Explore topic-wise InterviewSolutions in Current Affairs.

This section includes 7 InterviewSolutions, each offering curated multiple-choice questions to sharpen your Current Affairs knowledge and support exam preparation. Choose a topic below to get started.

1.

Replace the underlined phrasal verb with a single word. Yesterday evening I ran into my uncle in the bazaar. (a) forgot (b) remembered(c) lost (d) met

Answer»

Correct answer is (d) met

2.

Supply suitable modal/quasi-modal verb for the following sentences (a) You _________ not apologise for a flimsy mistake. (b) Ramu is so timid that he _________ not discuss major issues with his wife. (c) You _________ donate your old books to needy students. (d) Thou _________ not steal. (e) My _________ brother be happy to see you

Answer»

(a) need 

(b) dare 

(c) should 

(d) shall 

(e) will

3.

Supply suitable modal/quasi-modal verb for the following sentences (a) One _________ obey the traffic rules. (b) _________ I ask you a favour, Sir? (c) She _________ sing and dance. (d) Kapil Dev _________ play football initially. (e) You _________ wait as it is getting dark now.

Answer»

(a) must 

(b) Can 

(c) can 

(d) used to 

(e) need not

4.

Fill in the blanks with suitable modal auxiliary verbs: We .......... follow the traffic rules.

Answer»

We should follow the traffic rules.

5.

Spot the errors in the following sentences and correct them:1. He gave an evocative rendering to his recent poems. (a) thrilling (b) enchanting (c) emotional (d) incantatory2. David’s promotion is on the cards. (a) due (b) evident (c) certain (d) probable3. Professor Christopher asked the students to buckle down this semester. (a) work seriously (b) take it easy (c) drop a subject (d) take leave4. Rita is not cut out for this kind of work. (a) trained (b) suitable(c) recommended (d) considered5. The director hesitated to assign the job to the newcomer as he was wet behind the ears. (a) stupid and slow-witted (b) young and inexperienced (c) deaf (d) lethargic

Answer»

1. (c) emotional

2. (c) certain

3. (a) work seriously

4. (b) suitable

5. (b) young and inexperienced

6.

Choose the answer that you consider the most suitable according to the essay: 1. What can be achieved through co-existing and co-operation? (a) Conflict (b) Money (c) Social progress (d) None of them.2. Communism succeeds in some measure by giving content to man’s life but it fails because of (a) its rigidity and because it ignores certain essential needs of human nature. (b) Its flexibility and discipline (c) Its faith (d) Its increasing contradictions. 3. Nehru had great admiration for many of the achievements of the Soviet Union. Which of the following cannot be called its achievement? (a) Its system of education (b) The suppression of individual freedom (c) Its system of health (d) The value attached to the child and the common man.

Answer»

Correct option is 

1 (c) Social progress

2 (a) its rigidity and because it ignores certain essential needs of human nature.

3 (b) The suppression of individual freedom

7.

Choose the answer that you think correct according to the text:1. Indian Constitution was drafted(a) long before revolution(b) after a revolution(c) Both of the above (d) None of the above. 2. Who pays the leader of the opposition in England?(a) The government of England (b) the government of India (c) Both of the above (d) None of the above.

Answer»

Correct option is 

1 (b) after a revolution

2 (b) the government of India

8.

Find the antonym of word:Abbreviate A) lengthen B) shorten C) reduce D) cut

Answer»

Correct option is A) lengthen

9.

Find the antonym of word:Absurd A) silly B) foolish C) ridiculous D) sensible

Answer»

Correct option is D) sensible

10.

Find the antonym of word:Flimsy A) strong B) evil C) weak D) minute

Answer»

Correct option is A) strong

11.

Choose the answer from the following that you think correct according to the text: 1. Mesopotamian and Egyptian literatures are definitely(a) After the Vedas (b) prior to the Vedas (c) after the Bible (d) None of them. 2. Most of the Verses of the Sama Veda are (a) Original compositions (b) borrowings from the Rig Veda (c) Written in prose (d) ritualistic instructions. 3. Name the third compilation of the Vedas (a) RigVeda (b) Atharva Veda(c) Sama Veda(d) YajurVeda

Answer»

Correct option is 

1 (b) prior to the Vedas

2 (a) Original compositions

3 (d) Yajur Veda

12.

Choose the answer that you think correct according to the text: 1. Which is the highest virtue according to the Mahabharata?(a) Falsity (b) Cleverness (c) Truth (d) None.2. The scene of the Bhagvad Gita has been laid – (a) In Ayodhya (b) in Sumatra (c) in Gandhar (d) in Kurukshetra 3. Lord Krishna created this world by his own (a) Prakriti (b) sword (c) Chariot (d) None of them

Answer»

Correct option is 

1 (c) Truth

2 (d) in Kurukshetra

3 (a) Prakriti

13.

A particle having at least one dimension less than 10-7  metre, is known as1. Micro particle2. Milli particle3. Nano particle4. Macro particle

Answer» Correct Answer - Option 3 : Nano particle

The correct answer is Nanoparticle.

  • A particle having at least one dimension less than 10-7  metre, is known as Nanoparticle (10-9).
  • At least one means 10-8,10-9,10-10,10-11, etc.
  • Hence as per given options, 10-9 is Nanoparticle.

 

  • Milli particle-10-3
    • The millimetre or millimetre is a unit of length in the metric system, equal to one-thousandth of a metre, which is the SI base unit of length. Therefore, there are one thousand millimetres in a metre. There are ten millimetres in a centimetre. One millimetre is equal to 1000 micrometres or 1000000 nanometres​.
  • Micro particle-10-6
    • The micrometre or micrometre, also commonly known as a micron, is an SI derived unit of length equalling 1×10⁻⁶ metre; that is, one-millionth of a metre. The next smallest common SI unit is the nanometre, equivalent to one one-thousandth of a micrometre, or one billionth of a metre.
  • Nano particle-10-9
    • A nanoparticle or ultrafine particle is usually defined as a particle of matter that is between 1 and 100 nanometres in diameter. The term is sometimes used for larger particles, up to 500 nm, or fibres and tubes that are less than 100 nm in only two directions
  • Pico particle-10-12
    • The picometre or picometer is a unit of length in the metric system, equal to 1×10⁻¹² m, or one trillionth of a metre, which is the SI base unit of length.
    • The picometre is one-thousandth of a nanometre, one-millionth of a micrometre, and one trillionth of a meter.
    • The symbol μμ was once used for it. 
14.

Match column I with column II and choose the correct answer using the codes given below: Column - I Column – IIA. Grape Sugar (i) FructoseB. Fruit Sugar(ii) SucroseC. Milk Sugar(iii) Natural Sweetener D. Table Sugar(iv) GlucoseE. Stevia(v) Lactose1. A - (ii), B - (iv), C - (v), D - (iii), E - (i)2. A - (i), B - (iii), C - (iv), D - (v), E - (ii)3. A - (iv), B - (i), C - (v), D - (ii), E - (iii)4. A - (i), B - (iv), C - (v), D - (iii), E - (ii)

Answer» Correct Answer - Option 3 : A - (iv), B - (i), C - (v), D - (ii), E - (iii)

The correct answer is A - (iv), B - (i), C - (v), D - (ii), E - (iii).

The table below is correctly matched:

Column IColumn II
Grape SugarGlucose
Fruit SugarFructose
Milk SugarLactose
Table SugarSucrose
SteviaNatural Sugar

  • Sugar is the generic name for sweet-tasting, soluble carbohydrates, many of which are used in food.
  • Table sugar, granulated sugar, or regular sugar, refers to sucrose, a disaccharide composed of glucose and fructose.
  • Simple sugars, also called monosaccharides, including glucose, fructose, and galactose. Compound sugars, also called disaccharides or double sugars, are molecules composed of two monosaccharides joined by a glycosidic bond.
  • Common examples are sucrose (table sugar) (Sucrose=glucose + fructose), lactose (glucose + galactose), and maltose (two molecules of glucose).
  • In the body, compound sugars are hydrolysed into simple sugars. 
15.

Answer each of the following questions in one or two complete sentences.1. Why does according to author, communism fail in spite of its apparent success?2. In what way has communism succeeded?3. Is there suppression of individual freedom in democracy?4. Which approach does Nehru think wholly unscientific and unreasonable?5. What is socialism according to author?

Answer»

1. According to author, in spite of its apparent success communism fails because of its rigidity and because it ignores certain essential needs of human nature.

2. Communism has succeeded in some measure by giving content to man’s life.

3. No, there is no suppression of individual freedom in democracy.

4. Nehru thinks that having no approach to tolerance to the point of view of others is all unscientific and unreasonable.

5. According to the author it is very difficult to define socialism in exact terms but it lessens economic inequalities and adds to productive forces.

16.

Answer the questions given below in one complete sentence:1. What happened in 1857?2. What did the British say about their primary objective?3. When was the dream of having a national organization finally fulfilled?4. Write the two basic issues on which the congress petitioned to the British.5. Write the slogan which Tilak gave to the Indian masses.6. Write the response of Indians to the appeal made by the British.7. Why were the Rowlatt acts known as the “Black Acts”?

Answer»

1. In 1857 different sections of Indian society rose to challenge the British authority.

2. The British said that their primary objective in India was about the welfare of the Indian people.

3. In 1885, the dream of having a national organization was finally fulfilled by the establishment of All India National Congress.

4. The two basic issues on which the congress petitioned to the British were representation in government and Indianization of the administration.

5. The slogan which Tilak gave to the Indian masses was, “Swaraj is my birthright and I shall have it.”

6. The Indians agreed to help the British during the First World War because they thought that India would be rewarded in the form of self government.

7. Rowlatt Acts were known as ‘Black Acts’ because they curtailed Indian freedom

17.

Which of the following Indus Valley sites is presently in Pakistan? (a) Kalibangan (b) Harappa (c) Alamgirpur (d) Lethal

Answer»

Harappa Indus Valley sites is presently in Pakistan.

18.

Which important industry show signs of having thrived in Lothal? (a) Pottery (b) Shipbuilding (c) Terracota toys (d) Aircraft manufacture

Answer»

(b) Shipbuilding

19.

Answer the questions given below in one complete sentence:1. When did Gandhiji leave for Dandi?2. Why were the identities of the members joining the march to Dandi published?3. What did Gandhi exhort people to do when he addressed them on the way to Dandi?4. How many people finally reached Dandi along with Gandhiji?5. What were the words of Mrs. Sarojini Naidu when Gandhi broke the salt law?6. How did Subhash estimate that March?7. What had Gandhiji communicated by breaking the salt law?

Answer»

1. Gandhiji left for Dandi on Mrch 12, 1930.

2. The identities of the members joining the march to Dandi were published for the information of the police so as to make their arrest easier.

3. Mahatma Gandhi exhorted the people to wear Khadi, abjure alcohol and drugs, abandon child marriage, keep clean and lead a pure life on the way to Dandi.

4. Thousands of people reached Dandi when Ganhiji finally reached there.

5. Mrs. Sarojini Naidu cried, “Hail, Deliverer” when Gandhiji broke the salt law.

6. Subhash compared this march with that of Napolean’s march to Parish from Elba.

7. Gandhiji had communicated the idea of breaking laws of the British administration to the people of this country and as a symbol they were asked to break salt law.

20.

Complete the following sentences on the basis of your textual reading : (a) ................ is a neighbour of Choobookov.(b) The condition of the Negroes was ................ .

Answer»

(a) Lomov is a neighbour of Choobookov.

(b) The condition of the Negroes was appalling.

21.

The archaeologist to initially discover the Mohenjo-Daro site of the Indus Valley Civilization was (a) Sir John Marshall (b) Daya Ram Sahni (c) Sir Martimir Wheeler (d) Rakhal Das Banrji

Answer»

(d) Rakhal Das Banerji

22.

Write a short note on English as the second language of India.

Answer»

English is not our native language. English is used as the second language in India. It is the language of science and commerce. English is the language which bound us together during the freedom struggle. English has become a passport to jobs in India and abroad. It is a language in which Indians trying to make themselves proficient. As a link language, English has been bringing unity among the people of India, speaking different regional languages.

23.

Write a short note on English as the second language of India.

Answer»

English played a very important role in the past as it helped Indian writers to acquire the latest knowledge from the western countries. It worked in linking different parts of our country and helped to bring people of different regions together.

It was English gave the dimensions of a National Movement to our freedom struggle.Today science bring the world closer. Thus it is important for Indians to learn English in this age of globalisation. So, we can say that English is the second language of India.

24.

Write a short note on English as a world language.

Answer»

The most widely used language in the world today is English. There is hardly any country where English is not being taught. It is the language of science and commerce. In this globalised world, no country can progress without the knowledge of English. The United States has English as its main language. In the European countries like Russia, France and Germany etc. the language is spoken by a large number of people. It is the main language of the United Nations. It is used as the medium of instructions in schools and colleges almost in every countries. Really English is expanding by leaps and bonds as a world language.

25.

Write a short note on English as an International language.

Answer»

English is said to be an International language as no other language is read and spoken in so many countries as it is. Naturally, it has became the language of international trade and commerce. All the earlier colonies of the British Empire including India had the presence of this language. The united states has English as its main language. In the European countries like France, Russia, Germany etc. the language is spoken by a large number of people. It is the main language of the United Nations. Even the Chinese are learning English in order to compete with the world in the age of global market economy.

26.

A die is thrown once. What is the probability of getting a prime number?

Answer»

Total no of possible outcomes = 6 

There are 3 Prime numbers, 2,3,5. 

So, Probability of getting a prime number is \(\frac{3}{6}\) = \(\frac{1}{2}\)

27.

A car starts accelerating with an acceleration of `(1)/(3)m//s^(2)` when its speed was `18 km//h`. If the desplacement of car in `n^(th)` minute after it starts accelerating is `4.8km` then n is :

Answer» Correct Answer - 4
A car starts ………..
`Acc = (1)/(3)m//s^(2) = (60xx60)/(3)m//min^(2) = 1800m//min^(2)`
initially velocity `18 km//h = (18000)/(60) = 300m//min`
`4800=300+(1800)/(2)(2n-1)`
`(2xx4500)/(1800) = 2n-1`
`implies n = 3`
28.

Answer the following question in one complete sentence:1. What was the length of the Staff which Gandhi on this occasion carried? 2. On which date did Gandhi and his followers reach Dandi? 3. To whom do Mrs Sarojini Naidu’s words ‘Hail Deliverer’ refer? 4. Mark the statements True or False in the context of the text: (i) The Villagers gave the marchers a cold reception. (ii) Gandhi’s Dandi Salt March started on March 12,1930. 5. Choose the answer from the following questions that you think correct: (i) If Gandhi had gone by train to Dandi to make salt, it would have made (a) no impact at all (b) more of an impact (c) equal impact (d) less of an impact(ii) Who does ‘a sophisticated critic and some one fierce opponent’ of Gandhi refer to?(a) the illiterate peasant(b) Subhash Chandra Bose(c) Gandhi himself(d) Napoleon.

Answer»

1. The staff that Gandhiji carried with him was only 54 inches long and one inch thick.

2. On April 5, 1930 Gandhiji and his followers reached Dandi.

3. Mrs Sarojini Naidu’s words ‘Hail Deliverer’ refer to Gandhiji.

4.(i) False;(ii)True.

5. (i) (d) less of an impact. 

(ii) (c) Gandhi himself

29.

A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is red? 

Answer»

No. of red balls = 3, 

No. of black balls = 5 

Total number of balls = 5 + 3 = 8 

Probability of red balls = \(\frac{3}{8}\)

30.

If the perimeter and the area of a circle are numerically equal, then find the radius of the circle

Answer»

Perimeter = Area 

2πr = πr2 

r = 2 units

31.

1. What is the foundation of Constitution?2. What sort of executive is there in the United States?3. Of which fallibility is a democrat always conscious?4. Mark the Statements True or False in the context of Text:(i) The party system has been worked almost to its perfection in England.(ii) The government of England pays the leader of the opposition in England.(iii) The revolution which brought India freedom was a bloody one.5. Choose the correct alternative from the following:(i) Our constitution sets up(a) a secular state(b) a non-secular state (c) an orthodox state (d) none of them. (ii) Who surrendered a domination?(a) America (b) Canada (c) England (d) India

Answer»

1. The foundation of our constitution is based on all good things of the past.

2. In the United States the executive is divorced from the legislature.

3. A democrat is always conscious of human fallibility. He must always understand and appreciate the other man’s point of view or even opponents.

4. (i) True; (ii) True; (iii) False.

5. (i) a, (ii) c.

32.

Answer the following questions in one complete sentence:1. Give the No. of the article by which the lacuna regarding fundamental duties was removed form the Indian constitution.2. What does the first article of Fundamental Duties direct us?3. What is the fifth article of Fundamental Duties?4. Choose the most suitable alternative from the following questions:(i) What is the fourth article of Fundamental Duties:(a) to defend the country and render national service when called upon to do so. (b) to preserve our heritage. (c) Both of them (d) None of them. (ii) Which article tells to uphold and protect the sovereignty, unity and integrity of India? (a) First (b) Second (c) Third (d) None 5. Tell whether the following statements are True or False: (i) Indian Constitution boasts to ten Fundamental Duties. (ii) We should not value and preserve the rich heritage of our composite culture.

Answer»

1. By the article 52-A, the lacuna regarding Fundamental Duties was removed from the Indian constitution. 

2. The first article of the list of Fundamental Duties directs us to abide by the Constitution and respect its ideals and institutions, the National flag and the National Anthem. 

3. The fifth article of the Fundamental Duties directs us to promote harmony and the spirit of common brotherhood amongst all the people of India transcending religions, linguistic and regional or sectional diversities.

4. ( i) a; (ii) c. 

5. (i) True; (ii) False.

33.

Tell whether the following statements are True or False: 1. Communism can succeed only through violence, which is not a right means for a good end. 2. There are no contradictions within the framework of communism itself. 3. Democracy and socialism are means to an end.

Answer»

1. True 

2. False 

3. True

34.

Answer the following questions in one complete sentence each:1. How does communism succeed? 2. What hardly convey a clear and unambiguous meaning? 3. What does Nehru think about socialism?4. Tell whether the following statements are True or False:(i) Communism fails because of its flexibility and discipline.(ii) Nehru had great admiration for many of the achievements of the Soviet Union.(iii) Socialism does not lessen inequalities.5. Choose the correct alternative from the following:(i) Welfare state, democracy and socialism convey(a) an unclear meaning(b) a clear and unambiguous meaning(c) both of the above(d) none of the above(ii) Socialism aims at(a) conflict(b) equality(c) nothing(d) None of the above

Answer»

1. Communism succeeds only through violence, which is not a right means for a good end.

2. Welfare state, democracy and socialism hardly convey a clear and unambiguous meaning.

3. To Nehru, socialism can wipe out the wide gap between capitalism and common man in society.

4. (i) False; (ii) True; (iii) False.

5. (i) b; (ii) b

35.

For what values of p does the pair of equations 4x + py + 8 = 0 and 2x + 2y + 2 = 0 has unique solution?

Answer»

\(\frac{a1}{a2}\) ≠ \(\frac{b1}{b2}\) is the condition for the given pair of equations to have unique solution.

\(\frac{4}{2}\) ≠ \(\frac{p}{2}\)

p ≠ 4

Therefore, for all real values of p except 4, the given pair of equations will have a unique solution.

36.

Write the empirical relationship among mean, median and mode.

Answer»

3 median = mode + 2 mean

37.

To divide a line segment BC internally in the ratio 3 : 5, we draw a ray BX such that ∠CBX is an acute angle. What will be the minimum number of points to be located at equal distances, on ray BX?

Answer»

The minimum number of points to be located at equal distances, on ray BX is 8

38.

Tangents are drawn from a point P to the hyperbola `x^2-y^2= a^2` If the chord of contact of these normal to the curve, prove that the locus of P is `1/x^2 - 1/y^2 = 4/a^2`

Answer» Let locus of point `P(h,k)``Eqn.` of tangent from point `P` to the hyperbola is:`y = mx pm sqrt(a^2*m^2 - a^2)``y = mx pm a*sqrt(m^2 -1)``Eqn.` of normal to this hyperbola:`a^2*y_1(x-x_1) + a^2*x_1(y-y_1)`Tangents starts from point `P(h,k)` , so it is point of contact. And since this point is normal to the curve, it satisfies `Eqn.` of normal.`a^2*k(x-h) + a^2*h(y-k) = 0`Converting it to slope form:`kx - kh + hy - kh = 0``y = -k/h*x + 2k`And from equation of tangent:`k = mh pm a*sqrt(m^2-1)``rArr m = -k/h` ......`(eq.1)`and,`a*sqrt(m^2-1) = 2k`Squaring both sides`a^2*(m^2-1) = 4k^2``a^2*((-k/h)^2 - 1) = 4k^2``k^2/h^2 - 1 = 4k^2/a^2`Divinding above `Eqn.` by `k^2``1/h^2 -1/k^2 = 4/a^2``:.` locus of point P(h,k) is`1/x^2 -1/y^2 = 4/a^2`
39.

In the isosceles triangle `ABC, |vec(AB)| = |vec(BC)| = 8`,a point E divide AB internally in the ratio `1:3`, then the cosine of the angle between `vec(CE)` and `vec(CA)` is (where `|vec(CA)| = 12`)

Answer» `abs(AB)=abs(BC)=8`
`abs(vecb)=8`
`abs(vecc)=12`
`abs(vecb-vecc)=8`
`vec(CE)=vecE-vecC`
`=vecb/4-vecC`
Therefore, `abs(vecCE)=sqrt(148-(vecb*vecc)/2)`
`a/sina=b/sinb=c/sinc`
`cosA=3/4`
Therefore `vecb*vecc=72`
Put it in equation 1, we get,
`sqrt(148-36)=sqrt112`
In `triangle AEC`,
`(CE)/sinA=(AE)/sinalpha`
`sin A= sqrt7/4`
Therefore,
`(sqrt112*4)/sqrt3=2/sinalpha`
Hence, `sinalpha=1/8`
and `cosalpha=3sqrt7/8`
40.

A dish, in the shape of a frustum of a cone, has a height of 6 cm. Its top and its bottom have radii of 24 cm and 16 cm respectively. Find its curved surface area (in `cm^2`).

Answer» Lateral surface area=>`pi(R+r)sqrt((R-r)^2+h^2)`
substituting values we get=>`22/7xx40xx10=400pi(cm)^2`
41.

When the weather became colder, we put some ______ on fire to heat the living room. A) log B) blanket C) toys D) sofa

Answer»

Correct option is A) log

42.

Find the words which are out of the logic list:A) blanket B) quilt C) pillow D) sofa

Answer»

Correct option is D) sofa

sofa because here everything is related to bed except sofa So D sofa is correct




43.

दो संख्याओं के ल0स0 तथा म0 स0 का गुणनफल 24 है। उन संख्याओं के बीच का अंतर 2 है तो संख्याएं ज्ञात करें।A. 8 and 6B. 8 and 10C. 2 and 4D. 6 and 4

Answer» Correct Answer - D
LCM `xx` HCF `=24`
`:.` संख्याओं का गुणनफल `=24`
Let no `=x,y`
`xy =24`
and `x-y=2` (given)
Factors of `xy=24` are `(4,6)(12,2)(8,3)(24,1)`
`implies` संख्याओं का अंतर
`=(x-y)=2`
गुणक (4,6)
44.

Disciss tth enatureof solution ofofth quadratic equation x2-9x+20=0

Answer»

he equation x2 – 9x + 20 = 0 has real and unequal roots.

45.

How many 4 digit numbers greater than 2300 can be formed with the digits 0,1,2,3,4,5 and 6 no digit being repeated in any number?

Answer»

There are two cases, (i) First digit is greater than 2 (ii First digit is 2

Case : (i) First digit can be chosen from 3,4,5,6 so 4 possibilities, then next 3 digits can be selected from any 6 digits.

= 4 × 6 × 5 × 4 = 480

Case : (ii) First digit is 2, second digit starts from 3, so it can be 3,4,5,6 so 4 possibilities.

Now if any one digit of third and fourth digit can be 0 as no digit is repeated so it will be surely greater than 2300. So last two digits can be selected from any 5 digits.

= 1 × 4 × 5 × 4 = 80

∴ Total = 480 + 80 = 560.

46.

(b)

Answer»

now for this first we will begin with finding the area of rectangle or the total area= 60×30 cm^2 =1800 cm^2

now second step we will find that area of small quadrant i.e. 3.14 ×122 / 4 cm^2 = 3.14× 36 cm^2 =113.04 cm^2

Now area of unshaded portion is total - area of quadrant = (1800 - 113.04 )cm^2  = 1686.96 cm^2  = 1687 cm^2(approx )

47.

Prove that : \(\frac{1}{cosec θ- cot θ}\) = (cosecθ + cotθ).

Answer»

L.H.S = \(\frac{1}{cosec θ+ cot θ}\) = \(\frac{cosec\theta+cot\theta}{(cosecθ − cotθ)(cosecθ + cotθ)}\) 

(By multiplying numerator and denominator by cosecθ + cotθ) 

= \(\frac{cosecθ + cotθ}{cosec^2θ − cot^2θ}\) (∵ (a+b)(a–b) = a2 − b2

= \(\frac{cosecθ + cotθ }{1}\) = cosecθ + cotθ = R.H.S. 

(∵ cosec2θ = 1 + cot2θ ⇒ cosec2θ − cot2θ = 1

Hence proved

48.

Find the radius of curvature at (1,1) to the curve y2-x = x3.

Answer»

y2-x = x3 at (1, 1)

(2 - x) log y = 3 log x-----(i) (by taking log both sides)

(2-x)1/y (dy/dx) - log y  = 3/x-----(ii) (by differentiating (i) w.r.t x)

(2 - x)1/y(dy/dx) = 3/x + (3logx)/(2 - x)

\(\frac{dy}{dx}=y\left(\frac{3}{x(2-x)}+\frac{3logx}{(2-x)^2}\right)\)

\(\frac{dy}{dx}\) at (1, 1) = 1\((\frac{3}{1\times1}+\frac{3log 1}{1^2})\) = 3

\(\frac{(2-x)}y\frac{d^2y}{dx^2}-\frac{(2-x)}{y^2}(\frac{dy}{dx})^2\) - \(\frac1y\frac{dy}{dx}-\frac1y\frac{dy}{dx}=\frac{-3}{x^2}\)......(iii)

(By differentiating equation (ii) w.r.t. x)

Put x = 1, y = 1, \(\frac{dy}{dx}=3\) in equation (iii), we get

⇒ \(\frac{2-1}1\frac{d^2y}{dx^2}-\frac{(2-1)}1\times9-3-3=-3\) 

⇒ \(\frac{d^2y}{dx^2}=12\) 

radius of curvature at (1, 1) to given curve is

p = \(\cfrac{(1+(\frac{d^2y}{dx^2})^2)^{3/2}}{\frac{d^2y}{dx^2}}\) 

\(\frac{(1+3^2)^{3/2}}{12}=\frac{(10)^{3/2}}{12}\)

49.

The students in the three hostels are in a ratio of 2 : 3 : 5. If the students are increased by 5 in each hostel, the new ratio will become 5 : 7 : 11. The total number of students in the three hostels after the increase is?1. 1202. 1103. 1154. 1305. 100

Answer» Correct Answer - Option 3 : 115

Given:

Initial ratio of students in the hostel = 2 : 3 : 5

New ratio of students in the hostel = 5 : 7 : 11

Concept used:

Initial value + Increment = Final value

Calculation:

Let the number of students in the first hostel be 2x

Let the number of students in the second hostel be 3x

Let the number of students in the third hostel be 5x

(2x + 5)/(3x + 5) = 5/7

⇒ 14x + 35 = 15x + 25

⇒ 15x – 14x = 35 – 25

⇒ x = 10

Number of students initially = 20, 30 and 50

Increased number of students = 25, 35, 55

Total number of increased students = 25 + 35 + 55 = 115

The new total number of students in three hostels is 115.

50.

A diamond of weight 150 grams is cut into three pieces. The weight of the first piece and the second piece are in a ratio of 8 : 5. The weight of the third piece is 20 grams. Find the weight of the first piece.1. 80 grams2. 90 grams3. 100 grams4. 85 grams5. 95 grams

Answer» Correct Answer - Option 1 : 80 grams

Given:

Weight of diamond = 150 grams

The ratio of the weight of two pieces = 8 : 5

Weight of the third piece = 20 grams

Concept used:

Total weight = Weight of first piece + Weight of second piece + Weight of third piece

Calculation:

Let the weight of the first piece be 8x

Let the weight of the second piece be 5x

150 = 8x + 5x + 20

⇒ 13x = 130

⇒ x = 10

Weight of the first piece = 8 × 10 = 80

The weight of the first piece is 80 grams.