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INFOMATHSJAWAHARLAL 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. JNU-2015 SOL. 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 (c) symmetric and 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)

(c) (-3, 0) (d) (0, - 3) 5. If , then |z2 + 2z cos | is |

(a) less than 1 (b) (c) (d) N.O.T

6. If |z| = 3, then the points representing the complex number – 1 + 4z lies on a (a) line (b) circle (c) parabola (d) none of these

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) (c) –loga (logb b) (d) loga(logeb) – loga (logea)

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) (c) (d)

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(c) 2abx2 +(b2 + ac)x + bc = 0(d) N.O.T

14. The value of is

(a) 10 (b) 6 (c) 8 (d) 415. If x2 – 2x cos + 1 = 0, the value of x2n – 2xn cos n

+ 1 is equal to (a) cos2n (b) sin 2n(c) 0(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 (c) real roots (d) N.O.T

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

(c) n(n-1).2n-3 (d) N.O.T

1 INFOMATHS/MCA/MATHS/

INFOMATHS19. 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 (c) n + 2-n – 1 (d) 2n – 1

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)

(c) (d)

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 (c) |A|n-2 A (d) |A|n-3 A

27. If , then the value of Xn is

(a) (b)

(c) (d) N.O.T

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)

(c) (d) N.O.T

30. If A and B are two matrices such that AB = B and BA = A, then A2 + B2 is 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)

(c) (d) N.O.T

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 (c) one lies inside the 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 (c) x – y + 6 = 0 (d) 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(c) 2x2 + 5xy + 2y2 = 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

(c) does not exist because f is not one-one (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, ) (c) n (-, 0) (d) n = 0

41. Let g(x) be the inverse of the function f(x) and

. Then g’(x) is equal to


INFOMATHS

(a) (b)

(c) 1 + (g(x))3 (d) 1 + (f(x))3

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(c) cosec2 (d) N.O.T

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

is

(a) y = 2 (b) y = 2x (c) y = 2x – 4 (d) y = 2x2 – 4

53. The differential equation representing the family of

curves , where c is a positive

parameter, is of (a) order 1 (b) order 2 (c) degree 3 (d) both (a) and (c)

54. If and

, they x + y + z is equal to

(a) (b)

(c) (d) N.O.T

55. If the vector bisects the angle between the

vector and the vector then the unit vector

in the direction of is

(a) (b)

(c) (d)

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)

(c) (d)

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)

(c) (d)

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 (c) isosceles (d) N.O.T

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)

(c) f(x) g'(x) – f'(x) g(x) (d) f(x) g'(x) + f '(x) g(x)

64. is equal to


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

(c) - log (x4 + 1) + C (d) N.O.T

65. is equal to

(a) (b)

(c) (d) N.O.T

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

(c) (d)

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 (c) Ellipse (d) Parabola

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(c) = 2k + /2 (d) N.O.T

77. If , given y = 0 when x = 5 then the value

of X for y = 3 is (a) e5 (b) e6 + 1

(c) (d) bge 6

78.

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

(c) L = 1 (d) none 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

(c) GEOMETRY (d) ALGEBRA 83. Odd one out

(a) ARC (b) TANGENT (c) DIAGONAL (d) DIAMETER

84. INFLUENZA: VIRUS :: TYPHOID: ? (a) BACILLUS (b) PARASITE (c) PROTOZOA (d) BACTERIA

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

(c) (d)

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


INFOMATHS(c) 360 km (d) 400 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 (c) SATURDAY (d) SUNDAY

96. If Curd: milk, then which of the following show similar relationship? (a) Clot: Blood (b) Flow : River (c) Decant : wine (d) Coffee : Brew

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 (c) SOUTH-EAST (d) SOUTH-WEST

98. Solution of equation ,

then equals

(a) (b) n

(c) (d) N.O.T

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

(c) (d)

101. is equal to

(a) (b)

(c) (d) N.O.T

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 (c) is transitive (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 (c) discontinuous everywhere(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 (c) (p + 1)n – 1 (d) (n + 1)p – 1

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)

(c) (d)

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 (c) 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)

(c) (d) N.O.T

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 (c) uncle (d) None

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 (c) Son-in-law (d) none

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 (c) y = x3 + 3x + 1 (d) y = x2 + 3x – 1

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

, is


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

(c) sq. units (d) 27 sq. units

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 (c) 60, 60 (d) 30, 90

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! (c) 24P5 20! 2 (d) 24P5 24! 2

JNU-2015 SOLUTIONS1. 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 6 INFOMATHS/MCA/MATHS/

INFOMATHS

are in A.P.

11. Ans. (d) Product of roots = 31 2e2logk-1 = 31 2k2 – 1 = 31 2k2 = 32 k2 = 16 k = 4But 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)n nCnxn)= mC0 . nC0 – (mC0 nC1 – nC0 mC1) x + (mC0 nC2 + nC0 mC2 – mC1 nC1)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 mC0 nC2 + nC0 mC2 – mC1 nC1 = - 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

to n

terms

= n – 1 + 2-n.

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


INFOMATHS, 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


INFOMATHSf(x) is differentiable at x = 0 if

exists finitely

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


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


INFOMATHS51. 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]

or

But 0, Because = 0 implies that the given

vectors are parallel

and

Hence,


= 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


INFOMATHSClearly 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 = 0when = 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, 0where a is variable All the four option given are incorrectAns. (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


INFOMATHS 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]

80. Ans. (a)

abc – a3 – b3 + abc + abc – c3 = - (a + b + c) (a + bk + ck2) (a + bk2 + ck) +(a3 + b3 + c3 – 3abc) = (a + b + c) (a + bk + ck2) (a + bk2 + ck) (a + b + c) (a + b + c)2 (a + 2b + c) = (a + b + c) (a + bk + ck2) (a + bk2 + ck) Hence k = {where is cube root of unity 1 + + 2 = 0}

81. Ans. (a)

2

82. Ans. (a) All expect Maths are branches of Maths

83. Ans. (c) All expect Diagonal are terms associated with circle

84. Ans. (d) INFLUENZA is caused by VIRUS and TYPHOID is caused by BACTERIA.

85. Ans. (b) Angle traced by hour hand in 12 hour = 360

Angle traced by it 5 hrs. 10 min. i.e. hours

Angle traced by minute hand in 60 min = 360

Angle traced by it in 10 min

Required angle = (155 - 60) = 95

86. Ans. (c) Let the number of girls in the class be x then, the number of boys in class = 3x Total no, of students = x + 3x = 4x It means total no. of students must be a multiple of 4


INFOMATHS 42 cannot be total no. of students as it is not multiple of 4.

87. Ans. (a) First worker’s one day’s work

Second worker’s one day’s work

Both worker’s one day’s work

Therefore both the workers will finish the work in 6 days.

88. Ans. (b) Clearly, the given sequence follows the pattern +6, + 12, + 24, + 48, ….. Thus, 7 + 6 = 13, 13 + 12 = 25, ……

89. Ans. (d) M P U T E RM P U T E R SU TU T E RT E RR S

There are 6 such pairs

90. Ans. (c)

91. Ans. (c) 55 min spaces are covered in 60 min

60 min spaces are covered in min

min

Loss in 64 min min

Loss in 24 hrs. min

min

92. Ans. () (given) xy = 1 (given)then

Substituting the value of and y in the

expression,

93. Ans. () Let the train will meet after time t hours. Then, distance travelled by train before meet is equal to 90t

now,

90t = 80t + 40 10t = 40 t = 4 hrs. Hence distance travelled = 90 4 = 360 km.

94. Ans. (a) Let the number of coins of each type be x. The sum of money = Rs. 35 (given) Hence, sum of money = (x + 1) + (x 0.5) + (x 0.25)35 = x (1 + 0.5 + 0.25)35 = x 1.75

95. Ans. (wrong) The fourth day after 6th January is Saturday (Given) So, the day on 6th January will be Tuesday Number of days between 1st December and 6th

January are 36 days. Number of odd days = 1 day So, the day on 1st December will be 1 day before Tuesday i.e. Monday Correct Ans. Monday which is not given any of the four option.

96. Ans. (a) First is the product of the second.

97. Ans. (d) Clearly the man initially faces in the direction SOUTH On moving 135 anticlockwise he faces in the direction NORTH-EAST Finally moving 180 clockwise he faces in the direction SOUTH-WEST.

98. Ans. ()

R1 R1 + R3

2 cos [sin2+ cos2] = 02cos = 0 cos = 0

99. Ans. (a) Circumference of the wheel m

m (given) It makes 7 revolutions in 4 second (given)


INFOMATHSDistance travelled by wheel in 4 second

m

Distance travelled in 1 second m

Distance travelled in 1 hr. (in km)

km

100. Ans. (d)

Putting and – 1 respectively, we get

101. Ans. (a) Putting xn + 1 = t and n xn-1 dx = dt, we get

102. Ans. (d) From Definition: Equivalence Relation: A relation R on a set A is said to be an equivalence relation on A iff (i) it is reflexive i.e. (a, a) R for all a A (ii) it is symmetric i.e. (a, b) R (b, a) for all a,

b A (ii) it is transitive i.e. (a, b) R (a, c) for all a,

b, c A

103. Ans. (a) Here, function

For continuity of function Check for left hand limit

For Right Hand Limit

L.H.L. = R.H.L. = 0 Hence given function is continuous at x = 0 Fro differentiability

For

For R.H.L.

L.H.L R.H.L. Hence given function is not differentiable at x = 0

104. Ans. (c) In the case of each book we may take 0, 1, 2, 3, …., p copies; That is, we may deal with each book in p + 1 ways and therefore with all the books in (p + 1)n ways. But, this includes the case where all the books are rejected and no selection is made. Number of ways in which selection can be made = (p + 1)n – 1

105. Ans. (a) AAT = 9I

Equation a + 4 + 2b = 0 a + 2b = - 4 2a + 2 – 2b = 0 2a – 2b = - 2 & a2 + 4 + b2 = 0 a2 + b2 = 5 Solving a = - 2, b = - 1, a + b = - 3

106. Ans. (a) Check by option

(a) put in |z – 2 + 2i| = 1

107. Ans. () Set of natural number {1, 2, ……, 120}Divisible of 5 are {5, 10, 15, …., 120}Total divisible of 5 = 24 Divisible of 15 are {15, 30, 45, ….., 120}Total divisible of 15 are 8 (A B) = P(A) + P(B) – P(A B)

P(A) = divisible of

P(B) = divisible of


INFOMATHSP (A B) = divisible by 5 and

Put all above values in the above formula

P (A B)

P (A B)

108. Ans. () for (a)

y – xy = 1

Replace x = y It is clear that inverse of f(x) does not exist for x = 0 (b) f(x) = x2 for x inverse of f(x) y = x2

Replace x = y It is clear that inverse of f(x) does not exist for negative value of x. (c) f(x) = x2 for all x 0inverse of f(x) y = x2

Replace x = y Inverse of given function exist for all positive value of x Hence Answer is (c)

109. Ans. (c) sec 4 - sec 2 = 2

cos2 - 2cos22 + 1 = 2cos2(2cos22 - 1) -2cos22 + 2cos2 - cos2 + 1 = 2cos2 (2cos22-1) -2cos2 (cos 2- 1) – (cos2 - 1) = 2cos2 (2cos22 - 1) cos = 0

110. Ans. () If f(x) = x3 + 3x2 – 9x + c = (x – )2 (x - ), then (x - ) is a factor of order 2. So, x - is a factor of order one of f '(x) i.e. 3x2 + 7x – 9 = 3(x2 + 2x – 3) = 3(x + 3) (x – 1) f ' (x) = 0 x = 1 or x = - 3 If = 1, then as is a root of x3 + 3x2 – 9x + c = 0. Therefore, 1 + 3 – 9 + c = 0 c = 5 If = - 3, then as is a root of x3 + 3x2 – 9x + c = 0. Therefore, - 27 + 27 + 27 + c = 0 c = - 27

111. Ans. (b) Brother of woman’s mother is maternal uncle of woman hence, son of maternal uncle is cousin of the woman. Hence, answer is (b)

112. Ans. (b) Daughter of Nilesh’s wife is daughter of Nilesh also. And Husband of his daughter is man, whom he is pointing. Hence, Nilesh is father-in-law of the man.

113. Ans. (a) Out of 9 socks, 2 can be drawn in 9C2 ways Total number of elementary event = 9C2 Two socks drawn from the drawer will match if either both are brown or both are blue. Favourable number of elementary events = 5C2 + 4C2

Hence, required probability

114. Ans. (wrong) Given y2 (x2 + 1) = 2xy1

Integrating both sides, we get log y1 = log (x2 + 1) + log c y1 = c(x2 + 1) Given, y1 = 3 at x = 0 c = 3 y1 = 3(x2 + 1)

Again, integrating we get

This passes through (0, 1) c1 = 1 Equation of the curve is = x3 + 3x + 1 Correct ans. is x3 + 3x + 1 which is not given in any of the four option

115. Ans. () We have,

Let e be the eccentricity of this ellipse, Then,

The coordinates of the end-points of latusrecta are L (2, 5/3), M(-2, 5/3), M ' (-2, 5/3) and L '(2, -5/3) The equation of tangents at these points are 2x + 3y – 9 = 0 …(i)-2x + 3y – 9 = 0 …(ii)2x + 3y + 9 = 0 …(iii)-2x + 3y + 9 = 0 …(iv)Clearly, these tangents form a parallelogram whose are is given by

sq. units

A = 27 sq. units

116. Ans. (c) Consider the following events. E = Student A solves the problem F = Student B solves the problemG = Student C solves the problemH = Student D solves the problemClearly, E, F, G, H are independent events such that

Required probability is P(E F G H) Required probability

Required probability


INFOMATHS

117. Ans. (d) Equation of circle through origin and chord of contact is x2 + y2 + 2gx + 2fy + c + (gx + fy + c) = 0 = - 1 [By x = 0, y = 0]Therefore, equation is x2 + y2 + gx + fy = 0

Hence, circumcentre is

118. Ans. (c) Let E1, E2 and A be the event defined as follows reports that is a six

We have

Now, P(A/E1) = Probability that the man reports that there is a six on the die given that six has occurred on the die = Probability that man speak truth = ¾ and P(A/E2) = Probabilitouty that man reports that there six on the die given that six has not occurred on the die = Probability that man does not speak truth

By Baye’s Rule, we have

119. Ans. ()

120. Ans. (c)Total number of English alphabet = 26Total number of arrangements = 24P5 2012

JNU-2015 ANSWERS 1 2 3 4 5 6 7 8 9 10A B A B A B A C D A11 12 13 14 15 16 17 18 19 20D B B D C C D D C A21 22 23 24 25 26 27 28 29 30C C C B C C D C C C31 32 33 34 35 36 37 38 39 40C B B A A D B A B A41 42 43 44 45 46 47 48 49 50C A D A A C A A A D51 52 53 54 55 56 57 58 59 60A C D A D C A X A C61 62 63 64 65 66 67 68 69 70B C C B B A B D71 72 73 74 75 76 77 78 79 80A C B A B B C D D A81 82 83 84 85 86 87 88 89 90A A C D B C A B D C91 92 93 94 95 96 97 98 99 100C D C A X A D A D

101 102 103 104 105 106 107 108 109 110A D A C A A A C C

111 112 113 114 115 116 117 118 119 120A C D C C C


INFOMATHS


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