Jawaharlal Nehru University (Jnu)-2017

JAWAHARLAL NEHRU UNIVERSITY (JNU)-2017

Exam Month :Dec 27-30, Time : 3 Hrs., Question : 120

Eligibility:

Bachelor’s degree in any discipline with adequate competence in mathematics under 10+2+3 pattern of education with at least 55% marks.

Reservation of Seats:

22.5% (15% for SC and 7.5% for ST) seats in each Programme of Study are reserved for Scheduled Caste / Scheduled Tribe candidates respectively. All Scheduled Caste / Scheduled Tribe candidates who have passed the qualifying examination are eligible to appear in the entrance examination irrespective of their percentage of marks. 27% seats are reserved for OBC candidates (non creamy layer).

EXAM PATTERN

The question paper for the entrance examination will consist of multiple choice questions.

SYLLABUS

10+2 / Graduates level Mathematics and Probability and General Aptitude and reasoning. About 80% questions will be from mathematics and Probability and about 20% questions from General Aptitude and reasoning.

JNU-2015 SOL.

1 INFOMATHS/MCA/MATHS/

1.The relation R = {(1, 1), (2, 2), (3, 3), (1, 2) (2, 3), (1, 3)} on set A = {1, 2, 3} is

(a) reflexive but not symmetric

(b) reflexive but not transitive

(d) neither symmetric nor transitive

2.If R is a relation on a finite set having n elements, then the number of relations on A is

(a) 2n(b) (c) n2(d) nn

3.Let R be a reflexive relation on a finite set A having n –elements, and let there be m ordered pair in R. Then

(a) m  n (b) m  n (c) m = n (d) N.O.T

4.If = 325 (x + iy), where x and y are reals then the ordered pairs (x, y) is given by

(a) (0, 3) (b)

5.If , then |z2 + 2z cos | is |

(a) less than 1 (b)

6.If |z| = 3, then the points representing the complex number – 1 + 4z lies on a

(a) line (b) circle

7.If logxa, ax/2 and logb x are in G.P., then x is equal to

(a) loga (logba) (b) loga (logea) + loga (logeb)

8.Let a, b, c be in AP and |a| < 1, |b < 1, |c| < 1. If |

x = 1 + a + a2 + …. to 

y = 1 + b + b2 + …. to 

z = 1 + c + c2 + …. to 

then x, y, z are in

(a) AP(b) GP(c) HP(d) N.O.T

9.Let a1, a2, ……, a10 be in AP and h1, h2, ……, h10 be in HP. If a1 = h1 = 2 and a10 = h10 = 3, then a4h7 is

(a) 2(b) 3(c) 5(d) 6

10.If a, b, c are in GP, then the equations ax2 + 2bx + c = 0 and dx2 + 2ex + d = 0 have a common root if d/a, e/f, f/c are in

(a) AP(b) GP(c) HP(d) N.O.T

11.If the product of the roots of the equation is 31, then the roots of the equations are real for k equal to

(a) 1(b) 2(c) 3(d) 4

12.The roots of the equation

, where a2 – b = 1, are

(a) (b)

13.The quadratic equation whose roots are A.M. and between the roots of the equation ax2 + bx + c = 0

(a) abx2 +(b2 + ac)x + bc = 0

(b) 2abx2 +(b2 + 4ac)x + 2bc = 0

(d) N.O.T

14.The value of is

(a) 10(b) 6(c) 8(d) 4

15.If x2 – 2x cos  + 1 = 0, the value of x2n – 2xn cos n + 1 is equal to

(a) cos2n(b) sin 2n

(d) some real number other than 0

16.If a, b, c  R and a + b + c = 0, then the quadratic equation 4ax2 + 3bx + 2c = 0 has

(a) one positive and one negative roots

(b) imaginary roots

17.The number of ways in which one can post 5 letters in 2 letter boxes is

(a) 35(b) 7P5(c) 75(d) N.O.T

18.The value of 12 . C1 + 32 . C3 + 52 . C5 + … is

(a) n (n – 1)2n-2 + n.2n-1(b) n(n-1)2n-2

19.If in the expression of (1 + x)m (1 – x)n, the coefficients of x and x2 are 3 and – 6 respectively, then m is

(a) 6(b) 9(c) 12(d) 24

20.If g(f(x)) = [sin x] and f(g(x)) , then

(a)

(b)

(c)

(d) f and g cannot be determined

21.The sum o f n terms of the series is

(a) 2n – n – 1 (b) 1 – 2-n

22.If the equations x2 + px + q = 0 and x2 + p'x + q' = 0 have a common root, then it is equal to

(a) (b)

23.The number of ways in which n distinct objects can be put into two different boxes so that no box remains empty, is

(a) 2n – 1 (b) n2 – 1 (c) 2n – 2 (d) n2 – 2

24.The coefficient of x5 in the expression of (1 + x2 – x3)8 is

(a) 80(b) 84(c) 238(d) 92

25.If nC4, nC5, nC6 are in AP., then n is equal to

(a) 12(b) 11(c) 7(d) 8

26.If A is a square matrix of order n  n, then adj (adj A) is equal to

(a) |A|n A (b) |A|n-1 A

27.If , then the value of Xn is

(a) (b)

28.If  be one of the roots of unity, then is equal to

(a) (b) 2(c) 0(d) 1

29.If and then  is equal to

(a) (b)

30.If A and B are two matrices such that AB = B and BA = A, then A2 + B2is equal to

(a) 2AB (b) 2BA (c) A + B (d) AB

31.The circle whose equations are x2 + y2 + c2 = 2ax and x2 + y2 + c2 – 2by = 0 will touch on another externally if

(a) (b)

32.The two circles x2 + y2 – 2x – 3 = 0 and x2 + y2 – 4x – 6y – 8 = 0 are such that

(a) they touch each other

(b) they intersect each other

(d) each lies outside the other

33.The equation of the normal to the parabola y2 = 8x having slope 1 is

(a) x + y + 6 = 0 (b) x – y – 6 = 0

34.The line y = mx + 1 is a tangent to the parabola y2 = 4x if m = 1

(a) 1(b) 2(c) 3(d) 4

35.P is a variable point on the ellipse with AA' as the major axis. Then, the maximum value of the area of the  AP' A is

(a) ab(b) 2ab(c) ab/2(d) N.O.T

36.An ellipse is describe by using an endless string which is passed over two pins. If the axes are 6 cm and 4 cm, then necessary length of the string and the distance between the pins, respectively in cms, are

(a) (b) (c) (d) N.O.T

37.The e and e1, are the eccentricities of the hyperbolas xy = c2 and x2 – y2 = c2, then is equal to

(a) 1(b) 4(c) 6(d) 8

38.The combined equation of the asymptotes of the hyperbola 2x2 + 5xy + 2y2 + 4x + 5y = 0

(a) 2x2 + 5xy + 2y2 + 4x + 5y + 2 = 0

(b) 2x2 + 5xy + 2y2 + 4x + 5y – 2 = 0

(d) N.O.T

39.If f : R  R is given by f(x) = 3x – 5, then f-1 (x)

(a) is given by

(b) is given by

(d) does not exist because f is not onto.

40.Let

Then, f(x) is continuous but not differentiable at x = 0, if

(a) n  (0, 1)(b) n  [1, )

41.Let g(x) be the inverse of the function f(x) and . Then g’(x) is equal to

(a) (b)

42.If , then f '(1) equals

(a) – 1 (b) 1(c) log2 (d) – log2

43.If f(x) = logx (in (x)), then f'(x) = at x = e is

(a) e(b) -e(c) e2(d) e-1

44.If , then F '(4) equals

(a) (b) (c) (d) N.O.T

45.If xp yq = (x + y)p + q, then is equal to

(a) (b) (c) (d)

46.If is

(a) 1(b) (c) 0(d)

47.Let f(x) = x – [x], for every real number x, where [x] is integral part of x. Then is

(a) 1(b) 2(c) 0(d) 1/2

48., then

(a) I1 = I2(b) 2I1 = I2(c) I1 = 2I2(d) N.O.T

49.The value of the integral is

(a)  log2 (b) -log2 (c)  log 3 (d) N.O.T

50.If f(x) is a function satisfying for all non-zero x, then equals

(a) sin  + cosec (b) sin2

51.The order and degree the differential equation of all tangent lines to the parabola x2 = 4y is

(a) 1, 2 (b) 2, 2(c) 3, 1(d) 4, 1

52.A solution of the differential equation

(a) y = 2 (b) y = 2x

53.The differential equation representing the family of curves, where c is a positive parameter, is of

(a) order 1 (b) order 2

54.If and

, they x + y + z is equal to

(a) (b)

55.If the vector bisects the angle between the vector and the vector then the unit vector in the direction of is

(a) (b)

56.If , are unit vectors such that the vector is perpendicular to and is perpendicular to , then the angle between and is

(a) (b) (c) (d)

57.Twelve balls are distributed among three boxes. The probability that the first box contains 3 balls is

(a) (b)

58.If A and B are two events such that P(A) > 0 and P(B)  1, then is equal to

(a) 1 – P(A/B) (b)

59.A coin is tossed (m + n) times, (m > n). Then, the probability of at least m consecutive heads is

(a) (b) (c) (d) N.O.T

60.Let ABC be a triangle such that A = 45, B = 75, then is equal to

(a) 0(b) b(c) 2b(d) -b

61.In any ABC, if sin2A + sin2B = sin2C, then the triangle is

(a) equilateral (b) right-angled

62.The smallest angle of the triangle whose sides are is

(a (b) (c) (d) N.O.T

63. is equal to

(a) (b) f '(x) g(x) – f(x) g'(x)

64. is equal to

(a) log (x4 + 1) + C (b)

65. is equal to

(a) (b)

66.If , then the value of a is

(a) 1(b) 0(c) e(d) N.O.T

67.Evaluate the following limits :

(a) 0(b) 1(c) -1 (d) N.O.T

68.If (1 + x)n = C0 + C1x + C2x2 + ….. + Cnxn, then

is equal to

(a) 22n-2(b) 2n

69.If the radius of the circumcircle of an isosceles triangle PQR is equal to PQ = (=PR), then the angle P is

(a) (b) (c) (d)

70.Identify the curve y2 – x2 + 2ax – 1 = 0

(a) Pair of straight line (b) circle

71.The value of

(a) (b) (c) (d) N.O.T

72.Period of function f(x) = |sin x| is

(a) (b) 2(c) 1(d) 2

73.Domain of the function 2x + 2y = 2 is

(a) (-, 0](b) (-, 1)(c) (0, )(d) (1, )

74.The digit of unit place of

1! + 2! + 3! + 4! + …….

(a) 3(b) 4(c) 7(d) 8

75.The number of values of x in the interval [0, 5] satisfying the equation

3sin2x - 7 sin x + 2 = 0

(a) 5(b) 6(c) 0(d) 8

76.The solution of the trigonometric equation

1 – cos  = sin /2. sin , where  = ?

(a)  = k, k  I (b)  = 2k, k  I

77.If , given y = 0 when x = 5 then the value of X for y = 3 is

(a) e5(b) e6 + 1

78.

(a) (b) g(x) = logx

79.Range of the function |sin 2x – cos 2x|

(a) (-, ) (b) (-4, 4) (c) [-2, 2](d) [-1, 1]

80.If

then the value of k, where  is complex cube root of unity (a, b, c  R)

(a) (b) 2(c) 1(d) -

81.

(a) 2(b) -2(c) (d)

ODD ONE OUT

82.(a) MATHS (b) TRIGONOMETRY

83.Odd one out

(a) ARC (b) TANGENT

84.INFLUENZA: VIRUS :: TYPHOID: ?

(a) BACILLUS (b) PARASITE

85.What is the angle between hour hand and minute hand at 10 past 5?

(a) 90(b) 95(c) 98(d) 100

86.If number of boys in a class is 3 times the number of girls which cannot be the total number of students?

(a) 44(b) 48(c) 42(d) 40

87.If 2 workers complete work in 10 days and 15 days respectively, than in how many days will they complete the work together

(a) 6 days (b) 5 days (c) 7 days (d) 9 days

88.The next term is of the series is

7, 13, 25, 49, …..?

(a) 96(b) 97(c) 98(d) 99

89.How many such pairs of letter are there in the word "COMPUTERS" each of which have as many letters between them in the word as in the alphabet?

(a) 1(b) 2(c) 3(d) more than 3

90.In what ratio water and 66% solution of wine is mixed to get the 55% solution of wine?

(a) 2 : 5 (b) 1 : 4 (c) 1 : 5 (d) 1 : 6

91.How much does a watch loose per day, if its hands coincides every 64 min?

(a) 90 minute (b) 96 minutes

92.If and xy = 1 then find the value of , is

(a) 1(b) 0(c) (d) N.O.T

93.Two trains start from station A at 9:00 am and 8:30 am with the speed of 90 km/h and 80 km/h respectively then how much distance from station A the both trains will meet together?

(a) 270 km (b) 820 km

94.A bag contains coins of 25 paise, 50 paise and 1 rupees and the sum of money is rupees 35, then find the total number of coins of each type.

(a) 20(b) 25(c) 30(d) 33

95.If 4th day after 6th January is SATURDAY, then what will be the day on 1st December in the previous year?

(a) THURSDAY (b) FRIDAY

96.If Curd: milk, then which of the following show similar relationship?

(a) Clot: Blood (b) Flow : River

97.A man is facing south be turns 135 anticlock-wise than 180 clock-wise. How, in which direction is he facing?

(a) NORTH-EAST (b) NORTH-WEST

98.Solution of equation , then  equals

(a) (b) n

99.A wheel has circumference m and it makes 7 revolution in 4 seconds, than find the speed of car in km/h?

(a) 27(b) 67(c) 37(d) 47

100.The sum of the coefficient of all the integral powers of x in the expansion of , is

(a) 340 + 1 (b) 340- 1

101. is equal to

(a) (b)

102.In order that a relation R defined on a non-empty set A is an equivalence relation, it is sufficient, if R

(a) is reflexive

(b) is symmetric

(d) possess all the above three properties

103.If , then f(x) is

(a) continuous for all x but not differentiable, at x = 0

(b) neither differentiable nor continuous at x = 0

(d) continuous as well as differentiable at x = 0

104.There are n different books and p copies of each. The number of ways in which a selection can be made from them is

(a) np(b) pn

105.If is a matrix satisfying the equation AAT = 9I, where I is 3  3 identify matrix, then the a + b is equal to :

(a) – 3 (b) 3(c) 2(d) 1

106.If z is a complex number having least absolute value is |z – 2 + 2i| =1, then z equals

(a) (b)

107.A number is selected from a first 120 natural numbers. Then, the probability that the number is divisible by 5 or 15 is

(a) (b) (c) (d)

108.Inverse of which function is exist:

(a) for all x  R

(b) f(x) = x2 for all x  R

(d) f(x) = x2 for all x  R

109.sec 4 - sec2 = 2 then value of  equal to

(a) (b)

110.If x3 + 3x2 – 9x + c is of the from (x–)2 (x - ), then c equals

(a) – 5 (b) 27(c) -27(d) 0

111.Pointing to a man, a woman said that, "he is the, son of the brother of my mother”. How is that man related to woman?

(a) Brother (b) cousin

112.Pointing to a man, Nilesh said that, "his wife is the daughter of my uncle". How is Nilesh related to that man?

(a) Father (b) Father-in-law

113.A drawn contains 5 brown socks and 4 blue socks well mixed. A man reaches the drawer and pulls out 2 shock at random. What is the probability that they match?

(a) 4/9(b) 5/8(c) 5/9(d) 7/12

114.The equation of the curve satisfying the differential equation y2 (x2 + 1) = 2xy1 passing through the point (0, 1) and having slope of tangent at x = 0 as 3, is

(a) y = x3 + 3x + 2 (b) y = x3 – 3x – 2

115.The area of the quadrilateral formed by the tangents at the end-points of latusrectum to the ellipse , is

(a) sq. units (b) 9 sq. unit

116.A problem in Mathematics is given to four students A, B, C and D their respective probability of solving the problem are and . Probability that the problem is solved is

(a) (b) (c) (d) N.O.T

117.If O is the origin and OP, OQ are tangent to the circle x2 + y2 + 2gx + 2fy + c = 0, the circumcentre of OPQ, is

(a) (-g, -f) (b) (g, f) (c) (-f, -g) (d) N.O.T

118.A man is known to speak truth 3 out of 4 times. He throws of die and report that it is 6. Then, find the probability that it is actually 6.

(a) 1/4(b) 5/8(c) 3/8(d) 1/6

119.In ABC, if a = 2, b = 4 and C = 60 then A and B are respectively equal to.

(a) 90, 30(b) 45, 75

120.The number of ways of arrange the letters of the English alphabet, so the there are exactly 5 letters a and b, is

(a) 24P5(b) 24P5 20!

1 INFOMATHS/MCA/MATHS/

JNU-2015SOLUTIONS

1 INFOMATHS/MCA/MATHS/

1.Ans. (a) Since (1, 1), (2, 2), (3, 3)  R, therefore R is reflexive (1, 2)  R but (2, 1)  R, therefore R is not symmetric.

It can be easily seen that R is transitive.

2.Ans. (b)2n

3.Ans. (a)Since R is reflexive relation on A, therefore (a, a)  R for all a  A

 the minimum number of ordered pairs in R is n.

Hence, m  n.

4.Ans. (b)

 - (-)50 = x + iy

 -50 = x + iy  -)2 = x + iy

5.Ans. (a)|z2 + 2z cos|  |z2| + |2z cos |

= |z|2 + 2|z| |cos|  |z|2 + 2|z|

6.Ans. (b)Let  = - 1 + 4z. Then,

 + 1 = 4z  | + 1| = 4 |z| = 12

Thus,  lies on a circle with centre at – 1 and radius equal to 12.

7.Ans. (a)Since logxa, ax/2 and logb x are in G.P. Therefore, (ax/2)2 = logxa, logbx

 ax = logba  x = loga(logba)

8.Ans. (c)We have,

Now, a, b, c are in A.P.  1 – a 1 – b, 1 – c are in A.P.

are in H.P.  x, y, z are in H.P.

9.Ans. (d)Le d be the common difference of the AP, Then,

a10 = 3  a1 + 9d = 3 2 + 9d = 3  d

Let D be the common difference of . Then,

10.Ans. (a)ax2 + 2bx + c = 0

This satisfies dx2 + 2ex + f = 0

are in A.P.

11.Ans. (d)Product of roots = 31

 2e2logk-1 = 31

 2k2 – 1 = 31  2k2 = 32  k2 = 16 k =  4

But k > 0. Therefore, k = 4

Now, Disc = 8k2 – 8e2logk + 4 = 8k2 – 8k2 + 4 = 4 > 0 for all k, Hence, = 4

12.Ans. (b)We have,

by putting , the given equation becomes

 x2 – 15 = 1 or x2 – 15 = - 1  x =  4,

13.Ans. (b)Let ,  be the roots of the given equation. Then  +  = - b/a and  = c/a.

Required equation is

 2ax2 + (b2 + 4ac)x + 2bc = 0

14.Ans. (d)Let . Then,

 x = 4.

15.Ans. (c)x2 - 2x cos  + 1 = 0  x = cos  i sin

 x2n = cos 2n i sin 2n and xn = cos n  i sin n 

 x2n – 2xn cos n  + 1

= cos 2n  i sin 2n - 2 cos2n 2i sin n cos n + 1 = 0

16.Ans. (c)The Discriminant of the given quadratic is

D = 9b2 – 32 ac

= 9(-a – c)2 – 32 ac

= 9a2 + 9c2 – 14ac = c2 [9(a/c)2 – 14a/c + 9]

Since the discriminant of 9(a/c)2 – 14(a/c) + 9 is negative therefore the sign of the expression 9(a/c)2 – 14(a/c) + 9 is always positive.

Hence, the roots of the given equation are real.

17.Ans. (d)Each letter can be posted in any one of the 2 letter

So, required number of ways = 2  2  2  2  2 = 25

18.Ans. (d)We have :

Adding these two, we get

2[12C1 + 32C3 + 52C5 + …] = n(n – 1)2n-2 + n.2n-1

 12C1 + 32C3 + 52C5 + … = n(n – 1)2n-3 + n.2n-2

19.Ans. (c)(1 + x)m (1 – x)n = (mC0 + mC1x + mC2x2 + … + mCmxm) x (nC0 – nC1 x + nC2x2 …. + (-1)nnCnxn)

= mC0 . nC0 – (mC0nC1 – nC0mC1) x + (mC0nC2 + nC0mC2 – mC1nC1)x2 + ….

It is given that the coefficients of x and x2 in the expression of (1 + x)m (1 – x)n are 3 and – 6 respectively

Therefore,

-(mC0 . nC1 – nC0 . mC1) = 3

and mC0nC2 + nC0mC2 – mC1nC1 = - 6

 m – n = 3 and n(n – 1) + m(m – 1) – 2mn = - 12

 m – n = 3 and (m – n)2 + (m – n) = - 12

 m – n = 3 and m + n = 21  m = 12, n = 9

20.Ans. () and

21.Ans. ()We have,

to n terms

= n – 1 + 2-n.

22.Ans. (c)Let  be a common roots of x2 + px + q = 0 and x2 + px + q’ = 0

Then, 2 + p + q = 0 and 2p' + q' = 0

23.Ans. (c)Each object can be put either in box B1 (say) or in box B2(say). So, there are two choices for each of the n objects.

Therefore the number of choices for n distinct objects is 2  2 …  2 = 2n.

n–times

The of these choices correspond to either the first or the second box being empty.

Thus, there are 2n – 2 ways in which neither box is empty.

24.Ans. (b)The general term in the expansion of (1 + x2 – x3)8 is

, where r + s + t = 8,

For the coefficient of x6, we must have 2x + 3t = 6.

Now, r + s + t = 8 and 2s + 3t = 6

where 0  t  8.

For t = 0, r = 5, s = 3 For t = 2, r = 6, s = 0

 Coefficient of

25.Ans. (c)

26.Ans. (c)Four any square matrix X, we have

X (adj X) |x| In

Taking X = adj A, we get

(adj A) (adj (adj A)) = |adj A| In

 adj A (adj (adj A)) = |A|n – 1 In

 (A adj A) (adj (adj A)) = |A|n-1 A

 (|A|In) (adj (adj A)) = |A|n-1A

 adj (adj A) = |A|n-2A

27.Ans. (d), Clearly for n = 2, then matrices in (a), (b), (c) do not tally with

28.Ans. (c)

29.Ans. (c)

30.Ans. (c)We have AB = B and BA = A. Therefore,

A2 + B2 = AA + BB = A(BA) + B(AB) = (AB) A + (BA)B

= BA + AB = A + B

31.Ans. (c)The two circle are

x2 + y2 – 2ax + c2 = 0 and x2 + y2 – 2by + c2 = 0

Centres : C1 (a, 0), C2(0, b)

radii :

Since the two circle touch each other externally, therefore C1C2 = r1 + r2

 c4 = a2b2 – c2(a2 + b2) + c4 a2b2 = c2 (a2 + b2)

32.Ans. (b)The coordinates of centres C1 and C2 of two circles are (1, 0) and (2, 3) respectively. Let r1 and r2 be the radii of two circles. Then r1 = 2 and . Clearly r1 – r2 < c1c2 < r1 + r2

Hence the two circle intersect each other.

33.Ans. (b)Equation of normal to the parabola y2 = 8x at (x1, y1)

It is given that

y2 = 8x then

y1 = - 4

then x1 = 2

then equation of normal will be y + 4 = x – 2

x – y = 6

34.Ans. (a)

35.Ans. (a)

The maximum area corresponds to when P is at either of the minor axis and hence area for such a position of P is

36.Ans. (d)Given 2a = 6, 2b = 4. Therefore,

So, Distance between foci

and, length of the string = 2a + 2ae

37.Ans. (b)Eccentricity of rectangular hyperbola is

38.Ans. (a)Let the equation of asymptotes be

2x2 + 5xy + 2y2 + 4x + 5y = 0 …(i)

Te equation represents a pair of straight line. Therefore,

abc + 2fgh – af2 – bg2 – ch2 = 0

Here, a = 2, b = 2, h = 5/2, g = 2, f = 5/2 and c = 

Putting the value of  in (i), we get

2x2 + 5xy + 2y2 + 4x + 5y + 2 = 0

This is the equation of the asymptotes.

39.Ans. (b)Clearly, f : R  R is a one-one onto function. So, it is invertible.

Let f(x) = y. Then, 3x – 5 = y

Hence,

40.Ans. (a)Since f(x) is continuous at x = 0, therefore

f(x) is differentiable at x = 0 if

exists finitely

exists finitely.

 n – 1 > 0  n > 1.

If n  1, then does not exist and hence f(x) is not differentiable at x = 0 Hence, f(x) is continuous but not differentiable at x = 0 for 0 < n  1 i.e. n  (0, 1].

41.Ans. (c)Since g(x) is the inverse f(x), therefore

f(x) = y  g(y) = x

 g'(f(x)) = 1 + x3,  x

 g'(y) = 1 + {g(y))}3

[Using f(x) = yx =  x = g(y)]

 g'(y) = 1 + {g(y))}3 [replacing y by x]

42.Ans. (a)

43.Ans. ()

44.Ans. (a)We have . Therefore,

Differentiating both sides with respect to x, we get

2x F(x) + x2 F’(x) = 4x2 – 2F’(x)

Putting x = 4, we get

8F(4) + 16F'(4) = 64 – 2F'(4)  18F'(4) = 64

45.Ans. (a)We have xp yq = (x + y)p+q

 p log x + q log y = (p + q) log (x + y)

Diff. w.r.t. x, we get

46.Ans. (c)We have,

47.Ans. (a)

48.Ans. (a)Putting log x = t i.e. x = et in It we get

49.Ans. (a)

= - (-/2 log2) – (-/2 log2) =  log 2

50.Ans. (d)We have

where

2I = 0  I = 0

51.Ans. (a)The equation of any tangent to x2 = 4y is

; where m is an arbitrary constant.

Differentiating this w.r. to x, we get

Putting the value of m in we get

Which is differential equation of order 1 degree 2.

52.Ans. (c)Clearly, y = 2x – 4 satisfies the given differential equation.

53.Ans. (d)We have, …(i)

 2yy1 = 2x  yy1 = c …(iii)

Eliminating c from (i) and (ii), we get

Clearly, it is a differential equation of order one and degree 3.

54.Ans. (a)We have

Taking dot products with , we get

and

55.Ans. (d) be the unit vector along . Since, bisect the angle between and . Therefore,

and z = - 

Now, x2 + y2 + z2 = 1

[ is unit vector]

But  0, Because  = 0 implies that the given vectors are parallel and

Hence,

56.Ans. (c) We have,

= 7 + 16cos  - 15 = 0

57.Ans. (a)Since each ball can be placed in any one of the 3 boxes, therefore there are 3 ways in which a ball can e placed in any one of the three boxes. thus there are 312 ways in which 12 balls can be placed in 3 boxes. The number of ways in which 3 balls out of 12 can be put in the first box is 12C3. The remaining 9 balls can be placed in 2 boxes in 29 ways.

So, required probability

58.Ans. (wrong)Since .Therefore,

Correct answer is which is not given in any of the four option.

59.Ans. (a)

60.Ans. (c)As A = 45, B = 75, we have

C = 180 - (45 - 75) = 60

Now, b = k sin B  b = k sin 75

From (i) and (ii),

61.Ans. (b)

62.Ans. (c)Let

Clearly c is the smallest side. Therefore the smallest angle C is given by

63.Ans. (c)

64.Ans. (b)

65.Ans. (b)Let

Putting x + 1 = t2, dx = 2t, we get

66.Ans. (a)

Then is satisfied only when a = 1.

67.Ans. (b)

68.Ans. ()Given question is incomplete it should have been

If (1 + x)n = (C0 + C1x + C2x2 + …. + Cnxn) then

is equal to

We have,

(1 + x)n = (C0 + C1x C2x2 + ….. + Cnxn) …(i)

Also,

(1 + x)n = (C0xn + C1xn-1 +…. + Cn-1x + C) …(ii)

Multiplying (ii) and (iii), we get

(C0 + C1x + C2x2 + C3x3 + ….. + Cnxn)

 (C0xn + C1xn-1 + C2xn-2 + Cn xn-2 + … + Cn-1x + Cn) = (1 + x)2n…(iii)

Equating coefficient of xn on both sides of (iii) we get

(Given question is incomplete)

69.Ans. (d)In PQR, the radius of the circumcircle is given by . But it is the given the radius is

70.Ans. (wrong)We know that the equation second degree curve in

ax2 + by2 + 2xy + 2gx + 2fy + c = 0

for pair of straight line  = 0

when  = abc + 2fgh – af2 – bg2 – ch2 = 0

a – 1, b = 1, c = - 1, f = 0, g = a, h = 0

Put in equation (i)

- 1 + 2  0  a  0 – 1  0 + 1  a2 – 1  0 = 0

- 1 + a2 = 0,  0

where a is variable

All the four option given are incorrect

Ans. (wrong)

71.Ans. (a)We know that

Hence,

{where }

72.Ans. (c)We know that period of function |sin x| = 

Hence, X = 

 x = 1

73.Ans. ()Given that,

2y + 2x = 2

2y = 2 – 2x

taking log on both side

y log 2 = log (2 – 2x)

for log (2 – 2x) the necessary

condition that 2 – 2x > 0

2x < 2

Hence, x < 1

So, domain is (- , 1)

74.Ans. ()Given 1! + 2! + 3! + 4! + …..

Expand the given equation

1 + 2 + 6 + 24 + 120 + 720 + ......

Add upto 4!

Hence digit at unit place 3

75.Ans. (b)Given equation is 3 sin2x – 7 sinx + 2 = 0

By factoring

3 sin2x – 6sinx – sinx + 2 = 0

3sin x (sinx – 2) – 1 (sin x – 2) = 0

and sin x  2

(because sin x = [-1, 1]

From the above graph it is clear that in every (0, ), (2, 3), (4, 5) there are two solution, hence total no. of solution is 6 from [0, 5]

Hence there are 6 value of x which satisfy the given equation.

76.Ans. (b)Given that

1 – cos  = sin /2 . sin …(1)

We know that cos  = 1 – 2 sin2/2 put in eq. (1)

1 – (1 – 2 sin2/2) = sin/2 sin

2 sin2/2 = sin/2 sin

sin /2 = 0

sin /2 = kwhere k  I

 = 2k

77.Ans. (c)Given that

e2ydy = dx

on integrating both side

put x = 5. y = 0

78.Ans. (d)

Integrate by part

Compare with f(x) log (x + 1) + g(x) x2 + Lx + C

Hence, none of these is correct options.

79.Ans. (d)We have f(x) = |sin2x – cos2x|

We know that the maximum and minimum value of function a sin x + b cos x is

and respectively

Hence, a = 1, b = 1

the max. value of function is

and Minimum value is -

Hence, the range will be but it is not given in the option so approximately range is [-1, 1]