SSC CHSL 2025 Tier-1 (Held On: 12 Nov, 2025 Shift 3) (Previous Year Paper)

The correct answer is '2' i.e. were going on .

Key Points

• The subject of the sentence is "The group of friends", which is a collective noun.
• Collective nouns are generally treated as singular unless the context clearly indicates plural usage.
• Therefore, the correct verb form here should be "was" instead of "were".

Correct sentence: The group of friends was going on a hiking trip next weekend.

Additional Information

• Collective Nouns:
• When referring to a group as a single entity, use singular verbs (e.g., "The team is winning").
• When emphasizing the individuals in the group, use plural verbs (e.g., "The team are arguing among themselves").
• Verb Agreement Rules:
• Always ensure the verb agrees with the subject in both number and tense.
• For subjects joined by "and," use plural verbs (e.g., "John and Mary are coming").

The correct answer is '4' i.e. with .

Key Points

• The given sentence emphasizes the conditions under which a treaty was signed, which involves mutual understanding and cooperation.
• The most appropriate preposition to indicate the conditions or basis of an agreement is "with".
• "With" is commonly used to describe agreements or partnerships based on shared principles or mutual understanding.

Correct sentence: The treaty was signed with mutual understanding and cooperation..

Additional Information

• Usage of Prepositions:
• "By" is used to indicate the agent or means of an action (e.g., "signed by the president").
• "Over" is used to indicate disputes or discussions (e.g., "debated over the issue").
• "Under" is used to indicate conditions or rules (e.g., "under the agreement").
• "With" is used to indicate collaboration, partnership, or mutual agreement.

The correct answer is '2' i.e. Argumentation .

Key Points

• The word "Dialectic" refers to the art or practice of logical discussion and reasoning, particularly for investigating the truth of opinions.
• Among the given options:
• Monologue: A speech by one person, often without interaction (incorrect).
• Argumentation: Logical reasoning or debate used to investigate the truth (correct).
• Impressionism: A style of art focusing on visual impressions, unrelated to logical reasoning (incorrect).
• Fantasy: Imagination-based ideas, unrelated to logical discussion (incorrect).

Correct answer: Argumentation, which aligns with the meaning of the word "Dialectic."

Additional Information

• Dialectic is often associated with debates or discussions where reasoning is used to explore contradictions and reach the truth.
• It is a key concept in philosophy, particularly in the works of thinkers like Plato and Hegel.

The correct answer is '3' i.e. Gregarious .

Key Points

• The word Gregarious means someone who is sociable, fond of company, or enjoys being in groups.
• Among the given options, option 3 is the correctly spelt word.
• Other options—Gragarous, Gragerious, and Gregareous—are incorrectly spelt and do not exist in the English vocabulary.

Correct Usage: He is a gregarious person who loves socializing and meeting new people.

Additional Information

• Etymology of Gregarious:
• The word Gregarious comes from the Latin word gregarius, meaning "belonging to a flock or herd."
• It is often used to describe animals or people who prefer living in groups.
• Similar Words: Sociable, outgoing, convivial.

The correct answer is '2' i.e. The kids ran to the park to play on the swings.

Key Points

• In the given sentence, the word "park" refers to a place (an open area with grass, trees, and recreational facilities) where people can relax or play.
• Among the options, Option 2 uses "park" with the same meaning: "The kids ran to the park to play on the swings." Here, "park" also refers to a recreational space.
• Options 1, 3, and 4 use the word "park" to refer to the act of stopping and placing a vehicle in a specific location, which is a different meaning.

Correct sentence: We decided to spend the afternoon playing in the park, and the kids ran to the park to play on the swings.

Additional Information

• The word "park" can have multiple meanings depending on the context:
• Place for recreation: An open area for public use, such as a playground or garden.
• Vehicle parking: The act of stopping and leaving a vehicle in a designated location.
• It is essential to analyze the context of the sentence to determine the correct meaning.

The correct answer is '3' i.e. Receive .

Key Points

• The word "Receive" is correctly spelt.
• The spelling rule for words like "Receive" is based on the famous mnemonic: "I before E except after C."
• Among the given options, only "Receive" follows the correct spelling rule and is an actual word used in English.

Correct spelling: Receive.

Additional Information

• Commonly Misspelled Words:
• Recieve: Incorrect because it violates the "I before E except after C" rule.
• Ricieve: Incorrect as "Ci" is not a valid combination in English spelling for this word.
• Receve: Incorrect as it omits the "i" in the correct spelling.
• Mnemonic for Spelling: Use "I before E except after C" to remember the correct spelling of words like "Receive" and "Deceive."

The correct answer is Option 2.

Key Points

• The passage explicitly identifies the role of IOs in the very first sentence: "acting as key arenas for diplomacy, mediation, and humanitarian intervention."
• It further explains that these organisations provide "neutral environments" where states can communicate and resolve conflicts without resorting to violence.
• Why other options are incorrect:
• Option 1: The text states IOs help states resolve conflicts "without resorting to violence," which is the opposite of escalating it.
• Option 3: The passage says IOs allow states to "make treaties," not prevent them.
• Option 4: The text highlights that IOs provide a platform where "states can communicate," thereby facilitating rather than limiting communication.

Therefore, the correct answer is 2.

Analytical Insight: In exams, reading comprehension questions on international relations often focus on the "functional" purpose of institutions. The keyword here is "Neutral," which is the primary value proposition of an IO in a conflict scenario.

Additional Information

• Mediation: Intervention in a dispute in order to resolve it; arbitration.
• Multilateral: Agreed upon or participated in by three or more parties, especially the governments of different countries.
• Geopolitical: Relating to politics, especially international relations, as influenced by geographical factors.
• Veto Power: The power or right vested in one branch of a government or a specific member (like the P5 in the UN) to cancel or postpone the decisions of another.

The correct answer is Option 2.

Key Points

• The passage specifically addresses post-conflict stability by stating: "In order to stabilise unstable areas, IOs regularly dispatch peacekeeping operations to monitor ceasefires, protect people, and aid in post-conflict reconstruction."
• This direct evidence confirms that the contribution involves active monitoring and rebuilding efforts rather than punitive or purely military measures.
• Why other options are incorrect:
• Option 1: The passage does not mention "harsh sanctions" or the concept of a "defeated state."
• Option 3: While IOs "dispatch peacekeeping operations," the passage emphasizes "peaceful coexistence norms" and resolving conflicts "without resorting to violence," which contrasts with promoting general "military intervention."
• Option 4: The text asserts that IOs provide environments where "States can communicate," meaning they encourage rather than discourage communication.

Therefore, the correct answer is 2.

Analytical Insight: In exams, the distinction between "Peacekeeping" and "Military Intervention" is crucial. Peacekeeping is described here as a stabilizing force (monitoring and reconstruction) rather than a primary combatant force.

Additional Information

• Post-conflict reconstruction: The process of rebuilding the social, economic, and physical infrastructure of a region after a war.
• Ceasefire: A temporary suspension of fighting; a truce.
• Multilateral actions: Collaborative efforts involving three or more states or parties.
• Interdependent: Mutually reliant on each other; a key characteristic of the modern globalized world described in the text.

The correct answer is Option 2.

Key Points

• The passage identifies several factors that hinder the effectiveness of International Organisations (IOs).
• In the final sentence, it explicitly states: "Though their potential is undermined by geopolitical competition, veto powers, and resource constraints..."
• These three elements are presented as the primary hurdles that limit the capacity of IOs to fulfill their mandates.
• Why other options are incorrect:
• Option 1: "Increased cooperation" would be a facilitator, not a challenge. The passage mentions "geopolitical competition" as the obstacle.
• Option 3: The text describes IOs as "indispensable for organising multilateral actions," suggesting they are the vehicle for such interest rather than suffering from a lack of it.
• Option 4: The passage cites "resource constraints" (limited funding/manpower), which is the exact opposite of "excessive funding."

Therefore, the correct answer is 2.

Analytical Insight: Identifying "limiting factors" is a common comprehension task. In political passages, words like "undermined," "constrained," or "hindered" are your signposts to finding the challenges or weaknesses of a system.

Additional Information

• Veto Power: The legal power to unilaterally stop an official action. In the UN Security Council, this is held by the five permanent members.
• Geopolitical: Relating to politics, especially international relations, as influenced by geographical factors.
• Indispensable: Absolutely necessary; something that cannot be replaced or done without.
• Multilateral: Involving more than two nations or parties (e.g., an agreement between multiple countries).

The correct answer is Option 2.

Key Points

• The passage directly addresses the promotion of peace by stating that IOs "serve to enforce international law and human rights, making states accountable for their behaviour and encouraging peaceful coexistence norms."
• This highlights a dual approach: accountability (ensuring states follow laws) and norm-setting (encouraging standard behaviors for peace).
• Why other options are incorrect:
• Option 1: The text mentions "peacekeeping operations" rather than "military control," focusing on monitoring and protection rather than absolute military rule.
• Option 3: The passage asserts that IOs provide environments where "States can communicate," which is the opposite of restricting it.
• Option 4: While IOs may use sanctions in reality, this specific passage does not mention economic sanctions or "rogue states."

Therefore, the correct answer is 2.

Analytical Insight: In competitive exams, the concept of "Norms" is vital. Norms are shared expectations of behavior. By "encouraging peaceful coexistence norms," IOs help create a global culture where negotiation is preferred over conflict.

Additional Information

• Accountability: The obligation of an individual or organization to account for its activities and accept responsibility for them.
• Coexistence: The state or condition of living in harmony despite different ideologies or interests.
• Interdependent: A situation where countries rely on each other for economic, political, or social stability.
• Multilateralism: The principle of participation by several nations in an action.

The correct answer is Option 2.

Key Points

• In the passage, the word "indispensable" is used to describe International Organisations: "IOs are indispensable for organising multilateral actions..."
• The context emphasizes that despite challenges like veto powers and resource constraints, these organisations are essential or absolutely necessary for confronting complex emergencies and maintaining peace.
• Why other options are incorrect:
• Option 1 (Easily replaceable): This is an antonym. The passage calls their role "irreplaceable" in the very first sentence.
• Option 3 (Optional but useful): This diminishes the weight of the word; "indispensable" implies that without them, the system would fail.
• Option 4 (Unlikely to succeed): This refers to a lack of effectiveness, whereas "indispensable" refers to the necessity of their existence and function.

Therefore, the correct answer is 2.

Analytical Insight: The prefix 'in-' often acts as a negation. "Dispense" means to do without; "dispensable" means you can do without it. Therefore, "indispensable" means you cannot do without it.

Additional Information

• Multilateral: Involving more than two nations or parties.
• Avert: To prevent or ward off an undesirable occurrence.
• Interdependent: Mutually reliant on each other.
• Mediation: Intervention in a dispute in order to resolve it.

The correct answer is '3' i.e. a pregnancy .

Key Points

• The idiom "a bun in the oven" is a figurative expression used to describe pregnancy.
• It humorously refers to the idea of a baby growing inside the mother, akin to a bun baking in an oven.
• This idiom is commonly used in informal conversations.

Correct sentence: She announced she has a bun in the oven, which means she is expecting a baby..

Additional Information

• Origin of the Idiom:
• The phrase is believed to have originated in the mid-20^th century and gained popularity for its playful imagery.
• It likens the process of pregnancy to baking, where the "bun" symbolizes the baby and the "oven" represents the mother's womb.
• Usage in Pop Culture:
• The idiom has been featured in movies, TV shows, and literature as a lighthearted way to announce pregnancy.
• It is often used during casual conversations or celebrations related to expecting a child.

The correct answer is '1' i.e. Pharmacy .

Key Points

• The question asks for a one-word substitute for "A place where you can buy medicines."
• The correct answer is Pharmacy, which refers to a store or facility where medicinal drugs are prepared, sold, or dispensed.
• The other options are incorrect:
• Supermarket: A large store selling groceries, household items, and other goods, not specifically medicines.
• Hospital: A healthcare institution providing treatment for patients, not a place for buying medicines.
• Clinic: A facility for outpatient medical care, not focused on dispensing medicines for sale.

Correct sentence: A place where you can buy medicines is called a Pharmacy.

Additional Information

• Pharmacy: Also known as a chemist or drugstore in some regions, it is a place where prescription and over-the-counter medicines are sold.
• Pharmacist: A professional who prepares, dispenses, and provides advice about medicines in a pharmacy.

The correct answer is '1' i.e. Pleasant .

Key Points

• The word vexatious means causing annoyance, frustration, or worry.
• An antonym is a word that has the opposite meaning.
• Among the options provided:
• Option 1: Pleasant - This means enjoyable and satisfying, which is opposite to vexatious.
• Option 2: Annoying - Synonym for vexatious.
• Option 3: Troublesome - Synonym for vexatious.
• Option 4: Irritating - Synonym for vexatious.
• Hence, the most appropriate antonym is Option 1: Pleasant.

Correct sentence: The most appropriate antonym of the word vexatious is Pleasant.

Additional Information

• Synonyms: Annoying, troublesome, irritating.
• Antonyms: Pleasant, enjoyable, delightful.

The correct answer is '4' i.e. Repetition .

Key Points

• The question asks for a one-word substitute for a technique used to create narrative rhythm.
• Among the given options, Repetition is the correct choice.
• Repetition is a literary device that involves repeating words, phrases, or sentences to establish rhythm and emphasize ideas in a narrative.

Correct sentence: The technique of creating narrative rhythm is known as Repetition.

Additional Information

• Personification: A figure of speech where human qualities are given to non-human entities.
• Joke: A humorous statement or story intended to entertain.
• Story: A structured narrative describing events, characters, and ideas.
• Repetition: A literary technique where elements are repeated to create rhythm or emphasize ideas.

The correct answer is '1' i.e. Our team won the championship game.

Key Points

• The given sentence is in the passive voice. Passive voice emphasizes the action or the object receiving the action rather than the subject performing it.
• To change a sentence from passive to active voice, identify the subject performing the action and place it at the beginning of the sentence.
• The structure of active voice is Subject + Verb + Object.
• In the sentence, "The championship game was won by our team," the subject performing the action is "our team," the verb is "won," and the object is "the championship game."

Correct sentence: Our team won the championship game.

Additional Information

• Passive Voice:
• Structure: Object + to be (correct tense) + past participle + by + subject.
• Example: The book was read by the teacher.
• Usage: Used when the focus is on the action or the object receiving the action.
• Active Voice:
• Structure: Subject + Verb + Object.
• Example: The teacher read the book.
• Usage: Used when the focus is on the subject performing the action.

The correct answer is '2' i.e. He will tell her that he has already finished his work .

Key Points

• In indirect speech, the tense of the verb generally changes depending on the tense of the reporting verb.
• Since the reporting verb in the sentence is in the future tense ("He will tell her"), the tense in the indirect speech remains unchanged for the present perfect tense.
• The direct speech sentence contains the present perfect tense: "I have already finished my work." In indirect speech, it remains "has already finished his work."
• Option 2 is correct because the tense and meaning align accurately with the rules of indirect speech.

Correct sentence: He will tell her that he has already finished his work.

Additional Information

• Rules for Tense Change in Indirect Speech:
• Present Perfect → Remains Present Perfect if the reporting verb is in the future tense.
• Example: "I have completed the task" → He will say that he has completed the task.
• Common Errors:
• Option 1: "Had already finished" changes the tense unnecessarily.
• Option 3: "Already finished" omits the auxiliary verb "has," making the sentence grammatically incorrect.
• Option 4: "Would have finished" incorrectly uses a conditional form not present in the original sentence.

The correct answer is '3' i.e. is starting a new business venture .

Key Points

• The subject of the sentence, "Shyamlal, the businessman and philanthropist," is singular. Therefore, the verb must agree in singular form.
• The verb "were" is incorrect because it is plural and doesn't match the singular subject.
• The correct verb form is "is starting", which agrees with the singular subject and fits the context.

Correct sentence: Shyamlal, the businessman and philanthropist, is starting a new business venture for farmers.

Additional Information

• Subject-Verb Agreement:
• Singular subjects require singular verbs, while plural subjects require plural verbs.
• Example: He is going (singular). They are going (plural).
• Verb Tense:
• The sentence is in present continuous tense, indicating an action that is happening currently.
• Structure: Subject + is/are + verb-ing.
• Example: She is reading a book.

The correct answer is '2' i.e. are similar to those of any viral fever.

Key Points

• The sentence highlights a comparison between the initial symptoms of malaria and the symptoms of a viral fever. To ensure grammatical accuracy and clarity, the comparison must be correctly structured.
• The phrase "those of" is used to refer back to the symptoms mentioned earlier in the sentence, avoiding redundancy.
• The correct option, "are similar to those of any viral fever," ensures proper comparison by matching the plural subject "symptoms" with the plural verb "are."

Correct sentence: It is difficult to diagnose malaria because its first symptoms are similar to those of any viral fever.

Additional Information

• Why other options are incorrect:
• Option 1: "are similar to all fevers" - This is overly broad and incorrect, as the sentence specifically compares malaria symptoms to viral fevers, not all fevers.
• Option 3: "is similar to those of any other viral fever" - The use of "is" is incorrect as the subject "symptoms" is plural.
• Option 4: "is similar to those of any viral fevers" - The plural "fevers" is unnecessary and ungrammatical in this context.
• Key Grammar Rule: Ensure subject-verb agreement and maintain logical clarity when making comparisons.

The correct answer is '2' i.e. An elegant and efficient code has been written by the skilled programmer to solve the intricate problem. .

Key Points

• The given sentence is in the present perfect tense ("has written").
• To convert it to passive voice, the structure of the present perfect tense changes as follows:
• Active voice: Subject + has/have + past participle + object.
• Passive voice: Object + has/have + been + past participle + by + subject.
• The subject in the active voice ("The skilled programmer") becomes the agent in the passive voice ("by the skilled programmer").
• The object in the active voice ("an elegant and efficient code") becomes the subject in the passive voice.

Correct transformation: An elegant and efficient code has been written by the skilled programmer to solve the intricate problem.

Additional Information

• Incorrect options:
• Option 1: "An elegant and efficient code was written by the skilled programmer..." uses the past tense ("was written"), which is incorrect as the original sentence is in the present perfect tense.
• Option 3: "An elegant and efficient code has been being written..." is grammatically incorrect because "has been being" is rarely used and awkward in standard English.
• Option 4: "An elegant and efficient code is written by the skilled programmer..." uses the present simple tense ("is written"), which is incorrect as the original sentence is in the present perfect tense.

The correct answer is '2' i.e. What had been done by them to ensure the project's success before the official launch?

Key Points

• The original sentence is in the active voice: What had they done to ensure the project's success before the official launch?
• In passive voice transformation, the object of the active sentence becomes the subject of the passive sentence.
• The correct form for the past perfect tense in passive voice is: had been + past participle.
• Thus, the correct transformation is: What had been done by them to ensure the project's success before the official launch?

Correct sentence: What had been done by them to ensure the project's success before the official launch?

Additional Information

• Active Voice: The subject performs the action. Example: They completed the project.
• Passive Voice: The subject receives the action. Example: The project was completed by them.
• Use the passive voice when the focus is on the action or the recipient of the action, rather than the doer.

The correct answer is '3' i.e. "The old man said, 'I have visited that small town many times in my youth.'"

Key Points

• The given sentence is in indirect speech, where the original statement has been reported rather than quoted directly.
• To convert it into direct speech, we must accurately reconstruct the speaker's original words, using appropriate pronouns and verb tenses.
• In direct speech, the verb "said" is followed by the quoted statement enclosed in quotation marks.
• The pronoun "he" in indirect speech changes to "I" in direct speech, as the old man is referring to himself.
• The verb "had visited" in indirect speech changes to "have visited" in direct speech because the reporting verb "said" is in the past tense.

Correct sentence: The old man said, "I have visited that small town many times in my youth."

Additional Information

• Direct speech: The exact words of the speaker are quoted, enclosed in quotation marks, and the tense is kept as it was originally spoken.
• Indirect speech: The speaker's words are paraphrased or reported, often involving changes in pronouns, verb tenses, and time expressions.

The correct answer is Option 2.

Key Points

• The given sentence is in indirect speech, and it needs to be converted into direct speech.
• In direct speech, the manager addresses the team directly. Therefore, "their" changes to "your", "they" changes to "you", and "he" changes to "I."
• The reported phrase "had been impressive" is restored to "has been impressive" in direct speech.
• The phrase "they had to now focus" is appropriately expressed as "you have to now focus" in the direct statement.
• Similarly, "he was confident that they would succeed" becomes "I am confident that you will succeed."

Therefore, the correct answer is Option 2.

Correct Direct Speech: "Your recent performance has been impressive, but you have to now focus on the upcoming project which is a much bigger challenge, and I am confident that you will succeed in it," the manager said to the team.

Additional Information

• Option 1: Incorrect. "Had been impressive" is changed to "was impressive," which does not accurately restore the original tense.
• Option 2: Correct. It correctly changes the pronouns and restores the reported statement to direct speech while using "have to" for the obligation expressed by "had to."
• Option 3: Incorrect. Although "must" can express obligation, the expected direct form corresponding to "had to" in this question is "have to."
• Option 4: Incorrect. It retains the third-person pronouns and the indirect-speech structure instead of converting the statement into direct speech.

The correct answer is '3' i.e. P, Q, R, S .

Key Points

• The given sentence parts describe a sequential process of refurbishing an antique table.
• Option '3' i.e., P, Q, R, S is the correct order as it logically arranges the steps in the process:
• P: Removing the old, flaking paint is the first step.
• Q: Smoothing the wood with fine sandpaper is the next step to bring out the original grain.
• R: Adding varnish and a new stain follows as the final touch for protection and aesthetics.
• S: The concluding statement summarizes the result of the refurbishment process.

Correct arrangement: P, Q, R, S.

Additional Information

• This type of question tests your ability to logically sequence ideas or events.
• Pay attention to the context and keywords in each sentence part to determine the correct chronological or logical order.
• Practice with similar questions to improve your comprehension and sequencing skills.

The correct answer is '4' i.e. Q, R, P, S .

Key Points

• The given question asks us to rearrange the sentence parts (Q, R, P, S) to form a meaningful and grammatically correct paragraph.
• Q introduces the idea of a consumer-driven society where people face an overwhelming array of options. This makes it the opening statement.
• R follows logically by introducing the term "the paradox of choice," which explains the outcome of the scenario introduced in Q.
• P elaborates on the paradox by explaining how more choices lead to anxiety and dissatisfaction.
• S concludes the paragraph by providing a broader perspective, suggesting that excessive freedom can be psychologically harmful.

Correct sequence: Q, R, P, S.

Additional Information

• Step-by-Step Approach:
• Step 1: Identify the introductory sentence that sets the context. Q serves as the introduction as it talks about a consumer-driven society with numerous choices.
• Step 2: Find the sentence that defines or explains the phenomenon mentioned in Q. R introduces the term "the paradox of choice."
• Step 3: Look for the sentence that elaborates on the phenomenon. P explains the impact of excessive choices.
• Step 4: Conclude with a sentence that provides insight or a broader perspective. S ties the idea together by discussing the psychological effects of excess freedom.

Conclusions:

I. Remote work always leads to decreased productivity in organizations. - Does not follow (The statement says productivity decreased only in certain departments of some organizations, not always.)

II. Companies benefit financially from remote work arrangements. - Follows (The statement clearly says companies reported reduced overhead costs (less office space and utilities), which means financial benefit.)

III. Hybrid work models are superior to purely remote or office-based work. - does not follow (The statement says research indicates hybrid models may offer the best, but it does not conclusively prove they are superior.)

Thus, only II follow.

Hence, the correct answer is "Option 2".

The situation requires a balanced and strategic decision. Shruti should consider both financial growth and maintaining control over her brand, which is built on personal touch and customer trust.

Let's check each option:

1. Accept immediately for financial gain - Not appropriate Course of Action (Accepting immediately ignores the risk of losing control over key decisions, which could harm the brand identity.) 2. Decline outright to retain autonomy - Not appropriate Course of Action (Not the best choice. While autonomy is preserved, it rejects a potentially beneficial growth opportunity without exploring alternatives.) 3. Negotiate for a smaller stake and partial control - Appropriate Course of Action (Negotiating allows Shruti to secure financial investment while maintaining significant decision-making power, protecting the brand’s values and customer trust.) 4. Close her business and look for a new venture - Not appropriate Course of Action (Completely inappropriate. Her business is already successful, so shutting it down makes no logical sense.)

Hence, the correct answer is "Option 3".

BODMAS Rule:

Decoding the given information:

Given equation: 10?5^6^12=?

Now replacing the sign from left to right using BODMAS Rule, we get:

⇒ 10 - 5 × 6 × 12 = ?

⇒ 10 - 360 = ?

⇒ -350 = ?

Hence, the correct answer is "Option 4".

The goal is to bring the partner back into the group while maintaining teamwork and respect.

Now, let's check each option:

1. Ignore and continue without them - Not appropriate. (As, ignoring the issue may increase their feeling of exclusion and weaken team collaboration.) 2. Blame them for overreacting - Not appropriate. (As, blaming can damage trust and worsen the conflict.) 3. Replace them silently - Not appropriate. (As, this avoids communication and can harm relationships within the volunteer group.) 4. Acknowledge their feelings, explain reasons, and invite their input for future decisions - Appropriate. (As, this approach shows empathy, transparency, and collaboration, helping rebuild trust and encouraging them to re-engage with the group.)

Hence, the correct answer is "Option 4".

Given: In a queue of 45 people, Mohan is 16^th from the front and Sohan is 12^th from the rear. 

Total people in the queue = 45

Mohan’s position from the front = 16

Sohan’s position from the rear = 12

So, Sohan’s position from the front = 45 - 12 + 1 = 46 - 12 = 34

Now, number of people between Mohan and Sohan = 34 - 16 - 1

= 34 - 17

= 17

Thus, 17 people are standing between Mohan and Sohan.

Hence, the correct answer is "Option 2".

Statement: Reports indicate that children in rural areas are dropping out of school due to lack of mid-day meals.

Course of Action I: Introduce a robust mid-day meal program in rural schools. - Follows. (As, this directly addresses the problem mentioned in the statement. Providing mid-day meals would encourage children to stay in school and reduce dropout rates.)

Course of Action II: Make education optional for children in rural areas. - Does not follow. (As, this does not solve the problem. Instead of addressing the cause (lack of meals), it may worsen the situation by allowing more children to leave school.)

Thus, only I follows.

Hence, the correct answer is "Option 1".

The position and positional values or English Alphabets are:

Given:  A, D, G, J, ?

The logic followed here is:

Thus, M will come next in the following series.

Hence, the correct answer is "Option 1".

Given: All applicants meet technical criteria, but one also shows strong teamwork and leadership. 

1. Applicant with highest test score - No (As, technical ability is already satisfied for all candidates, so this is not the best differentiator.) 2. Applicant with most degrees - No (As, more degrees do not necessarily add practical value if technical requirements are already met.) 3. Applicant with strong teamwork and leadership - Yes (As, these are additional strengths that improve collaboration, project success, and potential leadership within the team.) 4. Applicant with longest resume - No (As, a longer resume does not guarantee better performance.)

Thus, 'Applicant with strong teamwork and leadership' should be hired.

Hence, the correct answer is "Option 3".

Given Word: 'CONSTRUCTION'

Now, let's check each option one by one:

1. SUCTION → CONSTRUCTION → Can be formed.
2. COINS → CONSTRUCTION → Can be formed.
3. TION → CONSTRUCTION → Can be formed.
4. MOTION → CONSTRUCTION → Cannot be formed as 'M' is missing.

Thus, MOTION cannot be formed from the ​letters.

Hence, the correct answer is "Option 4".

The correct mirror image of the given figure is:

Hence, the correct answer is "Option 2".

Given: During a cultural festival, two community groups claim the same venue for their performances.

1. Give it to the bigger group - Does not solves the problem (As, it is unfair to the smaller group and may create resentment.) 2. Split the venue time and coordinate schedules - It solves the problem (As, this allows both groups to perform, promotes fairness, and resolves the dispute peacefully.) 3. Ask one group to cancel - Does not solves the problem (It is unfair and may discourage participation.) 4. Hold a coin toss - Does not solves the problem (As, random decisions are not ideal for organized events and may seem unprofessional.)

Thus, 'Split the venue time and coordinate schedules' is the appropriate problem solving method that can be implemented.

Hence, the correct option is “Option 2”.

Given: A school bus route is inefficient causing delays. Logical improvement?

1. Redesign route using student locations and pick optimal path - It resolves the problem (As, this directly addresses the issue by optimizing the route, reducing delays and improving efficiency.)
2. Charge students extra - It does not resolves the problem (As, this does not solve the routing problem.) 3. Keep same route to avoid change - It does not resolves the problem (As, this keeps the inefficiency and delays unchanged.) 4. Add more buses regardless of cost - It does not resolves the problem (As, this is not a logical or cost-effective solution.)

Thus, 'Redesign route using student locations and pick optimal path' resolve the problem in statement.

Hence, the correct option is “Option 1”.

The position and positional values or English Alphabets are:

The logic followed here is:

TR : RT

Similarly,

UV : ?

Thus, UV is related to VU.

Hence, the correct answer is "Option 4".

Given: 6, 15, 51, 195, ?

The logic followed here is:

Logic: (Previous number × 4) - 9 = Next number

6

⇒ 6 × 4 - 9 = 24 - 9 = 15

⇒ 15 × 4 - 9 = 60 - 9 = 51

⇒ 51 × 4 - 9 = 204 - 9 = 195

⇒ 195 × 4 - 9 = 780 - 9 = 771

Hence, the correct answer is "Option 2".

The logic followed here is:

Logic: n, (n + 1)²

1. 9, 16 → 3², (3 + 1)² = 4² = 16
2. 16, 25 → 42, (4 + 1)2 = 52 = 25
3. 25, 36 → 52, (5 + 1)2 = 62 = 36
4. 36, 64 → 62, (6 + 1)2 = 72 = 49 ≠ 64

Thus, '36,64' is the odd one out.

Hence, the correct answer is "Option 4".

The paper after unfolded will be:

Hence, the correct answer is "Option 1".

The image embedded in given figure is as shown below:

Hence, the correct answer is "Option 1".

The logic followed here is:

In a row: (Sum of first three numbers) × 2 = Fourth number

Row 1: (7 + 6 + 3) × 2 → 16 × 2 = 32

Row 2: (9 + 5 + 8) × 2 → 22 × 2 = 44

Similarly,

Row 3: (11 + ? + 3) × 2 → (14 + ?) × 2 = 40

⇒ 28 + 2 × ? = 40

⇒ 2 × ? = 40 - 28

⇒ 2 × ? = 12

⇒ ? = 12 ÷ 2

⇒ ? = 6

Hence, the correct answer is "Option 1".

According to the given Venn diagram:

The shaded area show the total number of cows that are neither dogs nor cats.

So, total number of cows that are neither dogs nor cats = 10.

Hence, the correct answer is "Option 3".

Given: Engineer, Doctor, Lawyer

Hence, the correct answer is "Option 1".

The correct order of words as they would appear in the English dictionary is:

1.Archaeology

2.Archeologist

3.Archipelago

4. Architectural

5. Architecture

The correct order is '1, 2, 3, 4, 5'.

Hence, the correct answer is "Option 1".

The image that completes the pattern is:

Hence, the correct answer is "Option 1".

The logic followed here is:

1. 

 → The middle-most element Triangle in the first image shifts to the centre in the second image.

2. 

 → The middle-most element Quadrilateral in the first image shifts to the centre in the second image.

3.  

 → The middle-most element Rectangle in the first image shifts to the centre in the second image, but the triangle in first image is missing in second image.

4. 

→ The middle-most element Triangle in the first image shifts to the centre in the second image.

Thus, option 3 image is the odd one out.

Hence, the correct answer is "Option 3".

The logic followed here is:

Element with arrow-head is shifting clockwise.

Element with semi-circle is shifting anti-clockwise.

Note: Each element is rotating by an angle of 180 degree everytime they shift their position.

Hence, the correct answer is "Option 4".

The position and positional values or English Alphabets are:

The logic followed here is:

'AWLS' is coded as 'EBPX'

'ARCS' is coded as 'EWGX'

Similarly,

Thus, 'BATS' is coded as 'FFXX'.

Hence, the correct answer is "Option 3".

Shortcut Trick

Investment Ratio = (Profit Ratio) ÷ (Time Ratio)

Investment Ratio = (6/14) : (8/8) : (9/7)

Investment Ratio = (3/7) : 1 : (9/7)

Multiply the entire ratio by 7 to simplify: 3 : 7 : 9

∴ The correct answer is 3:7:9.

Alternate Method

Given:

Profit ratio of three partners = 6 : 8 : 9

Time of investment for each partner = 14 months, 8 months, and 7 months

Formula Used:

Profit = Investment × Time

⇒ Investment = Profit ÷ Time

Let the investments of the three partners be I₁, I₂, and I₃ respectively.

⇒ I₁ : I₂ : I₃ = (6/14) : (8/8) : (9/7)

⇒ I₁ : I₂ : I₃ = (3/7) : 1 : (9/7)

To eliminate the fractions, multiply the entire ratio by the LCM of denominators (7, 1, 7), which is 7:

⇒ I₁ : I₂ : I₃ = (3/7 × 7) : (1 × 7) : (9/7 × 7)

⇒ I₁ : I₂ : I₃ = 3 : 7 : 9

∴ The correct answer is 3:7:9.

Additional Information

Partnership Basic Rule

The profit earned in a business is distributed among partners in the ratio of the product of their capital and time duration (Profit = C × T).

Ratio of Capital/Investment

When the profit ratio and time ratio are given, the investment ratio is calculated as (P₁/T₁) : (P₂/T₂) : (P₃/T₃).

Working vs Sleeping Partner

A working partner manages the business and may receive a salary or extra profit share, while a sleeping partner only invests money.

Shortcut Trick

Combine the Pythagorean identity terms: sin² x + cos² x = 1.

Substitute this into the expression to get 1 + tan² x.

Recognize that 1 + tan² x is the standard identity for sec² x.

∴ The correct answer is sec² x.

Alternate Method

Given: sin² x + cos² x + tan² x

Formula Used:

1. sin² θ + cos² θ = 1
2. 1 + tan² θ = sec² θ

⇒ Expression = (sin² x + cos² x) + tan² x

⇒ Expression = 1 + tan² x

⇒ Expression = sec² x

∴ The correct answer is sec² x.

Additional Information

Primary Pythagorean Identity

The most fundamental identity in trigonometry is sin² θ + cos² θ = 1, which relates the sine and cosine of any angle.

Secant-Tangent Relationship

The identity sec² θ − tan² θ = 1 is derived by dividing the primary identity by cos² θ.

Cosecant-Cotangent Relationship

Similarly, dividing the primary identity by sin² θ yields the identity 1 + cot² θ = cosec² θ.

Given:

Profit to cost price ratio = 3:8

Formula Used:

Selling Price (SP) = Cost Price (CP) + Profit

Profit = (Profit to CP ratio) × CP

SP to CP ratio = SP ÷ CP

Calculation:

⇒ Profit = (3/8) × CP

⇒ SP = CP + Profit

⇒ SP = CP + (3/8) × CP

⇒ SP = (1 + 3/8) × CP

⇒ SP = (8/8 + 3/8) × CP

⇒ SP = (11/8) × CP

⇒ SP to CP ratio = 11:8

∴ The correct answer is 11:8.

Shortcut Trick

Ratio of apples to oranges = 7 : 9

According to the question, 7 units = 63 apples.

1 unit = 63 ÷ 7 = 9

Number of oranges = 9 units = 9 × 9 = 81

∴ The correct answer is 81.

Alternate Method

Given: Ratio of apples to oranges in a basket = 7 : 9

Given: Total number of apples = 63

⇒ Let the number of apples be 7x and the number of oranges be 9x.

⇒ Since the number of apples is 63, we can write:

⇒ 7x = 63

⇒ x = 63 ÷ 7

⇒ x = 9

⇒ Now, number of oranges = 9x

⇒ Number of oranges = 9 × 9 = 81

∴ The correct answer is 81.

Additional Information

Direct Proportion

Two quantities are in direct proportion if the ratio of their values remains constant. If x ∝ y, then x₁/y₁ = x₂/y₂.

Simplification of Ratios

A ratio a : b can be simplified by dividing both terms by their Highest Common Factor (HCF). For example, 63 : 81 = (63/9) : (81/9) = 7 : 9.

Compound Ratio

The compound ratio of two ratios a : b and c : d is defined as (a × c) : (b × d).

Given:

Difference between Compound Interest (CI) and Simple Interest (SI) = ₹360

Time = 2 years

Rate of Interest = 15% p.a.

Formula Used:

CI − SI = P × (r/100)²

Where P = Principal, r = Rate of Interest

Calculation:

⇒ 360 = P × (15/100)²

⇒ 360 = P × (225/10000)

⇒ P = 360 ÷ (225/10000)

⇒ P = 360 × (10000/225)

⇒ P = ₹16,000

∴ The correct answer is ₹16,000.

Given:

Principal = 200000

Rate of interest = 10%

Amount withdrawn after 1 year = 20000

Total time = 2 years

Formula Used:

Compound amount = Principal × (1 + Rate ÷ 100)

Calculation:

⇒ Amount after 1 year = 200000 × (1 + 10 ÷ 100)

⇒ Amount after 1 year = 200000 × 1.10

⇒ Amount after 1 year = 220000

⇒ Balance after withdrawal = 220000 − 20000

⇒ Balance = 200000

⇒ Amount after 2^nd year = 200000 × (1 + 10 ÷ 100)

⇒ Amount after 2^nd year = 200000 × 1.10

⇒ Amount after 2^nd year = 220000

∴ The correct answer is 220000.

Given:

Average of 50 numbers = 38

Wrongly taken numbers = 33 and 27

Correct numbers = 23 and 17

Total numbers = 50

Formula Used:

Total sum of numbers = Average × Total numbers

Corrected total sum = Original total sum - Wrong values + Correct values

Corrected average = Corrected total sum ÷ Total numbers

Calculation:

⇒ Total sum of numbers = 38 × 50 = 1900

⇒ Corrected total sum = 1900 - (33 + 27) + (23 + 17)

⇒ Corrected total sum = 1900 - 60 + 40

⇒ Corrected total sum = 1880

⇒ Corrected average = 1880 ÷ 50

⇒ Corrected average = 37.6

∴ The correct answer is 37.6.

Given:

First number = 13511

Second number = 13903

Third number = 14589

Same divisor = n

Same remainder = m

Formula Used:

If three numbers leave the same remainder when divided by a number, then that divisor is the H.C.F. of the pairwise differences of the numbers.

Remainder = Number mod Divisor

Calculation:

⇒ 13903 − 13511 = 392

⇒ 14589 − 13903 = 686

⇒ 14589 − 13511 = 1078

⇒ H.C.F. of 392 and 686

⇒ 686 = 392 × 1 + 294

⇒ 392 = 294 × 1 + 98

⇒ 294 = 98 × 3 + 0

⇒ H.C.F. = 98

⇒ n = 98

⇒ m = 13511 mod 98

⇒ 13511 = 98 × 137 + 85

⇒ m = 85

⇒ n + m = 98 + 85

⇒ n + m = 183

∴ The correct answer is 183.

Given:

Price increases by 20% in first year

Price decreases by 10% in second year

Final price = 59400

Formula Used:

New price = Original price × (1 + Increase%) × (1 − Decrease%)

Calculation:

⇒ Let original price = x

⇒ After 20% increase = x × 1.20

⇒ After 10% decrease = x × 1.20 × 0.90

⇒ Final price = x × 1.08

⇒ x × 1.08 = 59400

⇒ x = 59400 ÷ 1.08

⇒ x = 55000

∴ The correct answer is 55000.

Given:

Speed while going = 30 km/h

Speed while returning = 20 km/h

Total time = 5 hours

Formula Used:

Time = Distance ÷ Speed

Total time = Time going + Time returning

Calculation:

⇒ Let one-way distance = x

⇒ Time going = x ÷ 30

⇒ Time returning = x ÷ 20

⇒ x ÷ 30 + x ÷ 20 = 5

⇒ (2x + 3x) ÷ 60 = 5

⇒ 5x ÷ 60 = 5

⇒ x ÷ 12 = 5

⇒ x = 60

∴ The correct answer is 60.

Shortcut Trick

If A and B together take x days and B alone takes y days, then A alone takes:

Time = (x × y) ÷ (y − x)

Time taken by A = (10 × 15) ÷ (15 − 10)

Time taken by A = 150 ÷ 5 = 30 days

∴ The correct answer is 30 days.

Alternate Method

Given:

Time taken by A and B together = 10 days

Time taken by B alone = 15 days

Formula Used:

Total Work = LCM of individual days

Efficiency = Total Work ÷ Time Taken

Calculations:

⇒ Total work (LCM of 10 and 15) = 30 units

⇒ Efficiency of A + B = 3 units/day

⇒ Efficiency of B = 2 units/day

⇒ Efficiency of A = (Efficiency of A + B) − (Efficiency of B)

⇒ Efficiency of A = 3 − 2 = 1 unit/day

⇒ Time taken by A alone = Total Work ÷ Efficiency of A

⇒ Time taken by A = 30 ÷ 1 = 30 days

∴ The correct answer is 30 days.

Additional Information

Pipes and Cisterns

The concept is identical to time and work; filling pipes act like positive work, and drainage pipes (leaks) act like negative work.

Work and Wages

Wages are always distributed among workers in the direct ratio of their efficiencies or the share of work done by each.

Efficiency-Time Relation

Efficiency is inversely proportional to the time taken to complete the work (Efficiency ∝ 1 ÷ Time).

Given:

AB ∥ CD

AB = 2CD

Area of $\Delta COD$ = 8

Formula Used:

In a trapezium, diagonals divide each other in the ratio of the parallel sides

Area of triangles with same altitude is proportional to their bases

Calculation:

⇒ AB : CD = 2 : 1

⇒ AO : OC = 2 : 1

⇒ BO : OD = 2 : 1

⇒ Area($\Delta AOB$) : Area($\Delta COD$) = (AB / CD)²

⇒ Area ratio = 2² : 1

⇒ Area ratio = 4 : 1

⇒ Area($\Delta AOB$) = 4 × 8

⇒ Area($\Delta AOB$) = 32

∴ The correct answer is 32.

Given:

Discount offered = 10%

Selling price after discount for two books = 540

Profit after discount = 20%

Formula Used:

Selling Price = Cost Price × (1 + Profit%)

Discounted price = Marked Price × (1 − Discount%)

Calculation:

⇒ 540 = 120% of Cost Price

⇒ Cost Price = 540 × 100 ÷ 120

⇒ Cost Price = 450

⇒ 540 = 90% of Marked Price

⇒ Marked Price = 540 × 100 ÷ 90

⇒ Marked Price = 600

⇒ Profit without discount = 600 − 450

⇒ Profit = 150

∴ The correct answer is 150.

Given:

Simple interest = Principal ÷ 4

Number of years = Rate of interest

Formula Used:

Simple Interest = (Principal × Rate × Time) ÷ 100

Calculation:

⇒ Let Principal = P

⇒ Rate = r

⇒ Time = r

⇒ Simple Interest = (P × r × r) ÷ 100

⇒ P ÷ 4 = (P × r²) ÷ 100

⇒ 1 ÷ 4 = r² ÷ 100

⇒ r² = 25

⇒ r = 5

∴ The correct answer is 5%.

Given:

Length of sheet = 44 cm

Height of cylinder = 10 cm

Formula Used:

Circumference of base = 2 × π × r

Volume of cylinder = π × r² × h

Calculation:

⇒ 2 × (22 ÷ 7) × r = 44

⇒ (44 ÷ 7) × r = 44

⇒ r = 7

⇒ Volume = (22 ÷ 7) × 7² × 10

⇒ Volume = (22 ÷ 7) × 49 × 10

⇒ Volume = 22 × 7 × 10

⇒ Volume = 1540

∴ The correct answer is 1540.

Shortcut Trick

For two-digit numbers ab and cd, the property (ab)² + (cd)² = (ba)² + (dc)² holds true if a² + c² = b² + d².

In the given expression 14² + 87²: a = 1, b = 4, c = 8, d = 7.

Check the property: 1² + 8² = 1 + 64 = 65 and 4² + 7² = 16 + 49 = 65.

Since the sums are equal, the reversal of digits will yield the same value: 14 → 41 and 87 → 78.

∴ The correct answer is 41² + 78².

Alternate Method

Given:

Expression = 14² + 87²

Calculations:

⇒ 14² = 196

⇒ 87² = (90 − 3)² = 8100 − 540 + 9 = 7569

⇒ Value = 196 + 7569 = 7765

Evaluating Option (2):

⇒ 41² + 78²

⇒ 41² = 1681

⇒ 78² = (80 − 2)² = 6400 − 320 + 4 = 6084

⇒ Value = 1681 + 6084 = 7765

⇒ Since both sums are equal to 7765, the expressions are identical.

∴ The correct answer is 41² + 78².

Additional Information

Digit Reversal Square Property

A sum of two squares equals the sum of the squares of their reversed digits only when the sum of the squares of the first digits equals the sum of the squares of the second digits.

Algebraic Identity for Squaring

Use (a ± b)² = a² ± 2ab + b² for quick calculation of squares near base values like 10, 50, or 100.

Numbers ending in 1

To square a number ending in 1 (like 41), use (x+1)² = x² + 2x + 1. For example, 41² = 40² + 2(40) + 1 = 1600 + 80 + 1 = 1681.

Given:

Total alloy = 40 kg

Initial Zinc percentage = 10%

Required Zinc percentage = 20%

Formula Used:

Quantity of element = Total mixture × Percentage ÷ 100

Calculation:

⇒ Initial Zinc = 40 × 10 ÷ 100

⇒ Initial Zinc = 4

⇒ Let Zinc added = x

⇒ New total mixture = 40 + x

⇒ Total Zinc = 4 + x

⇒ $\frac{4 + x}{40 + x} = \frac{20}{100}$

⇒ $\frac{4 + x}{40 + x} = \frac{1}{5}$

⇒ $5(4 + x) = 40 + x$

⇒ $20 + 5x = 40 + x$

⇒ $5x - x = 40 - 20$

⇒ $4x = 20$

⇒ $x = 5$

∴ The correct answer is 5 kg.

Given:

Inner radius (r₁) = 12 cm

Outer radius (r₂) = 13 cm

Formula Used:

Volume of a hemisphere = (2/3)πr³

Volume of the shell = Volume of outer hemisphere - Volume of inner hemisphere

Calculation:

⇒ Volume of outer hemisphere = (2/3)π(13)³

⇒ Volume of outer hemisphere = (2/3)π × 2197

⇒ Volume of outer hemisphere = (4394/3)π

⇒ Volume of inner hemisphere = (2/3)π(12)³

⇒ Volume of inner hemisphere = (2/3)π × 1728

⇒ Volume of inner hemisphere = (3456/3)π

⇒ Volume of the shell = Volume of outer hemisphere - Volume of inner hemisphere

⇒ Volume of the shell = (4394/3)π - (3456/3)π

⇒ Volume of the shell = (938/3)π

∴ The correct answer is (938/3)π .

Given:

Initial filled part of tank = $\frac{3}{5}$

Final filled part of tank = $\frac{7}{10}$

Water added = 20

Formula Used:

Added water = Total capacity × (Final fraction − Initial fraction)

Calculation:

⇒ Increase in fraction = $\frac{7}{10} - \frac{3}{5}$

⇒ Increase in fraction = $\frac{7}{10} - \frac{6}{10}$

⇒ Increase in fraction = $\frac{1}{10}$

⇒ $\frac{1}{10}$ of capacity = 20

⇒ Capacity = 20 × 10

⇒ Capacity = 200

∴ The correct answer is 200 liters.

Given:

Coordinates of vertex A = (2, 3)

Coordinates of vertex B = (4, 5)

Coordinates of vertex C = (6, 7)

Formula Used:

Centroid (G) = ((x₁ + x₂ + x₃) ÷ 3, (y₁ + y₂ + y₃) ÷ 3)

Calculation:

⇒ x-coordinate of centroid = (2 + 4 + 6) ÷ 3

⇒ x-coordinate of centroid = 12 ÷ 3

⇒ x-coordinate of centroid = 4

⇒ y-coordinate of centroid = (3 + 5 + 7) ÷ 3

⇒ y-coordinate of centroid = 15 ÷ 3

⇒ y-coordinate of centroid = 5

⇒ Centroid (G) = (4, 5)

∴ The correct answer is (4, 5).

Given:

Ratio of the perimeters of two similar triangles = a : b

Formula Used:

Ratio of areas of two similar triangles = (Ratio of their perimeters)²

Calculation:

⇒ Ratio of areas = (a : b)²

⇒ Ratio of areas = a² : b²

∴ The correct answer is a² : b².

Given:

$\frac{4}{\sqrt{7} + \sqrt{5}}$

Formula Used:

$(a+b)(a-b) = a^2 - b^2$

Calculation:

⇒ $\frac{4}{\sqrt{7} + \sqrt{5}}$

⇒ $\frac{4}{\sqrt{7} + \sqrt{5}} \times \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} - \sqrt{5}}$

⇒ $\frac{4(\sqrt{7} - \sqrt{5})}{(\sqrt{7} + \sqrt{5})(\sqrt{7} - \sqrt{5})}$

⇒ $\frac{4(\sqrt{7} - \sqrt{5})}{7 - 5}$

⇒ $\frac{4(\sqrt{7} - \sqrt{5})}{2}$

⇒ $2(\sqrt{7} - \sqrt{5})$

∴ The correct answer is $2(\sqrt{7} - \sqrt{5})$.

Given:

$\dfrac{(x+y)^3+(y+z)^3+(z+x)^3-3(x+y)(y+z)(z+x)}{x^3+y^3+z^3-3xyz}$

Formula Used:

$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$

Calculation:

⇒ $(x+y)^3+(y+z)^3+(z+x)^3-3(x+y)(y+z)(z+x)$

⇒ $=[(x+y)+(y+z)+(z+x)]{(x+y)^2+(y+z)^2+(z+x)^2-(x+y)(y+z)-(y+z)(z+x)-(z+x)(x+y)}$

⇒ $=2(x+y+z){(x+y)^2+(y+z)^2+(z+x)^2-(x+y)(y+z)-(y+z)(z+x)-(z+x)(x+y)}$

⇒ $=(2x+2y+2z)(x^2+y^2+z^2-xy-yz-zx)$

⇒ $x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)$

⇒ $\dfrac{(2x+2y+2z)(x^2+y^2+z^2-xy-yz-zx)}{(x+y+z)(x^2+y^2+z^2-xy-yz-zx)}$

⇒ $\dfrac{2(x+y+z)}{(x+y+z)}$

⇒ $2$

∴ The correct answer is $2$.

Shortcut Trick

Apply the algebraic identity a² − b² = (a + b)(a − b) to the expression.

sin⁴(x) − cos⁴(x) = (sin²(x) + cos²(x)) × (sin²(x) − cos²(x)).

Using the Pythagorean identity, sin²(x) + cos²(x) = 1.

The expression simplifies directly to 1 × (sin²(x) − cos²(x)), which is sin²(x) − cos2(x).

∴ The correct answer is sin²(x) − cos2(x).

Alternate Method

Given: sin⁴(x) − cos⁴(x)

Formula Used: a² − b² = (a − b)(a + b) and sin²(x) + cos²(x) = 1

⇒ sin⁴(x) − cos⁴(x) = (sin²(x))² − (cos²(x))²

⇒ (sin²(x) + cos²(x)) × (sin²(x) − cos²(x))

⇒ 1 × (sin²(x) − cos²(x))

⇒ sin²(x) − cos²(x)

Comparing the result with the given choices, the expression is represented as:

⇒ sin²(x) − cos2(x)

∴ The correct answer is sin²(x) − cos2(x).

Additional Information

Pythagorean Identities

The most fundamental identity is sin²(x) + cos²(x) = 1. Other variations include 1 + tan²(x) = sec²(x) and 1 + cot²(x) = cosec²(x).

Double Angle Formulas

The formula for cos(2x) can be written in three ways: cos²(x) − sin²(x), 2cos²(x) − 1, or 1 − 2sin²(x).

Algebraic Decomposition

Higher-order powers like a⁴ − b⁴ are simplified by treating them as (a²)² − (b²)², allowing the use of the difference of squares method.

Given:

Point 1: (1, 3)

Point 2: (4, 9)

Slope-intercept form: y = mx + c

Formula Used:

Slope (m) = (y₂ - y₁) ÷ (x₂ - x₁)

To find c: y = mx + c

Calculation:

⇒ Slope (m) = (9 - 3) ÷ (4 - 1)

⇒ Slope (m) = 6 ÷ 3

⇒ Slope (m) = 2

Using point (1, 3):

⇒ 3 = 2(1) + c

⇒ 3 = 2 + c

⇒ c = 3 - 2

⇒ c = 1

Equation of the line: y = 2x + 1

∴ The correct answer is y = 2x + 1.

The correct answer is Article 39 A.

Key Points

• Article 39A of the Indian Constitution directs the state to ensure that the legal system promotes justice on the basis of equal opportunity and to provide free legal aid to economically weaker sections of society.
• This article was introduced by the 42^nd Amendment Act of 1976 under the Directive Principles of State Policy (DPSP).
• It mandates the state to ensure that no citizen is denied access to justice due to economic or other disabilities.
• The provision emphasizes the importance of creating a fair and equitable judicial system accessible to all, irrespective of their financial status.
• The implementation of free legal aid services has been institutionalized through the Legal Services Authorities Act, 1987, which established National Legal Services Authority (NALSA) to oversee legal aid programs.

Additional Information

• Directive Principles of State Policy (DPSP):
• DPSPs are non-justiciable rights enshrined in Part IV of the Indian Constitution, aimed at guiding the state in governance and policy-making to establish a welfare state.
• These principles are inspired by the Irish Constitution and aim to promote social and economic justice.
• Though not enforceable in a court of law, they are fundamental in the governance of the country and are intended to be implemented by legislative and executive actions.
• Legal Services Authorities Act, 1987:
• This act was enacted to provide free and competent legal services to weaker sections of society and to ensure justice for all.
• It led to the creation of institutions like National Legal Services Authority (NALSA) at the national level and State Legal Services Authorities (SLSA) at the state level.
• These institutions provide free legal aid, conduct Lok Adalats, and spread legal awareness.
• Article 38:
• Article 38 directs the state to strive to promote the welfare of the people by securing and protecting a social order based on justice, social, economic, and political.
• It focuses on minimizing inequalities in income, status, facilities, and opportunities among individuals and groups.
• Other Related Articles:
• Article 45: Originally directed the state to provide free and compulsory education for children below the age of 14; later amended to focus on early childhood care and education for children below six years.
• Article 42: Directs the state to make provisions for securing just and humane conditions of work and maternity relief.

Important Points

• Role of NALSA:
• NALSA was constituted under the Legal Services Authorities Act, 1987, and works to provide free legal services to the weaker sections of society.
• It organizes Lok Adalats to resolve disputes through alternative dispute resolution mechanisms.
• NALSA also promotes legal awareness through campaigns and programs at the grassroots level.
• Significance of Article 39A:
• It is a cornerstone of India's commitment to ensuring justice for all.
• It reflects the constitutional vision of building an inclusive judicial system that does not discriminate against marginalized groups.
• Free legal aid under this article is a step toward achieving the goal of social and economic justice.
• Global Perspective:
• Many countries have similar provisions to ensure access to justice, such as the Legal Aid, Sentencing and Punishment of Offenders Act, 2012 in the UK.
• The Universal Declaration of Human Rights (1948) also emphasizes the right to access justice without discrimination.