SSC CHSL 2025 Tier-1 (Held On: 15 Nov, 2025 Shift 1) (Previous Year Paper)

The correct answer is Option 4.

 

Key Points

• The issue in the sentence lies in part D: "and entertain."
• The word "entertain" is incorrectly used in this context.
• The correct form should be "entertaining" to match the grammatical structure of the sentence.
• Thus, the error is located in part D of the sentence.

Therefore, the correct answer is Option 4.

 

Correct Sentence: "The movie, which you recommended, was both interesting and entertaining."

 

Additional Information

• A: "The movie," – This part of the sentence is grammatically correct and does not require any changes.
• B: "which you recommended," – This part of the sentence is grammatically correct and does not require any changes.
• C: "was both interesting" – This part of the sentence is grammatically correct and does not require any changes.

The correct answer is: Option 3 (Repeal).

Key Points

• The word "Abrogate" means to formally put an end to a law, right, or agreement; to annul or revoke something officially. (अधिनियम रद्द करना)
• Example: The government decided to abrogate the outdated law that no longer served its purpose.
• "Repeal" is the correct synonym for "abrogate." It means to officially cancel or revoke a law, regulation, or act. (निरसन करना)
• Example: Parliament voted to repeal the controversial tax law after nationwide protests.
• Hence, we can conclude that the most appropriate synonym of "Abrogate" is "Repeal".

Therefore, the correct answer is: Option 3 (Repeal).

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Institute (संस्थापित करना): To establish, organize, or introduce something, typically a system, rule, or law.
• Example: The school decided to institute a new dress code policy.
• Enact (अधिनियमित करना): To make a bill or proposal into a law or to perform a role in a play.
• Example: The government plans to enact legislation to protect endangered species.
• Approve (अनुमोदन करना): To officially agree to or accept something.
• Example: The board approved the proposal to expand the company’s operations overseas.

The correct answer is 'hand over'.

 

Key Points

• The phrasal verb "hand over" means to pass or transfer control or responsibility to someone else.
• In the context of the sentence, the assistant is being given some responsibilities, making "hand over" the most appropriate option.
• Options 1, 2, and 4 do not fit the context of transferring responsibilities.

Therefore, the correct answer is Option 3.

 

Complete Sentence: "She was so overwhelmed with tasks that she had to hand over some responsibilities to her assistant."

 

Additional Information

• Option 1: "hand out" is incorrect as it refers to distributing items, not transferring responsibilities.
• Option 2: "hand down" is incorrect as it implies passing something to someone of the next generation, not delegating tasks.
• Option 4: "hand in" is incorrect as it means submitting something, not delegating responsibilities.

The correct answer is 'on a shoestring'.

 

Key Points

• The phrase "on a shoestring" means operating with a very limited budget or resources.
• The sentence refers to budget cuts, implying that the organization had to manage with minimal resources, which aligns with the meaning of "on a shoestring."
• Options 1, 3, and 4 are not relevant in this context as they do not convey the idea of limited budget or resources.

Therefore, the correct answer is Option 2.

 

Complete Sentence: "Due to the budget cuts, the organization had to operate strictly on a shoestring."

 

Additional Information

• Option 1: "in a nutshell" is incorrect because it means summarizing something briefly, which does not fit the context.
• Option 3: "with flying colors" is incorrect because it refers to achieving something with great success, which is unrelated to the idea of operating on a limited budget.
• Option 4: "under the radar" is incorrect because it means doing something secretly or unnoticed, which does not align with the sentence context.

The correct answer is: Option 2 (Knowledge-related).

Key Points

• The word "Epistemic" relates to knowledge or the study of knowledge. It is often used in contexts involving understanding, justification, and the nature of knowledge. (ज्ञान या ज्ञान के अध्ययन से संबंधित)
• Example: The discussion about scientific theories often involves epistemic questions about how we can know what is true.
• "Knowledge-related" refers to anything that is directly associated with or pertains to the concept of knowledge. (ज्ञान-संबंधित)
• Example: His research is primarily focused on knowledge-related issues in artificial intelligence.
• Hence, we can infer that the synonym of 'Epistemic' is 'Knowledge-related'.

Therefore, the correct answer is: Option 2 (Knowledge-related).

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Physical (भौतिक): Related to the body or physical existence, as opposed to mental or spiritual aspects.
• Example: Physical fitness plays an essential role in overall health.
• Symbolic (प्रतीकात्मक): Representing something else, especially an idea, concept, or quality.
• Example: The dove is often used as a symbolic representation of peace.
• Spatial (स्थानिक): Relating to space and the relationship of objects within it.
• Example: Architects need a strong understanding of spatial awareness to design functional buildings.

The correct answer is Option 2: Circle.

 

Key Points

• The term "mandala" originates from Sanskrit, meaning "circle."
• Mandalas are often used in spiritual and ritual contexts, symbolizing wholeness, balance, and unity.
• They are significant in Hinduism and Buddhism, representing the universe and various spiritual concepts.
• Mandalas are frequently used as tools for meditation and introspection.

Therefore, the correct answer is Option 2.

 

Additional Information

• Mandalas in Art: Mandalas are often depicted as intricate geometric patterns in visual art.
• Therapeutic Use: Mandalas are used in art therapy to encourage mindfulness and self-expression.
• Global Influence: The concept of mandalas has influenced cultures worldwide, extending beyond its original religious contexts.

The correct answer is Option 2: One’s identity or a heavenly essence.

 

Key Points

• The center of a mandala represents one’s identity or a heavenly essence, which is often symbolic of the self or a spiritual core.
• Mandalas are spiritual and ritual symbols in Hinduism and Buddhism, representing the universe and inner harmony.
• The intricate designs radiating from the center symbolize the connectedness of all life and the cosmos.
• By focusing on the center, one can meditate and reflect on their inner self or spiritual journey.

Therefore, the correct answer is Option 2.

 

Explanation: The center of the mandala is often considered the most sacred and significant part, symbolizing the essence of being or the universe's soul. It is a focal point for meditation and self-awareness.

 

Additional Information

• The universe – While the mandala as a whole may represent the universe, the center specifically signifies one’s identity or spiritual core.
• Earthly possessions – Mandalas are not typically associated with materialistic or worldly possessions.
• The continuous flow of life – The design of the mandala as a whole may symbolize this, but the center specifically pertains to identity or spiritual essence.

The correct answer is Option 2: To symbolize the fleeting nature of life.

 

Key Points

• Monks create mandalas out of colored sand to symbolize the fleeting nature of life.
• Once the intricate artwork is completed, it is typically destroyed to demonstrate the impermanence of existence.
• This practice aligns with Buddhist teachings about the transient and ever-changing nature of life.

Therefore, the correct answer is Option 2.

 

Additional Information

• Creating mandalas is also a meditative process that helps monks focus and attain mindfulness during the activity.
• The destruction of the mandala reminds individuals to detach from material possessions and ego-driven desires.
• This tradition has become a cultural and spiritual practice recognized worldwide.

The correct answer is Option 3: It helps reduce tension, and calm the mind.

 

Key Points

• Creating a mandala has therapeutic benefits, particularly in helping to reduce tension and calming the mind.
• The practice of making mandalas is often used in art therapy and mindfulness exercises.
• It promotes relaxation, self-expression, and mental clarity.
• Mandalas are often associated with meditation and spirituality, helping individuals focus and center their thoughts.

Therefore, the correct answer is Option 3.

 

Explanation: Creating a mandala does not directly improve physical health, increase artistic creativity, or help build wealth. However, it is a powerful tool for reducing mental tension and achieving calmness.

 

Additional Information

• Mandalas have been used in various cultures for centuries as symbols of wholeness, balance, and harmony.
• They are often incorporated in meditative practices to aid in stress relief and focus.
• Artistic creation of mandalas can also be a satisfying and reflective activity, even for individuals who do not consider themselves artists.

The correct answer is Option 2: To symbolize the cosmos and the individual’s journey.

 

Key Points

• In Buddhist and Hindu traditions, mandalas are used to symbolize the cosmos and the individual’s spiritual journey toward enlightenment.
• They are intricate geometric designs that represent balance, unity, and harmony.
• Mandalas are often used in meditation practices and rituals to focus the mind and aid in spiritual growth.

Therefore, the correct answer is Option 2.

 

Additional Information

• Mandala designs vary across cultures, but they typically share a circular pattern and spiritual significance.
• In Tibetan Buddhism, mandalas are created during rituals and destroyed afterward to symbolize the impermanence of life.
• Hindu mandalas often depict deities and are used in pujas (worship ceremonies).

The correct answer is: Option 3.

Key Points

• The word "close" has multiple meanings based on context. It can mean to shut or secure something (e.g., a door, window, or container), or it can refer to proximity or nearness. (Hindi meaning: बंद करना / पास होना)
• Example: Please close the door when you leave. (यहां "close" का मतलब है "दरवाजा बंद करना।")
• In the given sentence, the word "close" is used to mean "to shut or secure something," specifically the door. (Hindi meaning: बंद करना)
• Example: It’s a good idea to close all the windows before it starts to rain.
• Hence, the correct answer is Option 3, as it uses "close" with the same meaning as the given sentence.

Therefore, the correct answer is: Option 3.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Option 1: She lives very close to the library. (पास)
• Explanation: Here, "close" refers to physical proximity or nearness, not shutting something.
• Example: The school is close to the market.
• Option 2: The museum is close to the park. (पास)
• Explanation: Similar to Option 1, this usage of "close" indicates nearness or being nearby.
• Example: The restaurant is close to my house.
• Option 4: He keeps his family photos close to his bed. (पास)
• Explanation: In this context, "close" again refers to proximity or nearness in terms of placement, not shutting or securing something.
• Example: She keeps her diary close to her at all times.

The correct answer is option 1.

Key Points

The correctly spelt word is 'Jaundice'.

• Jaundice refers to a medical condition where the skin and eyes turn yellow due to high bilirubin levels in the blood. (पीलिया)
• For Example -
• The doctor diagnosed her with jaundice and recommended immediate treatment.

Therefore, the correct answer is 'Jaundice'.

Additional Information

Explanation of the other options - 

• Jaundise - Incorrect spelling.
• Joundice - Incorrect spelling.
• Jaundiece - Incorrect spelling.

The correct answer is option 1.

Key Points

The misspelled words are 'Psycology', 'Psycholoy', and 'Psycholgy'.

• The correct spelling is 'Psychology'.
• The word 'Psychology' is a noun that refers to 'the scientific study of the human mind and its functions, especially those affecting behavior in a given context'. (मनोविज्ञान)
• For Example -
• She is studying psychology to understand human behavior better.

Therefore, the correct answer is option 1: Psychology.

Additional Information

Let's understand the meanings of some related terms:

• Psychiatry - the branch of medicine focused on the diagnosis, treatment, and prevention of mental, emotional, and behavioral disorders (मनोचिकित्सा)
• Neurology - the branch of medicine dealing with disorders of the nervous system (तंत्रिका विज्ञान)
• Behavioral Science - the scientific study of human and animal behavior (व्यवहार विज्ञान)

The correct answer is: Option 2.

Key Points

• The idiom "sit on the fence" means to refuse to take a side or give an opinion on a matter. (किसी मुद्दे पर पक्ष लेने या राय देने से इनकार करना)
• Example: During the debate, she chose to sit on the fence and did not clearly support either side.
• The correct meaning refers to staying neutral or undecided in a discussion, avoiding committing to one side or the other. (तटस्थ या अनिर्णीत रहना)
• Example: Politicians often sit on the fence to avoid upsetting any voter group.

Therefore, the correct answer is: Option 2.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Option 1: To be very decisive about a matter (किसी मामले में बहुत निर्णायक होना) - This means to quickly and firmly make a decision.
• Example: She was very decisive about quitting her job and starting her own business.
• Option 3: To carefully choose a position (सावधानीपूर्वक स्थिति चुनना) - This implies selecting a stance or opinion with a lot of thought and consideration.
• Example: He carefully chose his position during the negotiation to ensure the best deal.
• Option 4: To be in a neutral, relaxed position (तटस्थ, आरामदायक स्थिति में होना) - This refers to being in a calm or neutral physical state, not necessarily related to decision-making.
• Example: After a long day, she sat in a relaxed position on the couch.

The correct answer is: Effective.

Key Points

• The word "Futile" means something that is incapable of producing any useful result or is pointless. (बेकार, व्यर्थ)
• Example: His attempts to fix the broken clock were futile because it was beyond repair.
• "Effective" refers to something that is successful in producing a desired or intended result. (प्रभावी)
• Example: The new marketing strategy was highly effective in increasing sales.
• Hence, we can infer that the opposite of 'Futile' is 'Effective'.

Therefore, the correct answer is: Effective.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Useless (बेकार): Not fulfilling or not able to fulfill any purpose or role.
• Example: The old phone is useless because it no longer works.
• Pointless (निरर्थक): Having little or no sense, use, or purpose.
• Example: Continuing the debate seemed pointless since no one was willing to compromise.
• Vain (व्यर्थ): Producing no result or effect; futile.
• Example: Despite her efforts, all her attempts to win the contest were in vain.

The correct answer is: Option 2 - Astronomer.

Key Points

• The word "Astrologer" refers to a person who studies the positions and movements of celestial bodies to interpret their influence on human affairs and the natural world. (ज्योतिषी)
• Example: The astrologer predicted a bright future for her based on her birth chart.
• An "Astronomer" is a scientist who studies celestial objects, space, and the physical universe as a whole. (खगोलशास्त्री)
• Example: The astronomer spent countless nights observing the stars through his telescope.
• Hence, an astronomer is the correct term for someone who studies stars and space.

Therefore, the correct answer is: Option 2 - Astronomer.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Astrologer (ज्योतिषी): A person who predicts the future by studying the positions and movements of celestial bodies.
• Example: The astrologer advised her to wear an emerald ring for good luck.
• Meteorologist (मौसम विज्ञानी): A scientist who studies weather patterns and atmospheric phenomena.
• Example: The meteorologist warned about a potential storm in the coming week.
• Geographer (भूगोलवेत्ता): A person who studies the Earth's physical features, human societies, and their interactions.
• Example: The geographer specialized in studying climate change and its effects on geography.

The correct answer is: Conservative.

Key Points

• "Conservative" refers to a person who supports traditional practices and resists changes or innovations. (रूढ़िवादी)
• Example: Many conservative politicians argue against rapid reform in the education system.
• "Liberal" refers to someone open to new ideas and willing to discard traditional values, especially in political or social matters. (उदारवादी)
• Example: Liberals advocate for equal rights and freedoms for everyone in society.
• "Reactionary" indicates someone who strongly opposes progress or change and desires to return to a previous state of society. (प्रतिक्रियावादी)
• Example: The reactionary group protested against the new laws aiming for social reform.
• "Progressive" refers to someone who supports social reform or new, innovative changes for society’s betterment. (प्रगतिशील)
• Example: Progressive thinkers often push for advancements in science and technology to improve human life.

Therefore, the correct answer is: Conservative.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Liberal (उदारवादी): Someone open to new ideas and willing to discard traditional values.
• Example: He is a liberal thinker who supports gender equality in all aspects of life.
• Reactionary (प्रतिक्रियावादी): Someone opposing progress or change and desiring a return to a previous state.
• Example: The reactionary group resisted adopting modern technology in their workplaces.
• Progressive (प्रगतिशील): A person who supports innovation and social reform.
• Example: Progressive leaders often focus on improving the quality of education and healthcare.

The correct answer is: Quiet.

Key Points

• The word "Garrulous" means excessively talkative, especially on trivial matters. (बातूनी)
• Example: The garrulous old man spent hours recounting his childhood tales.
• "Quiet" refers to a state of being silent or not making noise; it is the opposite of being talkative. (शांत)
• Example: She prefers quiet evenings at home rather than loud parties.
• Hence, we can infer that the opposite of 'garrulous' is 'quiet'.

Therefore, the correct answer is: Quiet.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Talkative (बातूनी): Someone who speaks a lot, often excessively.
• Example: The talkative child kept interrupting the teacher during the lesson.
• Loquacious (बहुत बातूनी ): Another word for being excessively talkative.
• Example: Her loquacious nature made her a favorite at social gatherings.
• Chatty (गपशप करने वाला): Informal term for someone who enjoys chatting or talking casually.
• Example: He has a chatty personality and loves striking up conversations with strangers.

The correct answer is Option 3.

 

Key Points

• Active voice involves a subject performing the action, while in passive voice, the subject receives the action.
• The original sentence is in passive voice: "The report is expected to be delivered by the committee next week."
• To convert it into active voice, the subject "The committee" is placed at the beginning of the sentence.
• The verb "is expected to be delivered" is transformed into the active form "expects to deliver."
• The time phrase "next week" remains at the end of the sentence for clarity.

Therefore, the correct answer is Option 3.

 

Correct Sentence: "The committee expects to deliver the report next week."

 

Additional Information

• Option 1: "The committee is expecting to deliver the report next week." – The sentence uses the present continuous tense, which is incorrect as it changes the meaning.
• Option 2: "The committee is expected to deliver the report next week." – This sentence is still in passive voice, not active.
• Option 4: "The committee will deliver the report next week." – This sentence changes the tense to future simple, which alters the original meaning.

The correct answer is Option 1.

 

Key Points

• Indirect Speech: The tour guide said that they had been walking for hours, but the ruins they had been looking for were just around that corner, and they would be there in a few minutes.
• Key Conversions:
• "We have been walking" becomes "they had been walking"
• Pronoun "we" changes to "they" in indirect speech.
• Present perfect continuous tense "have been walking" is converted to past perfect continuous tense "had been walking."
• "are just around this corner" becomes "were just around that corner"
• Present tense "are" is converted to past tense "were" in indirect speech.
• The demonstrative "this" becomes "that" in indirect speech.
• "we will be there" becomes "they would be there"
• Pronoun "we" changes to "they" in indirect speech.
• Future tense "will" changes to conditional "would" in indirect speech.
• Punctuation and Format:
• Direct speech is converted to indirect speech, so no quotation marks are used.
• Reporting clause "The tour guide said" introduces the indirect speech correctly.

Therefore, the correct answer is Option 1.

 

Correct Sentence: The tour guide said that they had been walking for hours, but the ruins they had been looking for were just around that corner, and they would be there in a few minutes.

 

Additional Information

• Option 2: "The tour guide said that we have been walking for hours, but the ruins we have been looking for are just around this corner, and we will be there in a few minutes."
• Fails to convert the pronouns "we" to "they" and does not change tenses correctly.
• Option 3: "The tour guide said that they had been walking for hours, but the ruins they had been looking for are just around that corner, and they will be there in a few minutes."
• Incorrect tense usage: "are" and "will" are not converted to "were" and "would" respectively.
• Option 4: "The tour guide said that they were walking for hours, but the ruins they were looking for were just around that corner, and they would be there in a few minutes."
• Incorrect tense conversion: "were walking" implies past continuous, but the original sentence uses present perfect continuous.

The correct answer is 'in a quick, detailed, and thorough manner'.

 

Key Points

• The sentence requires parallelism in the list of adjectives to maintain grammatical correctness and readability.
• The phrase "in a quick, detailed, and thorough manner" follows proper parallel structure, which means all the adjectives are consistent in form.
• Options 2, 3, and 4 disrupt the parallel structure, making them incorrect.

Therefore, the correct answer is Option 1.

 

Complete Sentence: "The manager was asked to write a report in a quick, detailed, and thorough manner."

 

Additional Information

• Option 2: "in a quick, detailed and in thorough manner" is incorrect because the repetition of the preposition "in" is redundant.
• Option 3: "in a quick, detailing and thorough manner" is incorrect because "detailing" is not consistent with the other adjectives.
• Option 4: "quickly, detailed and thoroughly" is incorrect because "quickly" is an adverb and does not match the adjective form of "detailed" and "thoroughly" disrupts the structure.

The correct answer is 'Of which only six are officially recognized'.

 

Key Points

• The phrase "of which" is used to refer to things or objects (colleges in this case), making Option 4 grammatically correct.
• The original sentence uses "of whom," which is used for people and is therefore incorrect here.
• The sentence requires the plural verb "are" to match the plural subject "colleges." Option 4 satisfies this condition.

Therefore, the correct answer is Option 4.

 

Complete Sentence: "The country has at least twenty colleges of which only six are officially recognized."

 

Additional Information

• Option 1: "of whom only six are officially recognized" is incorrect because "whom" refers to people, not things.
• Option 2: "of whom only six have been officially recognized" is incorrect for the same reason as Option 1.
• Option 3: "of which only six has official recognition" is incorrect because the subject "six" is plural, and the singular verb "has" does not agree with it.

The correct answer is Option 2.

 

Key Points

• Passive voice structure: The object of the active voice sentence becomes the subject of the passive voice sentence.
• In the active voice, the sentence is: "Will the company launch the new product?"
• To convert to passive voice: Place the object "the new product" at the beginning.
• Change the active verb form "will launch" to "will be launched" (future simple passive).
• Add the preposition "by" followed by the original subject "the company."

Therefore, the correct answer is Option 2.

 

Correct Sentence: "Will the new product be launched by the company?"

 

Additional Information

• Option 1: "The new product will be launched by the company?" – This is incorrect because it does not follow the interrogative sentence structure in passive voice.
• Option 3: "Is the new product launched by the company?" – Incorrect tense (present simple passive).
• Option 4: "The new product would be launched by the company?" – Incorrect use of modal verb "would" instead of "will."

The correct answer is Option 1.

 

Key Points

• Indirect Speech: He said to her that the film they had watched the previous night was not as good as he had expected it to be.
• Key Conversions:
• "the film they had watched the previous night" becomes "The film we watched last night"
• Time expression "the previous night" is converted to "last night" to reflect direct speech.
• Pronoun "they" is changed to "we" to match the perspective of the speaker in direct speech.
• "was not as good as he had expected it to be" becomes "was not as good as I expected it to be"
• Pronoun "he" is replaced with "I" to match the speaker's point of view.
• Verb tense "had expected" is simplified to "expected" as per direct speech rules.
• Reporting verb:
• Indirect: "He said to her"
• Converted to: "He said to her," maintaining the same reporting clause.
• Punctuation and Format:
• Direct speech is enclosed in quotation marks.
• Starts with a capital letter "The" and ends with a full stop ".".
• Reporting clause is separated from the quoted speech by a comma and quotation marks.

Therefore, the correct answer is Option 1.

 

Correct Sentence: "He said to her, "The film we watched last night was not as good as I expected it to be.""

 

Additional Information

• Option 2: "He said to her, "The film we had watched last night was not as good as I had expected it to be.""
• Incorrect because the verb tense "had watched" and "had expected" does not match the direct speech style.
• Option 3: "He said to her, "The film we watched the previous night was not as good as I expected it to be.""
• Incorrect because "the previous night" is not typically used in direct speech where "last night" is preferred.
• Option 4: "He said to her, "The film we had watched last night was not as good as I expected it to be.""
• Incorrect because "had watched" is unnecessarily complex for direct speech.

The correct answer is Option 4.

 

Key Points

• The paragraph begins with Sentence Q, which introduces the restoration of the library.
• Sentence Q is the first sentence as it describes the preservation of valuable texts and historic architecture during the library's restoration:
• "The valuable texts and historic architecture of the old library were preserved throughout its restoration."
• Sentence S follows, as it elaborates on the collaborative efforts of local historians and preservationists:
• "To guarantee the restoration’s correctness, local historians and preservationists collaborated."
• Sentence P comes next, as it highlights the success of the restoration project:
• "The restoration project was hailed as a triumph of community and expert cooperation."
• Sentence R is the last sentence, as it discusses the library's international recognition after restoration:
• "The library quickly gained international recognition as a cultural icon."
• The correct sequence is Q, S, P, R.

 

Coherent paragraph after rearrangement:

"The valuable texts and historic architecture of the old library were preserved throughout its restoration. To guarantee the restoration’s correctness, local historians and preservationists collaborated. The restoration project was hailed as a triumph of community and expert cooperation. The library quickly gained international recognition as a cultural icon."

Given:

A bus travels at a speed of 72 km/hr.

We need to find the distance it covers in 50 minutes.

Now, Convert 50 minutes into hours:

50 minutes = 50 ÷ 60 = 5/6 hours

Distance = Speed × Time

Distance = 72 × (5/6)

Distance = 72 × 0.8333 ≈ 60 km

Hence, the correct answer is Option 2.

Given:

A car travels from town A to B at 60 km/hr and returns from B to A at 40 km/hr.

We need to find the average speed for the entire trip.

Now,When the distance for both trips is the same, the average speed is calculated as:

Average speed

= (2 × speed1 × speed2) ÷ (speed1 + speed2)

Here,

Average speed  = (2 × 60 × 40) ÷ (60 + 40)

= 4800 ÷ 100

= 48 km/hr

Hence, the correct answer is Option 2.

Given:

A piece of cloth is 8 meters long and 2.5 meters are cut from it.

Now, Subtract the cut portion from the total length:

Cloth left = 8 − 2.5 = 5.5 meters

Hence, the correct answer is Option 2.

Given:

Find the LCM of 6, 9, and 12.

Now,

6 = 2 × 3

9 = 3 × 3

12 = 2 × 2 × 3

So, LCM = 2² × 3² = 4 × 9 = 36

Hence, the correct answer is Option 3.

Given Statement:

The TCS company has decided to lay off 1000 employees due to financial losses.

Conclusions:

I. The company is facing financial difficulties – True (The layoffs are due to financial losses, which indicate financial difficulties).

II. The employees who were laid off were not performing well –

False (No information about employee performance is given; the reason is financial).

III. The company will be able to recover from its financial losses after the layoffs –

False (There is no certainty about recovery mentioned in the statement).

Hence, only conclusion I follows.

The correct answer is "Option 1".

The logic followed here is:

Logic: The second number is the first number minus 1, then cube.

Here,

10 → (10 − 1)³

= 9³

= 729 

Similarly,

7 → (7 − 1)³

= 6³

= 216 

Hence, the correct answer is "Option 1."

The logic followed here is:

thus, "68" will replace the question mark in the given series.

Hence, the correct option is "Option 3".

Given:

Series: (3, 5), (5, 10), (7, 15), ?, (13, 25)

First number increase by consecutive prime number.

First numbers: 

Second numbers:

So, the missing term is (11, 20).

Hence, the correct answer is Option 2.

The logic followed here is:

thus, "33" will replace the question mark in the given series.

Hence, the correct option is "Option 3".

Given items: Plotter, Joystick, Stylus, Trackball

Plotter – Output device

Joystick – Input device

Stylus – Input device

Trackball – Input device

Thus, Three items (Joystick, Stylus, Trackball) are input devices, while Plotter is an output device.

So, the odd one out is Plotter.

Hence, the correct answer is Option 1.

The logic followed here is:

12, 9, and 56 are Composite Numbers (numbers with more than two factors),

Whereas,

67 is a Prime Number, as it is only divisible by 1 and itself.

So, 67 is the odd one out from the given options.

Hence, the correct answer is "Option 2".

Would you like me to find the prime factors for the other numbers in this list?

The logic followed here is:

thus, "19" will replace the question mark in the given series.

Hence, the correct option is "Option 3".

The logic followed here is:

Given Sequence: m_pq_rpqm_pqmrp_

Now, putting the letters in the blank spaces given in the options one by one:

Option 1) rmrq → mrpq/mrpq/mrpq/mrpq → Repeat the pattern "mrpq".

Option 2) rmrp → mrpq/mrpq/mrpq/mrpp→ Does not follow any pattern.

Option 3) mmrq →mmpq/mrpq/mrpq/mrpq→Does not follow any pattern.

Option 4) mrpq → mmpq/rrpq/mppq/mrpq→ Does not follow any pattern.

Hence, the correct answer is Option 1.

The logic followed here is:

So, 'U169' will replace the question mark in the given series.

Hence, the correct answer is "Option 2".

The logic followed here is:

thus, "V" will replace the question mark in the given series.

Hence, the correct option is "Option 1".

The logic followed here is:

Given Sequence: a_ _fh_c_f_a_e_h

Now, putting the letters in the blank spaces given in the options one by one:

Option 1) ceahecf → a c e f h a c h f e a c e f h (Does not repeat any pattern)

Option 2) caeehcf → a c a f h e c e f h a c e f h (Does not repeat any pattern)

Option 3) ceaehcf → a c e f h/ a c e f h/ a c e f h (Repeat the pattern "a c e f h")

Option 4) eceafch → a e c f h e c a f f a c e h h (Does not repeat any pattern)

Hence, the correct answer is Option 3.

BODMAS Table: 

Given equation: (12 + 3) × 4 − (8 ÷ 2) + 6 = ?

Now, if the operators are interchanged as per the rule, then:

⇒ (12 × 3) − 4 ÷ (8 + 2) × 6 = ?

⇒ 36 − 4 ÷ 10 × 6 = ?

⇒ 36 − 2.4 = ?

⇒ 33.6

Hence, the correct answer is Option 3.

Given:

{{(20 - 4) $\times 3 + 6$} / 2} = ?

By using the BODMAS rule:

By, decoding the given symbols we get.

Substituting the given operator in the given expression:

⇒ {{(20 - 4) × 3 + 6} / 2}

⇒ ((20 × 4) ÷ 3) - 6 + 2

⇒ (80 ÷ 3) - 6 + 2

⇒ 26.67 - 6 + 2

⇒ 26.67 + 2 - 6

⇒ 28.67 - 6

⇒ 22.67

Hence, the correct answer is 22.67.

Given statements:

All novels are books.

Some books are boring.

Option 1) Some novels are boring – Not  true (Some books are boring, but it doesn’t mean any novels are boring.)

Option 2) All boring things are books – Not true (Some books are boring, but there could be boring things that are not books.)

Option 3) Some books are novels – True (Since all novels are books, at least some books are definitely novels.)

Option 4) Some boring things are novels – Not true (The boring books may or may not include novels.)

Hence, the correct answer is Option 3.

The logic followed here is:

Logic: The first word is a part of the second word or object.

Here,

Valve : Trumpet → A valve is a part of a trumpet that controls airflow and sound.

Similarly,

Ignition : Car → Ignition is a part of a car that starts the engine.

Hence, the correct answer is Option 4.

The logic followed here is:

Given:

‘table chair lamp’ → ek, mp, su

‘lamp clock pen’ → su, yz, op

‘table pen book’ → ek, op, uv

So, the code for “clock” is yz.

Hence, the correct answer is Option 2.

Given:

Option 1) Jharkhand

– Does not share an international border with any country

Option 2) Sikkim

– Shares international borders with Nepal, Bhutan, and China

Option 3) Gujarat

– Shares an international border with Pakistan

Option 4) West Bengal

– Shares international borders with Bangladesh, Nepal, and Bhutan

So, Jharkhand does not belong to the group.

Hence, the correct answer is Option 1.

The given word: IMPLEMENTATION

Now, If we arrange all the alphabets of IMPLEMENTATION in reverse alphabetical order, we get:

T T P O N N M M L I I E E A

The 13^th alphabet from the right end of the new word is T.

The 6^th alphabet to the right of this alphabet is M.

Thus, the answer is "M".

Hence, the correct answer is Option 3.

The logic followed here is:

thus, "1890" will replace the question mark in the given series.

Hence, the correct option is "Option 2".

The logic followed here is:

thus, "3599" will replace the question mark in the given series.

Hence, the correct option is "Option 1".

Given:

Option 1: Every integer is a real number

Option 2: Every natural number is an integer

Option 3: Every irrational number is rational

Option 4: Rational and irrational numbers are real numbers

Formula used:

Definitions of number sets:

1. Real numbers include both rational and irrational numbers.
2. Natural numbers are a subset of integers.
3. Rational numbers and irrational numbers are mutually exclusive.

Calculation:

1. Option 1: Every integer is a real number. (True, as integers are part of real numbers)
2. Option 2: Every natural number is an integer. (True, as natural numbers are a subset of integers)
3. Option 3: Every irrational number is rational. (False, as irrational numbers cannot be expressed as a ratio of two integers and are not rational)
4. Option 4: Rational and irrational numbers are real numbers. (True, as both are subsets of real numbers)

∴ The correct answer is option (3).

Shortcut Trick

Every rational number expressed as a fraction p/q results in one of two decimal types.

If the denominator q has only 2 or 5 as prime factors, the decimal terminates (stops).

If the denominator q has any other prime factors, the decimal is non-terminating but repeating.

Since these are the only two possibilities for rational numbers, the answer is the combination of both.

∴ The correct answer is Terminating or repeating.

Alternate Method

Given: A rational number of the form p/q, where p and q are integers and q ≠ 0.

Concept: Decimal representation of Real Numbers.

⇒ Let the rational number be x = p/q.

⇒ If the prime factorization of q is of the form 2ⁿ5^m (where n, m are non-negative integers), then x has a terminating decimal expansion.

⇒ Example: 7/8 = 7/(2³) = 0.875.

⇒ If the prime factorization of q is not of the form 2ⁿ5^m, then x has a non-terminating repeating decimal expansion.

⇒ Example: 1/3 = 0.333... or 0.3̅.

⇒ In contrast, irrational numbers like √2 or π have non-terminating and non-repeating expansions.

∴ The correct answer is Terminating or repeating.

Additional Information

Irrational Numbers

Numbers that cannot be written as p/q. Their decimal expansion is always non-terminating and non-repeating.

Denominator Rule for Termination

A rational number p/q in its simplest form terminates if and only if the prime factors of q are only 2 and 5.

Pure vs Mixed Recurring Decimals

A decimal is pure recurring if all digits after the decimal point repeat (e.g., 0.444...), and mixed recurring if some digits do not repeat (e.g., 0.1222...).

Given:

Profit on the first item = 25%

Loss on the second item = 10%

Cost price of the first item = 60% more than the second

Overall profit = ₹225

Formula used:

Profit/Loss = Selling Price - Cost Price

Overall Profit = Profit from the first item + Loss from the second item

Cost price relation: Cost price of the first item = 1.6 × Cost price of the second item

Calculations:

Let cost price of the second item = ₹x

⇒ Cost price of the first item = 1.6 × x

Profit on the first item = 25% of 1.6x = 0.25 × 1.6x = 0.4x

Loss on the second item = 10% of x = 0.10 × x = 0.1x

Overall profit = Profit from the first item - Loss from the second item

⇒ 0.4x - 0.1x = ₹225

⇒ 0.3x = ₹225

⇒ x = ₹750

Cost price of the second item = ₹750

Cost price of the first item = 1.6 × ₹750 = ₹1,200

∴ The correct answer is option (2): ₹1,200 and ₹750.

Given:

Number of pressure cookers = 25

Cost price (CP) per cooker = ₹800

Sold 15 cookers at 20% profit

Sold remaining (10 cookers) at 10% loss

Formula used:

1. Total CP = Number of items × CP per item
2. SP for profit = CP × (1 + Profit%)
3. SP for loss = CP × (1 - Loss%)
4. Overall profit or loss % = ^{(Total SP - Total CP)}÷_{Total CP} × 100

Calculations:

Total CP = 25 × 800 = ₹20,000

SP of 15 cookers at 20% profit:

⇒ SP = 15 × (800 × 1.20) = 15 × 960 = ₹14,400

SP of 10 cookers at 10% loss:

⇒ SP = 10 × (800 × 0.90) = 10 × 720 = ₹7,200

Total SP = 14,400 + 7,200 = ₹21,600

Overall profit %:

⇒ Profit % = ^{(21,600 - 20,000)}÷_20,000 × 100

⇒ Profit % = ^1,600÷_20,000 × 100

⇒ Profit % = 8%

∴ The correct answer is option (2).

Given:

Cost Price (CP) of 12 books = 12 × ₹360 = ₹4320

Selling Price (SP) of 12 books = ₹3600

Formula used:

Discount Percentage = $\dfrac{CP - SP}{CP} \times 100$

Calculations:

Discount Percentage = $\dfrac{4320 - 3600}{4320} \times 100$

⇒ Discount Percentage = $\dfrac{720}{4320} \times 100$

⇒ Discount Percentage = 16.67%

∴ The correct answer is option (1).

Shortcut Trick

Apply the Successive Percentage formula: Net Profit % = M − D − (M × D)/100

Where M is Markup % (40) and D is Discount % (10).

Net Profit % = 40 − 10 − (40 × 10)/100

Net Profit % = 30 − 4 = 26%

∴ The correct answer is 26%.

Alternate Method

Given:

Cost Price (CP) = ₹10000

Markup = 40%

Discount = 10%

Formula Used:

Marked Price (MP) = CP × (100 + Markup %)/100

Selling Price (SP) = MP × (100 − Discount %)/100

Profit % = [(SP − CP)/CP] × 100

⇒ Marked Price (MP) = 10000 + (40/100) × 10000 = ₹14000

⇒ Selling Price (SP) = 14000 − (10/100) × 14000

⇒ SP = 14000 − 1400 = ₹12600

⇒ Total Profit = SP − CP = 12600 − 10000 = ₹2600

⇒ Profit Percentage = (2600/10000) × 100 = 26%

∴ The correct answer is 26%.

Additional Information

Markup Percentage

It is the extra price added to the Cost Price to arrive at the Marked Price. Formula: Markup % = [(MP − CP)/CP] × 100.

Discount and Selling Price

Discount is always calculated on the Marked Price (MP). SP = MP − Discount.

Effective Profit with Successive Changes

When a price is increased by x% and then decreased by y%, the net change is [x − y − (xy/100)]%.

Given:

Average marks of 15 students = 72.4

Incorrect marks of one student = 68.9

Correct marks of the same student = 86.3

Formula used:

Corrected Average = $\dfrac{\text{(Sum of marks) + (Correction)}}{\text{Number of students}}$

Correction = Correct marks - Incorrect marks

Calculation:

Correction = 86.3 - 68.9

⇒ Correction = 17.4

Sum of marks = Average × Number of students

Sum of marks = 72.4 × 15

⇒ Sum of marks = 1086

Corrected Sum of marks = 1086 + 17.4

⇒ Corrected Sum of marks = 1103.4

Corrected Average = $\dfrac{1103.4}{15}$

⇒ Corrected Average = 73.56

∴ The correct answer is option (2).

Shortcut Trick

Ratio of marks in Science to Math = 100 : 120 = 5 : 6

Total marks = Average × 2 = 66 × 2 = 132

Total ratio units = 5 + 6 = 11 units

11 units = 132 ⇒ 1 unit = 12

Marks in Science = 5 units = 5 × 12 = 60

∴ The correct answer is 60.

Alternate Method

Given:

Average marks in Math and Science = 66

Math marks = 20% more than Science marks

Formula Used:

Average = (Sum of all observations) ÷ (Total number of observations)

Calculations:

Let the marks in Science be x.

⇒ Marks in Math = x + 20% of x = 1.2x

⇒ Average = (x + 1.2x) ÷ 2

⇒ 66 = 2.2x ÷ 2

⇒ 66 = 1.1x

⇒ x = 66 ÷ 1.1

⇒ x = 660 ÷ 11

⇒ x = 60

∴ The correct answer is 60.

Additional Information

Percentage Increase

To increase a value by a%, multiply the original value by (1 + a/100). Here, a 20% increase means multiplying by 1.2 or 6/5.

Relationship between Average and Sum

The sum of n observations is always equal to the Average × n. For two subjects, Sum = Average × 2.

Ratio and Proportion in Marks

When one value is k% more than another, they can be expressed as a ratio (100 + k) : 100 to simplify division of the total sum.

Given:

Percentage of students who passed in English = 60%

Percentage of students who passed in Math = 70%

Percentage of students who passed in both = 45%

Formula used:

Percentage of students who passed in at least one subject = Passed in English + Passed in Math - Passed in both

Calculations:

Percentage of students who passed in at least one subject

⇒ 60% + 70% - 45%

⇒ 130% - 45%

⇒ 85%

∴ The correct answer is option (3).

Given:

A invests ₹30,000 for 12 months

B invests ₹50,000 for 12 months

C invests ₹60,000 for 6 months

Formula used:

Profit share is proportional to the product of investment and time.

Profit share = Investment × Time

Calculation:

Profit share of A = 30,000 × 12 = 360,000

Profit share of B = 50,000 × 12 = 600,000

Profit share of C = 60,000 × 6 = 360,000

Total profit share = 360,000 + 600,000 + 360,000

⇒ Ratio of profits = A : B : C

⇒ 360,000 : 600,000 : 360,000

⇒ 36 : 60 : 36

⇒ 3 : 5 : 3

∴ The correct answer is option (4).

Given:

A, B, and C invest ₹20,000, ₹30,000, and ₹50,000, respectively.

Total Profit = ₹50,000

Formula used:

Profit share of a partner = $\dfrac{\text{Partner's Investment}}{\text{Total Investment}} \times \text{Total Profit}$

Calculations:

Total Investment = ₹20,000 + ₹30,000 + ₹50,000 = ₹1,00,000

B's Share = $\dfrac{30,000}{1,00,000} \times 50,000$

⇒ B's Share = $\dfrac{3}{10} \times 50,000$

⇒ B's Share = ₹15,000

∴ The correct answer is option (1).

Shortcut Trick

Equate the Total Surface Area to the Volume of the cylinder: 2πr(r + h) = πr²h.

Simplify by canceling π and one r from both sides: 2(r + h) = rh.

Expand and rearrange the terms: 2r + 2h = rh ⇒ 2r = h(r − 2).

Isolate the height (h) to find the relation: h = 2r / (r − 2).

∴ The correct answer is h = (2r)/(r−2).

Alternate Method

Given that the Total Surface Area (TSA) of a cylinder is equal to its Volume (V).

Formula Used:

Total Surface Area of Cylinder = 2πr(r + h)

Volume of Cylinder = πr²h

Calculations:

⇒ 2πr(r + h) = πr²h

⇒ 2πr² + 2πrh = πr²h

Dividing both sides by πr:

⇒ 2(r + h) = rh

⇒ 2r + 2h = rh

⇒ 2r = rh − 2h

⇒ 2r = h(r − 2)

⇒ h = 2r / (r − 2)

∴ The correct answer is h = (2r)/(r−2).

Additional Information

Curved Surface Area (CSA)

The lateral surface area of a cylinder excluding the top and bottom circles is given by CSA = 2πrh.

Volume of a Cylinder

The amount of space occupied by a cylinder is V = πr²h, where r is the radius and h is the height.

Relation logic

For the relation h = 2r/(r−2) to be physically valid, the radius (r) must always be greater than 2 units.

Given:

A circle is inscribed within a square with a side length of 18 cm.

Formula used:

Area of square = side²

Area of circle = π × (radius)²

Area not covered by the circle = Area of square - Area of circle

Calculations:

Side of square = 18 cm

Radius of circle = side/2 = 18/2 = 9 cm

Area of square = 18² = 324 cm²

Area of circle = 3.14 × 9²

⇒ Area of circle = 3.14 × 81 = 254.34 cm²

Area not covered by the circle = Area of square - Area of circle

⇒ Area not covered = 324 - 254.34 = 69.66 cm²

∴ The correct answer is option (2).

Shortcut Trick

Apply the direct Algebraic Identity: (a + b − c)² = (a² + b² + c²) + 2(ab − bc − ca)

Given a² + b² + c² = 240 and ab − bc − ca = 64.

Substitute the values: (a + b − c)² = 240 + 2 × 64

Calculation: 240 + 128 = 368

∴ The correct answer is 368.

Alternate Method

Given:

Sum of squares: a² + b² + c² = 240

Mixed term expression: ab − bc − ca = 64

Formula Used:

The general expansion of a trinomial squared is: (x + y + z)² = x² + y² + z² + 2(xy + yz + zx)

Calculations:

⇒ Let x = a, y = b, and z = −c

⇒ (a + b + (−c))² = a² + b² + (−c)² + 2(a×b + b×(−c) + (−c)×a)

⇒ (a + b − c)² = a² + b² + c² + 2(ab − bc − ca)

⇒ (a + b − c)² = 240 + 2 × 64

⇒ (a + b − c)² = 240 + 128

⇒ (a + b − c)² = 368

∴ The correct answer is 368.

Additional Information

Trinomial Square Expansion

(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca. The signs of the mixed terms depend on the signs of individual variables.

Alternate Sign Variation

(a − b − c)² = a² + b² + c² − 2ab + 2bc − 2ca. Notice that (−b)×(−c) results in a positive 2bc.

Sum of Squares Formula

a² + b² + c² = (a + b + c)² − 2(ab + bc + ca). This is frequently used to find individual components.

Given:

Curved Surface Area of Cone = Curved Surface Area of Hemisphere

Radius (r) = r

Formula used:

Curved Surface Area of Cone = πrl

Curved Surface Area of Hemisphere = 2πr²

Calculations:

πrl = 2πr²

⇒ l = 2r

∴ The correct answer is option (3).

Given:

4 cosA + 3 secA = 4√3

0° ≤ A ≤ 90°

Formula used:

secA = 1/cosA

Calculations:

4 cosA + 3 secA = 4√3

⇒ 4 cosA + 3(1/cosA) = 4√3

⇒ 4 cosA + 3/cosA = 4√3

⇒ Multiply through by cosA:

⇒ 4 cos²A + 3 = 4√3 cosA

⇒ Rearrange:

⇒ 4 cos²A - 4√3 cosA + 3 = 0

⇒ Let x = cosA, the equation becomes:

⇒ 4x² - 4√3 x + 3 = 0

⇒ Solve the quadratic equation using the quadratic formula:

x = [-(-4√3) ± √{(-4√3)² - 4(4)(3)}] / (2 × 4)

⇒ x = [4√3 ± √{48 - 48}] / 8

⇒ x = (4√3 ± 0) / 8

⇒ x = √3 / 2

So, cosA = √3 / 2

⇒ A = 30°

∴ The correct answer is option (3).

Given:

2 sin²A + 2√2 cosA − 3 = 0

Formula used:

sin²A = 1 − cos²A

Calculation:

⇒ 2(1 − cos²A) + 2√2 cosA − 3 = 0

⇒ 2 − 2cos²A + 2√2 cosA − 3 = 0

⇒ −2cos²A + 2√2 cosA − 1 = 0

Multiply by −1

⇒ 2cos²A − 2√2 cosA + 1 = 0

⇒ (√2 cosA − 1)² = 0

⇒ √2 cosA = 1

⇒ cosA = 1/√2

⇒ sinA = 1/√2

tanA = sinA ÷ cosA

⇒ 1

∴ tanA = 1.

Given:

Total wages for 5 days = Rs. 600

Man's efficiency = 2 × Boy's efficiency

Let Boy's daily wage = x

Man's daily wage = 2x

Formula used:

Total daily wage = Man's daily wage + Boy's daily wage

Wages for 5 days = 5 × Total daily wage

Calculation:

Total daily wage = Man's daily wage + Boy's daily wage

⇒ Total daily wage = 2x + x

⇒ Total daily wage = 3x

Wages for 5 days = 5 × Total daily wage

⇒ 600 = 5 × 3x

⇒ 600 = 15x

⇒ x = 600 ÷ 15

⇒ x = 40

∴ The daily wage of the boy is Rs. 40. The correct answer is option (4).

Given:

A solid sphere is cut into two equal hemispheres.

Formula used:

Total Surface Area (TSA) of a Sphere = 4πr²

TSA of 1 Hemisphere = 2πr² (curved surface) + πr² (base area) = 3πr²

Percentage Increase = $\frac{\text{Increase in Surface Area}}{\text{Original Surface Area}} \times 100$

Calculation:

Original TSA of Sphere = 4πr²

TSA of 2 Hemispheres = 2 × 3πr² = 6πr²

Increase in Surface Area = 6πr² - 4πr² = 2πr²

⇒ Percentage Increase = $\frac{2πr^2}{4πr^2} \times 100$

⇒ Percentage Increase = $\frac{2}{4} \times 100 = 50\%$

∴ The correct answer is 50%.

Given:

Base diagonal = 17 cm

Height = 15 cm

Area of base = 120 cm²

Formula used:

Let sides of base = a, b

a² + b² = (diagonal)²

Area = a × b

Total surface area = 2(ab + bc + ca)

Calculation:

a × b = 120

a² + b² = 17² = 289

(a + b)² = a² + b² + 2ab

⇒ (a + b)² = 289 + 2 × 120

⇒ (a + b)² = 289 + 240

⇒ (a + b)² = 529

⇒ a + b = 23

Total surface area:

⇒ 2(ab + h(a + b))

⇒ 2(120 + 15 × 23)

⇒ 2(120 + 345)

⇒ 2 × 465

⇒ 930

∴ Total surface area = 930 cm².

Given:

If x = √2 + √3, find x³ - 3x

Formula used:

(a + b)³ = a³ + b³ + 3ab(a + b)

Calculation:

Let x = √2 + √3

x³ = (√2 + √3)³

Using formula (a + b)³, where a = √2 and b = √3:

⇒ x³ = (√2)³ + (√3)³ + 3(√2)(√3)(√2 + √3)

⇒ x³ = 2√2 + 3√3 + 3(√6)(√2 + √3)

⇒ x³ = 2√2 + 3√3 + 3√6 × √2 + 3√6 × √3

⇒ x³ = 2√2 + 3√3 + 3×√12 + 3×√18

⇒ x³ = 2√2 + 3√3 + 6√3 + 9√2

⇒ x³ = (2√2 + 9√2) + (3√3 + 6√3)

⇒ x³ = 11√2 + 9√3

Now, calculate x³ - 3x:

x = √2 + √3

3x = 3(√2 + √3) = 3√2 + 3√3

⇒ x³ - 3x = (11√2 + 9√3) - (3√2 + 3√3)

⇒ x³ - 3x = (11√2 - 3√2) + (9√3 - 3√3)

⇒ x³ - 3x = 8√2 + 6√3

∴ The correct answer is option (3).

Given:

Base side length (a) = 16 cm

Height (h) = 30 cm

Formula used:

1. Slant height (l) = √(h² + (a/2)²)
2. Total Surface Area (TSA) = Base Area + Lateral Surface Area

Where,

Base Area = a²

Lateral Surface Area = 2 × a × l

Calculations:

1. Calculate slant height (l):

⇒ l = √(h² + (a/2)²)

⇒ l = √(30² + (16/2)²)

⇒ l = √(900 + 8²)

⇒ l = √(900 + 64)

⇒ l = √964

⇒ l ≈ 31.05 cm

2. Calculate Base Area:

⇒ Base Area = a²

⇒ Base Area = 16²

⇒ Base Area = 256 cm²

3. Calculate Lateral Surface Area:

⇒ Lateral Surface Area = 2 × a × l

⇒ Lateral Surface Area = 2 × 16 × 31.05

⇒ Lateral Surface Area ≈ 993.6 cm²

4. Calculate Total Surface Area (TSA):

⇒ TSA = Base Area + Lateral Surface Area

⇒ TSA = 256 + 993.6

⇒ TSA ≈ 1249.6 cm²

∴ Slant height is approximately 31.05 cm and Total Surface Area is approximately 1249.6 cm². The correct answer is option (1).

Shortcut Trick

Volume of water displaced = Volume of the submerged pyramid.

Rise in water level (h) = 28 − 25 = 3 cm.

Volume of cylinder (rise) = π × r² × h = 3.14 × 10² × 3 = 942 cm³.

Volume of pyramid = (1/3) × Base Area × Height = 942.

Base Area (Equilateral Triangle) = (√3/4) × 18² ≈ 0.433 × 324 ≈ 140.3 cm².

Height = (942 × 3) ÷ 140.3 ≈ 20.14 cm.

∴ The correct answer is 20.14 cm.

Alternate Method

Given:

Side of equilateral triangular base (a) = 18 cm

Radius of cylindrical tank (R) = 10 cm

Initial water height (H₁) = 25 cm

Final water height (H₂) = 28 cm

Formula Used:

Volume of Cylinder = πR²H

Volume of Pyramid = (1/3) × Area of Base × Height

Area of Equilateral Triangle = (√3/4) × a²

Rise = 3 cm

h = ? Vol. of Pyramid = Vol. of Water Rise

Calculations:

⇒ Rise in water level (h_rise) = 28 − 25 = 3 cm

⇒ Volume of water displaced = π × R² × h_rise

⇒ Volume = 3.14159 × 10² × 3 = 3.14159 × 300 = 942.477 cm³

⇒ Area of pyramid base = (√3/4) × 18² = (1.732/4) × 324

⇒ Area of base = 1.732 × 81 = 140.292 cm²

⇒ Volume of pyramid = (1/3) × Area of base × Height (H_p)

⇒ 942.477 = (1/3) × 140.292 × H_p

⇒ 942.477 = 46.764 × H_p

⇒ H_p = 942.477 ÷ 46.764 ≈ 20.15 cm

⇒ Using more precise √3 ≈ 1.73205 and π ≈ 3.14159, H_p ≈ 20.14 cm

∴ The correct answer is 20.14 cm.

Additional Information

Archimedes' Principle in Mensuration

When an object is completely submerged in a liquid, the volume of the liquid displaced is exactly equal to the volume of the submerged object.

Volume of a Regular Pyramid

The volume is always V = (1/3) × Base Area × Perpendicular Height, regardless of the shape of the base (triangular, square, etc.).

Equilateral Triangle Properties

Area = (√3/4)a² and Height = (√3/2)a. These are essential for solving pyramid problems with triangular bases.

Cylinder Volume

Volume = πr²h. When calculating water rise, 'h' represents only the change in height, not the total height of the tank.

Given:

Two vessels contain mixtures of milk and water in the ratios 3:2 and 7:3 respectively.

Resulting mixture should have milk and water in the ratio 5:3.

Formula used:

To find the ratio of mixing, apply the rule of alligation:

$\dfrac{\text{Milk in mixture 1 (3/5)} ;-; \text{Milk in resulting mixture (5/8)}}{\text{Milk in resulting mixture (5/8)} ;-; \text{Milk in mixture 2 (7/10)}}$

Calculations:

Milk in mixture 1 = $\dfrac{3}{5}$, Milk in mixture 2 = $\dfrac{7}{10}$, Milk in resulting mixture = $\dfrac{5}{8}$

⇒ Difference 1 = $\dfrac{3}{5} - \dfrac{5}{8}$

⇒ Difference 1 = $\dfrac{24}{40} - \dfrac{25}{40} = \dfrac{-1}{40}$

⇒ Difference 2 = $\dfrac{5}{8} - \dfrac{7}{10}$

⇒ Difference 2 = $\dfrac{50}{80} - \dfrac{56}{80} = \dfrac{-6}{80}$

⇒ Ratio = $\dfrac{-1}{40} / \dfrac{-6}{80}$

⇒ Ratio = $\dfrac{1 \times 80}{6 \times 40} = \dfrac{80}{240} = 3:1$

∴ The mixtures should be mixed in the ratio 3:1.