SSC CHSL Tier-1 2021: Official Paper (Held On : 1 June 2022 Shift 2) (Previous Year Paper)

The correct answer is: Ardour

Key Points

• Ardour means a strong, intense feeling of love or enthusiasm for someone or something. (किसी के लिए या कुछ के लिए प्यार या उत्साह की एक मजबूत, तीव्र भावना)
• So, it is clear that 'Ardour' can be used as a one-word substitute for 'Zeal; burning enthusiasm; passion'

Therefore, the correct answer is option 2.

Additional Information 

• Arid means land or a climate having little or no rain; too dry or barren to support vegetation. (भूमि या जलवायु जिसमें बहुत कम या कोई वर्षा न हो; वनस्पति का समर्थन करने के लिए बहुत सूखा या बंजर)
• Arcuate means bent or curved like a bow. (धनुष की तरह मुड़ा या मुड़ा हुआ)
• Arena means an enclosed area used for public entertainment. (सार्वजनिक मनोरंजन के लिए उपयोग किया जाने वाला एक संलग्न क्षेत्र)

The correct answer is "BCDA"

Key Points 

• Let's look at the correct sequence of the passage.
• Statement B is the opening sentence of the passage which explains that every field in the world has transformed because to COVID-19. No field was left unaffected by it.
• Statement C is the second sentence of the passage which explains that education is one of the most severely impacted fields. Millions of youngsters worldwide were unable to receive a quality education.
• Statement D is the third sentence of the passage which explains that during the epidemic, millions of teachers who are currently employed—particularly those in private educational institutions were not paid their full salaries.
• Statement A is the last sentence of the passage which shows that It is without a doubt clear that this illness caused suffering and anguish in the lives of persons who worked in other professions as well.

Hence, the correct sequence is B, C, D, A

The correct answer is option 2.

The correct passage is: B. Covid-19 changed the form and face of every field in the world. It did not leave any field untouched. C. One of the most affected areas is education. Millions of students all over the world could not get a proper education. D. Millions of teachers who are working, specifically in private educational institutes, were not paid a full salary during the pandemic. A. It is unquestionably evident that this virus brought misery and grief into the lives of people of other professions too.

The correct answer is: are looking into

Key Points

• Remove the preposition 'around' to correct the sentence.
• Let's look at the meaning of the given phrasal verbs.
• "Looking around" means going through a place in order to see what is there: to explore a place.
• "Looking into" means trying to find out about something.
• "Looking upto" means to respect and admire someone.
• The sentence is talking about a possibility, so "looking into" should be used.
• We cannot use two prepositions (around and into) simultaneously in a single sentence. It doesn't make any sense.
• The implied meaning of the sentence is: Vishnu and his family are considering relocating to Canada.

Therefore, the correct answer is option 3, which can substitute the underlined segment in the given sentence.

Corrected sentence: Vishnu and family are looking into the possibility of moving to Canada.

The correct answer is: To study very hard

Key Points 

• The idiom "Hit the books" means to study seriously with intensity and concentration. (तीव्रता और एकाग्रता के साथ गंभीरता से अध्ययन)

• Example - I hit the books all week so now it's time to take a break.

• So, it is clear that the most appropriate meaning of the given idiom "Hit the books" is "To study very hard"

Hence, the correct answer is option 4.

The correct answer is: Destroy

Key Points

• Devastate means to bring to ruin or desolation by violent action. (हिंसक कार्रवाई से बर्बाद या उजाड़ लाने के लिए)
• Example - The city was devastated by a huge earthquake.
• Marked option Destroy means to end the existence of something by damaging or attacking it. (किसी चीज के अस्तित्व को नुकसान पहुँचाने या उस पर हमला करने से समाप्त करना)
• Example - The room had been destroyed by fire.
• It is clear that 'Devastate' and 'Destroy' are similar in meaning.

Therefore the correct answer is option 1.​

Additional Information 

• Produce means to make something or bring something into existence. (कुछ बनाने या कुछ अस्तित्व में लाने के लिए)
• Recite means to repeat aloud or declaim a poem or passage from memory before an audience. (दर्शकों के सामने जोर से दोहराना या किसी कविता या स्मृति से पारित होने की घोषणा करना)
• Suppress means to prevent something from being expressed or known. (किसी चीज को व्यक्त या ज्ञात होने से रोकना)

The correct answer is: innocent

Key Points

• Let's look at the meaning of the given words.
• Innocent means free from moral wrong; without sin; pure. (मासूम)
• Shy means nervous and uncomfortable with other people. (शर्मीला)
• Fragile means something easily broken or damaged.  (नाजुक)
• Cunning means characterized by cleverness and trickery. (चालाक)
• So, as per the meaning of the given words "innocent" is the most appropriate option.
• The implied meaning of the sentence is: We wished we could have lived out our entire lives as naive children.

Therefore, the correct answer is option 1.

Complete sentence: We would have loved to stay innocent like children all our lives.

The erroneous part of the sentence is: I remember vivid

Key Points

• Replace the adjective "vivid" with the adverb "vividly" to correct the sentence.
• Adding -ly to an adjective makes an adverb.
• An adjective is a word that describes nouns, such as large or beautiful.
• An adverb is a word that describes verbs, adjectives, and other adverbs, such as silently or really.
• In the given sentence an adverb should be used because it is describing the verb "remember**.**"
• Vividly means in a way that produces powerful feelings or strong, clear images in the mind. (एक तरह से जो मन में शक्तिशाली भावनाओं या मजबूत, स्पष्ट छवियों को उत्पन्न करता है)
• Example - Margaret remembers vividly the day she received the grim news.
• So, from here we can easily say that using "vividly" instead of "vivid" is the correct choice for the given sentence.

Therefore, the correct answer is option 4, which contains an error.

Corrected sentence: I remember vividly the last road trip we had before the pandemic hit.

The erroneous part of the sentence is: gone to

Key Points 

• Replace "gone" with "go" to correct the sentence.
• Gone is the third form of the verb "Go"
• The given sentence is written in simple past tense and in negative form.
• The structure should be: Subject + didnot+ v1+ object
• Example - I didn't want this to happen.
• So, the first form of the verb "go" should be used here.

Therefore, the correct answer is option 1, which contains an error in it.

Corrected sentence: The manager did not go to Bombay yesterday.

The correct answer is: Clean

Key Points 

• Squalid means a place extremely dirty and unpleasant. (एक जगह बेहद गंदी और अप्रिय)

• Example - Conditions in the camp were squalid.

• Marked option Clean means free from dirt, marks, or stains. (गंदगी, निशान या दाग से मुक्त)

• Example - The room was spotlessly clean.

• It is clear that 'Squalid' and 'Clean' are opposite in meaning.

Therefore the correct answer is option 4.​

Additional Information 

• Contemporary means belonging to the same period. (इसी काल से संबंधित)
• Bare means without covering or clothing; naked. (बिना ढके या कपड़े के)
• Advanced means modern and recently developed. (आधुनिक और हाल ही में विकसित)

The correct answer is: Conceit

Key Points 

• Vanity means excessive pride in or admiration of one's own appearance or achievements. (किसी की अपनी उपस्थिति या उपलब्धियों पर अत्यधिक गर्व या प्रशंसा)
• Example - It flattered his vanity to think I was in love with him.
• Marked option Conceit means excessive pride in oneself. (अपने आप में अत्यधिक अभिमान)
• Example - He was puffed up with conceit.
• It is clear that 'Vanity' and 'Conceit' are similar in meaning.

Therefore the correct answer is option 4.​

Additional Information 

• Beauty means the quality attributed to whatever pleases or satisfies the senses or mind. (सुंदरता)
• Empty means containing nothing; having nothing in it. (जिसमें कुछ भी न हो)
• Humility means freedom from pride or arrogance or the quality or state of being humble. (विनम्रता)

The correct answer is: Our grandmother was preparing Diwali sweets.

Key Points 

• The given sentence is in the passive form and we need to change it into the active form.
• The given sentence is written in the passive form of the past continuous tense.
• The past continuous tense in the passive voice has the following structure: Object+ was/were +being+V3+ by + subject
• The past continuous tense in the active voice has the following structure: Subject + was/were + v1+ing+ object
• Example - 
• Passive voice - The wall was being painted by Rahul.
• Active voice - Rahul was painting the wall.
• Rules of Conversion from Passive to Active Voice:
• Identify the subject, the verb, and the object.
• Change the object into the subject.
• Change the verb into the past participle form of the verb.
• Change the subject into an object.

​Therefore, the correct answer is option 2.

Correct Active Voice: Our grandmother was preparing Diwali sweets.

The correct answer is: will be

Key Points

• We know that when two singular nouns joined by "AND" denote the same person or thing, the pronoun used for them must be singular in number.
• The article ‘THE’ is placed before the first Noun.
• Here, in the given sentence "This cat and mouse game" is a singular subject.
• But the sentence is talking about a situation that will happen in the future, as "in the future" is already mentioned.
• So, we have to write the sentence in the simple future tense.
• For that reason, "will be" should be used in the given sentence.
• If the sentence was written in the present tense then we had to use "is" in place of "are."

Therefore, the correct answer is option 3, which can substitute the underlined segment in the given sentence.

Corrected sentence: This cat and mouse game will be famous with all children in the future.

The correct answer is: livelihood

Key Points

• Let's look at the meaning of the given words.
• Balanced means keeping or showing a balance; in good proportions. (संतुलन रखना या दिखाना; अच्छे अनुपात में)
• Total means comprising the whole number or amount. (पूरी संख्या या राशि शामिल)
• Strict means demanding that rules concerning behaviour are obeyed and observed. (यह मांग करना कि व्यवहार से संबंधित नियमों का पालन और पालन किया जाता है)
• Livelihood means a means of securing the necessities of life. (जीवन की आवश्यकताओं को सुरक्षित करने का एक साधन)
• The implied meaning of the passage is: MGNREGA seeks to increase the living standards of rural households in the nation.
• ​So, as per the meaning of the given words "livelihood" is the most appropriate option to fill in the blank.

Therefore, the correct answer is option 4.

Complete passage: The Mahatma Gandhi National Rural Employment Guarantee Act 2005 (Mahatma Gandhi NREGA) aims at enhancing livelihood security of households in rural areas of the country by providing at least one hundred days of guaranteed wage employment in a financial year to every household whose adult members volunteer to do unskilled manual work.

The correct answer is: An easy task

Key Points 

• The idiom "A piece of cake" means something that is very easy to do. (कुछ ऐसा जो करना बहुत आसान है)
• Example - The boy found the project to be a piece of cake because it was incredibly easy.
• So, it is clear that the most appropriate meaning of the given idiom "A piece of cake" is "An easy task"

Hence, the correct answer is option 2.

The correct answer is: Declare

Key Points 

• Profess means to claim something. (कुछ दावा करने के लिए)
• Example - They professed to have no knowledge of the event.
• Marked option Declare means to announce or express something clearly and publicly. (स्पष्ट रूप से और सार्वजनिक रूप से कुछ घोषित या व्यक्त करना)
• Example - The courts declared the policy unconstitutional.
• It is clear that 'Profess' and 'Declare' are similar in meaning.

Therefore the correct answer is option 4.​

Additional Information 

• Convince means to cause someone to believe something or to do something. (किसी को कुछ विश्वास करने या कुछ करने के लिए प्रेरित करना)
• Assume means to accept something to be true without question or proof. (बिना किसी प्रश्न या प्रमाण के किसी बात को सत्य मान लेना)
• Predict means to say that an event or action will happen in the future. (यह कहना कि भविष्य में कोई घटना या कार्य होगा)

The correct answer is: Queeuee

Key Points 

• The correct spelling should be "Queue"
• Queue means a line or sequence of people or vehicles awaiting their turn to be attended to or to proceed. (लोगों या वाहनों की एक पंक्ति या अनुक्रम में शामिल होने या आगे बढ़ने के लिए अपनी बारी का इंतजार कर रहे हैं)

Therefore, the correct answer is option 4, which is incorrectly spelt.

Additional Information 

• Glamour means an attractive or exciting quality that makes certain people or things seem appealing. (एक आकर्षक या रोमांचक गुण जो कुछ खास लोगों या चीजों को आकर्षक बनाता है)
• Quarantine means to stay away from others for a period of time because you have a disease, or may have one, in order to prevent the spread of the disease. (कुछ समय के लिए दूसरों से दूर रहना क्योंकि आपको कोई बीमारी है, या हो सकती है, ताकि बीमारी को फैलने से रोका जा सके)
• Guarantee means a promise that something will be done or will happen. (एक वादा है कि कुछ किया जाएगा या होगा)

The correct answer is: Spychology

Key Points 

• The correct spelling should be "Psychology"
• Psychology means the study of mind and behavior in relation to a particular field of knowledge or activity. (मनोविज्ञान)

Therefore, the correct answer is option 4, which is incorrectly spelt.

Additional Information 

• Philosophy means the study of the fundamental nature of knowledge, reality, and existence, especially when considered as an academic discipline. (दर्शनशास्र)
• Geography means the study of the physical features of the earth and its atmosphere, and of human activity as it affects and is affected by these, including the distribution of populations and resources and political and economic activities. (भूगोल-शास्र)
• Economics means the social science that studies the production, distribution, and consumption of goods and services. (अर्थशास्त्र)

The correct answer is: heedless

Key Points 

• Solicitous means showing care and helpful attention to someone. (किसी की देखभाल और मददगार ध्यान दिखाना)
• Example - He made a solicitous enquiry after her health.
• Marked option Heedless means showing a reckless lack of care or attention. (देखभाल या ध्यान की लापरवाह कमी दिखा रहा है)
• Example - Heedless of the hurricane warnings, they took the small boat out into open water.
• It is clear that 'Solicitous' and 'Heedless' are opposite in meaning.

Therefore the correct answer is option 2.​

Additional Information 

• Worried means anxious or troubled about actual or potential problems. (वास्तविक या संभावित समस्याओं के बारे में चिंतित या परेशान)
• Knowledgeable means intelligent and well informed. (बुद्धिमान और अच्छी तरह से सूचित)
• Interesting means engaging or exciting and holding the attention or curiosity. (आकर्षक या रोमांचक और ध्यान या जिज्ञासा को पकड़ना)

The correct answer is: Heart skipped a beat

Key Points 

• The idiom "Heart skipped a beat" is used to say that someone is suddenly very surprised, excited, or nervous about something. (कोई अचानक किसी बात को लेकर बहुत हैरान, उत्साहित या घबराया हुआ है)
• Example - The shocking news made her heart skip a beat.
• So, it is clear that the given idiom "Heart skipped a beat" means "extremely nervous"

Hence, the correct answer is option 1.

Additional Information

• "Like a cakewalk" means that something is very easy to do or achieve. (यह करना या हासिल करना बहुत आसान है)
• "Crying Wolf" means not be believed because of previous false warnings. (पिछली झूठी चेतावनियों के कारण विश्वास नहीं किया गया)
• "Raining cats and dogs" means to rain very heavily. (बहुत भारी बारिश होना)

The correct answer is: Children welcome summer vacation with zeal.

Key Points

• The error here lies in the spelling of "vacation"
• In option 1 it is written as "vecation"
• In option 2 it is written as "vacashun"
• In option 3 it is written as "vakation"
• In option 4 it is written as "vacation" which is the correct spelling.
• Vacation means a time when someone does not go to work or school but is free to do what they want, such as travel or relax. (एक समय जब कोई काम या स्कूल नहीं जाता है, लेकिन वह जो चाहे करने के लिए स्वतंत्र होता है, जैसे कि यात्रा करना या आराम करना)

From here we can easily conclude that the sentence written in the option number 4 has no spelling error.

Therefore, the correct answer is option 4.

The correct answer is: adverse

Key Points

• Let's look at the meaning of the given words.
• Contemptuous means feeling or expressing deep hatred or disapproval. (महसूस करना, या गहरी घृणा या अस्वीकृति व्यक्त करना)
• Adverse means having a negative or harmful effect on something. (किसी चीज पर नकारात्मक या हानिकारक प्रभाव पड़ना)
• Affirm means to state something as true. (किसी बात को सच बताना)
• Disperse means to distribute or spread over a wide area. (एक विस्तृत क्षेत्र में वितरित या फैला हुआ)
• The implied meaning of the sentence is: Certain unfavorable health issues have been linked to immunizations, according to certain claims.
• So, as per the meaning of the given words "adverse" is the most appropriate option.

Therefore, the correct answer is option 2.

Complete passage: Claims have been made that vaccines are responsible for certain (1) adverse health conditions, particularly autism, speech disorders, and inflammatory bowel disease. Some of those claims focused on thimerosal, a mercury-containing compound used as a preservative in vaccines. Some people believed that autism (2)__________ a form of mercury poisoning, caused specifically by thimerosal in childhood vaccines. Those claims have been discredited. Still, misinformation and fear generated by false claims about associations (3)__________ autism and vaccines had a significant impact on individuals’ perceptions about vaccine safety. (4), most individuals in countries where vaccination is widespread have never personally experienced the vaccine-preventable disease. Thus, the focus of concern for some people shifted from the negative effects of the vaccine-preventable disease to the possible negative effects of the vaccines (5).

The correct answer is: was

Key Points

• Let's look at the meaning of the given auxiliary verbs (helping verbs).
• Auxiliary or Helping verbs are used together with a main verb to show the verb's tense.
• 'Will' is used in all other persons (you, he/she, they).
• 'Shall' is used in the first person to form sentences in the future tense. (I, we)
• 'Was' is the first and third-person singular past tense of the verb to be.
• 'Were' is the second-person singular past and plural past of to be.
• In the given passage "autism" is a singular subject and the passage is written in the past tense.
• So, as per the usage of the given auxiliary verbs "was" is the most appropriate option.

Therefore, the correct answer is option 3.

Complete passage: Claims have been made that vaccines are responsible for certain (1) adverse health conditions, particularly autism, speech disorders, and inflammatory bowel disease. Some of those claims focused on thimerosal, a mercury-containing compound used as a preservative in vaccines. Some people believed that autism (2) was a form of mercury poisoning, caused specifically by thimerosal in childhood vaccines. Those claims have been discredited. Still, misinformation and fear generated by false claims about associations (3)__________ autism and vaccines had a significant impact on individuals’ perceptions about vaccine safety. (4), most individuals in countries where vaccination is widespread have never personally experienced the vaccine-preventable disease. Thus, the focus of concern for some people shifted from the negative effects of the vaccine-preventable disease to the possible negative effects of the vaccines (5).

The correct answer is: between

Key Points

• Let's look at the usage of the given words.
• 'One another' is used for each of two or more people, animals, etc, who are doing something together or in relationship to the other or others in the group.
• 'Between' is used to refer to the time, space, or differences that separate two things, people, places, ideas, etc.
• 'Each other' is used to show that each person in a group of two or more people does something to the others.
• 'Through' is used as a function word to indicate movement into at one side or point and out at another and especially the opposite side.
• In the given passage the difference of both autism and vaccine is shown here.
• The implied meaning of the sentence is: Nevertheless, misleading assertions regarding links among both autism and vaccinations created disinformation and anxiety that significantly influenced people's opinions of vaccine safety.
• So, as per the usage of the given words "between" is the most appropriate option.

Therefore, the correct answer is option 2.

Complete passage: Claims have been made that vaccines are responsible for certain (1) adverse health conditions, particularly autism, speech disorders, and inflammatory bowel disease. Some of those claims focused on thimerosal, a mercury-containing compound used as a preservative in vaccines. Some people believed that autism (2) was a form of mercury poisoning, caused specifically by thimerosal in childhood vaccines. Those claims have been discredited. Still, misinformation and fear generated by false claims about associations (3) between autism and vaccines had a significant impact on individuals’ perceptions about vaccine safety. (4), most individuals in countries where vaccination is widespread have never personally experienced the vaccine-preventable disease. Thus, the focus of concern for some people shifted from the negative effects of the vaccine-preventable disease to the possible negative effects of the vaccines (5).

The correct answer is: For instance

Key Points

• Let's look at the usage of the given words.
• 'For instance' is used especially as an example of something that happens generally.
• 'Therefore' is used as a result; because of that; for that reason.
• 'In addition' is used as an extra piece of information.
• 'Likewise' is used in the same way or manner; similarly.
• The implied meaning of the sentence is: "For example, the majority of people in nations with high vaccination rates have never directly encountered the sickness that vaccines can prevent."
• So, as per the usage of the given words "For instance" is the most appropriate option.

Therefore, the correct answer is option 1.

Complete passage: Claims have been made that vaccines are responsible for certain (1) adverse health conditions, particularly autism, speech disorders, and inflammatory bowel disease. Some of those claims focused on thimerosal, a mercury-containing compound used as a preservative in vaccines. Some people believed that autism (2) was a form of mercury poisoning, caused specifically by thimerosal in childhood vaccines. Those claims have been discredited. Still, misinformation and fear generated by false claims about associations (3) between autism and vaccines had a significant impact on individuals’ perceptions about vaccine safety. (4) for instance, most individuals in countries where vaccination is widespread have never personally experienced the vaccine-preventable disease. Thus, the focus of concern for some people shifted from the negative effects of the vaccine-preventable disease to the possible negative effects of the vaccines (5)__________.

The correct answer is: themselves.

Key Points

• Let's look at the usage of the given reflexive pronouns.
• A reflexive pronoun is a pronoun used as an object that refers to the same person or thing as the subject.
• 'Themselves' is used as the object of a verb or preposition to refer to a group of people or things previously mentioned as the subject of the clause.
• 'Yourself' is used to refer to the person being addressed as the object of a verb or preposition when they are also the subject of the clause.
• 'Ourselves' is used as the object of a verb or preposition when this is the same as the subject of the clause and the subject is the speaker and one or more other people considered together.
• 'Itself' is used as the object of a verb or preposition to refer to a thing or animal previously mentioned as the subject of the clause.
• Since 'themselves' is an empathetic pronoun, it has the same connotations as "vaccines". "themselves" refer to the "vaccines" in this context.
• So, as per the usage of the given pronouns "themselves" is the most appropriate option.

Therefore, the correct answer is option 1.

Complete passage: Claims have been made that vaccines are responsible for certain (1) adverse health conditions, particularly autism, speech disorders, and inflammatory bowel disease. Some of those claims focused on thimerosal, a mercury-containing compound used as a preservative in vaccines. Some people believed that autism (2) was a form of mercury poisoning, caused specifically by thimerosal in childhood vaccines. Those claims have been discredited. Still, misinformation and fear generated by false claims about associations (3) between autism and vaccines had a significant impact on individuals’ perceptions about vaccine safety. (4) for instance*, most individuals in countries where vaccination is widespread have never personally experienced the vaccine-preventable disease. Thus, the focus of concern for some people shifted from the negative effects of the vaccine-preventable disease to the possible negative effects of the vaccines* (5) themselves.

Here, the alphabets and their positions are shown as below:

The logic here is: Each word's first and last letters interchange their position and add 1 to the 2^nd letter, 2 to the 3^rd letter, 3 to the 3^rd letter to get the final answer.

For the word POINT ∶ TPKQP,

For the word BONES ∶ SPPHB, 

Similarly, for the word SEVEN:

Hence, the option 'NFXHS' is related to the word SEVEN,

 

Confusion Point: Keep in mind the positions of all the letters.

Here, we need to find the pattern used to get the missing numbers.

The logic here is: Each odd terms are added with odd numbers  to get the next consecutive numbers and each even terms are multiplied with next odd number to get the next consecutive number in the given series.

So, we have:

33 = 11 $\times$ 3

38 = 33 + 5

266 = 38 $\times$ 7

275 = 266 + 9

Thus, the missing number we will get, 275 $\times$ 11 = 3025

Hence, the correct  answer is 3025.

Here, the alphabets and their positions are shown as below:

The corresponding of alphabets are:

The logic here is:

First we need to start writing the given word from right to left and then see the corresponding of each letter is the code language.

For the word FLUTE, reverse is ETULF then it is written as VGFOU:

For the word EXILE, the reverse is ELIXE then it is written as VORCV

Similarly, for the word BLOOD, the reveres is DOOLB then it is written as WLLOY.

 

Hence, the given word can be written as 'WLLOY'.

Here, we will interchange the signs to get the balance equation.

Given: 115 * 25 * 15 * 2 * 3

First we will check options one by one.

1) –, +, =, ×

$\Rightarrow$ 115 - 25 + 15 = 2 × 3

Applying BODMAS rule, we will get,

$\Rightarrow$ 115 - 25 + 15 = 6

$\Rightarrow$ 105 = 6, which is not possible. This option is eliminated.

2) –, =, ×, ×

$\Rightarrow$ 115 - 25 = 15 × 2 × 3

Applying BODMAS rule, we will get,

$\Rightarrow$ 115 - 25 = 90

$\Rightarrow$ 90 = 90, which is possible. This option is true. 

3) =, +, +, ×

$\Rightarrow$ 115 = 25 + 15 + 2 × 3

Applying BODMAS rule, we will get,

$\Rightarrow$ 115 = 25 + 15 + 6

$\Rightarrow$ 115 = 46, which is not possible. This option is eliminated.

4) =, ×, ×, ×

$\Rightarrow$ 115 = 25 × 15 × 2 × 3

Applying BODMAS rule, we will get,

$\Rightarrow$ 115 = 2250, which is not possible. This option is eliminated.

Hence, the correct option is 2.

Here, we have some information given and using that, we need to find out the answer.

According to the given information:

1) Pari has scored more than Kirti but less than Siya:

$\Rightarrow$ Siya > Pari > Kirti $\Rightarrow$ Kirti < Pari < Siya

2) Deepika has scored less than Preeti but more than Amita.

$\Rightarrow$ Amita < Deepika < Preeti

3. Kirti has scored more than Preeti.

$\Rightarrow$ Preeti < Kirti

4. Siya is not the highest scorer: This means Jeet is the highest scorer.

​$\Rightarrow$ Siya < Jeet​

Next, we will combine all the statements and we will get,

$\Rightarrow$ Amita < Deepika < Preeti < Kirti < Pari < Siya < Jeet

Thus, Jeet and Siya have scored more than Pari. But in option there is only Jeet.

So, Jeet scored more than Pari.

Hence, the correct answer is Jeet.

Here, we need to find the pattern to get the missing number.

The logic is: The 2^nd term of first row is multiplied by 4 to get the 3^rd term. Then the 2^nd term of second row is multiplied by 3 to get the 3^rd term.

 

Logic: Second Number x 4 = Third Number

In 1^st row: 16, 12, 48 → 12 $\times$ 4 = 48

 

Logic: Second Number x 3 = Third Number

In 2^nd row: 9, 8, 24 → 8 $\times$ 3 = 24

 

Logic: Second Number x 2 = Third Number

In 3^rd row: 4, 24, ? → 24 $\times$ 2 = 48

 

Hence, the missing number is 48.

The logic followed here is:

Logic: (3^rd number - 1^st number - 1) ÷ 4 = 2^nd number.

Now,

First row- 14, 4, 31

→ (31 - 14 - 1) ÷ 4

→ 16 ÷ 4 = 4

Second row- 38, 10, 79

→ (79 - 38 - 1) ÷ 4

→ 40 ÷ 4 = 10.

Third row- 30, 8, ?

→ (63 - 30 - 1) ÷ 4

→ 32 ÷ 4 = 8.

Hence, the correct answer is "Option 3".

Here, we are given some signs in the options and we need to check the correct option.

Given: 54 * 14 * 5 * 72 * 18 * 31 * 11

First, we will check the options one by one.

1) =, ×, −, ÷, ×, +

$\Rightarrow$54 = 14 × 5 - 72 ÷ 18 × 31 + 11

Applying BODMAS rule, we will get, 

$\Rightarrow$ 54 = 14 × 5 - 4 × 31 + 11

$\Rightarrow$ 54 = 70 - 124 + 11

$\Rightarrow$ 54 = -43, which is not possible. This option is true.

 

2) +, −, =, ×, −, ×

$\Rightarrow$ 54 + 14 - 5 = 72 × 18 - 31 × 11

Applying BODMAS rule, we will get, 

$\Rightarrow$ 54 + 14 - 5 = 1296 - 341

$\Rightarrow$ 63 = 955, which is possible. This option is true.

 

3) =, ×, +, ÷, −, +

$\Rightarrow$ 54 = 14 × 5 + 72 ÷ 18 - 31 + 11

Applying BODMAS rule, we will get, 

$\Rightarrow$ 54 = 14 × 5 + 4 - 31 + 11

$\Rightarrow$ 54 = 70 + 4 - 31 + 11

$\Rightarrow$ 54 = 54, which is possible. This option is true.

 

4) -, +, =, ×, ÷, -

$\Rightarrow$ 54 - 14 + 5 = 72 × 18 ÷ 31 - 11

Applying BODMAS rule, we will get, 

$\Rightarrow$ 54 - 14 + 5 = 72 × 0.5 - 11

$\Rightarrow$ 54 - 14 + 5 = 36 - 11

$\Rightarrow$ 45 = 25, which is not possible. This option is not true.

Hence, the correct option is 3.

The logic followed here is:

Logic: (1^st number - 2^nd number) × 1^st number = 3^rd number.

Now,

For, (15, 5, 150)

→ (15 - 5) × 15

⇒ 10 × 15 = 150.

For, (7, 5, 14)

→ (7 - 5) × 7

⇒ 2 × 7 = 14.

Now, check the options:

Option 1: (16, 8, 128)

→ (16 - 8) × 16

⇒ 8 × 16 = 128.

Option 2: (12, 6, 40)

→ (12 - 6) × 12

⇒ 6 × 12 = 72 ≠ 40.

Option 3: (11, 8, 37)

→ (11 - 8) × 11

⇒ 3 × 11 = 33 ≠ 37.

Option 4: (14, 7, 108)

→ (14 - 7) × 14

⇒ 7 × 14 = 98 ≠ 108.

Hence, the correct answer is "Option 1".

Here, we have some statements and we need find out if the given conclusion is true or not.

First, we will see the statements as below:

1. All shoes are boots:

2) All boots are heels:

3) Some slippers are shoes:

Next, we will combine all the three statements and we will get the venn diagram as below:

 

Thus, from this above diagram we will see the conclusions as below:

1) All shoes are heels: Since, combining statement 1 and 2, we will get this conclusion. This is true.

2) No slipper is a boot: Since, some slippers are shoes and all shoes are boots. Then some slippers are boot and so this is not true.

3) All shoes are slippers: We are given some slippers are shoes and so some shoes can be slippers but not all. This is not true.

Hence, from the above statements, only conclusion 1 follows.

Here, we need to find the pattern and get the missing number.

The logic here follows is as:

11 $\times$ 2 = 22, 

71 $\times$ 6 = 426

Thus, the pattern is each odd term is multiplied by multiple of 2 to get the answer.

So, 19 $\times$ 4 = 76

Hence, the correct answer is 76.

Here, we need to find the pattern and get the missing number.

The logic here is: (1^st number × 3) + (1^st number ÷ 3) = 2^nd number.

1. 93 $\times$ 3 + 31 = 310,

2. 54 $\times$ 3 + 18 = 180

3. 48 $\times$ 3 + 16 = 160 $\ne$ 165

4. 33 $\times$ 3 + 11 = 110

Thus, the pattern is: Each 1^st is multiplied by 3 and then 2^nd term first two digits are added to the answer to get the final answer.

Hence, the correct option is 48 - 165

Alternate MethodLogic: (1^st number ÷ 3) × 10 = 2^nd number.

1. (93 ÷ 3) × 10 → 31 × 10 = 310,

2) (54 ÷ 3) × 10 → 18 × 10 = 180,

3. (48 ÷ 3) × 10 → 16 × 10 = 160 ≠ 165

4) (33 ÷ 3) × 10 → 11 × 10 = 110.

The figure is given below:

The shaded portion is embedded in the figure.

Hence, 'option 3' is the correct answer.

Here, we have a paper which is folded three times and then cut it out to get some design on it.

First, we will open the 1^st fold and see as below:

So, we can see that we will get the mirror image of 1^st part in the 2^nd part.

Next, we will open the last part and see, we will get the water image of the 1^st half part on to the 2^nd half part.

Hence, the correct answer is option 1.

Here, the alphabets and their positions are shown as below:

The logic follows here is as: The last letter of each term is subtracted by 2, to get the first letter of the next term.

Hence, the missing place will have 'CBA' as the correct answer.

Here, the alphabets and their positions are shown as below:

The logic follows here is:

    Each alphabet is coded as two letters +1 and -1 of its positional value.

Similarly,

Hence the correct answer is option 4)

Here, we are given the word and we need to find the mirror image of it.

First keep in mind that, in mirror image the first letter will be the last letter of the given word. Also, the size of the letters will not change i.e. capital letters will appear capital only.

The correct mirror image of the given combination when the mirror is placed at XY is as shown below :-

Detailed Explanation :- 

• The line XY is the mirror.
• The word "

"  ends with "

" . So, the mirror image will start with "

" → Option 4 is eliminated.

• The last letter  "

" is used . So, the first letter of the mirror image will be "

"→ Options 1 and 3 are eliminated.

 

Hence, 'Option 2' is the correct answer.

 

Additional Information

• In a mirror image Left changes to rightand theright changes to leftwheretop and bottom remains the same.
• When placement of mirror not given, by default we will assume it in vertical direction.
• The starting letter word will be at last after the mirror image drawn with left changes to right and right changes to left.

Here, the alphabets and their positions are shown as below:

The logic follows here is:

Hence, the correct  answer is 'LYZJ'.

Confusion Point: Keep in mind the positions of all the letters.

Here we are given two dices and faces 1 and 3 are common in both.

So, we have:

 

So, we will rotate the 2^nd dice in such a way, that it will have the same positions of the common faces.

Thus, the face opposite to 4 will be 2.

Hence, the face opposite to number 4 will be number 2 .

Here, we have given a day and need to find the next day of the given date.

The year 2000 is a leap day. So, it has 2 odd days.

So, 19^th Sep of 1999 was Sunday and so 19^th Sep of 2000 will be Tuesday.

As we know, Sep has 30 days. So, 30 - 19 = 11 days.

Then, Oct = 31 days, Nov = 30 days and Dec = 31 days.

Total = 11 + 31 + 30 + 31 = 103 days

Next, for 19^th Jan 2001 = 19 days.

Thus, the total = 103 + 19 = 122 days.

So, 122 $\div$ 7 = 17 weeks = 3 odd days.

This means Tuesday + 3 days = Friday

So, the Friday is the day of the week on 19^th January 2001.

Hence, the correct option is 1.

Confusion Point: Keep in mind what to you mean by number of odd days, leap year and ordinary years.

Here, we have an image in the square and it is rotating.

Given image is:

As we can see that each image is moving at an angle of 90 degree in clockwise direction. 

From the images, we can see that, the arrow is changing its place and the image in the center is also changing.

So, we can say that the arrow will be facing upwards at the right side on the top. This means we will get the 1^st image as out answer.

Thus, we get:

Hence, the correct answer is option 4.

Here, we will apply the options one by one to check which option gives the correct equation.

Given: 16 * 8 * 10 * 2 * 4 * 28

1) ×, ÷, –, +, =

$\Rightarrow$16 $\times$ 8 $\div$ 10 - 2 + 4 = 28

Applying BODMAS rule, we will get,

$\Rightarrow$ 16 $\times$ 0.8 - 2 + 4 = 28

$\Rightarrow$ 12.8 - 2 + 4 = 28

$\Rightarrow$ 14.8 $\ne$ 28, this option is eliminated.

 

2. ÷, ×, +, –, =

$\Rightarrow$ 16 $\div$ 8 $\times$ 10 + 2 - 4 = 28

Applying BODMAS rule, we will get,

$\Rightarrow$ 2 $\times$ 10 + 2 - 4 = 28

$\Rightarrow$ 20 + 2 - 4 = 28

$\Rightarrow$ 18 $\ne$ 28, this option is eliminated

 

3)  –, +, ÷, ×, =

$\Rightarrow$ 16 - 8 + 10 $\div$ 2 $\times$ 4 = 28

Applying BODMAS rule, we will get,

$\Rightarrow$ 16 - 8 + 5 $\times$ 4 = 28

$\Rightarrow$ 16 - 8 + 20 = 28

$\Rightarrow$ 28 = 28, this option is true.

 

4) ÷, ×, –, +, =

$\Rightarrow$ 16 $\div$ 8 $\times$ 10 - 2 + 4 = 28

Applying BODMAS rule, we will get,

$\Rightarrow$ 2 $\times$ 10 - 2 + 4 = 28

$\Rightarrow$ 20 - 2 + 4 = 28

$\Rightarrow$ 18 $\ne$ 28, this option is eliminated

Hence, the correct option is 3.

Confusion Point: Keep in mind to follow BODMAS rule while solving these type of problem.

The figure is given below:

The shaded portion is embedded in the figure.

Hence, 'option 3' is the correct answer.

Here, we need to arrange the given options in alphabetical order:

So, the arrangement is done as shown below:

4.  Analogies 2.  Analyze 1.  Ancestors 5.  Android 3.  Antibiotic

Hence, the correct answer  is '4, 2, 1, 5, 3'.

Here, we need to interchange the numbers to make the given equation correct.

Given: (2)3 ÷ 4 + (72 ÷ 6) × 9 – (21 × 3) + (15 × 7) = 92

First we will apply each options one by one as shown below:

1. 6 and 3

$\Rightarrow$ (2)3 ÷ 4 + (72 ÷ 3) × 9 – (21 × 6) + (15 × 7) = 92

Applying BODMAS rule, we will get,

$\Rightarrow$ 8 ÷ 4 + 24  × 9 – 126 + 105 = 92

$\Rightarrow$ 2 + 216 - 126 + 105 = 92

$\Rightarrow$ 197 = 92, is not possible. This option is not true.

 

2. 3 and 7

$\Rightarrow$ (2)3 ÷ 4 + (72 ÷ 6) × 9 – (21 × 7) + (15 × 3) = 92

Applying BODMAS rule, we will get,

$\Rightarrow$ 8 ÷ 4 + 12  × 9 – 147 + 45 = 92

$\Rightarrow$ 2 + 108 - 147 + 45 = 92

$\Rightarrow$ 8 = 92, is not possible. This option is not true.

 

3. 3 and 9

$\Rightarrow$ (2)3 ÷ 4 + (72 ÷ 6) × 3 – (21 × 9) + (15 × 7) = 92

Applying BODMAS rule, we will get,

$\Rightarrow$ 8 ÷ 4 + 12 × 3 – 189 + 105 = 92

$\Rightarrow$ 2 + 36 - 189 + 105 = 92

$\Rightarrow$ -46 = 92, is not possible. This option is not true.

 

4. 6 and 9 

$\Rightarrow$ (2)3 ÷ 4 + (72 ÷ 9) × 6 – (21 × 3) + (15 × 7) = 92

Applying BODMAS rule, we will get,

$\Rightarrow$ 8 ÷ 4 + 8 × 6 – 63 + 105 = 92

$\Rightarrow$ 2 + 48 - 63 + 105 = 92

$\Rightarrow$ 92 = 92, is possible. This option is true.

Hence, the correct answer is '6 and 9'.

Confusion Point: Keep in mind to apply BODMAS rule, while solving these type of problems.

Given:

Discount percentage is 35%

Selling price of the toy is 975Rs

Calculation:

Let the marked price be M

The selling price is 35% less than marked price   

Which implies selling price is 100-35=65% of the marked price

=$\frac{65}{100}\times M=975$

Solving this we get M as 1500Rs.

The marked price of the toy is Rs. 1500

Theory:

The third proportional means the three numbers are in same proportion to each 

Let three numbers a, b, c

Then a:b::b:c

Calculation:

$\frac{a}{b}=\frac{b}{c}$

Thus we can write  $c=\frac{b^2}{a}$

Here a=14 , b=28

c=$=\frac{28^2}{14}=56$

∴ The third proportional is 56

Given:

x² + 2x + 3

x = –2

Calculation:

x = – 2

⇒ x + 2 = 0

Now we have to expand only options 2 and 3,

As in options 1 and 4, the first term is (x - 2)

Expanding option 2, 

⇒ (x + 2)2 – 2(x + 2) + 3

⇒ x² + 4x + 4 - 2x - 4 + 3

⇒ x² + 2x + 3

Hence it gives the given expression, 

∴ Option 2 is correct.

Additional Information As option 2 has given the desired result, 

We don't have to go for 3^rd option. 

Mistake Points

x = – 2

⇒ x + 2 = 0

If we consider option 3, substituting x + 2 instead of x, 

⇒ (x + 2)2 + 2(x + 2) + 3

⇒ x2 + 4x + 4 + 2x + 4 + 3

⇒ x2 + 6x + 11 

Hence it is not equal to the x2 + 2x + 3 (given expression)

∴ Option 3 can not be the answer.

Shortcut Trick

Ratio of MP to CP = 70 ÷ 42 = 5/3

Ratio of SP to MP = 21 ÷ 25

Multiply both ratios to find SP/CP: (SP/MP) × (MP/CP) = (21/25) × (5/3) = 7/5

Profit % = [(SP − CP) / CP] × 100 = [(7 − 5) / 5] × 100 = 40%

∴ The correct answer is 40%.

Alternate Method

Given:

42 × Marked Price (MP) = 70 × Cost Price (CP)

25 × Selling Price (SP) = 21 × Marked Price (MP)

Formula Used:

Profit = SP − CP

Profit % = (Profit / CP) × 100

CP = 15

SP = 21

MP = 25

Profit = +6

Markup

Discount

Calculations:

⇒ MP / CP = 70 / 42 = 5 / 3

⇒ SP / MP = 21 / 25

⇒ To equalise the ratio value of MP in both conditions, multiply the first ratio by 5.

⇒ MP / CP = 25 / 15

⇒ By combining the ratios, we establish CP = 15, MP = 25, and SP = 21

⇒ Profit = SP − CP = 21 − 15 = 6

⇒ Profit % = (6 / 15) × 100 = 40%

∴ The correct answer is 40%.

Additional Information

Marked Price and Discount

Discount is fundamentally derived from the Marked Price. Formula: SP = MP − (Discount% × MP / 100).

Successive Discounts

If two successive discounts of d₁% and d₂% are applied, the overall equivalent discount is computed as [d₁ + d₂ − (d₁ × d₂) / 100]%.

Relation between CP and MP

A rapid formula to interlink the direct ratio between Cost Price and Marked Price is: CP / MP = (100 − Discount%) / (100 + Profit%).

Given:

The following pie chart describes the number distribution of research scholars department-wise in a university.

Calculations:

According to the question,

Physics= 14%

Mathematics= 22%

Zoology = 22%

Hindi = 12%

Chemistry= 12%

Education= 17%

Now, Physics+ Mathematics = 14 + 22 = 36 %

Only, Zoology + Education= 22 + 18 = 40 % exceeds the combination of Physics and Mathematics.

Hence, The Required answer is zoology and education.

Shortcut Trick

When equal distances are covered at speeds a and b, the average speed is given by (2 × a × b) ÷ (a + b).

⇒ Average Speed = (2 × 21 × 17) ÷ (21 + 17)

⇒ Average Speed = 714 ÷ 38

⇒ Average Speed = 357 ÷ 19

⇒ Average Speed = 18¹⁵/₁₉ km/h

∴ The correct answer is 18¹⁵/₁₉ km/h.

Alternate Method

Given:

Speed during onward journey (S₁) = 21 km/h

Speed during return journey (S₂) = 17 km/h

Let the one-way distance between the two points be d km.

Point A Point B

Distance = d

Onward Speed = 21 km/h

Return Speed = 17 km/h

Formula Used:

Average Speed = Total Distance ÷ Total Time

Time = Distance ÷ Speed

Calculations:

⇒ Total Distance = d + d = 2d

⇒ Time taken for onward journey = d ÷ 21

⇒ Time taken for return journey = d ÷ 17

⇒ Total Time = (d ÷ 21) + (d ÷ 17)

⇒ Total Time = (17d + 21d) ÷ (21 × 17)

⇒ Total Time = 38d ÷ 357

⇒ Average Speed = 2d ÷ (38d ÷ 357)

⇒ Average Speed = (2d × 357) ÷ 38d

⇒ Average Speed = 714 ÷ 38

⇒ Average Speed = 357 ÷ 19

⇒ Average Speed = 18¹⁵/₁₉ km/h

∴ The correct answer is 18¹⁵/₁₉ km/h.

Additional Information

Average Speed for Three Equal Distances

When three equal distances are covered at speeds x, y, and z, the average speed formula is: (3 × x × y × z) ÷ (xy + yz + zx).

Average Speed for Equal Time Intervals

When different distances are covered at varying speeds for equal amounts of time, the average speed is simply the arithmetic mean of all the speeds: (S₁ + S₂ + ... + S_n) ÷ n.

Relative Speed Concept

When two objects move in opposite directions at speeds x and y, their relative speed is (x + y). When they move in the same direction, the relative speed is (x − y).

Shortcut Trick

Net change in total weight = Total members × Increase in average

Net change = 8 × 2 = 16 kg

Since the average increases, the new man is heavier than the replaced man.

Weight of new man = Weight of replaced man + Net change

Weight of new man = 90 + 16 = 106 kg

∴ The correct answer is 106 kg.

Alternate Method

Given:

Total number of men = 8

Increase in average weight = 2 kg

Weight of replaced man = 90 kg

Old Man (90 kg)

• 16 kg

8 men × 2 kg

New Man = 106 kg

Formula Used:

Average = Sum of observations ÷ Number of observations

Calculations:

⇒ Let original average weight of 8 men be A.

⇒ Total original weight = 8 × A = 8A

⇒ After replacing, the new average becomes (A + 2).

⇒ Total new weight = 8 × (A + 2) = 8A + 16

⇒ Increase in total weight = (8A + 16) − 8A = 16 kg

⇒ This 16 kg increase is strictly due to the weight difference between the new and replaced man.

⇒ Weight of new man = Weight of replaced man + Increase in total weight

⇒ Weight of new man = 90 + 16 = 106 kg

∴ The correct answer is 106 kg.

Additional Information

Replacement in Averages

When one member is replaced, the weight of the new member can be calculated directly:

New Value = Old Value + (Total Members × Increase in Average)

New Value = Old Value − (Total Members × Decrease in Average)

Addition of a New Member

When a new member joins the group:

New Member = Old Average + (Total Members after addition × Increase in Average)

Removal of a Member

When a member leaves the group:

Removed Member = Old Average − (Remaining Members × Increase in Average)

Given:

percentage  change in population is 10%

present population is 21000

formula used:

Population after n years = $P=p(1+\frac{r}{100})^n$

where P is the population to be found

p is the population now

r is the rate of increase

n is the number of years

Calculation

$P=21000(1+\frac{10}{100})^3$

⇒ P = 21,000 × (11/10)³

⇒ P = 21,000 × 11/10 × 11/10 × 11/10

⇒ P = 21 × 11 × 11 × 11 = 27,951 

The population 3 years from now is 27,951.

Shortcut Trick⇒ 10% = (+1/10)

⇒ 1000x = 21000

⇒ x = 21

∴ 1331x = 1331 × 21 = 27,951.

Given:

Average speed is 30km/h

Speed which covers 1^st 50 km is 25km/h

Formula used:

average speed = $\frac{2\times x \times y}{x+y}$

where x and y are speeds

Calculation:

x = 25km/h

substituting the equation

average speed = 30 = $\frac{2\times 25 \times y}{25+y}$

solving we get y = 37.5km/h

The speed for the next 50km will be 37.5km/hr.

Given:

Four number are given.

Formula used:

If the sum of the digits of a number are divisible by three, then the number is divisible by 3.

If the digit at one's place of a number is either 5 or 0, then the number is divisible by 5.

Calculation:

The factors of 15 are 3 and 5. Therefore, for a given number to be divisible by 15, it must be divisible by both, 3 and 5.

Since the last digit of all the given options is 5, all of them are divisible by 5.

To check which of the numbers is divisible by 3.

Option 1: 1+6+5+4+8+5 = 28, which is not divisible by 3.

Option 2: 3+0+6+0+4+5 = 18, which is divisible by 3.

Option 3: 2+1+2+6+9+5 = 25, which is not divisible by 3.

Option 4: 2+9+5+1+4+5 = 2, which is not divisible by 3. 

Therefore, the number which is divisible by 15 is 3,06,045.

Given:

Four sets of three side lengths are given.

Formula used:

For a given triangle, the sum of the lengths of any two sides must be greater than the third side.

Calculation:

Option 1: 12 + 9 > 15, 12 +15 > 9, 15 + 9 > 12. Therefore, the given three lengths are the sides of a triangle. 

Option 2: 20 + 20 > 20. Therefore, the given three lengths are the sides of a triangle. 

Option 3: 3 + 5 > 4, 3 + 4 > 5, 5 + 4 > 4. Therefore, the given three lengths are the sides of a triangle. 

Option 4: 3 + 5 > 1, 5 +1 > 3, but 1 + 3 < 5. Therefore, the given three lengths are not the sides of a triangle. 

Therefore, 3 cm, 5 cm and 1 cm are not sides of a triangle.

Given:

Arun can finish a work in 12 days and Shama can finish in 15 days.

Formula used:

Time taken for A and B together, to do a work is $\frac{xy}{x+y}$, where

x is the time taken by A alone, and

y is the time taken by B alone.

Calculation:

Here, x = 12 days and y = 15 days.

Therefore, time taken to finish the work, if Arun and Shama work together is $\frac{12 \times 15}{12+15} = \frac{12 \times 15}{27} =\frac{20}{3} =6 \frac{2}{3}$ days.

Shortcut Trick

Directly apply the Product-to-Sum formula: cos A cos B = 1/2 [cos(A + B) + cos(A − B)]

Here, A = 75° and B = 15°.

Value = 1/2 × [cos(75° + 15°) + cos(75° − 15°)] = 1/2 × [cos 90° + cos 60°]

Since cos 90° = 0 and cos 60° = 1/2, the result is 1/2 × (0 + 1/2) = 1/4.

∴ The correct answer is 1/4.

Alternate Method

Given: Angles A = 75° and B = 15°. Formula: 2cosA cosB = cos(A + B) + cos(A − B).

Calculations:

⇒ cos 75° cos 15° = 1/2 × [2 cos 75° cos 15°]

⇒ Applying the identity: cos 75° cos 15° = 1/2 × [cos(75° + 15°) + cos(75° − 15°)]

⇒ cos 75° cos 15° = 1/2 × [cos 90° + cos 60°]

⇒ Substitute standard values: cos 90° = 0 and cos 60° = 1/2

⇒ cos 75° cos 15° = 1/2 × [0 + 1/2]

⇒ cos 75° cos 15° = 1/2 × 1/2 = 1/4

∴ The correct answer is 1/4.

Additional Information

Other Product-to-Sum Identities

2 sin A sin B = cos(A − B) − cos(A + B)

Mixed Sine and Cosine Products

2 sin A cos B = sin(A + B) + sin(A − B)

Standard Values for Reference

For standard trigonometric calculations: sin 30° = 1/2, cos 30° = √3/2, sin 60° = √3/2, and sin 90° = 1.

Given:

${x^4} + \left( {\frac{1}{{{x^4}}}} \right) = 322$

Formula used:

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

$a^3-b^3=(a-b)(a^2+b^2+ab)$

Calculation:

Using $a^3-b^3=(a-b)(a^2+b^2+ab)$,

${x^3} - \left( {\frac{1}{{{x^3}}}} \right)=\left( {x-\frac{1}{{{x}}}} \right)\left( {x^2+\frac{1}{{{x^2}}}} +1 \right)$    ....(1)

Now, using $(a+b)^2=a^2+b^2+2ab$,

${x^4} + \left( {\frac{1}{{{x^4}}}} \right) = \left( { x^2+\frac{1}{{{x^2}}}} \right)^2 -2=322$

∴ $\left( { x^2+\frac{1}{{{x^2}}}} \right)^2 =324$

∴ $\left( { x^2+\frac{1}{{{x^2}}}} \right)=18$      .... (2)

Again, using $(a+b)^2=a^2+b^2+2ab$,

${x^2} + \left( {\frac{1}{{{x^2}}}} \right) = \left( { x-\frac{1}{{{x}}}} \right)^2 +2=18$

∴ $\left( { x-\frac{1}{{{x}}}} \right)^2 =16$

∴ $\left( { x-\frac{1}{{{x}}}} \right)=4$       ....(3)

Substituting (2), (3) in (1),

${x^3} - \left( {\frac{1}{{{x^3}}}} \right)=4 \times (18+1) = 76$

∴ If ${x^4} + \left( {\frac{1}{{{x^4}}}} \right) = 322$, then what is the value of ${x^3} - \left( {\frac{1}{{{x^3}}}} \right)=$ 76.

Given:

3\frac{6}{7} - \left{ {5\frac{1}{7} - \left( {2\frac{3}{7} - 3\overline {\frac{5}{7} - 4} } \right)} \right}

Calculation:

⇒ 3\frac{6}{7} - \left{ {5\frac{1}{7} - \left( {2\frac{3}{7} - 3\overline {\frac{5}{7} - 4} } \right)} \right}=3\frac{6}{7} - \left{ {5\frac{1}{7} - \left( {2\frac{3}{7} - \overline{\frac{26}{7} - 4} } \right)} \right}

⇒ 3\frac{6}{7} - \left{ {5\frac{1}{7} - \left( {2\frac{3}{7} - {\frac{-2}{7}} } \right)} \right}

⇒ 3\frac{6}{7} - \left{ {5\frac{1}{7} - \left( {\frac{17}{7} + {\frac{2}{7}} } \right)} \right}=3\frac{6}{7} - \left{ {5\frac{1}{7} - {\frac{19}{7} }} \right}

⇒ $\frac{27}{7}-\frac{36-19}{7}=\frac{27}{7}-\frac{17}{7}$

⇒ $\frac{27-17}{7}=\frac{10}{7}$

Hence, the simplified value of the expression is $\frac{10}{7}$

Given:

a + b = 2c

Formula used:

x - y = -(y - x)

Calculation:

If a + b = 2c, then a - c = c - b.

Then, $\frac{a}{{a - c}} + \frac{c}{{b - c}} = \frac{a}{{c - b}} + \frac{c}{{b - c}} = -\frac{a}{{b- c}} + \frac{c}{{b - c}}$

∴ $\frac{a}{{a - c}} + \frac{c}{{b - c}} = \frac{-a+c}{{b - c}} = \frac{b - c}{{b - c}} = 1$

∴ If a + b = 2c, then the value of $\frac{a}{{a - c}} + \frac{c}{{b - c}}$ is 1.

Shortcut TrickWe can use Value putting method here.

Keep the value of 'c' = 0

So,

⇒ a + b = 0

∴ a = -b

then, the question being asked shrinks down to → a/a = 1.

Given:

The length of a cuboid is twice more than its breadth and its breadth is 8 cm, which is twice its height.

Formula used:

Volume of cuboid = length × breadth × height  

Calculation:

If breadth is 8cm, then the height is $\frac{8}{2}=4cm$.

If length is thrice its breadth, the length is 8 × 3 = 24cm.

∴ The volume of the cuboid is 24 × 8 × 4 = 768cm³.

Given:

A graph representing the data for the percentage of the population affected by flue and cold is given.

Formula used:

The shortest bar is the minimum value and the longest bar is the maximum value.

Calculation:

According to the given graph, the year which has the shortest blue bar is 1990.

∴ The year in which the percentage of the population affected by flue is the minimum is 1990.

Mistake PointsRead the graph carefully. Value in the graph indicates the percentage of people infected with flu and cold.

We have to calculate the percentage out of the total people(100%) which is already given in the graph.

Formula used:

Discounted price = $\frac{\text {100 - discount}}{100} \times \text {marked price}$

Calculation:

Let the marked price be x.

The amount paid in cash with 24% discount is 74% of 40% of x,

⇒ $0.76 \times 0.4 \times x$.

The amount paid with credit card with charge of 1.5% is 101.5% of 60% of x

⇒ $1.015 \times 0.6 \times x$.

Then, the total amount paid is 

⇒ $0.76 \times 0.4 \times x$ + $1.015 \times 0.6 \times x$ = 6391.

Solving for x,

$0.304x+0.609x=6391$

$x=\frac{6391}{0.913} = 7000$

∴ The marked price of the item is Rs. 7000.