DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE TOLD TO DO SO

T.B.C. : B-QGC-O-REV Test Booklet Series

TEST BOOKLETSerial No.

B04 O~\202MATHEMATICS

Time Allowed : Two Hours and Thirty Minutes Maximum Marks : 300

INSTRUCTIONS

1. IMMEDIATELY AFTER THE COMMENCEMENT OF THE EXAMINATION, YOU SHOULDCHECK THAT THIS TEST BOOKLET DOES NOT HAVE ANY UNPRINTED OR TORN ORMISSING PAGES OR ITEMS, ETC. IF SO, GET IT REPLACED BY A COMPLETE TESTBOOKLET.

2. Please note tbat it is the candidate's responsibility to encode and fill in the Roll Number andTest Booklet SerlesA, B, C or D carefully and without any omission or discrepancy at theappropriate places in the OMR Answer Sheet. Any omission/discrepancy will render theAnswer Sheet liable for rejection.

3. You have to enter your Roll Number on the TestBooklet in the Box provided alongside. DO NOTwrite anything else on the Test Booklet. .

4. This Test Booklet contains 120 items (questions). Each item is printed both in Hindi and English.Each item comprises four responses (answers). You will select the response which you want tomark on the Answer Sheet. In case you feel that there is more than one correct response, mark theresponse which you consider the best. In any case, choose ONLY ONE response for each item.

S. You have to mark all your responses 0 LYon the separate Answer Sheet provided. See directionsin the Answer Sheet.

6. All items carry equal marks.7. Before you proceed to mark in the Answer Sheet the response to various items in the Test Booklet,

you have to fill in some particulars in the Answer Sheet as per instructions sent to you with yourAdmission Certificate. . .

8. After you have completed filling in all your responses on the Answer Sheet and the examination hasconcluded, you should hand over to the Invigilator only the Answer Sheet. You are permitted to takeaway with you the Test Booklet.

9. Sheets for rough work are appended in the Test Booklet at the end.10. Penalty for wrong Answers :

THERE WILL BE PENALTY FOR WRONG ANSWERS MARKED BY A CANDIDATE IN THEOBJECTIVE TYPE QUESTION PAPERS.

(i) There are four alternatives for the answer to every question. For each question for whicha wrong answer has been given by the candidate, one-third of the marks assigned to thatquestion will be deducted as penalty.

(ii) If a candidate gives more than one answer, it will be treated as a wrong answer even if oneOf the given answers happens to be correct and there will be same penalty as above to thatquestion.

(iii) If a question is left blank: i.e., no answer is given by the candidate, there will be no penaltyfor that question.

DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE TOLD TO DO SO

r

1. The area of the figure formed bythe lines ax + by + c = 0, ax - by + c = 0,

_ax + by - c = 0 and ax - by - c = 0 is

(a) 10 unit

(b) 9 unit

c2(a)

ab

(c) 8 unit

.-

(d) 6 unit

2c2

(b) ab4. A straight line x =y + 2 touches the

circle 4(x2 +/) =? The value of r is

c2(c) 2ab

(a) .fi

(b) z-Ii

c2(d) 4ab

(c) 2

(d) 1

2. If a line is perpendicular to the line5x - y = 0 and forms a triangle of area5 square units with co-ordinate axes,then its equation is

5. The three lines 4x+4y= 1, 8x-3y=2,y=O are

(a) .the sides of an isosceles triangle

(a) x+5y±5.fi = 0 (b) concurrent

(c) mutually perpendicular(b) x-5y±5.fi =0

(d) the sides of an equilateral triangle

(c) 5x+ y±5.fi = 06. The line 3x + 4y - 24 = 0 intersects the

x-axis at A and y-axis at B. Then thecircumcentre of the triangle OAB whereo is the origin is

(d) 5x- y±5.fi = 0

3. Consider any point P on the ellipsex2 2- +L= 1 in the first quadrant. Let r25 9and s represent its distances from (4, 0)and (-4, 0) respectively, then (r+ s) isequal to

(a) (2, 3)

(b) (3, 3)

(c) (4, 3)

(d) None of the above

B-QGC-O-REV - B 2

7. TIle eccentricity of the hyperbola16.i - 91 = 1 is

(a) 35

(b) 53

4(c) 5

5(d) 4

8. The product of the perpendicularsfrom the two points (±4, 0) to the line3x cos q, + 5y sin q, = 15 is

(a) 25

(b) . 16

(c) 9

(d) 8

9. If the centre of the circle passingthrough the origin is (3, 4), then theintercepts cut off by the circle on x-axisand y-axis respectively are

(a) 3 unit and 4 unit

(b) 6 unit and 4 unit

(c) 3 unit and 8 unit

(d) 6 unit and 8 unit

B-QGC-O-REV - B

10. The lines 2x = 3y ='-z and 6x = -y = -4z

(a) are perpendicular

(b) are parallel

(c) intersect at an angle 45°

(d) intersect at an angle 60°

11. Two straight lines passing through thepoint A(3, 2) cut the line 2y = x + 3and x-axis perpendicularly at P and Qrespectively. The equation of the linePQ is

(a) 7x+y-21 =0

(b) x + 7y + 21 = 0

(c) 2x+y-8=0

(d) x + 2y + 8 = 0

12. The radius of the sphere3.i + 31 + 3i - 8x + 4y + 8z- 15= 0 .IS

(a) 2

(b) 3

(c) 4

(d) 5

13. The direction ratios of the line per-pendicular to the lines with directionratios < 1, -2, -2 > and < 0, 2, 1 > are,

(a) < 2, -1, 2 >

(b) < -2, 1, 2 >

(c) < 2, 1, -2 >

(d) < -2, -1, -2 >

4

14. What are the co-ordinates of the foot ofthe perpendicular drawn from the point(3, 5, 4) on the plane z = 0 ?

(a) (0, 5, 4)

(b) (3, 5, 0)

(c) (3, 0, 4)

(d) (0, 0, 4)

15. The lengths of the intercepts on theco-ordinate axes made by the plane5x + 2y + z -13 = 0 are

(a) 5, 2, 1 unit

(b) 13 13 13 it- - um5' 2'

(c) 5 2 1 "- - -umt13'13'13

(d) 1, 2, 5 unit

For the next three (03) items that follow :

Consider the expansion of (1 + Xin+1

16. If the coefficients of xr and X,-+I areequal in the expansion, then r is equal to

(a) n

(b) 2n-l2

2n+l(c) 2

(d) n + l'

B-QGC-O-REV - B

17. The average of the coefficients of thetwo middle terms in the expansion is

(a) 2n+1 Cn+2

(c) 2n+1Cn-l

(d) 2nCn+l

18. The sum of the coefficients of all theterms in the expansion is

(a) 22n-1

(c) 2 x 4n

(d) None of the above

19. The nth term' of an A.P. is 3+n, then4

the sum of first 105 terms is

(a) 270

(b) 735

(c) 1409

(d) 1470

6

(d) Neither in G.P. nor in A.P. nor inH.P. (d) 8000

20. A polygon has 44 diagonals. The numberof its sides is

(a) 11

(b) 10

(c) 8

(d) 7

21. If p, q, r are in one geometricprogression and a, b, c are in anothergeometric progression, then ap, bq, crare III

(a) Arithmetic progression

(b) Geometric progression

(c) Harmonic progression

(d) None of the above

For the next two (02) items that follow:

Consider a triangle ABC satisfying

2a sin2( ~) + 2c sin2( ~) = 2a + 2c- 3b

22. The sides of the triangle are in

(a) GP.

(b) A.P.

(c) H.P.

B-QGC-O~REV - B 8

23. sin A, sin B, sin C are in

(a) GP.

(b) A.P.

(c) H.P.

(d) Neither in G.P. nor in A.P. nor inH.P.

(un) (21n)24. If p=tan -6 ,q=tan 4 and

(283n) .r = cot -6- , then WhIChof the fol-

lowing is/are correct?

1. The value of p x r is 2.

2. p, q and r are in GP.

Select the correct answer using thecode given below :

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

25. The number of ways in which 3 holidaytickets can be given to 20 employees ofan organization if each employee iseligible for anyone or more of thetickets, is

(a) 1140

(b) 3420

(c) 6840

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26. What is the sum of n terms of the

series F2 +.J8 +.f18 +m+....?29. The number of 3-digit even numbers that

can be formed from the digits 0, 1, 2, 3,4 and 5, repetition of digits being notallowed, is

n(n-l)(a) -Ii (a) 60

(b) -lin(n + 1)(b) 56

(c) 52

(c) n(n + 1)-Ii (d) 48

(d) n(n -1)2 30. If 10g8m + 10g8.!.=~, then m is equal

6 3to

27. The coefficient of x99 in the expansionof (x-I)(x-2)(x-3) ..... (x-IOO) is (a) 24

(a) 5050 (b) 18

(b) 5000 (c) 12

(c)· -5050 (d) 4

(d) -5000 31. If different words are formed with allthe letters of the word 'AGAIN' and arearranged alphabetically among themselvesas in a dictionary, the word at the 50thplace will be

28. z Z+ (3 - i)z + (3 + i)z + 1= 0 represents acircle with

(a) centre (-3, -I) and radius 3 (a) NAAGI

(b) centre (-3, 1) and radius 3 (b) NAAIG

(c) centre (-3, -1) and radius 4 (c) IAAGN

(d) centre (-3, 1) and radius 4 (d) IAANG

~QGC-O-REV - B

f r

10

(A- \(j.-'rc., ~ ) (b) 5

r~'l ')'1- 16~ ~()~'r).r~

};fi- \. 't (c) !y;. ~ rr'-~()/ /? J>, S/~ _<1:- - 7,f.J -i- (d)

f)'-_4'r':..-<:' 17\0 .....~T'/ •...\C.

37. Two poles are 10 m and 20 m high.The line joining their tops makes anangle of 15° with the horizontal. Thedistance between the poles is approxi-mately equal to

(a) 36·3 m ~\0

I'-: 9£fJ (b) 37·3 m 'l~~ -{.1 ~ ,()~Z- \0 (c) 38·3 mL.~ ;-

32. The number of ways in which a cricketteam of 11 players be chosen out of abatch of 15 players so that the captainof the team is always included, is

(a) 165

(b) 364

(c) 1001

(d) 1365

(1 )10

33. In the expansion of .j;+ 3x2 the

value of constant term (independentof x) is

(a) 5

(b) 8

(c) 45

(d) 90

34. The value ofsin2 5° + sin2100 + sin215° +

sin2 20° +.....+ sin2 90°IS

(a). 7

35. On simplifying

sirr' A + sin3A cos' A - cos3A---------+---------sinA cosA

we get

(a) sin3A

(b) cos3A

(c) sinA + cosA

(d) 3

36. The value of tan (2tan-1~ - :) is

(a)717

(b) 8

(c) 9

(d) 192 (d) 39·3 m

B-QGC-O-REV - B

138. If g(x) = --" and f(x) = x, x *0, then

f(x)which one of the following is correct?

(a) f(f(f(g(g(f(x»»»= g(g(f(g(f(x»») .

(b) f(f(g(g(g(f (x»»»=g(g(f(g(f(x»»)

(c) f(g(f(g(g(f(g(x»»»)= g(g(f(g(f(x»»)

(d) f(f(f(g(g(f(x»»»=f(f(f(g(f(x»»)

39,. Consider the following:

. -i 4 . -i 3 tt1. sm - + sm - = -5 5 2

Which of the above is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

40. If A is an orthogonal matrix of order 3

[1 2 3]

and B = -3 0 2 , then which of the2 5 0

following is/ are correct?

1. IABI = ±47

2. AB = BA

B-QGC-O-REV - B

Select the correct answer using thecode given below:

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

41. If a, b, c are the sides of a triangle1. 1. 1."

ABC, then a" +bP -cP where p > 1, is

(a) always negative

(b) always positive

(c) always zero

(d) positive if 1< P < 2 and negative ifp>2

42. If a, b, c are real numbers, then thevalue of the determinant

I-a a-b-cI-b b-c-al-c c-a-b

b+cc+a isa+b

(a) 0

(b) (a - b)(b - c)(c - a)

(c) (a + b + C)2

(d) (a + b + C)3

43. If the point z\ = 1 + i where i = R isthe reflection of a point Z2 =x + iy in theline ii -iz = 5, then the point Z2 is

(a) 1+ 4i

(b) 4 + i

(c) 1 - i

(d) -1-i

14

(c) is less than -8 g ,~~1-J)

(d) must be greater than 8 \ If, ?~> I~~~ --J<f3 ,- $) \j --

~ 1'~~,'\I*-'] e- 1-13 -J:fl-}- ~ t--V ~

44. If sin x + sin y = a and cos x + cos Y = b,

then tan 2 ( x; y) + tan 2 ( x; y) is equal

to

(a) an equivalence relation

(b)' reflexive and transitive but notsymmetric

(c) symmetric and transitive but notreflexive

(d) reflexive -and symmetric but nottransitive

47. The matrix A = [~. 2

3x-I

7

will have inverse for every real number xexcept for

I

(d) None of the above II±.J5(a) x=--

2

(b)9.±.JS

x=2

(c)II±.J3

x=2

45. A vertical tower standing on a levelledfield is mounted with a vertical flag staffof length 3 m. From a point on the field,the angles of elevation of the bottom andtip of the flag staff are 30° and 45°respectively. Which one of the followinggives the best approximation to theheight of the tower? 9±.J3

x=2I \..~d)<>7

~III.~S '/ <),

,-;~x'" a~ , 48. If the value of the determinant I

~r,.¥) I

h-~l\<)G ~----

k~:t~~46. Let X be the set of all persons living in

Delhi. The persons a an~ b it) X aresaid to be related if the difference intheir ages is at most 5 years. Therelation is

(a) 3·90 m

is positive, where a * b * c, then the valueof abc

I 1b II c

(b) 4·00 m

(c) 4·lOm

(d) 4·25 m(a) cannot be less than I .

(b) is 'greater than -8

B-QGC-O-REV - B

Select the correct'\..code given below:

~~L\...'L~ Q. 'J, (a) 1 only

'l...-A-V\~\ ~2 only

(c) 3 only

(d) None

49. Consider the following statements mrespect of the determinant

acos2-2

. 2 asm -2

where a, f3 are complementary angles

1. The value of the determinant is

1 (a - f3)-ficos -2- '.

2. The maximum value of the deter-. . 1

mmant IS -fi'

Which of the above statements is/arecorrect?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

50. What is (1000000001)2 - (0·0101)2 equalto?

(a) (512·6775)10

(b) (512·6875)10

(c) (512·6975)10

(d) (512·0909)10

B-QGC-O-REV - B

[1 0 -2]51. If A = 2 -3 4 ' then the matrix X

for which 2X + 3A = 0 holds true is

[-~0 -3](a)9

-3 -- -62

,

[! 0 -3](b) 9-- -62

[! 0

:J(c) 9-2

[-~0

~J(d) 9-3 -

2

52. If ZI and Z2 are complex numbers with

Izil = Iz21, then which of the followingis /are correct?

1. ZI = Z2

2. Real part of ZI = Real part of Z2

3. Imaginary part of ZI = Imaginarypart of Z2

answer using the

18

-Z, Z. ::::0 Z

~ ,+',b,)(g,,~itv~ ~'2--\II ~l/(q z, -'I t

q,lL _ 0~I -t <Ajt;, -+ b

[

1 1

53. If A= 2 -33 -2

56. If A = {x E 1R: X2 + 6x - 7.< O} and

B={XElR:X2 +9x+14>O}, then

which of the following is/are correct?then whichcorrect?

1. (AnB)=(-2, 1)

-1] [-1 -2 -1]4 andB= 6 12 6 ,3 5 10 5

of the following is/are

1. A and B commute.2. (A\B)= (-7, -2)

2. AB is a null matrix.

Select the correct answer using thecode given below :

Select the correct answer using thecode given below:

(a) 1 only (a) 1 only

(b) 2 only (b) 2 only

(cJ Both 1 and 2 (c) Both 1 and 2

(d) Neither 1 nor 2 (d) Neither 1 nor 2

54. The number of real roots of the equation

x2 - 31 x 1+ 2 = 0 is57. A, B, C and D are four sets such

that An B = C n D = l/J. Consider thefollowing:

I\.....-

C\... 0-->cz ~ "Lr 1

C).r/-.J "r- Ur

l)- <t.,~ \l'1-' 'f\.-~ /"

('f'- roWc-t...\:f- <)..,\ ••,...,

\ '- y \..- 2. AnCandBnD are always dis-\...-x<')1'/... J.,<fo....\ joint.

<"l- V\'\Y- ~.ty... C'"x

55. If the sum of the roots of the equation ~ax2 + bx + c = 0 is equal to the sum oftheir squares, then ~ \,>

.c~ :,tk. (a) 1 onlyW ~+~=d /~ ~

(b) a2 + b2 = a + b cJ.....(),X~ (b),2 only\) -:.- (),. /~.//&. k~ (c) Both 1 and 2

~ C,.? q,./" (d) Neither 1 nor 2fl,./ o,%'V '),,4-- Ll},.,/ <2-~/\7 '/ 20 ~<t,., 0..'v~V ~\n/"~

~ /" r;. ~ 1J /' -':> 'oC).,~UY /"

(a) 4

(b) 3· 1. AuCandBuD are always dis-joint.

(c) 2

(d) 1

Which of the above statements is/arecorrect?

(

(c) ab + b2 = 2acp

(d) ab - b2 = 2ac

B-QGC-O-REV - B

58. If A is an invertible matrix of order nand k is any positive real number, thenthe value of [det(kA)rl det A is

(d) nk

~\~(a)ty' (b) 36

(c) 216

1AJ'!( 62. Two events A and !I are such that

P(not B) = 0·8, P(AuB) = 0·5 and

p(AIB) = 0·4. Then P(A) is equal to

,....o·'\.

~~) -- 1-.W~D'd'S'"/- ....6'a~

6·6~.fL.\-6- .JJ-

~ l-60. If the roots of the equation

r-nx+m=Odiffer by 1, then ,J r;k\

""'-, ~\Y

XrJ:r:j. \ ->\t\ 2~ (a) 3crp~):

61. Three digits are chosen at random from1, 2, 3, 4, 5, 6, 7, 8 and -9 without 7repeating any digit. What is the proba- (c) -

bility ~hat the product is odd ? ~ / ifA<~I

(a) "3 ./ ~ k* . (d) 16\A~~

B-QGC-O-REV- 8 22

(d) ex>

(d)

7(b) 48

5(c) 42

,...

5(d) 108

(a) 0·28

(b) 0·32

(c) 0·38

(d) None of the above

63. If mean and variance of a Binomialvariate X are 2 and 1 respectively, thenthe probability that X takes a valuegreater than 1 is

64. Seven unbiased coins are tossed 128times. In how many throws would youfind at least three heads ?

(a) 99

3(d) 32

~ Ef3 +Sp",,),St err, n2(n+1)2$' \ :>\ .-\:, (d)

::::-Q(t ~ \ 4

> 0 l6)\j;). ("\ 68. An unbiased coin is tossed until the first?> ~).. r) . head appears or until four tosses are

. ~ "\\ 'Y\ 'I'v- completed, whichever happens earlier.r Cf.-.,'If-.'L. ::') <"\ Which of the following statements is/are'-- ) correct? .,

1. The probability that no head IS

observed is _1 .16

n(n + 1)2(c) 4

'~+~>i)"

~Jc

(b) 102

(c) 103

67. The arithmetic mean of 1, 8, 27, 64, .....up to n terms is given by

(d) 104n(n + 1)

(a) 2

65. A coin is tossed five times. What is theprobability that heads are observed morethan three times ?

n(n + 1)2(b) 2

(a) 316

5(b) 16

1(c) 2

66. The geometric mean of the observationsxl' x2' x3' .... xn is Gl. The geometricmean of the observations Yl' Y2' Y3' .... Yn

is G2• The geometric mean of observa-

. Xl X2 X3 Xn .. 110ns -, -, -, .... - IS

Y\ Y2 Y3 Yn

2. The probability that the experiment

ends with three tosses is .!..8

Select the correct answer usmg thecode given below :

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

B-QGC-O-REV - B 24

- --

69. If x E [0, 5], then what is the proba-bility that :x! - 3x + 2 ~ 0 ?

(a)4 2- 5 ,- \ ~+1- I( 0-5

1 4~(-+'Lr"r--6

(b) -5

I'

(c) 2-5

'(d)3-5

70. A bag contains 4 white and 2 black ballsand another bag contains 3 white and 5black balls. If one ball is drawn fromeach bag, then the probability that oneball is white and one ball is black is

lt~ . (a) 5-24

(b)13-24

(c)1-4

(d) 2-3

71. A problem in statistics is given to threestudents A, Band C whose chances ofsolving it independently are t, t and ±respectively. The probability that theproblem will be solved is

, I~~+~L~q"'"

~\n~ '3

, ~~~26

(a) 112

(b) 1112

B:'QGC~O-REV - B

(c) 12

(d) 34

72. An insurance company insured 2000scooter drivers, 4000 car drivers and6000 truck drivers. The probabilities ofan accident involving a scooter 'driver,car driver and a truck driver are 0·01,0·03 and 0·15 respectively. One of theinsured persons meets with an accident.The probability that the person is ascooter driver is

(a) 52

(a) 12

(b) 13

2(c) 5

(d) 15

. -- -- ---

74. The regression coefficients of a bivariatedistribution are -0·64 and -0·36. Thenthe correlation coefficient of the distribu-tion is

(a) 0-48

(b) -0·48

(c) 0·50

(d) -0·50

75. What is the probability that the sum ofany two different single digit naturalnumbers is a prime number?

- .,., ~ \,','------~r- ~\ ,~ . ~

~ "-Y~I 78. The value of k is ~ y~=>: ~~ t» o/~»> 51 '$)'/ (a) 5 ~ X 'Y

,,~:o X 'V ~ 3 J( 0 x~D . ()~ 3 \0 /;(d) None of the above cy~ ~llo (b) 5 '1- "V

X~ ~ ~/ ~/>For the next two (02) items that follew ; 9., • .J.l-~) ~ ~ f/ '\ _I'l.

Consider the function \ "'.1 d 'Z 2-'1- \ ~) < \,,'f-) \It-"l'-

(1)2X2 k~ \c1J ~ ) 3 <1/V ,Lr/ P]

f(x) = - , where x> 0 \ 1\1 ~ J\..fcQ 5 / 'V \r'x 'i'() \£~. V-

~ /~~76. At what value of x does the function t'f. f"(-~) is equal to \ 0

attain maximum value? 3

'1Z'. ~

5

(' 27

7(b) 18

(c) 13

(a) e ~

(b) * -:(c) 1

* ..-LX"A-

(d) I d-e

B-QGC-O-REV - B 28

77. The maximum value of the function is

(a) e

1(d) e

For the next two (02) items that follow:

x2 •Consider f' (x) =- - kx + 1 such that

21(0) = 0 and 1(3) = 15

(a) -1

(b) 13

For the next two (02) items that follow: 83. What is the value of B?

Consider the functionf(x) = _2x3 - 9x2 - 12x + 1

tt(a) 4

(b) n2

9.- ,~f}<..-t 'L~C> 3n- 0')(. - (c)

1- ~ ')'-..k~I'::6 4_F'''' -\ -. (d) 1t

~ For the next two (01) items tbot fallew,

"" '4. >1A.:-1'" 1.;:::" Consider the function

81. The function f (x) is a decreasing func- ')<..L-\" l:lt.k ').,.1.l ~ )

tion in the interval '>t L)..k L) ~ : ~ --, \

~A-'l/~\')~ tn.? f(x)= Asinx+B

~

80. The function f(x) is an increasing func-tion in the interval

(a) (-2, -1)

(b) (-00, -2)

(c) (-1, 2)

(d) (-1, (0)

(a) (-2, -1)

cosx

-2sinx .f n1 x~--2

·f 1r n1 --<X<-2 2·f n1 xii!:-

2(b) (-00, -2) only

(c) (-1, (0) onlywhich is continuous everywhere.

(d) (-00, -2) u(-1, (0)

For the next two (02) items that follow:

84. The value of A is

~~uJ"cJt-0-=Afli{~+~)+~

~ £M~~85. The value of B is =-A ;J~Ih.)

(a) 1 ~

t~

(a) 1

Consider the integrals

A=] sinxdx andB=] sinxdxo sin x + cos x 0 sin x - cos x

(b) 0

,~~I:i (c) -1..£11"- "- ••••7~ 0~t).~r (d)-2

(a) A = 2B

(b) B = 2A

~

AtJ·{~-~)tf\AN, k tf,/ -Wh z: CPD (:t+~

(} ~ -,

(c) A = B (c) -1

(d) A = 3B ~.7'~",\

B-QGC-O-REV - B \ >f:-t}"

(d) -2

-=-l. -----'--- ~ __ L

______________________________ -L ~ ~ J

86. The degree of the .differential equation 89. The order and degree of the differentiald ( d )-4 equati~n. of parabolas having vertex at~ - x = y - x~ is \.. k- ,the ongm and focus at (a, 0) wheredx dx \ r "-\ / "I-.~r a > 0, are respectively

:: : r -]:r) \Q (a) 1, 1

~ (b) 2,1 \.9,19}(c) 4

(d) 5(c) 1, 2

(d) 2, 2

87. The solution of

dy = ~ 1- x2 _ y2 + x2 y2 isdx

90. f(xy) =f(x) +fey) is true for all

(a) Polynornialfunctions j" ~6~;h~(b) Trigonometric functions f \~

'Cl..'T~'

(c) Exponential functions ffL 4'~

(d) Logarithmic functions f 0 -:;~91. Consider the following statements :

where c is an arbitrary constant 1. The function f(x) = x2 + 2cos x IS. increasing in the interval (0, n)

88. The differential equation of the family of W q )2, +~_~)L ~ q 1..... /'

circles passing through the origin and . . (~)having centres on the x-axis is 'X.L-+ t-- 2..XQ 2. The function j'(r) = In ,,1+ x- - x

+~'L;:..1i~decreasing in the interval (-00, (0)

Which of the above statements is/arecorrect?

B-QGC-O-REV - B

~--~--------~-~

92. The derivative of In(x + sin x) withrespect to (x + cos x) is

93 If_ -1[..J1+ sin x +..Jl-sinx ]

• y - cot ,""1+ sin x - ""1~ sin x

h n dy.. Iwere 0 < x < -, then - IS equa to2 dx

(a)ls- cos x

(x + sin x)(l- sin x)

(b)l-cosx

(x + sin x)(l + sin x)

(c)l-cosx

(x - sin x)(1 + cos x)

(d)I+cos x

(x - sin x)(l- cos x)

(a)1-2

(b) 2

(c) sin x + cos X

I (d) sin x-cos x

cally increasing if

(a) x < 0 only

(b) x> 2 only

(c) 0<x<2

(d) x E (-00, 0) U (2, (0)

B-QGC-O-REV - B

1

(a) (X;7)3

( X~7)~

( X_~)3

95. If xai = (x - -r: then the value:~bf .

dy y . I '___ IS equa to b--

( , j.. r 0 ~ 'l( ') dx x I ~r\7 0~ a ~\;

~ <a) b . tr\ ~ X. .'tJ1~\1 .

~ (b) % r "

\L+ Cv~'r 1~(c) 1

~) Itr~'\~)(d) 0~~.

96. If I: 1R ~ 1R, g: 1R ~ 1R be two func-tions given byI(x) = 2x- 3 and g(x) = x3 + 5,then (fogrl(x) is equal to

1(d~J .-

~ (~-\-S) - ')

CJ~g -t 1-:::: ~

~7-)v1"

'.~~~

34

1. f(x) = In x is an increasing functionon (0, (0).

98. ISin'( ~) dx is equal to

(a)8

15

(b)1615

32(c) 15

~o

99. If f(x) = sin (ex-2

-1) , thenIn(x-l)

Lim f(x) is equal tox~2

(a) -2

(b) -1

(c) 0

(d) 1

ts=: d2s101. If s = vt2 + 1, then -2 is equal todt

(a) .!s

Statement 1: The function f: IR ~ IRsuch that f(x) = x3 for allx E IR is one-one.

Statement 2 : f(a) =f(b) => a =0 for'1 all a, b E IR if the func-

~ )r/1- \ ~ )iion f is one-?ne~ ~)X\\J~ t\$~~ ~Jct:P()-

Which one of the following is correctin respect of the above statements?

2. f(x) = l! - x (In x) is an increasingfunction on (1, (0). .

(a) Both the statements are true andStatement 2 is the correct explana-tion of Statement 1.

Which of the above statements is/arecorrect?

(9J Both the statements are true andStatement 2 is not the correctexplanation of Statemerlt 1.

(a) 1 only

(b) 2 only (c) Statement 1 is true but Statement 2is false.

(c) Both 1 and 2\~(d) Statement 1 is false but Statement 2

(d) Neither 1 nor 2 a~~~ is true.,

B-QGC-O-REV - B 36C).. '1-~

e-- e

(c) In (1 + eX) + c

~e.,f)l..A- ,.z->:~~

e9-- 6.,'X..(d) 21n (1 + e-X

) +c ~

where c is the constant of integration

s~ J,s::~y

~\K'1ik!J1r~

b:. ' L.\: ~1..- ~ .

(d) 156·25 square unit 5 '/-, ~, q.. 'l~~~~~i~~;. o..~

'" '~1r z-: 'l...-k- 107. ABCD is a parallelogram and P is the

% point of intersection of the diagonals. If\/' ---+ ---+ ---+ --+\ k 1\ 0 is the origin, then OA + OB + OC + OD .

\ \ -\ \) is equal to\6~

106. The area of the square, one of whose

diagonals is 3{+4j is

(a) 0

X~ ') '1--\'1'-• ~ '1-. \'" \

(aJ ---+40P

2 (b) ---+

(b) - 20P,3 4-/

~~.(c) ---+

(c) 2 OP

(d) -2 (d) Null vector

103. J dx is equal tol+e-x

(a) 1+ eX + c (a) 12 square unit

(b) 12·5 square unit

(c) 25 square unit

1

104. Jx\x\dx is equal to-1

1OS. The area bounded by the coordinate

axes and the curve ,J; +..JY = 1, is

(d) .!. square unit6

108. If b and ~ are the position vectors of the~['II.. points Band C respectively, then therd~\ It..POsition vector of the point D such that\,,-~ --+ --+1~~BD=4BC is

L(a) 1 square unit

(b) .!. square unit2

(c) .!. square unit3

B-QGC-O-REV - B

~109. If the position vector a of the point

(5, n) is such that \ C; \ = 13, then thevalue/values of n can be

(a) ±8

(b) ±12

(c) 8 only

(d) 12 only

110. If \ C; \ = 2 and \ b \ = 3, then

\ {; xb \2 +\ "d.b \2 is equal to

(a) 72

(b) 64

(c) 48

(d) 36

111. Consider the following inequalities m. -+ -+

respect of vectors a and b :

Which of the above is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

B-QGC-O-REV - B

112. If the magnitude of difference of two

unit vectors is -J3, then the magnitudeof sum of the two vectors is

(a) 1 .- umt2

(b) 1 unit

(c) 2 unit

(d) 3 unit

113. If the vectors ai + aj + yk, { + k and

y{+ ti+ 13k lie on a plane, whe;e a, {3and r are distinct non-negative numbers,then r is

(a) Arithmetic mean of a and f3

(b) Geometric' mean of a and f3

(c) Harmonic mean of a and f3

(d) None of the above

---+ ---+ ---+ ---+114. The vectors a, b, c and d are such that

---+ ---+ ---+ ---+ '---+ ---+ ---+ ---+a x b = ex d and a x c = b x d . Whichof the following is/are correct?

1. (1-d)X(b-1)=0

2. (1x b) x ( 7x d) = 0Select the correct answer usmg thecode given below:

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

40

(a) 2b - asin(b - a)

(b) a + 3bcos(b - a)(a) 1

2

. ,

115. The value of

JbX7 +sin r .---dx where a+b= 0 IS

cosxa

. 1- sin x +cos x118. The function f (x) =. is1+SIDX+COSX

not defined at x == n. The value of fen)so that f(x) is continuous at x = tt is

(c) sin a - (b - a)cos b1

(b) 2

(c) -1

(d) 1

(d) 0

116. If f(x)=~25-x2, then ~hat is

Lim f(x) - f(l) equal to?x-.I x-I

119. Consider the following functions:

(a) 1-5

(b)1-

24

(c) -54

(d)1

--54

1. f(x) = {! if x:t 0

o if x =0

2. f(x) = {2X+5 if x >0x2 + 2x + 5 if x:::::0

Which of the above functions is/arederivable at x = 0 ?

117. Consider the function

{

ax - 2 for - 2 < x < '71f (x) = -1 . for -,-I:s: x s 1

a + 2(x _1)2 for 1< x < 2

(a) 1 only

(b) 2 only.

(c) Both 1 and 2

(d) Neither 1 nor 2

What is the value of a for which f(x) iscontinuous at x = -1 and x = 1 ?

120. The domain of the function1

f (x) = ~I x 1_x IS

(a) -1

(b) 1

(c) 0

(d) 2

(a) [0, 00)

(b) (-«>, 0)

(c) [I, 00)

(d) (-«>, 0]

B-QGC-O-REV - B 42

r-