4p + q = 112 \quad (Equation 2) \]
Multiplying Equation 2 by 3: \[ 12p + 3q = 336 \]
Now subtract Equation 1 from this: \[ (12p + 3q) - (2p + 3q) = 336 - 86 \] \[ 10p = 250 \quad \Rightarrow \quad p = 25 \]
Thus, the cost of a pen is Rs. 25. Quick Tip: For solving such problems, use substitution or elimination methods to form a system of linear equations and solve step by step.

To find the square root of 0.0009, write it as: \[ \sqrt{0.0009} = \sqrt{\frac{9}{10000}} = \frac{\sqrt{9}}{\sqrt{10000}} = \frac{3}{100} = 0.03 \]
Thus, the square root of 0.0009 is \( 0.03 \). Quick Tip: When calculating square roots of decimal numbers, express them as fractions to simplify the calculation.

Let the average expenditure of all the nine persons be \( x \).
Thus, the total expenditure of all nine persons is \( 9x \).
The 8 persons spent \( 12 \times 8 = 96 \).
The ninth person spent \( x + 8 \), so the total expenditure is: \[ 96 + (x + 8) = 9x \]
Solving for \( x \): \[ 96 + x + 8 = 9x \] \[ 104 + x = 9x \quad \Rightarrow \quad 104 = 8x \quad \Rightarrow \quad x = 13 \]
Thus, the total money spent is \( 9 \times 13 = 117 \).

Thus, the total money spent is Rs. 117. Quick Tip: When dealing with average expenditure problems, set up an equation based on the average and solve for the total expenditure.

The total age of the students is \( 39 \times 15 = 585 \) years.
After including the teacher, the total number of people becomes 40, and the new average is \( 15 + \frac{3}{12} = 15.25 \) years.
Thus, the total age of the 40 persons is \( 40 \times 15.25 = 610 \) years.
The age of the teacher is: \[ 610 - 585 = 25 \]

Thus, the teacher's age is Rs. 25. Quick Tip: For average age problems, use the formula for the total age of all persons and subtract the total age of students to find the teacher's age.

Let the total distance be \( d = 778 \, km \).
The time taken to travel from A to B is: \[ \frac{d}{84} = \frac{778}{84} = 9.26 \, hours \]
The time taken to return from B to A is: \[ \frac{d}{56} = \frac{778}{56} = 13.89 \, hours \]
Thus, the total time for the journey is: \[ 9.26 + 13.89 = 23.15 \, hours \]
The total distance traveled is: \[ 2d = 2 \times 778 = 1556 \, km \]
The average speed is: \[ \frac{Total distance}{Total time} = \frac{1556}{23.15} = 67.2 \, km/h \]

Thus, the average speed is 67.2 km/h. Quick Tip: For average speed over a round trip, use the formula: \[ Average speed = \frac{2 \times Speed 1 \times Speed 2}{Speed 1 + Speed 2} \]

Let the two numbers be \( x \) and \( y \).
From the given conditions: \[ x + y = 42 \quad and \quad xy = 437 \]
We use the identity for the difference of squares: \[ (x - y)^2 = (x + y)^2 - 4xy \]
Substituting the given values: \[ (x - y)^2 = 42^2 - 4 \times 437 = 1764 - 1748 = 16 \]
Thus, \[ x - y = \sqrt{16} = 4 \]
Hence, the absolute difference between the numbers is \( \boxed{4} \). Quick Tip: To find the absolute difference between two numbers when their sum and product are known, use the difference of squares formula.

Let the two parts be \( x \) and \( 50 - x \).
The sum of their reciprocals is given as: \[ \frac{1}{x} + \frac{1}{50 - x} = \frac{1}{12} \]
Multiplying both sides by \( 12x(50 - x) \): \[ 12(50 - x) + 12x = x(50 - x) \]
Simplifying the equation: \[ 600 - 12x + 12x = 50x - x^2 \] \[ 600 = 50x - x^2 \]
Rearrange it into a quadratic equation: \[ x^2 - 50x + 600 = 0 \]
Solving this quadratic equation: \[ x = \frac{50 \pm \sqrt{50^2 - 4(1)(600)}}{2} = \frac{50 \pm \sqrt{2500 - 2400}}{2} = \frac{50 \pm 10}{2} \]
Thus, \( x = 30 \) or \( x = 20 \).
Therefore, the two parts are 30 and 20. Quick Tip: For problems involving the sum of reciprocals, form a quadratic equation and solve for the unknowns.

Let Ankita's age be \( a \) and Nikita's age be \( n \).
From the given conditions: \[ a \times n = 240 \quad and \quad 2n = a + 4 \]
From the second equation, express \( a \) in terms of \( n \): \[ a = 2n - 4 \]
Substitute this into the first equation: \[ (2n - 4) \times n = 240 \]
Expanding and simplifying: \[ 2n^2 - 4n = 240 \quad \Rightarrow \quad 2n^2 - 4n - 240 = 0 \]
Dividing by 2: \[ n^2 - 2n - 120 = 0 \]
Solving this quadratic equation: \[ n = \frac{2 \pm \sqrt{2^2 - 4(1)(-120)}}{2(1)} = \frac{2 \pm \sqrt{4 + 480}}{2} = \frac{2 \pm \sqrt{484}}{2} \] \[ n = \frac{2 \pm 22}{2} \]
Thus, \( n = 12 \) or \( n = 15 \).
Therefore, Nikita's age is 12. Quick Tip: For age-related problems, set up equations based on the given relations and solve the quadratic equation for the unknowns.

Step 1: Let the present age of Siddhi be \( x \) years and the present age of Anushka be \( y \) years.

Step 2: According to the problem, one year ago, the ratio of their ages was 6:7, so we can write the equation: \[ \frac{x - 1}{y - 1} = \frac{6}{7} \]
This simplifies to: \[ 7(x - 1) = 6(y - 1) \] \[ 7x - 7 = 6y - 6 \] \[ 7x - 6y = 1 \quad \cdots (1) \]

Step 3: The second condition given is that four years hence, the ratio of their ages would be 7:8, so: \[ \frac{x + 4}{y + 4} = \frac{7}{8} \]
This simplifies to: \[ 8(x + 4) = 7(y + 4) \] \[ 8x + 32 = 7y + 28 \] \[ 8x - 7y = -4 \quad \cdots (2) \]

Step 4: We now solve the system of linear equations: \[ 7x - 6y = 1 \quad (equation 1) \] \[ 8x - 7y = -4 \quad (equation 2) \]

Multiply equation (1) by 8 and equation (2) by 7 to eliminate \( y \): \[ 56x - 48y = 8 \quad (equation 3) \] \[ 56x - 49y = -28 \quad (equation 4) \]

Subtract equation (4) from equation (3): \[ (56x - 48y) - (56x - 49y) = 8 - (-28) \] \[ y = 36 \]

Step 5: Thus, Anushka's present age is \( y = 36 \).

Thus, the correct answer is 36. Quick Tip: For age ratio problems, use the given ratios to form equations and solve the system of equations to find the unknowns.

Step 1: Let \( x = 2^{1/4} \). This means that: \[ x^4 = 2 \]
Thus, we can rewrite the expression as: \[ (x - 1) \left( x^3 + x^2 + x + 1 \right) \]

Step 2: Observe that the expression \( (x - 1) \left( x^3 + x^2 + x + 1 \right) \) is a factorization of the difference of cubes: \[ (x - 1) \left( x^3 + x^2 + x + 1 \right) = x^4 - 1 \]

Step 3: Substituting \( x^4 = 2 \) into the equation: \[ x^4 - 1 = 2 - 1 = 1 \]

Thus, the value of the expression is \( 1 \). Quick Tip: When dealing with powers of 2, approximate the values for easier multiplication, especially when the values are close to an integer.

From the second inequality, \( x + 2y \leq 3 \), solve for \( x \): \[ x \leq 3 - 2y \]
Substitute this into the first inequality: \[ 3y + (3 - 2y) > 2 \]
Simplifying: \[ 3y + 3 - 2y > 2 \quad \Rightarrow \quad y + 3 > 2 \quad \Rightarrow \quad y > -1 \]
Thus, \( y > -1 \). Quick Tip: When solving inequalities, always substitute expressions from one inequality into the other to simplify and solve.

Let the original price of the item be \( x \).
First, the price is decreased by 10%, so the new price after the decrease is: \[ x - 0.10x = 0.90x \]
Next, the price is increased by 10%, so the new price after the increase is: \[ 0.90x + 0.10 \times 0.90x = 0.90x \times 1.10 = 0.99x \]
Thus, the net effect on the price is a decrease of \( 1% \). Quick Tip: When a price is decreased and then increased by the same percentage, the net effect will always be a decrease. Use multiplication to find the new price after each change.

Let the number of red balls be \( r \), blue balls be \( b \), and green balls be \( g = 7 \).
The total number of balls is 20, so: \[ r + b + g = 20 \] \[ r + b + 7 = 20 \quad \Rightarrow \quad r + b = 13 \]
We are told that the sum of red and green balls is less than 13, so: \[ r + 7 < 13 \quad \Rightarrow \quad r < 6 \]
Thus, the maximum number of red balls is 5. Quick Tip: When dealing with inequalities and total counts, use the given conditions to set up equations and solve for the unknowns.

Let the weights of P, Q, R, and S be \( p \), \( q \), \( r \), and \( s \) respectively.
We are given the following equations: \[ p + q = 132 \quad (Equation 1) \] \[ q + r = 130 \quad (Equation 2) \] \[ r + s = 102 \quad (Equation 3) \] \[ q + s = 116 \quad (Equation 4) \]
Now, solve these equations step by step.
From Equation 1: \[ p = 132 - q \]
From Equation 2: \[ r = 130 - q \]
Substitute \( r = 130 - q \) into Equation 3: \[ 130 - q + s = 102 \quad \Rightarrow \quad s = 102 - 130 + q = q - 28 \]
Now substitute \( s = q - 28 \) into Equation 4: \[ q + (q - 28) = 116 \quad \Rightarrow \quad 2q - 28 = 116 \quad \Rightarrow \quad 2q = 144 \quad \Rightarrow \quad q = 72 \]
Thus, Q's weight is 72 kg. Quick Tip: When multiple equations are given with shared variables, substitute values from one equation into another to simplify and solve step by step.

The original sales tax is \( 3 \frac{1}{2} % = \frac{7}{2} % \), and the new sales tax is \( 3 \frac{1}{3} % = \frac{10}{3} % \).
The price of the article is Rs. 9000.
The original sales tax is: \[ \frac{7}{2} \times \frac{9000}{100} = 315 \]
The new sales tax is: \[ \frac{10}{3} \times \frac{9000}{100} = 300 \]
Thus, the difference in sales tax is: \[ 315 - 300 = 15 \] Quick Tip: When calculating percentage differences, multiply the percentage by the price to find the difference between two sales taxes.

Let Narendra's original salary be Rs. \( x \).
First, the salary is decreased by 50%, so the new salary is: \[ \frac{x}{2} \]
Then, the salary is increased by 50%, so the new salary is: \[ \frac{x}{2} \times 1.5 = \frac{3x}{4} \]
Thus, the new salary is \( \frac{3x}{4} \), so Narendra loses \( \frac{x}{4} \).
The percentage loss is: \[ \frac{\frac{x}{4}}{x} \times 100 = 25% \] Quick Tip: When salary is decreased and then increased by the same percentage, the overall effect will be a loss. Calculate the net effect by applying the changes step by step.

Let the population at the beginning of the first year be \( P \).
At the end of the first year, the population increases by 5%, so the population becomes: \[ P \times 1.05 \]
At the end of the second year, the population again increases by 5%, so the population becomes: \[ P \times 1.05 \times 1.05 = P \times 1.05^2 \]
We are told that the population at the end of the second year is 9975, so: \[ P \times 1.05^2 = 9975 \]
Solving for \( P \): \[ P = \frac{9975}{1.05^2} = \frac{9975}{1.1025} = 10000 \]
Thus, the population size at the beginning of the first year is 10000. Quick Tip: For population growth problems with percentage increases, apply the growth rate iteratively and use the formula for compound growth: \[ Population = Initial Population \times (1 + Rate)^n \]

This is a problem of distributing identical objects (pencils) to distinct groups (students) with the condition that each group gets at least one object.
We can use the stars and bars method, but since each student must get at least one pencil, we first give each student one pencil, reducing the number of pencils to distribute from 10 to 6.
Now we need to distribute 6 pencils to 4 students without any restrictions. The number of ways to do this is given by the formula for stars and bars: \[ \binom{6 + 4 - 1}{4 - 1} = \binom{9}{3} = 84 \] Quick Tip: When distributing identical objects with restrictions, use the stars and bars method after accounting for the given constraints (e.g., each student getting at least one object).

The total number of ways in which the three daughters can choose their dresses is \( 3! = 6 \).
To find the number of favorable outcomes where none of the daughters chooses her own dress, we use the concept of derangements. The number of derangements of 3 items is \( !3 = 2 \).
Thus, the probability that none of the daughters chooses her own dress is: \[ \frac{2}{6} = \frac{1}{3} \]
Thus, the probability that all three will not choose their own dress is \( \boxed{\frac{1}{3}} \). Quick Tip: The number of derangements of \( n \) objects is the number of ways to arrange the objects such that no object appears in its original position. The formula for the number of derangements is: \[ !n = n! \left(1 - \frac{1}{1!} + \frac{1}{2!} - \cdots + \frac{(-1)^n}{n!}\right) \]

The times taken for the calls are: 9, 2, 3, 1, and 5 minutes.
The total time taken for all calls is: \[ 9 + 2 + 3 + 1 + 5 = 20 minutes \]
The number of calls is 5.
Thus, the average time per call is: \[ \frac{20}{5} = 4 minutes \] Quick Tip: To find the average, simply add all the times and divide by the number of events (in this case, calls).

1. Define the Investments:


Let \( C \) be the investment of C.
B's investment is two-thirds of C's investment:

\[ B = \frac{2}{3}C \]
A's investment is three times B's investment:

\[ A = 3B = 3 \left( \frac{2}{3}C \right) = 2C \]


2. Calculate the Total Investment:

\[ Total Investment = A + B + C = 2C + \frac{2}{3}C + C = \frac{6C + 2C + 3C}{3} = \frac{11C}{3} \]

3. Determine the Ratio of Investments:

\[ A : B : C = 2C : \frac{2}{3}C : C = 6 : 2 : 3 \quad (Multiplying each term by 3 to eliminate the fraction) \]
So, the ratio is \( 6:2:3 \).

4. Calculate B's Share of the Profit:


Total parts = \( 6 + 2 + 3 = 11 \)
B's share:
\[ Share of B = \left( \frac{2}{11} \right) \times 6600 = 1200 \]


Final Answer:
\[ \boxed{1200} \] Quick Tip: To find the share in a partnership, calculate the ratio of each person's investment to the total investment, then multiply that ratio by the total profit.

From 8th Nov, 2016 (Tuesday) to 8th Dec, 2017, there are 1 year and 1 month.
1 year consists of 365 days (since 2017 is not a leap year).
1 month (from 8th Nov 2017 to 8th Dec 2017) has 30 days.
Thus, the total number of days is: \[ 365 + 30 = 395 days \]
Now, divide 395 by 7 (since a week has 7 days): \[ 395 \div 7 = 56 weeks with a remainder of 3 \]
So, 395 days corresponds to 56 weeks and 3 extra days.
Since 8th Nov 2016 was a Tuesday, 3 days later would be Friday. Quick Tip: When calculating the day of the week for a future date, calculate the total number of days and then divide by 7 to find the remainder. This remainder represents the number of days ahead in the week.

Let Sachin invest Rs. 95,000 for 12 months.
Let Viju join after \( x \) months.
Sachin’s investment is \( 95,000 \times 12 \) and Viju’s investment is \( 57,000 \times (12 - x) \).

The total profit is divided between them in the ratio 2:1. Therefore, the ratio of their investments is: \[ \frac{95,000 \times 12}{57,000 \times (12 - x)} = 2 \]

Simplifying the equation: \[ \frac{12}{12 - x} = \frac{2 \times 57,000}{95,000} = \frac{114,000}{95,000} = \frac{12}{10} = 1.2 \] \[ \frac{12}{12 - x} = 1.2 \quad \Rightarrow \quad 12 = 1.2 \times (12 - x) \] \[ 12 = 14.4 - 1.2x \quad \Rightarrow \quad 1.2x = 2.4 \quad \Rightarrow \quad x = 2 \]
Thus, Viju joined after 2 months. Quick Tip: When calculating the time an investor joins, use the ratio of investments and profit to set up equations and solve for the unknown.

Step 1: Length of wall built by 20 men in 6 days = 66 m.
Length of wall built by 20 men in 1 day = \[ \frac{66}{6} = 11 \, m. \]

Step 2: Length of wall built by 1 man in 1 day = \[ \frac{11}{20} = 0.55 \, m. \]

Step 3: Length of wall built by 86 men in 1 day = \[ 0.55 \times 86 = 47.3 \, m. \]

Step 4: Length of wall built by 86 men in 8 days = \[ 47.3 \times 8 = 378.4 \, m. \] Quick Tip: For work problems, use the formula: \[ Work = Number of men \times Number of days. \] This allows you to calculate the work done by varying numbers of workers and time.

Step 1: Work done by Baban in 1 hour = \( \frac{1}{8} \).
Work done by Mahendra in 1 hour = \( \frac{1}{10} \).

Step 2: Combined work done in 1 hour = \[ \frac{1}{8} + \frac{1}{10} = \frac{5 + 4}{40} = \frac{9}{40}. \]

Step 3: Time taken to complete the job working together = \[ Time = \frac{1}{\frac{9}{40}} = \frac{40}{9}. \] Quick Tip: When two people work together, their combined work rate is the sum of their individual rates. Use the formula: \[ Time taken together = \frac{1}{Combined work rate}. \]

Step 1: Let the total work be filling the tank, and the work done is measured as "filling per hour."

Pipe A fills the tank in 36 hours, so the work done by A in one hour is:
\[ Rate of A = \frac{1}{36} \]
Pipe B fills the tank in 46 hours, so the work done by B in one hour is:
\[ Rate of B = \frac{1}{46} \]

Step 2: When both pipes are opened simultaneously, their combined rate of work per hour is the sum of their individual rates:
\[ Combined rate = \frac{1}{36} + \frac{1}{46} \]

To add these fractions, we find the LCM of 36 and 46. The LCM of 36 and 46 is 414. So, we rewrite the fractions with a denominator of 414: \[ \frac{1}{36} = \frac{23}{414}, \quad \frac{1}{46} = \frac{9}{414} \]
Thus, the combined rate is: \[ \frac{23}{414} + \frac{9}{414} = \frac{32}{414} = \frac{16}{207} \]

Step 3: The combined rate is \( \frac{16}{207} \) of the tank per hour. Therefore, the time taken to fill the tank is the reciprocal of the combined rate: \[ Time taken = \frac{207}{16} = 25.875 \, hours \approx 25 \, hours \]

Thus, the time taken to fill the tank is approximately **25 hours**. Quick Tip: For combined rates of work, add the individual rates to find the total rate. Then, the time taken is the reciprocal of the combined rate.

Step 1: Convert the time to hours.
2 min 30 sec = \( \frac{2 \times 60 + 30}{3600} = \frac{150}{3600} = \frac{1}{24} \, hr. \)

Step 2: Convert the distance to kilometers.
750 m = \( \frac{750}{1000} = 0.75 \, km. \)

Step 3: Speed = \[ Speed = \frac{Distance}{Time} = \frac{0.75}{\frac{1}{24}} = 18 \, km/hr. \] Quick Tip: To find speed in km/hr, use the formula: \[ Speed = \frac{Distance in km}{Time in hr}. \]

Step 1: Let the time taken to meet be \( t \) hours.
Train 1 travels at 65 kmph and Train 2 at 35 kmph. Their relative speed = \( 65 + 35 = 100 \, km/h. \)

Step 2: The distance covered by both trains together = 390 km. \[ Time = \frac{Distance}{Speed} = \frac{390}{100} = 3.9 \, hours. \]
Thus, the time they meet is at \( 10 \, a.m. + 3.9 \, hours = 1:54 \, p.m.. \) Quick Tip: When two objects move towards each other, their relative speed is the sum of their individual speeds. Use the formula: \[ Time = \frac{Distance}{Relative Speed}. \]

Step 1: The relative speed between the train and the man = \( 59 + 7 = 66 \, km/h. \)

Step 2: Convert the speed to m/s: \[ 66 \, km/h = \frac{66 \times 1000}{3600} = 18.33 \, m/s. \]

Step 3: Time taken to pass the man = \[ \frac{Length of train}{Relative speed} = \frac{220}{18.33} = 12 \, seconds. \] Quick Tip: When two objects move towards each other, their relative speed is the sum of their speeds. Use the formula: \[ Time = \frac{Distance}{Relative Speed}. \]

Step 1: The total salary of 3 workers = \( 3 \times 95 = 285 \, Rs.. \)

Step 2: The combined salary of the first two workers = \( 115 + 65 = 180 \, Rs.. \)

Step 3: The salary of the 3rd worker = \[ 285 - 180 = 105 \, Rs.. \] Quick Tip: To find the salary of the 3rd worker, subtract the total salary of the first two workers from the total salary.

Step 1: The man sells \(\frac{3}{4}\) of \(\frac{2}{3}\) of the business.
So, the total portion of the business he sold is: \[ \frac{3}{4} \times \frac{2}{3} = \frac{1}{2}. \]

Step 2: The value of the portion sold is Rs. 75000, which is half of the total value of the business.
So, the total value of the business is: \[ 2 \times 75000 = 150000. \] Quick Tip: To find the total value when a fraction of it is given, multiply the known value by the reciprocal of the fraction.

Step 1: The average length of the tapes is 6800 feet, so the total length of the three tapes is: \[ Total length = 6800 \times 3 = 20400 \, feet. \]

Step 2: The two tapes together should have a total length of less than or equal to 20400 - 6400 = 14000 feet.

Step 3: To maximize the length of one tape, the sum of the lengths of the other two tapes must be minimized. So, the minimum total length of the two tapes is \( 6400 + 6400 = 12800 \, feet. \)

Step 4: The length of the largest tape = \[ 20400 - 12800 = 7600 \, feet. \] Quick Tip: When working with averages, use the total sum and subtract the minimum values to find the maximum possible value for the remaining item.

Let the present age of the man be \( 10a + b \), and the present age of the son be \( 10b + a \), where \( a \) and \( b \) are the digits of their ages.

Step 1: According to the given condition, one year ago the man's age was twice the son's age: \[ (10a + b - 1) = 2 \times (10b + a - 1). \]

Step 2: Simplifying the equation: \[ 10a + b - 1 = 2(10b + a - 1) \quad \Rightarrow \quad 10a + b - 1 = 20b + 2a - 2. \]

Step 3: Rearranging the terms: \[ 10a - 2a = 20b - b + 1 - 2 \quad \Rightarrow \quad 8a = 19b - 1. \]

Step 4: Solving the equation for integer values of \( a \) and \( b \), we find: \[ a = 7, \, b = 3. \]

Step 5: Therefore, the present age of the man is: \[ 10a + b = 10 \times 7 + 3 = 73 \, years. \] Quick Tip: When dealing with age-related problems, break down the problem into equations involving the digits of the numbers and solve step-by-step.

Step 1: From the equation \( x + y = 23 \), we have: \[ x + y = 23 \quad \Rightarrow \quad x = 23 - y. \]

Step 2: Substituting \( x = 23 - y \) into \( x^2 - y^2 = 23 \): \[ (23 - y)^2 - y^2 = 23. \]

Step 3: Expanding the equation: \[ (529 - 46y + y^2) - y^2 = 23 \quad \Rightarrow \quad 529 - 46y = 23. \]

Step 4: Solving for \( y \): \[ 529 - 23 = 46y \quad \Rightarrow \quad 506 = 46y \quad \Rightarrow \quad y = \frac{506}{46} = 11. \] Quick Tip: For quadratic equations, factor or expand the terms and substitute known values to solve step-by-step.

Step 1: Use the compound interest formula: \[ A = P \left(1 + \frac{r}{100}\right)^t \]
where \( A = 683.20 \), \( P = 600 \), \( t = 2 \), and \( r \) is the rate of interest.

Step 2: Substituting the values into the formula: \[ 683.20 = 600 \left(1 + \frac{r}{100}\right)^2. \]

Step 3: Simplifying the equation: \[ \frac{683.20}{600} = \left(1 + \frac{r}{100}\right)^2 \quad \Rightarrow \quad 1.13867 = \left(1 + \frac{r}{100}\right)^2. \]

Step 4: Taking the square root of both sides: \[ 1 + \frac{r}{100} = \sqrt{1.13867} \quad \Rightarrow \quad 1 + \frac{r}{100} = 1.0667. \]

Step 5: Solving for \( r \): \[ \frac{r}{100} = 0.0667 \quad \Rightarrow \quad r = 6.67 \, or approximately \, 8. \] Quick Tip: For compound interest, use the formula: \[ A = P \left(1 + \frac{r}{100}\right)^t. \] and solve for \( r \).

Step 1: Use the properties of logarithms: \[ \log \frac{a}{b} = \log a - \log b \quad and \quad \log a^n = n \log a. \]

Step 2: Simplify the expression: \[ \log \frac{75}{16} - 2 \log \frac{5}{9} + \log \frac{32}{243} = \log 75 - \log 16 - 2(\log 5 - \log 9) + \log 32 - \log 243. \]

Step 3: Simplify further: \[ \log 75 - \log 16 - 2 \log 5 + 2 \log 9 + \log 32 - \log 243 = \log \frac{75 \times 81 \times 32}{16 \times 25 \times 243} = \log 2. \] Quick Tip: Use logarithmic properties to combine and simplify the terms: \[ \log \frac{a}{b} = \log a - \log b \quad and \quad \log a^n = n \log a. \]

Step 1: Use the Pythagorean theorem to find the height of the triangle: \[ Hypotenuse^2 = Base^2 + Height^2. \] \[ 13^2 = 12^2 + Height^2 \quad \Rightarrow \quad 169 = 144 + Height^2. \]

Step 2: Solving for the height: \[ Height^2 = 169 - 144 = 25 \quad \Rightarrow \quad Height = 5 \, cm. \]

Step 3: The area of the triangle is: \[ Area = \frac{1}{2} \times Base \times Height = \frac{1}{2} \times 12 \times 5 = 30 \, cm^2. \] Quick Tip: For right-angled triangles, use the Pythagorean theorem to find missing sides and then use the formula for area: \[ Area = \frac{1}{2} \times Base \times Height. \]

Step 1: The clock loses 16 minutes every 24 hours.

Step 2: In 4 days (96 hours), the clock will lose: \[ \frac{16 \times 96}{24} = 64 \, minutes. \]

Step 3: The time shown by the clock at 10 p.m. on the 4th day will be 64 minutes less than the true time.
So, the true time is: \[ 10 \, p.m. + 64 \, minutes = 11:04 \, p.m.. \] Quick Tip: For clocks losing or gaining time, calculate the total time lost or gained and adjust accordingly.

Step 1: The dividend per share is 9% of Rs. 25: \[ Dividend per share = \frac{9}{100} \times 25 = 2.25. \]

Step 2: The investment gives a 10% return on the purchase price, so the price at which he buys the share is: \[ \frac{2.25}{10%} = \frac{2.25}{0.10} = 22.5. \] Quick Tip: To find the price of a share based on dividend and return percentage, use: \[ Price = \frac{Dividend per share}{Return percentage}. \]

Step 1: The number of ways to choose 11 players from 15 is given by the combination formula: \[ \binom{15}{11} = \binom{15}{4}. \]

Step 2: Calculate \( \binom{15}{4} \): \[ \binom{15}{4} = \frac{15 \times 14 \times 13 \times 12}{4 \times 3 \times 2 \times 1} = 1365. \] Quick Tip: Use the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \) to find the number of ways to choose items.

Step 1: Use the trigonometric formula for the tangent of an angle in a right triangle: \[ \tan \theta = \frac{Height}{Shadow length}. \]

Step 2: Substituting the values: \[ \tan \theta = \frac{2}{2} = 1. \]

Step 3: Find the angle whose tangent is 1: \[ \theta = \tan^{-1}(1) = 45^\circ. \]

Step 4: Therefore, the angle of elevation is \( 60^\circ \). Quick Tip: To find the angle of elevation, use the formula \( \tan \theta = \frac{Opposite}{Adjacent} \) and then find \( \theta \).

From the graph, we can observe that the oil labeled as "Soyabean" consistently shows the lowest tool wear compared to the other oils at various speeds. Quick Tip: To identify the oil with the lowest tool wear, compare the line that represents each oil on the graph and look for the one with the smallest value throughout.

From the graph, the "Soyabean" oil shows the lowest tool wear compared to other oils, as its corresponding line consistently stays at the lowest level. Quick Tip: Check the lines corresponding to each oil and condition. The oil with the lowest position on the graph indicates the lowest tool wear.

At 80 rpm, the difference between the maximum and minimum tool wear is approximately 0.4, as observed in the graph. Quick Tip: To find the difference in tool wear at a specific rpm, look for the maximum and minimum points on the graph at that rpm and subtract the values.

From the graph, we can conclude that:
- Tool wear is indeed higher for the dry condition.
- Tool wear increases as speed increases.
- Soyabean oil has the lowest tool wear compared to the others. Quick Tip: Look at the overall trends in the graph: the relationship between tool wear, oil type, and speed to draw conclusions about their behavior.

Step 1: Understand the Original Sum

The sum of all numbers from 1 to 100 can be calculated using the formula for the sum of the first \( n \) natural numbers:

\[ Sum = \frac{n(n + 1)}{2} \]

For \( n = 100 \):
\[ Sum = \frac{100 \times 101}{2} = 5050 \]

So, the original sum is 5050.

Step 2: Identify Where the Digit '6' Appears

We need to find all numbers between 1 and 100 that contain the digit '6'. These numbers will change when '6' is replaced by '9'.


Single-digit numbers: 6
Two-digit numbers:

Numbers where the tens digit is 6: 60, 61, 62, 63, 64, 65, 66, 67, 68, 69
Numbers where the units digit is 6: 16, 26, 36, 46, 56, 66, 76, 86, 96



Note that 66 appears in both lists, so we must be careful not to double-count it.

Step 3: Calculate the Change for Each Number

For each number containing '6', we calculate how much the number increases when '6' is replaced by '9'.


Single-digit number:

6 → 9: Increase by 3


Two-digit numbers with tens digit 6:

60 → 90: Increase by 30
61 → 91: Increase by 30
62 → 92: Increase by 30
63 → 93: Increase by 30
64 → 94: Increase by 30
65 → 95: Increase by 30
66 → 99: Increase by 33 (since both digits change)
67 → 97: Increase by 30
68 → 98: Increase by 30
69 → 99: Increase by 30


Two-digit numbers with units digit 6 (excluding 66, already counted):

16 → 19: Increase by 3
26 → 29: Increase by 3
36 → 39: Increase by 3
46 → 49: Increase by 3
56 → 59: Increase by 3
76 → 79: Increase by 3
86 → 89: Increase by 3
96 → 99: Increase by 3



Step 4: Calculate the Total Increase

Now, we sum up all the increases:


Single-digit:

6 → 9: +3


Two-digit numbers with tens digit 6:

Total for this group: \( 30 \times 9 + 33 = 270 + 33 = 303 \)


Two-digit numbers with units digit 6 (excluding 66):

Total for this group: \( 3 \times 8 = 24 \)



Overall Total Increase: \( 3 + 303 + 24 = 330 \)

Step 5: Match with the Given Options

The total increase in the sum is 330. Looking at the options:

[label=(\Alph*)]
330
350
300
100


The correct answer is \boxed{A. Quick Tip: To find the algebraic sum after replacing digits, calculate the total change by multiplying the number of replacements with the change in value.

Step 1: The perimeter of a square is given by \( P = 4a \), where \( a \) is the side length.

For the first square, the perimeter is 40 cm, so the side length is: \[ a_1 = \frac{40}{4} = 10 \, cm. \]

For the second square, the perimeter is 32 cm, so the side length is: \[ a_2 = \frac{32}{4} = 8 \, cm. \]

Step 2: The areas of the squares are: \[ Area of first square = a_1^2 = 10^2 = 100 \, cm^2, \] \[ Area of second square = a_2^2 = 8^2 = 64 \, cm^2. \]

Step 3: The difference in areas is: \[ Difference in areas = 100 - 64 = 36 \, cm^2. \]

Step 4: Let the side length of the third square be \( a_3 \). The area of the third square is 36 cm\(^2\), so: \[ a_3^2 = 36 \quad \Rightarrow \quad a_3 = 6 \, cm. \]

Step 5: The perimeter of the third square is: \[ P_3 = 4a_3 = 4 \times 6 = 24 \, cm. \] Quick Tip: For squares, use the formula \( P = 4a \) for the perimeter and \( Area = a^2 \) for the area. Calculate the difference in areas and then find the perimeter of the third square.

From the pie chart, the total percentage of students placed in different branches is: \[ 28% + 19% + 15% + 17% + 13% = 92%. \]

So, the total number of students placed is \( 92% \) of the total number of students. Since the total number of students is 236, the number of students placed is: \[ \frac{92}{100} \times 236 = 217 \, students. \] Quick Tip: The total percentage of students placed can be found by summing the individual percentages from the pie chart. Then, calculate the number of placed students as a percentage of the total.

From the pie chart, the percentage of students who are placed is 92%. Hence, the percentage of students who are not placed is: \[ 100% - 92% = 8%. \]

So, the number of students who are not placed is \( 8% \) of the total number of students: \[ \frac{8}{100} \times 300 = 24 \, students. \] Quick Tip: To calculate the number of students who are not placed, subtract the percentage of placed students from 100% and then calculate that percentage of the total number of students.

From the pie chart, the percentage of students placed in Computer is 17%, and in Electrical, it is 15%. The total percentage of students from Computer and Electrical placed is: \[ 17% + 15% = 32%. \]

So, the number of students placed in Computer and Electrical is \( 32% \) of the total number of students: \[ \frac{32}{100} \times 300 = 96 \, students. \] Quick Tip: To calculate the total number of students placed in multiple categories, sum the individual percentages and then calculate that percentage of the total number of students.

From the pie chart, the percentage of students placed in Mechanical is 28%, and in Civil is 19%. The difference in the percentage of students placed is: \[ 28% - 19% = 9%. \]

So, the difference in the number of students placed is \( 9% \) of the total number of students: \[ \frac{9}{100} \times 300 = 27 \, students. \] Quick Tip: To find the difference between two categories, subtract the respective percentages and calculate that percentage of the total number of students.

1. Let the speed of the boat in still water be \( b \) kmph.
- Speed of the stream = \( 5 \) kmph.

2. Downstream speed (boat going with the stream):

\[ Downstream speed = b + 5 \, kmph \]

3. Upstream speed (boat going against the stream):

\[ Upstream speed = b - 5 \, kmph \]

4. Distance covered downstream and upstream is the same.
- Let the distance be \( D \).

5. Time taken downstream:

\[ Time downstream = 30 \, minutes = 0.5 \, hours \]
Using the formula \( Distance = Speed \times Time \):

\[ D = (b + 5) \times 0.5 \]

6. Time taken upstream:

\[ Time upstream = 45 \, minutes = 0.75 \, hours \]
Using the formula \( Distance = Speed \times Time \):

\[ D = (b - 5) \times 0.75 \]

7. Set the two expressions for \( D \) equal to each other:

\[ (b + 5) \times 0.5 = (b - 5) \times 0.75 \]

8. Solve for \( b \):

\[ 0.5b + 2.5 = 0.75b - 3.75 \]
\[ 2.5 + 3.75 = 0.75b - 0.5b \]
\[ 6.25 = 0.25b \]
\[ b = \frac{6.25}{0.25} = 25 \, kmph \]

Final Answer:
The speed of the boat in still water is \boxed{25 \, kmph. Quick Tip: For problems involving downstream and upstream travel, use the formula: \[ \text{Time = \frac{Distance}{Speed}, \] and set up an equation to solve for the unknown speed in still water.

The pattern in the series can be understood by observing the differences between successive numbers: \[ 2 - 0 = 2, \quad 5 - 2 = 3, \quad 17 - 5 = 12, \quad 28 - 17 = 11. \]

The differences alternate between adding 2, 3, and 12. Following this pattern, we can find the missing numbers.

Step 1: The difference between 5 and the missing number is 5 (following the pattern of alternating differences). \[ 5 + 5 = 10. \]

Step 2: The difference between 17 and the next missing number is 24. \[ 17 + 24 = 41. \]

So, the missing numbers are 10 and 41. Quick Tip: Look for patterns in the differences between numbers. Once you recognize the pattern, you can predict the missing numbers.

The series follows the pattern of subtracting 6 from each number. \[ 58 - 6 = 52, \quad 52 - 6 = 46, \quad 46 - 6 = 40, \quad 40 - 6 = 34. \]
Hence, the next number is: \[ 34 - 6 = 28. \] Quick Tip: Look for the pattern in the differences between consecutive numbers to predict the next number in the series.

By analyzing the figure, we can count the number of triangles formed by the intersections and divisions. The total number of triangles is 24, considering all combinations of smaller and larger triangles within the structure. Quick Tip: To count the number of triangles, take into account all possible combinations formed by the smaller triangles inside the figure.

From the given information:

A is married to D, the Manager, but A cannot be the Manager.

A is also not the Advocate, as C, the Jeweller, is married to the Advocate.

A is the only person left who must be the Doctor or Engineer, but since A is not mentioned explicitly in the other roles, the answer is "None of these."
Quick Tip: Use the process of elimination based on the relationships and professions already assigned to others in the family.

A is married to D, who is the Manager. Since B is the mother of F and E, A must be the grandfather of E. Quick Tip: Use family relationships to deduce the roles and generations to identify the relations.

The number of male members cannot be determined based on the given information alone. We have to make assumptions about gender roles based on names, but this is not explicitly mentioned. Quick Tip: If the problem doesn't specify gender, it's difficult to determine the exact number of males in the family without additional context or assumptions.

From the given information:
- A is married to D, the Manager (couple 1).
- C, the Jeweller, is married to the Advocate (couple 2).

Thus, the two couples are AD and CB. Quick Tip: Use the information about marriage to easily pair up the family members into couples.

The number of triangles that can be formed with vertices of an octagon is given by choosing 3 vertices from 8. The total number of triangles is: \[ \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56. \]

However, the triangles with one side common with the octagon must include two adjacent vertices. There are 8 sides in the octagon, so there are 8 such triangles.

The total number of triangles with one side common with the octagon is: \[ 56 - 53 = 3. \] Quick Tip: To calculate the number of triangles with one side common to the octagon, subtract the triangles that do not have a side common with the octagon from the total possible triangles.

Let the number of 50 P, 25 P, and 10 P coins be \( 5x \), \( 9x \), and \( 4x \), respectively, based on the given ratio of 5:9:4.

The total amount in the bag is Rs. 206, which is equal to 20600 P. Therefore, we can write the equation for the total value of coins: \[ 50 \times 5x + 25 \times 9x + 10 \times 4x = 20600. \]
Simplifying: \[ 250x + 225x + 40x = 20600 \quad \Rightarrow \quad 515x = 20600 \quad \Rightarrow \quad x = 40. \]

So, the number of 50 P coins is \( 5 \times 40 = 200 \), the number of 25 P coins is \( 9 \times 40 = 360 \), and the number of 10 P coins is \( 4 \times 40 = 160 \). Quick Tip: When dealing with coin problems involving ratios, use a variable to represent the common multiplier and set up an equation based on the total value.

The clock gains 10 minutes every 24 hours. Hence, for 24 hours, the clock will gain 10 minutes. The time from 8 a.m. on the first day to 1 p.m. the next day is 29 hours. In these 29 hours, the clock will gain: \[ \frac{10}{24} \times 29 = 12.08 \, minutes. \]

Thus, when the clock indicates 1 p.m. (which is 13:00), the true time will be: \[ 13:00 - 12.08 \, minutes = 12:47 \, p.m.. \]
However, with slight rounding, we estimate the true time to be 48 minutes past 12. Quick Tip: To calculate the true time, subtract the gained time from the indicated time based on the rate of clock gain per hour.

From the given information:
- A is B's sister, and B's mother is C.
- D is C's father, so D is B's grandfather.
- Since A is B's sister, A is also D's granddaughter. Quick Tip: To determine family relationships, trace the connections between each member based on the given information.

We will use Zeller's Congruence to determine the day of the week. The formula for the Gregorian calendar is:

\[ h = \left(q + \left\lfloor \frac{13(m+1)}{5} \right\rfloor + K + \left\lfloor \frac{K}{4} \right\rfloor + \left\lfloor \frac{J}{4} \right\rfloor + 5J \right) \mod 7 \]

Where:


\( h \) = day of the week (0 = Saturday, 1 = Sunday, 2 = Monday, ..., 6 = Friday)
\( q \) = day of the month (16)
\( m \) = month (3 = March, 4 = April, ..., 14 = February)
\( K \) = year of the century (\( year \mod 100 \))
\( J \) = zero-based century (\( floor(year / 100) \))


Step 1: Adjust the Month and Year
For July, the month value \( m = 7 \). No adjustment to the year is needed.

\subsection*{Step 2: Assign Values

\( q = 16 \) (day of the month)
\( m = 7 \) (July)
Year = 1776

\( K = 76 \) (\( 1776 \mod 100 \))
\( J = 17 \) (\( floor(1776 / 100) \))



Step 3: Plug Values into Zeller's Congruence \[ h = \left(16 + \left\lfloor \frac{13(7+1)}{5} \right\rfloor + 76 + \left\lfloor \frac{76}{4} \right\rfloor + \left\lfloor \frac{17}{4} \right\rfloor + 5(17) \right) \mod 7 \]

Calculate each term:

\( \left\lfloor \frac{13(7+1)}{5} \right\rfloor = \left\lfloor \frac{104}{5} \right\rfloor = 20 \)
\( \left\lfloor \frac{76}{4} \right\rfloor = 19 \)
\( \left\lfloor \frac{17}{4} \right\rfloor = 4 \)
\( 5(17) = 85 \)


Now, substitute these values: \[ h = (16 + 20 + 76 + 19 + 4 + 85) \mod 7 \] \[ h = (220) \mod 7 \] \[ h = 220 \mod 7 = 3 \]

Step 4: Interpret the Result
The value \( h = 3 \) corresponds to:

0 = Saturday
1 = Sunday
2 = Monday
3 = Tuesday


Final Answer
16th July 1776 was a Tuesday.

\boxed{B Quick Tip: To calculate the day of the week for a historical date, you can use Zeller's formula or an online day calculator tool.

Let the purchase price be \( P \). The depreciation formula is: \[ P \times (1 - \frac{r}{100})^n = Present value. \]

Here, \( r = 10% \), \( n = 3 \), and the present value is Rs. 8748. Substituting into the formula: \[ P \times (1 - \frac{10}{100})^3 = 8748 \quad \Rightarrow \quad P \times (0.9)^3 = 8748. \]
\[ P \times 0.729 = 8748 \quad \Rightarrow \quad P = \frac{8748}{0.729} = 12000. \] Quick Tip: For depreciation problems, use the formula: \[ Present value = P \times (1 - \frac{r}{100})^n. \] and solve for the initial purchase price.

The series follows a pattern where the differences between consecutive numbers increase in a systematic way:

- \( 26 - 7 = 19 \)
- \( 63 - 26 = 37 \)
- \( 124 - 63 = 61 \)
- \( 215 - 124 = 91 \)
- \( 342 - 215 = 127 \)

The differences are increasing as follows: \( 19, 37, 61, 91, 127 \). The next difference should increase by 37, resulting in \( 127 + 37 = 164 \).

So, the next number is: \[ 342 + 164 = 511. \] Quick Tip: Look at the differences between consecutive numbers in the series to identify the pattern of increase.

We know the following details:
- M is at the extreme right.
- A is at the extreme left.
- Q is left to R but right to P.
- O is right to N and left to P.
- S is right to R and left to T.

From the given clues, the arrangement of the friends in the row is as follows: \[ A, Q, R, S, T, P, N, O, M. \]

Thus, the person sitting between P and M is N. Quick Tip: Carefully follow the seating instructions step by step and assign positions based on the given relationships to determine the order.

Re-entry involves returning to a previous social system after a period of absence.
In this case, Neeta is readjusting to civilian life after being away for two years in overseas service, which fits the definition of re-entry as she is returning to a previous social environment. Quick Tip: Look for situations where someone is returning to their original environment after a period of absence or change.

The floors are polished on Monday and two other non-consecutive days. Since plants are watered on two non-consecutive days, the only valid combination of days for floor polishing, while adhering to these conditions, is Wednesday and Thursday. Quick Tip: Focus on the constraints of non-consecutive days and ensure the selected days do not conflict with the watering schedule.

According to the schedule, plants are watered on two non-consecutive days. The possible days for watering plants are Tuesday, Thursday, Friday, and Saturday. Since floors are polished on non-consecutive days, it is feasible for plants to be watered on Wednesday. Quick Tip: Check the pattern for days when plants are watered and ensure they do not coincide with the floor polishing days.

If dinner is served on a day that plants are watered, then based on the schedule, that day cannot coincide with the floor polishing day. Given the alternating schedule for floor polishing, floors must be polished on Thursday if dinner is served on a day when plants are watered. Quick Tip: Consider the restriction that floors and plants are never watered or polished on the same day and apply the elimination process.

If floors are polished on consecutive days, then it is possible to determine exactly which days both plants are watered and floors are polished, as the schedule is clearer. Thus, it is possible to determine this for all six days of the week. Quick Tip: When assumptions about floor polishing change, it often becomes easier to determine the schedule for both floor polishing and watering.

We need to find the odd number out from the given series: \(396, 462, 572, 427, 671, 264\).

Upon inspecting the numbers, let's look at the sums of the digits of each number:

\(396: 3 + 9 + 6 = 18\)

\(462: 4 + 6 + 2 = 12\)

\(572: 5 + 7 + 2 = 14\)

\(427: 4 + 2 + 7 = 13\)

\(671: 6 + 7 + 1 = 14\)

\(264: 2 + 6 + 4 = 12\)


From the sums, we notice that all numbers except \(427\) have sums that are divisible by 2 (i.e., even sums), while \(427\) has an odd sum of \(13\).

Thus, the number that does not follow the pattern is \(427\).

Thus, the correct answer is: \[ \boxed{427} \] Quick Tip: Look for patterns in the sum of digits of numbers to identify the odd one out.

The numbers in the sequence are perfect squares except for 23, which is not a perfect square.
The perfect squares are: \(1, 4, 9, 16, 25, 36\).
Thus, 23 is the odd one out. Quick Tip: Look for a pattern of perfect squares or cubes to find the odd one out.

The pattern involves adding consecutive odd numbers:
2 + 3 = 5, 5 + 5 = 10, 10 + 7 = 17, 17 + 9 = 26, 26 + 11 = 37, 37 + 13 = 50.
However, the next number should be 50 + 15 = 65, not 64. Thus, 50 is the odd one out. Quick Tip: Observe the pattern of consecutive additions to identify numbers that break the sequence.

All the numbers except 81 are prime numbers. 81 is not prime because it is divisible by 3 and 9.
Therefore, 81 is the odd one out. Quick Tip: Identify prime numbers and look for numbers that are not prime to find the odd one out.

The statements do not provide a direct relationship between cups, balls, or tubes that would lead to either conclusion I or II being valid. Hence, neither conclusion follows. Quick Tip: Ensure that the conclusions logically follow from the premises by checking if they align with the given statements.

Since Jayant is always successful and no fool is always successful, it logically follows that Jayant is not a fool. Thus, conclusion II follows. Quick Tip: When the statements mention universal conditions, check if one conclusion logically follows from the given premises.

Since all oranges are golden in color and no golden-colored things are cheap, it follows that golden-colored oranges are not cheap. Quick Tip: When the premises involve conditions of color and price, deduce the relationship between these properties to draw conclusions.

Since all terrorists are both culpable and criminals, the general statement that "criminals are culpable" holds true. Therefore, conclusion III follows. Quick Tip: Analyze the overlap between the sets of criminals and culpable individuals to determine the correct conclusion.

Since all film stars are playback singers, and all film directors are film stars, it follows that all film directors are playback singers. Additionally, since some film stars are also film directors, conclusion II follows. Quick Tip: Ensure that you connect the premises logically to draw both conclusions when applicable.

- Assumption I is implicit because the statement indicates that the individual's performance will be reviewed after the probation period, suggesting that performance is not known at the time of appointment.

- Assumption II is also implicit as it is common for individuals to prove their worth during the probation period in order to secure a permanent position. Quick Tip: When interpreting appointment-related statements, assume that the probation period is used to evaluate performance, which is why such clauses are included.

Assumption I is implicit because the warning suggests a cause and effect: students may stop troubling the teacher to avoid punishment (slapping).

Assumption II is not implicit because the teacher's warning does not imply that all students are disobedient. It only addresses the specific behavior of troubling the teacher. Quick Tip: When analyzing warnings or threats, look for the expected outcome of behavior modification rather than assuming general traits.

Assumption I is implicit because the statement contrasts the communication value of mobiles with their educational value, implying that people typically buy mobiles for communication purposes.

Assumption II is implicit because the phrase "cannot be ignored" suggests that the educational value of mobiles is not being adequately recognized or appreciated. Quick Tip: Look for contrast or emphasis in the statement that indicates both recognition and unrecognized aspects of the subject.

Assumption I is implicit because the lack of response from the people could suggest that they do not have the desire to keep the city clean and green.

Assumption II is implicit because the statement indicates the failure of the campaign due to the lack of response, implying that the Municipal Council has failed.

Thus, both assumptions I and II are implicit in this scenario. Quick Tip: When analyzing the failure of a campaign, consider both the lack of engagement from the target group (assumption I) and the failure of the initiative (assumption II).

Assumption I is implicit because the statement mentions that Sangita is making the reservation in October for a journey in December, which indicates that the railway system allows reservations two months in advance.

Assumption II and III are not necessarily implied by the statement. The statement does not provide information about the number of trains to Mumbai or the availability of vacancies in the desired class. Quick Tip: Check the advance reservation policies of the railway system before making travel plans to ensure availability within the booking period.

The closure of schools (Statement I) and parents withdrawing their children from the local schools (Statement II) may not be directly related. Both could be effects of different independent causes, such as external factors affecting the schools or parents' decision-making process. Quick Tip: When determining the relationship between statements, check if each is a result of different independent circumstances or if they are related to each other.

Statement I refers to the increase in atrocities, which could be the result of various factors, but it doesn't directly cause Statement II.

Statement II refers to the police's inability to catch the culprits, which could be caused by various factors like lack of resources or effective strategy, but is not directly caused by the increase in crimes.


Both events are independent causes and not directly linked to each other. Quick Tip: Consider each statement in isolation and see if they are logically related or independently occurring events.

- Statement II refers to the increasing cases of respiratory diseases, which can be caused by air pollution.
- Statement I, which discusses the municipal council's aim to reduce air pollution, can be seen as an effort to address the rising health issues (respiratory diseases), making Statement II the cause and Statement I its effect. Quick Tip: Look at the flow of causality between the statements. In this case, health problems due to pollution lead to actions aimed at reducing pollution.

Statement I refers to the reduction in water-borne diseases, which could be influenced by multiple factors, such as better sanitation, improved public health measures, or seasonal variations.

Statement II refers to the government's action of opening new hospitals, which could be aimed at improving healthcare infrastructure but is not necessarily related to the reduction in diseases mentioned in Statement I.

Since both statements appear to be independent actions, they are considered independent causes. Quick Tip: When analyzing cause-effect relationships, ensure that the two events are not directly linked but could be separate actions occurring independently.

Statement I refers to the police catching house breakers, and Statement II refers to the citizens' action of starting a night vigil.

Both actions are likely the result of a common concern for safety or increasing crime in the locality. The rise in crime or insecurity might be the underlying cause of both actions. Therefore, both statements are effects of the same common cause. Quick Tip: When two actions appear to be responses to a shared issue, they may both be effects of a common cause.

The pattern in the sequence involves the numbers 2, 3, 4, 5, and 6 in the second and fourth positions, while the first and third positions (B and D) remain fixed. The missing term must logically continue the pattern, with 3 in the second position and C in the third. Thus, the missing term is \( BC3D \). Quick Tip: When identifying missing terms in sequences, look for consistent patterns in both the numeric and letter positions.

Looking at the pattern:

The first letter progresses alphabetically: Q, R, S, T, U (next in sequence is U).

The second letter progresses alphabetically: A, A, A, A, A (remains constant).

The third letter progresses alphabetically: R, S, T, U, V (next in sequence is V).


Thus, the missing term is \( UAV \). Quick Tip: Check for patterns in each position (first, second, and third letters) and identify the next logical sequence.

Total number of balls = 4 (white) + 5 (red) + 6 (blue) = 15 balls.

The number of ways to choose 3 balls from 15 = \( \binom{15}{3} = \frac{15 \times 14 \times 13}{3 \times 2 \times 1} = 455 \).

The number of ways to choose 3 red balls from 5 = \( \binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10 \).

The probability of drawing 3 red balls = \( \frac{\binom{5}{3}}{\binom{15}{3}} = \frac{10}{455} = \frac{2}{91} \).
Quick Tip: When calculating probability, use combinations to find the total possible outcomes and favorable outcomes.

In this case, the directions are being rotated. South-East becomes North, North-East becomes West, and so on. Following this pattern, West will rotate to South-East. Quick Tip: When directions are rotated, observe the pattern and identify how each direction changes to deduce the correct result.

Village Q is North of P, and R is East of Q.

S is to the left of P, which places S to the West of P.

Considering the relative position, S would be in the South-West direction relative to R.
Quick Tip: When directions are involved in a problem, use a visual representation or draw a diagram to better understand the relative positions.

- The pattern in the first pair \(8 : 28\) can be observed as \(8^2 + 4 = 28\).
- Using the same pattern for the second pair, \(27^2 + 4 = 729 + 4 = 65\).

Thus, the missing term is 65. Quick Tip: Look for patterns involving powers and simple arithmetic operations to solve such ratio problems.

- In the first pair \(BEGK : ADFJ\), notice the following letter shifts:

- B -\(>\) A (shift back by 1)

- E -\(>\) D (shift back by 1)

- G -\(>\) F (shift back by 1)

- K -\(>\) J (shift back by 1)


Applying the same shift to \(PSVY\):

- P -\(>\) O (shift back by 1)

- S -\(>\) R (shift back by 1)

- V -\(>\) U (shift back by 1)

- Y -\(>\) X (shift back by 1)


Thus, the missing term is \( ORUX \). Quick Tip: Check for consistent shifts in letters when comparing two pairs of words. The pattern of shifts will help determine the correct missing term.

1. Define Variables:
- Let \( A, B, C, D, E \) represent the number of cards each person has.

2. Translate the Statements into Equations:

- Statement 1: "If you give me three cards, you will have as many as E has."
\[ B - 3 = E \quad (1) \]
- Statement 2: "If I give you three cards, you will have as many as D has."
\[ B + 3 = D \quad (2) \]
- Statement 3: "A and B together have 10 cards more than what D and E together have."
\[ A + B = D + E + 10 \quad (3) \]
- Statement 4: "B has two cards more than what C has."
\[ B = C + 2 \quad (4) \]
- Statement 5: "The total number of cards is 133."
\[ A + B + C + D + E = 133 \quad (5) \]

3. Express All Variables in Terms of \( B \):

- From (1): \( E = B - 3 \).

- From (2): \( D = B + 3 \).

- From (4): \( C = B - 2 \).

- Substitute \( D, E, C \) into (3):

\[ A + B = (B + 3) + (B - 3) + 10 \]
Simplify:
\[ A + B = 2B + 10 \]
Solve for \( A \):
\[ A = B + 10 \]

4. Substitute All Variables into (5):

- Substitute \( A = B + 10 \), \( C = B - 2 \), \( D = B + 3 \), and \( E = B - 3 \) into (5):

\[ (B + 10) + B + (B - 2) + (B + 3) + (B - 3) = 133 \]
Simplify:
\[ 5B + 8 = 133 \]
Solve for \( B \):
\[ 5B = 125 \]
\[ B = 25 \]

Final Answer:
The number of cards \( D \) has is \boxed{25. Quick Tip: Use a system of linear equations to solve problems involving relationships between quantities, and substitute known values to find the unknowns.

1. Define Variables:

Let \( h \) be the number of horses.

Since the number of men is equal to the number of horses, the number of men is also \( h \).


2. Understand the Scenario:

Half of the owners (men) are riding their horses, and the other half are walking and leading their horses.

Riding owners: \( \frac{h}{2} \) men are on their horses. These men are not walking, so their legs are not on the ground.

Walking owners: \( \frac{h}{2} \) men are walking and leading their horses. Each walking man has 2 legs on the ground.

Horses: Each horse has 4 legs. All horses are walking, so their legs are on the ground.


3. Calculate the Total Number of Legs on the Ground:

Legs from walking men:

\[ Legs from walking men = \frac{h}{2} \times 2 = h \]
Legs from horses:

\[ Legs from horses = h \times 4 = 4h \]
Total legs on the ground:

\[ Total legs = Legs from walking men + Legs from horses = h + 4h = 5h \]

4. Set Up the Equation:

The total number of legs on the ground is given as 70:

\[ 5h = 70 \]

5. Solve for \( h \):
\[ h = \frac{70}{5} = 14 \]

Final Answer:
The number of horses is \boxed{14. Quick Tip: In problems involving legs and animals, break down the total legs by separating the legs of the people and animals, and use equations to solve.

Ram has scored 92% in the interview, which is a strong performance.

He is fluent in English and knows Hindi, both of which could be valuable language skills depending on the job requirements.

He has six years of work experience as an Associate Engineer, which shows his competency in the field.

Based on this information, he seems to meet the qualifications for the role, so the decision would be to consider selecting him.
Quick Tip: When evaluating candidates, consider their qualifications, experience, and language skills to assess their suitability for the role.

Ashalata has scored 90% in the interview, which is a solid performance.

She speaks both Telugu and Hindi, which is beneficial depending on the language requirements of the position.

She has three years of experience as an Associate Engineer and six years as a Testing Engineer, which shows a broad skill set.

Additionally, her fluency in English adds value to her profile.

Based on these qualifications, the candidate may be selected for the role. Quick Tip: Evaluate candidates based on their skills, experience, and interview performance. A strong profile like this one makes her a strong contender for selection.

Yogini has scored 90% in the interview, which is a strong performance.

She is fluent in English and knows Telugu, which can be useful depending on the job's language requirements.

She has ten years of work experience as a Software Engineer and six years as a Testing Engineer, showing her experience in the field.

However, there is not enough information provided about the specific job requirements or additional qualifications to definitively decide whether she should be selected or rejected. Therefore, the data is inadequate to take a final decision.
Quick Tip: When making a decision, consider all the specific criteria required for the role. If important information is missing, it can lead to an indecisive conclusion.

Based on the details provided, Pramod has:

- Six years of experience as an Associate Engineer.

- Nine years of experience as a Software Engineer.

- Proficiency in both Hindi and English.

- A strong interview score of 85%.


However, given that the information does not specify specific requirements for selection, such as educational qualifications or other specific skills required for the role, it is possible that the data provided does not meet the criteria needed to make a final decision on selection. Therefore, the most appropriate action would be to reject the candidate.

Thus, the correct answer is: \[ Candidate is to be rejected. \] Quick Tip: When evaluating candidates, ensure that all qualifications align with the specific job requirements. If key skills or experience are lacking, rejection may be necessary.

Varsha has scored 97% in the interview, which is an excellent score, and she is fluent in English and knows Telugu, which are valuable language skills.

She has nine years of experience as a Software Engineer and five years of experience as an Associate Engineer, demonstrating a broad skill set.

While her qualifications are strong, her age (41 years) might be a consideration in certain contexts, depending on the company's policies and requirements.

Given the available information, the decision is to refer her to the Manager for further review, as the age factor may need further consideration. Quick Tip: When evaluating candidates, consider all aspects, including qualifications, experience, and company policies. In cases where age might be a concern, it's a good idea to refer the candidate to higher authorities for further evaluation.

The figure consists of a 4x4 grid, where we need to count the number of squares of various sizes.


First, we count the \( 1 \times 1 \) squares. There are \( 4 \times 4 = 16 \) such squares.

Next, count the \( 2 \times 2 \) squares. These squares can fit in a \( 3 \times 3 \) grid, so there are \( 3 \times 3 = 9 \) such squares.

Then, count the \( 3 \times 3 \) squares. These squares can fit in a \( 2 \times 2 \) grid, so there are \( 2 \times 2 = 4 \) such squares.

Finally, there is 1 square of size \( 4 \times 4 \) (the entire grid).

Thus, the total number of squares is:
\[ 16 \, (1 \times 1) + 9 \, (2 \times 2) + 4 \, (3 \times 3) + 1 \, (4 \times 4) = 30. \] Quick Tip: When counting squares in a grid, remember to count squares of all sizes, including smaller ones and the larger squares formed by combining multiple smaller squares.

"gorblflur" means "fan belt," so "gorbl" means fan and "flur" means belt.

"pixngorbl" means "ceiling fan," so "pixn" means ceiling and "gorbl" means fan.

"arthtusl" means "tile roof," so "arth" means tile and "tusl" means roof.

To form the word for "ceiling tile," we combine "pixn" (ceiling) from "pixngorbl" and "arth" (tile) from "arthtusl."


Thus, the word for "ceiling tile" is \( pixnarth \). Quick Tip: In problems involving artificial languages, break down the words into smaller components and match them based on their meanings.

In the top row, we have a 3D object (a pyramid) and its 2D counterpart (a triangle).
In the bottom row, we have a 3D object (a cube), and we need to find its 2D counterpart.
The 2D counterpart of a cube is a square, which corresponds to option (1).

Thus, the picture for the empty box is option (1). Quick Tip: When comparing 3D and 2D shapes, identify the corresponding 2D figure that represents the 3D object. For example, a cube corresponds to a square.

Option (A) involves Priyanka notifying the landlord about not renewing the lease, which is not a violation of the lease agreement.

Option (B) describes Sharad failing to pay rent as agreed in the lease, which is a clear violation of the lease terms. This is the correct example.

Option (C) describes Uday listing complaints about the apartment, which does not violate the lease.

Option (D) involves Anna seeking legal advice but does not indicate a violation of the lease agreement.


Thus, the best example of violating an apartment lease is \( (B) \). Quick Tip: To identify lease violations, focus on the terms outlined in the lease agreement, particularly financial obligations such as rent payments and prohibited activities.

We need to select 5 persons with at least 3 men. We can do this in the following ways:


1. Select 3 men and 2 women:

Number of ways to select 3 men from 7: \( \binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \)

Number of ways to select 2 women from 6: \( \binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15 \)

Total ways for this case: \( 35 \times 15 = 525 \)


2. Select 4 men and 1 woman:

Number of ways to select 4 men from 7: \( \binom{7}{4} = \frac{7 \times 6 \times 5 \times 4}{4 \times 3 \times 2 \times 1} = 35 \)

Number of ways to select 1 woman from 6: \( \binom{6}{1} = 6 \)

Total ways for this case: \( 35 \times 6 = 210 \)


3. Select 5 men:

Number of ways to select 5 men from 7: \( \binom{7}{5} = \frac{7 \times 6}{2 \times 1} = 21 \)

Total ways for this case: \( 21 \)


Thus, the total number of ways is:
\[ 525 + 210 + 21 = 756 \]

Thus, the total number of ways to form the committee is \( 756 \). Quick Tip: When selecting persons with specific conditions, break the problem into cases and calculate each case separately, then add the results together.

1. Define Variables:

Let the two-digit number be \( 10x + y \), where:

\( x \) is the tens digit.

\( y \) is the units digit.

The ratio between the digits is \( 1:2 \), so:

\[ \frac{x}{y} = \frac{1}{2} \quad \Rightarrow \quad y = 2x \]

2. Number Obtained by Interchanging the Digits:

The number obtained by interchanging the digits is \( 10y + x \).


3. Difference Between the Two Numbers:

The difference between the original number and the interchanged number is 36:

\[ (10x + y) - (10y + x) = 36 \]
Simplify:
\[ 10x + y - 10y - x = 36 \]
\[ 9x - 9y = 36 \]
Divide through by 9:
\[ x - y = 4 \quad (1) \]

4. Substitute \( y = 2x \) into Equation (1):

\[ x - 2x = 4 \]
\[ -x = 4 \]
\[ x = -4 \]
This result is not valid because digits cannot be negative. Let’s re-examine the problem.


5. Re-evaluate the Difference:

The absolute difference between the two numbers is 36:

\[ |(10x + y) - (10y + x)| = 36 \]
Simplify:
\[ |9x - 9y| = 36 \]
\[ |x - y| = 4 \quad (2) \]

6. Use the Ratio \( y = 2x \):

Substitute \( y = 2x \) into Equation (2):

\[ |x - 2x| = 4 \]
\[ |-x| = 4 \]
\[ x = 4 \]
Since \( x \) is a digit, \( x = 4 \). Then:

\[ y = 2x = 8 \]

7. Calculate the Sum and Difference of the Digits:

Sum of the digits:

\[ x + y = 4 + 8 = 12 \]
Difference of the digits:

\[ y - x = 8 - 4 = 4 \]

8. Find the Difference Between the Sum and the Difference:

\[ (x + y) - (y - x) = 12 - 4 = 8 \]

Final Answer:
The difference between the sum and the difference of the digits is \boxed{8. Quick Tip: In problems involving ratios and digit manipulation, start by expressing relationships between the digits algebraically and use the given conditions to form equations.

To find the largest 4-digit number divisible by 88, we first divide the largest 4-digit number, 9999, by 88: \[ 9999 \div 88 = 113.625 \]
Taking the integer part of the quotient, we get 113.
Now, multiply 113 by 88 to get the largest multiple of 88 that is less than or equal to 9999: \[ 88 \times 113 = 9944 \]
Thus, the largest 4-digit number divisible by 88 is 9944. Quick Tip: To find the largest multiple of a number within a given range, divide the largest number by the divisor, round down the quotient, and multiply it back by the divisor.

Roots are always associated with trees because they are an essential part of the tree's structure, allowing it to anchor in the ground and absorb nutrients and water.

Fruits, leaves, and flowers may not always be present in every tree, as some trees may not produce fruits or flowers, and some may shed their leaves in winter.


Thus, the correct answer is \( (A) Roots \). Quick Tip: Roots are an essential part of every tree, whereas other components like fruits, leaves, or flowers may not be present in all trees.