Maths

ENGINEERING MATHEMATICS

ENGINEERING MATHEMATICS

CONTENTS

Unit Chapters Page No.
1 LinearAlgebra 2 - 17
2 Calculus 18 - 32
3 Vector Calculus 33 - 38
4 Differential Equations 39 - 49
5 Transform Theory 50 - 53
6 Complex Variables 54 - 57
7 Numerical Methods 58 - 63
8 Probability and Statistics 64 - 73

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Published at : Ascent Gate Academy

“Shraddha Saburi”, Near Gayatri Vidyapeeth,
Rajnandgaon (Chhattisgarh) Mob : 09993336391

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MECHANICAL ENGINEERING

1 LINEAR ALGEBRA

0 0α (GATE - 93)
1. The eigen vector (s) of the matrix 0 0 0 , α = 0 is (are)

0 00

a) (0, 0, α) a) (α, 0, 0) a) (0, 0, 1) a) (0, α, 0)

1 001

2. If A = 0 -1 0 -1 the matrix A4, calculated by the use of Cayley - Hamilton theorem
00i i

0 0 0 -i (GATE - 93)

(or) otherwise is

1 01 (GATE - 94)
3. The inverse of the matrix -1 1 1 is

0 10

0 22 (GATE - 94) [ME]
4. Rank of the matrix 7 4 8 is 3

-7 0 -4

a) True b) False

1 00

5. Find out the eigen value of the matrixA= 2 3 1 for nay one of the eigen values, find out

0 24

(GATE - 94) [ME]

the corresponding eigen vector ?

6. Solve the system 2x + 3y + z = 9, 4x + y = 7, x - 3y - 7z = 6 (GATE - 95) [ME]

7. Among the following, the pair of vectors orthogonal to each other is (GATE - 95) [ME]

a) [3, 4, 7] , [3, 4, 7]
b) [1, 0, 0] , [1, 1, 0]
c) [1, 0, 2] , [0, 5, 0]
d) [1, 1, 1] , [-1, -1, -1]

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ENGINEERING MATHEMATICS

1 11 (GATE - 96)[ME]
8. The eigen values of 1 1 1 are d) 1, 1, 1

1 11

a) 0, 0, 0 b) 0, 0, 1 c) 0, 0, 3

9. In the Gauss - elimination for a solving system of linear algebraic equtaions, triangularization leads to

a) diagonal matrix b) lower traingular matrix (GATE-96)[ME]

c) upper triangular matrix d) singular matrix

10. For the following set of simultaneous equations (GATE-97)[ME]

1.5x - 0.05y + z = 2

4x + 2y + 3z = 9

7x + y + 5z = 10

a) the solution is unique

b) infinitely many solutions exist

c) the equations are incompatible

d) finite many solutions exist

11. The sum of the eigen values of the matrix 1 13 (GATE - 04)[ME]
1 5 1 is d) 18
3 11

a) 5 b) 7 c) 9

12. For what value of x will the matrix given below become singular ? 8x 0
40 2
12 6 0

(GATE - 04)[ME]

13. A is a 3 x 4 matrix and AX = B is an inconsistent system of equations. The highest possible rank of

Ais (GATE - 05)[ME]

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

5 000
14. Which one of the following is an eigen vector of the matrix 0 5 0 0 is

0 021
0 031

a) [1 -2 0 0]T b) [0 0 1 0]T (GATE - 05)[ME]
c) [1 0 0 -2]T d) [1 -1 2 1]T

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MECHANICAL ENGINEERING

15. Eigen values of a matrix S = 3 2 are 5 and 1 (GATE - 06)[ME]
23 d) 2, 10

What are the eigen values of the matrix s2 = SS ?

a) 1 amd 25 b) 6, 4 c) 5, 1

16. Multiplications of matrices E and F is G. Matrices E and G are (GATE - 06)[ME]

cosθ -cosθ 0 1 00 What is the matrix F ?
E = sinθ cosθ 0 and E= 0 1 0
1
00 0 01

cosθ -cosθ 0 cosθ cosθ 0
(a) sinθ cosθ 0 (b) -cosθ sinθ 0
1 1
00 00
0 0
cosθ sinθ 0 sinθ -cosθ 0
(c) -sinθ cosθ 1 (d) cosθ sinθ 1

00 00

17. If a square matrixAis real and symmetric then the eigen values (GATE-07)[ME]

a) are always real b) are always real and positive

c) are always read and non-negative d) occur in complex conjugate pairs

18. The number of linearly independent eigen vectors of 1 1 is (GATE-07)[ME]
0 2 d) infinite

a) 0 b) 1 c) 2

1 24

19. The matrix 3 0 6 has one eigen value to 3. The sun of the other two eigen values is

1 1p (GATE-08)[ME]

a) p b) p - 1 c) p - 2 d) p - 3
20. The eigen vectors of the matrix 1
2 are written in the form 1 1 . What is a + b ?
0 2 a &b
a) 0 b) 1
(GATE-08)[ME]
2 c) 1 d) 2

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ENGINEERING MATHEMATICS

21. For what values of ‘a’ if any will the following system of equations in x, y and z have a solution ?

2x + 3y = 4, x + y + z = 4, x + 2y - z = a (GATE-08)[ME]

a) any real number b) 0

c) 1 d) there is no such value

22. For a matrix [M] = 3 4
5
5 . The transpose of the matrix is equal to the inverse of the matrix,
x 3
5 (GATE-09)[ME]

[M]T = [M]-1. The value of x is given by

a) - 4 b) - 3 c) 3 d) 4
5 5 55

22 (GATE-10)[ME]
23. One of the eigen vector of the matrix A = 1 3 is

2 2 41
a) -1 b) 1 c) 1 d) -1

24. Eigen values of a real symmetric matrix are always (GATE-11)[ME]
d162. [A] is a square
a) positive b) negative c) real

25. Consider the following system of equations 2x1 + x2 + x3 = 0, x2 - x3 = 0 and x1 + x2 = 0.

This system has (GATE-11)[ME]

a) a unique solution b) no solution

c) infinite number of solutions d) five solutions

26. The eigen values of a symmetric matrix are all (GATE-13)[ME]
a) Complex with non-zero positive imaginary part (GATE-13)[ME]
b) Complex with non-zero negative inaginary part
c) reak
d) Pure imaginary

27. Choose the CORRECT set of functions, which are linearly depedent
a) sin x, sin2x and cos2x
b) cos x, sin x and tan x
c) cos 2x, sin2 x and cos2 x
d) cos 2x, sin x and cos x

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MECHANICAL ENGINEERING

28. Given that the determinant of the matrix 13 0
26 4 is -12 , the determinant of the matrix
-1 0
2 (GATE-14-ME-Set 1)

26 0
4 12 8 is
-2 0 4

a) -96 b) -24 c) 24 d) 96

29. ^Wh^ich^one^of th^e fo^llowing^des^cribe^s the relationship among the three vectors,

i + j + k, 2i + 3j + k and 5i + 6j + 4k ? (GATE-14-ME-Set 1)

a) The vectors are mutually perpendicular

b) The vectors are linearly dependent

c) The vectors are linearly independent

d) The vectors are unit vector

30. One of the eigen vectors of the matrix -5 2 is (GATE-14-ME-Set 2)
-9 6

{ }-1 { }-2 { }2 { }1

a) 1 b) 9 c) 1 d) 1

31. Consider a 3x3 real symmetric matrix S such that two of its eigen values are a ≠ 0, b ≠ 0 with

x1 y (GATE-14-ME-Set 3)
1 If a ≠ b then x1y1 + x2y2 + x3y3 equals

respective eigen vectors x2 , y2

x3 y3

a) a b) b c) ab d) 0

32. Which one of the following equations is a correct identify for arbitrary 3x3 real matrices P, Q and R ?

(GATE-14-ME-Set 4)

a) P(Q +R) = PQ + RP b) (P - Q)2 = P2 - 2PQ + Q2

c) det (P + Q) = det P + det Q d) (P +Q)2 = P2 + PQ + QP + Q2

33. At least one eigen value of a singular matrix is (GATE-15-ME-Set 2)
d) imaginary
a) positive b) zero c) negative

34. The lowest eigen value of the 2x2 matrix 4 2 is _________ (GATE-15-ME-Set 3)
13

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ENGINEERING MATHEMATICS

4 + 3i i , where i = √ -1, the inverse of matrix P is
35. For a given matrix P = i 4 - 3i (GATE-15-ME-Set 3)

a) 1 4 - 3i i b) 1 i 4 - 3i
24 -i 4 + 3i 25 4 + 3i i

c) 1 4 + 3i -i d) 1 4 + 3i -i
25 i 4 - 3i
24 i 4 - 3i

1 32
36. If the determinant of the matrix 0
5 -6 is 26, then t he determinant of the matrix
2
78

2 78 (GATE-97[CE])
0 5 -6 is
1 32

a) -26 b) 26 c) 0 d) 52

37. If A and B are two matrices and AB exists then BA exists, (GATE-97[CE])
a) only if A has as many rows as B has columns
b) only if both A and B are square matrtices
c) only if A and B are skew matrices
d) only if both Aand B are symmetric

0 10 (GATE-97[CE])
38. Inverse of matrix 0 0 1 is

1 00

0 01 1 00
a) 1 0 0 b) 0 0 1

0 10 0 10

c) 1 0 0 d) 0 0 1
0 10 0 10
0 01 1 00

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MECHANICAL ENGINEERING

39. IfAis a real square matrix then AAT is (GATE-98[CE])
c) skew - symmetric d) some times symmetric
a) un symmetric b) always symmetric

40. In matrix algebra AS = AT (A, S, T are matrices of appropriate order) implies (GATE-98[CE])

a)Ais symmetric b)Ais singular c)Ais non-singular d)Ais skew-symmetric

41 The real symmetric matrix C corresponsind to the quadratic form Q = 4x1x2 - 5 x2x2 is
(GATE-98[CE])

a) 1 2 b) 2 0 c) 1 1 d) 0 2
2 -5 0 -5 1 -2 2 -5

42. Obtain the eigen values and eigen vectors of (GATE-98[CE])
A = 8 -4 (GATE-99)
22
d) 5
1 -2 -1 (GATE-99)
43. If A = 2 3 1 and

0 5 -2

-11 -9 1
adj(A) 4 -2 -3 then K

10 k 7

a) -5 b) 3 c) -3

44. Find the eigen values and eigen vectors of the matrix 3 -1
-1 3

45. If A is any nxn matrix and k is a scalar then | kA| = α |A| where α is (GATE-99)[CE]
k
a) kn b) nk c) kn
d)
n

46. The number of terms in the expansion of general determinant of order n is

(GATE-99)[CE]

a) n2 b) n! c) n d) (n +a)2

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ENGINEERING MATHEMATICS

2 11
47. The equation 1
1 -1 = 0 represents a parabola passing through the points.
y
x2 x (GATE-99)[CE]

a) (0, 1), (0, 2), (0, 1) b) (0, 0), (-1, 1), (1, 2)
c) (1, 1), (0, 0), (2, 2) d) (1, 2), (2, 1), (0, 0)

48. If A, B, C are square matrices of the same order then(ABC)-1 is equal be (GATE-2000)[CE]

a) C-1A-1B-1 b) C-1B-1A-1 c) A-1B-1C-1 d) A-1C-1B-1

49. Consider the following two statements : (GATE-2000)[CE]

(I) The maximum number of linearly independent column vectors of a matrixAis called the rank ofA

(II) If A is nxn square matrix then it will be non-singular if rank of A= n

a) Bothe the statements are false b) Both the statements are true

c) (I) is true but (II) is false d) (I) is false but (II) is true

50. The determinant of the following matrix (GATE-01)[CE]

5 32 d) 72
1 26 (GATE-01)[CE]
3 5 10 d) (10.16, 3.84)

a) -76 b) -28 c) 28
are
51. The eigen values of the matrix 5 3
29

a) (5.13, 9.42) b) (3.85, 2.93) c) (9.00, 5.00)

52. The product [P] [Q]T of the following two matrices [P] and [Q] where (GATE-01)[CE]

[P] = 2 3 , [Q] 4 8
4 9 2
5=

a) 32 24 b) 46 56 c) 35 22 d) 32 56
56 46 24 32 61 42 24 46

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MECHANICAL ENGINEERING

53. Eigen values of the following matrix -1 4 are (GATE-02)[CE]
4 -1 d) 3, 5

a) 3, -5 b) -3, 5 c) -3, -5

4 213 (GATE-03)[CE]
54. Given matrix [A] = 6 3 4 7 , the rank of the matrix is

2 101

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

55. Asystem of equations represented byAX = 0 whre X is a column vector of unknown andAis a matrix

containing coefficients has non-trivial solution whenAis (GATE-03)

a) non-singular b) singular c) symmetric d) Hermitian

56. What values of x, y, z satisfy the following system of linear equations (GATE-04)

1 23x 6

1 3 4 y =8

2 23z 12

a) x = 6, y = 3, z = 2 b) x = 12, y = 3, z = -4
c) x = 6, y = 6, z = -4 d) x = 12, y = -3, z = 4

a1 and X2 - X + 1 = 0 then the inverse of X is (GATE- 04)
57. If matrix X = -a2 + a - 1 1 - a

a) 1 - a -1 1 - a -1
a2 a b)

a2 - a + 1 a

- a -1 a2 - a + 1 a
c) a2 + a + 1 a d) 1 1 - a

58. Real matrices [A]3x1, [B]3x3, [C] 3x5, [D]5x3, [E]5x5, [F]5x1 are given. Matrices [B] and [E] are
symmetric. Following statements are made with respect to their matrices. (GATE-04)[CE]

(I) Matrix product [F]T [C]T [B] [C] [F] is a scalar

(II) Matrix product [D]T [F] [D] is always symmetric.

a) statement (I) is true but (II) is false b) statement (I) is false but (II) is true

c) both the statements are true d) both the statements are false

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ENGINEERING MATHEMATICS

59. The eigen values of the matrix -1 4 are (GATE-04)[CE]
4 -1 d) can not be determined

a) 1, 4 b) -1, 2 c) 0, 5

60. Consider the following system of equations in three real variable x1, x2 and x3 : (GATE-05)[CE]
2x1 - x2 + 3x3 = 1
3x1 + 2x2 + 5x3 = 2
-x1 - 4x2 + x3 = 3
This system of equations has

a) no solution b) a unique solution

c) more than one but a finite number of solutions d) an infinite number of solutions

61. Consider a non-homogeneous system of linear equations represents mathematically an over

determined system. Such a system will be (GATE-05)[CE]

a) Consistent having a unique solution

b) Consistent having many solutions

c) Inconsistent having a unique solution

d) Inconsistent having no solutions

62. Consider the marices X , Y and P . The odre of [P(XT Y)PT)T will be (GATE-05)[CE]
4x3 4x3 2x3

a) 2x2 b) 3x3 c) 4x3 d) 3x4

63. The determinant of the matrix given below is (GATE-05)

0 102
-1 1 1 3
0 001
1 -2 0 1

a) -1 b) 0 c) 1 d) 2

64. Consider the system of equations,

Anxn Xnx1 = λ Xnx1 where λ is a scalar.
Let (λi, Xi) be an eigen value and its corresponding eigen vector for real matrix A. Let Inxn be unit

matrix. Which one of the following statement is not correct ? (GATE-05[CE]

a) for a homogeneous nxn system of linear equations (A -λI)X = 0, having a non trivial solution, the

rank of (A -λI) is less than n.

b) For matrixAm, m being a positive integer, (λim,Xim) will be eigen pair for all i.
c) if AT = A-1 then |λi| = 1for all i.
d) if AT = A then λi are real for all i.

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MECHANICAL ENGINEERING

65. Solution for the system defined by the set of equations 4y + 3z = 8 2x - z = 2 & 3x + 2y = 5 is

a) x = 0, y = 1, z = 4/5 b) x = 0, y = 1/2, z = 2 (GATE-06)[CE]

c) x = 1, y = 1/2, z = 2 d) non existent

2 -2 3
66. For a given matrix A= -2 -1 6 , one of the eigen value is 3. The other two eigen values are
0 (GATE-06)[CE]
12

a) 2, -5 b) 3, -5 c) 2, 5 d) 3, 5

1 13

67. The minimum and maximum eigen values of matrix 1 5 1 are -2 and 6 respectively.

3 11 (GATE-07)[CE]

What is the other eigen value ?

a) 5 b) 3 c) 1 d) -1

68. For what values of α and β the following simultaneous equations have an infinite number of solutions

x + y + z = 5, (GATE-07)[CE]
x + 3y + 3z = 9, x + 2y + αz = β

a) 2, 7 b) 3, 8 c) 8, 3 d) 7, 2

69. The inverse of 2 x 2 matrix 1 2 is (GATE-07)[CE]
57

a) 1 -7 2 17 2 c) 1 7 -2 1 -7 -2
3 5 -1 b) 3 5 1 35 1 d) 3 -5 1

70. X = [ x1 x2 .......... xn]T is an n-tuple non-zero vector. The n x n matrix V = XXT(GATE-07)[CE]

a) has rank zero b) has rank 1 c) is orthogonal d) has rank n

71. The product of marices (PQ)-1 P is (GATE-08)[CE]
d) P Q P-1
a) P-1 b) Q-1 c) P-1 Q-1 P
(GATE-08)[CE]
72. The eigen values of the matrix [P] =1 2 are
57 d) 1 and 2

a) -7 and 8 b) -6 and 5 c) 3 and 4

73. The following system of equations

x + y + z = 3, x + 2y + 3z = 4, x + 4y + kz = 6 will not have a unique solution for k equal to

a) 0 b) 5 c) 6 d) 7 (GATE-08)[CE]

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ENGINEERING MATHEMATICS

74. A square matrix B is symmetric if ________ (GATE-09)[CE]
d) B-1 = BT
a) BT = -B b) BT = B c) B-1 = B

75. In the solution of the following set of linear equations by Gauss-elimination using partial pivoting

5x + y + 2z = 34, 4y - 3z = 12 and 10x - 2y + z = -4.

The pivots for elimination of x and y are (GATE-09)[CE]

a) 10 and 4 b) 10 and 2 c) 5 and 4 d) 5 and -4

3 + 2i i (GATE-10)[CE]
76. The inverse of the matrix - i 3 -2i is
a) 1 3 + 2i -1
2 i 3 - 2i b) 1 3 - 2i -1
12 i 3 + 2i

c) 1 3 + 2i -1 d) 1 3 - 2i -1
14 i 3 - 2i 14 i 3 + 2i

77. The eigen values of matrix 9 5 are c) 4.70 and 6.86 (GATE-12)[CE]
58 d) 6.86 and 9.50

a) -2.42 and 6.86 b) 3.48 and 13.53

78. What is the minimum number of multiplications involved in computing the matrix product PQR ?

Matrix P has 4 rows and 2 columns, matrix Q has 2 rows and 4 columns and matrix R has 4 rows and

1 column _____________ (GATE-13)[CE]

3 21 1

79. Given the matrix J = 2 4 2 and K = 2 , the product KT JK is ____

1 26 -1 (GATE-14- CE - Set 1)

80. The sum of Eigen values of the matrix, [M] i whre [M] = 215 650 795
655 150 835
485 355 550

a) 915 b) 1355 c) 1640 (GATE-14- CE - Set 1)
d) 2180

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MECHANICAL ENGINEERING

0 123

81. The rank of matrix 1 0 3 0 is _____________ (GATE-14- CE - Set 2)
2 301

3 012

6 0 44 is _____________ (GATE-14- CE - Set 2)
82. The rank of matrix -2 14 8 18

14 -14 0 -10

83. For what value of ‘P’ the following set of equations will have no solution ?

2x + 3y = 5 GATE-14- CE - Set 1)

3x + py = 11

3 -2 2
84. The smallest and largest Eigen value of the following matrix are : 4 -4 6
-3 5
2
(GATE-15- CE - Set 1)
a) 1.5 and 2.5 b) 0.5 and 2.5 c) 1.0 and 3.0 d) 1.0 and 2.0

85. Let A = [aij], l<i, j<n with n>3 and aij = i.j. The rank of A is : (GATE-15- CE - Set 2)
d) n
a) 0 b) 1 c) n -1

86. The two Eigen Values of the matrix 2 1 have a ratio of 3 : 1 for p = 2. What is another value
1p

of ‘p’ for which the Eigen values have the same ratio of 3 : 1 ? (GATE-15- CE - Set 2)

a) -2 b) 1 c) 7/3 d) 14/3

87. The matrix 1 -4 is an inverse of the matrix 5 -4 (GATE-94)[PI]
1 -5 1 -1

a) True b) False

88. If for a matrix, rank equals both the number of rows and number of columns, then the matrix is called

a) Non-singular b) singular c) transpose d) minor (GATE-94)[PI]

1 49 is (GATE-94)[PI]
89. The value of the following determinant 4 9 16 d) -8
16 25
9 c) -12
a) 8 b) 12

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ENGINEERING MATHEMATICS

90. For the following matrix 1 -1 the number of positive characteristic roots is
2 3

(GATE-94)[PI]

a) one b) two c) four d) cannot be found

21 32

91. Given matrix L = 3 2 and M = 0 1 then Lx M is (GATE-95)[PI]
45

81 65 1 8 6 2
a) 13 2 b) 9 8 13 d) 9 4
c) 2 22 5
22 5 12 13 5 0

92. The eigen values of the matrix M given below are 15, 3 and 0. (GATE-05)[PI]
d) -10
21
M = 3 2 , the value of the determinant of a matrix is

45

a) 20 b) 10 c) 0

93. IfAis square symmetric real valued matrix of dimension 2n, then the eigen values ofAare

a) 2n distinct real values (GATE-07)[PI]

b) 2n real values not necesarily distinct

c) n distinct pairs of complex conjugate numbers

d) n pairs of complex conjugate numbers, not necesarily distinct

34 (GATE-08)[PI]
94. The eigen vector pair of the matrix 4 -3 is d) -2 1

a) 2 1 b) 2 1 c) -2 1 12
1 -2 12 1 -2

0 10 (GATE-08)[PI]
95. The inverse of matrix 1 0 0 is

0 01

0 10 0 -1 0 0 10 0 -1 0
a) 1 0 0 b) -1 0 0 c) 0 0 1 d) 0 0 -1

0 01 0 0 -1 1 00 -1 0 0

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MECHANICAL ENGINEERING

1 32

96. The value of the determinant 4 1 1 is (GATE-09)[PI]
2 13 d) 36

a) -28 b) -24 c) 32

97. The value of x3 obtained by solving the following system of linear equations is (GATE-09)[PI]

x1 + 2x2 - 2x = 4

2x1 + 2 + x3 = -2

-x1 + x2 + x3 = 2

a) -12 b) -2 c) 0 d) 12

98. The value of q for which the following set of linear equations 2x + 3y = 0, 6x + qy = 0 can have

non-trivial solution si (GATE-10)[PI]

a) 2 b) 7 c) 9 d) 11

99. If {1, 0, -1}T is an eigen vector of the following matrix 1 -1 0
-1 2 -1 then the corresponding
eigen value is 0 -1 1 (GATE-10)[PI]

a) 1 b) 2 c) 3 d) 5

100. The eigen values of the following matrix 10 -4 are (GATE-11)[PI]
18 -12 d) -6, 8

a) 4, 9 b) 6, -8 c) 4, 8

2 4 and matrix B = 4 6 the transpose of product of these two
101. If a matrix A = 1 3 5 9

matrices i.e, (AB)T is equal to (GATE-11)[PI]

a) 28 19 19 34 48 33 28 19
34 47 b) 47 28 c) 28 19 d) 48 33

102. For the matrix A = 10 -4, ONE of the normalized eigen vectors is given as
18 -12
(GATE-12)[ME, PI]

1 1 3 1
2 √10
a) √3 2 c) -1 2
2 b) -1 √10 d) 2

√2 √5

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ENGINEERING MATHEMATICS

103. x + 2y + z = 4, 2x + y + 2z = 5, x - y - z = 1. The system of algebraic equations given above has

a) a nunique solution of x = 1, y = 1 and z = 1 (GATE-12)[ME, PI]

b) only the two solutions of (x = 1, y = 1, z = 1) and x = 2, y = 1, z= 0)

c) infinite number of solutions

d) no feasible solution.

104. The system of equations, given below, has c) No solution (GATE-14)[PI-Set 1)
x + 2y + 4z = 2 d) More than two solutions
4x + 3y + z = 5
3x + 2y + 3z = 1
a)Aunique solution b) Two solution

**********************

“Perfect numbers like perfect men are very rare”

1. LINEAR ALGEBRA (ANS). :

1 - b&d 27 - c 53 - a 79 - 23
2 - A4 = I 28 - a 54 - c 80 - a
3 - sol. 29 - b 55 - b 81 - 88
4-b 30 - d 56 - c 82 - 2
5 - sol. 31 - d 57 - b 83 - 4.5
6-1 32 - d 58 - a 84 - d
7-c 33 - b 59 - c 85 - b
8-c 34 - 2 60 - b 86 - d
9-c 35 - a 61 - not correct 87 - a
10 - c 36 - a 62 - a 88 - a
11 - b 37 - a 63 - a 89 - d
12 - 4 38 - a 64 - b 90 - b
13 - b 39 - b 65 - d 91 - b
14 - a 40 - c 66 - b 92 - c
15 - a 41 - d 67 - b 93 - b
16 - c 42 - sol. 68 - a 94 - a
17 - a 43 - a 69 - a 95 - a
18 - b 44 - 4 70 - b 96 - b
19 - c 45 - c 71 - b 97 - b
20 - b 46 - b 72 - b 98 - c
21 - b 47 - b 73 - d 99 - a
22 - a 48 - b 74 - b 100 - b
23 - a 49 - b 75 - a 101 - d
24 - c 50 - b 76 - b 102 - b
25 - c 51 - d 77 - b 103 - c
26 - c 52 - a 78 - 16 104 - a

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MECHANICAL ENGINEERING

2 CALCULUS

1. The function f(x, y) = x2y - 3xy + 2y + x has (GATE-93[ME])
a) No local extermum b) One local maximum but no local minimum
c) One local minimum but no local maximum d) One local minimum and one local maximum

2. Limx(ex - 1) + 2(cosx - 1) = __________ (GATE-93[ME])
x→0 x (1 - cosx) (GATE-94[ME])

∞ dy is _________

∫3. The value of e -y3 . y1/2

0

4. The area bounded by the parabola 2y = x2 and the lines x = y - 4 is equal to ______

a) 6 b) 18 (GATE-95[ME])
c) ∞ d) None

a (GATE-05[ME])

∫5. [sin6 x + sin7 x] dx is equal to a
-a
∫b) 2 Sin7x dx
a
0
∫a) 2 Sin6x dx

0

a d) zero

∫c) 2 [sin6 x + sin7 x]

0

6. The minimum value of function y = x2 in the interval [1, 5] is (GATE- 07[ME])
d) undefined
a) 0 b) 1 c) 25

x2
ex - 1 + x + 2

7. Lim x3 = (GATE- 07[ME])
d) 1
x→0

a) 0 b) 1 c) 1
6 3

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ENGINEERING MATHEMATICS

8. If y = x + √x + √x + √x + ................. α then y(2) = ___________ (GATE-07[ME])

a) 4 (or) 1 b) 4 only c) 1 only d) Undefined

∫∫9. Consider the shaded triangular region P shown in the figure. What is xy dx dy ?

a) 1 b) 2 y P (GATE-08[ME])
6 9 >

x + 2y = 2

c) 7 d) 1 >x
16

10. In the Taylor series expansion of ex about x = 2, the coefficient of (x -2)4 is (GATE-08[ME])

a) 1 b) 24 c) e2 d) e4
4! 4! 4! 4!

11. The value of Lim x1/3 - 2 is (GATE-93[ME])
a) 1 x→0 x -8
16 c) 1 d) 1
b) 1 8 4
12

12. Which of the following integrals is unboundedf ? (GATE-08[ME])

π/4 α α 1
∫b) ∫d)
∫a) tan x dx 1 dx ∫c) x.e-x dx 1 dx
1 + x2 1 - x2
0 0
0 0

13. The length of the curve y = 2 x3/2 between x = 0 & x = 1 is (GATE-08[ME])
3

a) 0.27 b) 0.67 c) 1 d) 1.22

14. The parabolic are y = √ x, 1< x < 2 is revolved around the x-axis. The volume of the solid of revolution
is (GATE-10[ME])

π π 3π 3π
a) b) 2 c) 4 d)

4 2

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MECHANICAL ENGINEERING

15. The distance between the origin and the point nearest to it on the surface z2 = 1 + xy is

√3 (GATE-09[ME])
b) 2
a) 1 c) √3 d) 2

16. The area enclosed between the curves y2 = 4x and x2 = 4y is (GATE-09[ME])

a) 16 b) 8 c) 32 d) 16
3 3

17. The infinite series (GATE-10[ME])
d) ex
x3 x5 x7
f(x) = x - 3! + 5! - 7! + - - - Converges to

a) cos (x) b) sin (x) c) sinh(x)

α (GATE-10[ME])
d) π
∫ dx

18. The value of the interal 1 + x2

−α

a) -π b) -π π
2 c) 2

19. The function y = |2 - 3x| (GATE-10[ME])

a) is continuous A x ∈ Rand differentiable A x ∈ R

b) is continuous A x ∈ Rand differentiable A x ∈ R except at x= 3
2

A x ∈ R and differentiable A x∈R except at x= 2
b) is continuous 3

d) A x ∈ R and except at x= 3 and differentiable A x∈R
is continuous

20. A series expansion for the function sinθ is ________ (GATE-11[ME])

a) 1- θ2 + θ4 - ..... b) θ - θ3 + θ3 - .....
2! 4! 3! 5!

c) 1 + θ- θ + θ3 - ..... d) θ + θ3 + θ5 - .....
2! 3! 3! 5!

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ENGINEERING MATHEMATICS

a

∫21. If f(x) is even function and a is a positive real number, then f(x)dx equals _________
(GATE-11[ME])
-a
a
a) 0 b) 2 c) 2 a
∫d) 2 f(x)dx

0

22. What is Lim Sinθ equal to ? (GATE-11[ME])
θ d) 1
θ→0

a) θ b) Sinθ c) 0

e (GATE-13[ME])

∫23. The value of the definite intergral √x ln(x)dx is
1

a) 4 √e3 + 2 b) 2 √e3 - 4 c) 2 √e3 + 4 d) 4 √e3 - 2
9 9 9 9 9 9 9 9

24. Lt x - sin x is (GATE-14 - ME-Set 1)
θ → 0 1 - cos x d) not defined

a) 0 b) 1 c) 3

25. Lt e2x - 1 is equal to (GATE-14 - ME-Set 2)
θ → 0 sin(4x) b) 0.5 c) 1 d) 2

a) 0

26. If a function is continuous at a point, (GATE-14 - ME-Set 3)

a) The limit of the function may not exist at the point

b) The function must be derivable at the point

c) The limit of the function at the point tends to infinity

d) The limit must exist the point and the value of limit should be same as the value of the function at

the point.

2
∫27. The value of the integral
(x - 1)2 sin (x - 1) (GATE-14 - ME-Set 4)
dx is
(x - 1)2 + cos (x - 1)

0

a) 3 b) 0 c) -1 d) -2

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MECHANICAL ENGINEERING

2x (GATE-14 - ME-Set 4)

∫ ∫28. The value of the integral ex + y dy dx is

00

1 (e - 1) b) 1 (e2 - 1)2 c) 1 (e2 - 1) 1 1 2
2 2 2 2 e
a) d) e-

29. The value of lim - sin x is _______ (GATE-15 - ME-Set 3)
2 sin x + x cos x
θ→0

30. The value of lim 1 - cos(x2) is _______ (GATE-15 - ME-Set 1)
θ → 0 2x4 c) 1 d) undefined
4
a) 0 b) 1
2

31. Consider a spatial curve in three - dimensional space given in parametric form

2π
by x(t) = cost, y(t) = sin , z(t) =π t, 0 < t < ,.
2

The length of the curve is (GATE-15 - ME-Set 1)

32. Consider an ant crawling along the curve (x -2)2 + y2 = 4, where x and y are in meters. The ant starts

at the point (4, 0) and moves counter - clockwise with a speed of 1.57 meter per second. The time

taken by the ant to reach the point (2, 2) is (in seconds) (GATE-15 - ME-Set 1)

33. At x = 0, the function f(x) = |x| has (GATE-15 - ME-Set 2)
a) a minimum b) a maximum
c) a point of inflection d) neither a maximum nor minimum

34. Number of inflection points for the curve y = x + 2x4 is ________ (GATE-99[CE])
d) (n+1)2
a) 3 b) 1 c) n

p (GATE-2000[CE])
35. The Taylor series expansion of sin x about x = 6 is given by

1 + √3 2 √3 3 x5 x7
5! 7!
a) 22 x- Π -1 x- Π - x- Π + ---- b) x- x3 + - + ----
6 4 6 12 6 3!

x- Π - x- Π 3 x- Π 5 x- Π 7 d) 1
66 6 2
c) 6 +-
7! + ----
1! 3! 5!

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ENGINEERING MATHEMATICS

1 1/x
∫ ∫36. The value of the double integral
x (GATE - 93)
1 + y2 dx dy = _______
(GATE - 94)
0x d) None of the above

37. The intergration of ∫ logx dx has the value

a) ( x log x - 1) b) log x - x c) x (logx - 1)

38. The volume generated by revolving the area bounded by the parabola y2 = 8x and the line x = 2 about

y-axis is (GATE-94)

a) 128π 5 c) 127π d) None of the above
5 b) 128π 5

250 (GATE-94)
39. The function y = x2 + x at x = 5 attains

a) Maximum b) Minimum c) Neither d) 1

40. The value of ε in the mean value theorem of f(b) - f(a) = (b-a) f’(ε) for

f(x) = Ax2 + Bx C in (a, b) is (GATE - 94)

a) b + a b) b - a c) b + a d) b - a
2 2

2 2x

∫ ∫41. By reversing the order of integration f(x,y)dy dx may be represented as _______
0 x2 2 √y (GATE-95)
2 2x
∫ ∫b) f(x,y)dy dx
∫ ∫a) f(x,y)dy dx
0 x2 0 y

4 √y 2x 2

∫ ∫c) f(x,y)dy dx ∫ ∫d) f(x,y)dy dx
x2 0
0 y/2

42. The third term in the taylor’s series expansion of ex about ‘a’would be _________ (GATE-95)

a) ea (x - a) b) ea (x - a)2 c) ea d) ea (x - a)3
2 2 6

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MECHANICAL ENGINEERING

43. Lim x sin 1/x = __________ (GATE - 95)
c) 1 d) Does not exist
x→0

a) α b) 0

44. The function f(x) = |x +1| on the interval [-2, 0] is ________ (GATE - 95)
a) continous and differential
b) continuous on the interval but not differentiable at all points
c) Neigher continuous nor differentiable
d) Differentiable but not continous

45. The function f(x) = x3 - 6x2 + 9x + 25 (GATE - 95)
a) a maximum at x = 1 and a minima at x = 3
b) a maxima at x = 3 and a minima at x = 1
c) no maxima, but a minima at x = 3
d) a maxima at x = 1, but no minima

46. If f(0) = 2 and f’(x) = 1 , the lower and upper bounds of f(1) estimated by the mean value
5 - x2

therorem are _______ (GATE-95)

a) 1.9, 2.2 b) 2.2, 2.25 c) 2.25, 2.5 d) None of the above

47. If a function is continuous at a point its first derivative (GATE-96)

a) may or may not exist b) exists always

c) will not exist d) has a unique value

48. Area bounded by the curve y = x2 and the lines x = 4 and y = 0 is given by (GATE-97)

a) 64 64 c) 128 128
b) 3 3 d) 4

49. The curve given by the equation x2 + y2 = 3axy is (GATE-97)

a) Symmetrical about x-axis b) Symmetrical about y-axis

c) Symmmetrical about the line y = x d) Tangential to x = y = a/3

50. Lim sin m θ , where m is an integer, is one of the following : (GATE-97)
θ d) 1
x→0

a) m b) m π c) mθ

51. If y = |x| for x < 0 and y = x for x > 0 then (GATE-97)
a) dx is discontinous at x = 0
dy b) y is discontinous at x = 0
c) y is not defined at x = 0
d) Both y and dy are discontinous at x = 0
dx

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ENGINEERING MATHEMATICS

x2
∫52. If φ(x) =
√t dt then dφ = --------- (GATE-97)
dx d) 1

0

a) 2 x2 b) √x c) 0

53. The continuous function f(x, y) is said to have saddle point at (a, b) if (GATE-98)

a) fx (a, b) = fy (a, b) = 0 b) fx (a, b) = 0, fy (a, b) = 0
f2 - f f < 0 at (a, b) f2 - f f > 0 at (a, b)

xy xx yy xy xx yy

c) fx (a, b) = 0, fy (a, b) = 0 d) fx (a, b) = 0, fy (a, b) = 0
fxx and fyy < 0 at (a, b) f2xy - fxxfyy = 0 at (a, b)

54. The taylor’s series expansion of sin x is _____ (GATE-98)

a) 1 - x2 + x4 - .......... b) 1 + x2 + x4 + ..........
2! 4! 2! 4!

c) x + x3 + x5 + .......... d) x- x3 + x5 - ..........
3! 5! 3! 5!

55. Lim 1 1 - e-j5x = _______ (GATE-99[IN])
10 1 - e-jx
x→0

a) 0 b) 1.1 c) 0.5 d) 1
(GATE-99)
56. Limit of the function, Lim n is ________
n→∞ √n2 + n

a) 1/2 b) 0 c) ∞ d) 1

57. The function f(x) = ex is ______________ (GATE-99)
c) Neigher even nor odd d) None
a) Even b) odd

58. Value of the function, Lim (x - a)x-a is ________ (GATE-99)
d) a
x→a

a) 1 b) 0 c) ∞

π/2π/2 c) π/2 (GATE-2000)
d) 2
∫ ∫59 Sin(x+ y) dy dx

00

a) 0 b) π

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MECHANICAL ENGINEERING

60. Limit of the function (GATE-2000)
d) 0
1 - a4 as x → ∞ is given by
f(x) = x4

a) 1 b) e-a4 c)

61. If -1 ∂2f + ∂2f + ∂2f is equal to _______ (GATE-2000)
∂x2 ∂y2 ∂z2
f(x, y, z) = (x2 + y2 + z2) 2 ,

-5

a) 0 b) 1 c) 2 d) -3(x2 + y2 + z2) 2

a
62. Consider the following integral Lim
x-4 dx ____ (GATE-2000)
∫x→0

a) diverges 1

c) converges to - - 1 b) converges to 1/3
a3
d) converges to 0

63. Limit of the following series as x approaches π is (GATE-01[CE])
2
d) 1
f(x) = x - x3 + x5 - x7 + - - (GATE-01)
3! 5! 7! d) −π + 1

a) 2π π π 84
2 b) c) 3

2

π

4

∫64. The value to the integral is I = cos2 x dx

0

a) π + 1 b) π - 1 c) −π 1
84 8 4 - 4
8

65. The following function has local minima at which value of x, f(x) = x√5 - x2 (GATE-02[CE])

a) − √5 b) √5 c) 5 d) - 5
2 2 2

1 (GATE-02[CE])
d) - ∞
66. Limit of following sequences as n → ∞ is _______ xn = n n

a) 0 b) 1 c) ∞

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ENGINEERING MATHEMATICS

67. Which of the following functions is not differentiable in the domain [-1, 1] ? (GATE-02)

a) f(x) = x2 b) f(x) = x - 1 c) f(x) = 2 d) f(x) = maximum (x, -x)

π

2
∫68. The value of the following definite integral in
Sin2x dx = ____ (GATE-02)
1 + cos x
−π

2

a) - 2log 2 b) 2 c) 0 d) None

1

∫69. The value of the following improper integral is log x dx = ____0 (GATE-02)
d) 1
1 b) 0 c) -1
a) 4 4

70. The function f(x, y) = 2x2 + 2xy - y2 has (GATE-02)
a) Only one stationary point at (0, 0)
c) Two stationary points at (0, 0) and (1, -1) b) Two stationary points at (0, 0) and 1 , -1
63

d) no stationary point.

71. Lim Sin2x (GATE - 03)
x = _____ c) 1 d) -1
x→0

a) 0 b) ∞

72. The area enclosed between the parabola y = x2 and the straight line y = x is _____ (GATE-04)

1 1 0 1
a) 8 b) 6 d)
c) 1
3 2

73. If x = a (θ + Sinθ) and y = a(1 - Cosθ) then (GATE-04)
θ
dy θ θ
dx = ______ b) Cos 2 c) Tan 2 d) Cot 2

θ
a) Sin 2

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MECHANICAL ENGINEERING

74. The volume of an object expressed in spherical co-ordinates is given by (GATE-04)

2π π/3 1

∫ ∫ ∫V = r2 Sinθ drdφ dφ. The value of the intergral
0 00

π π 2π π
a) 3 b) 6 c) 3 d) 4

75. The value of the function, (GATE-04)

f(x) = Lim x3 + x2 is _____ d) ∞
x→0 2x3 - 7x2 (GATE-04)
d) both x = -2 and x = 3
a) 0 b) -1 c) 1
77

76. The function f(x) = 2x3 - 3x3 - 36x + 2 has its maxima at

a) x = -2 only b) x = 0 only c) x = 3 only

77. Changing the order of intergration in the double integral (GATE-05)

82 sq

∫ ∫I = f (x, y)dy dx leads to , ∫ ∫I = f (x, y)dy dx. What is q ?
0 x/4 rp

a) 4y b) 16 y2 d) x d) 8

78. By a chnage of variables x (u, v) = uv, y(u, v) = v/u in a double integral, the integral f(x, y) changes to

uv, u/v). Then φ(u, v) is ____ (GATE-05)

a) 2v b) 2 uv c) v2 d) 1
u

79. A parabolic cable is held between two supports at the same level. The horizontals span between the

x2
supports is L. The sage at the mid-spam is h. The equation of the parabola isy = 4h L2 , where

x is the horizontal coordinate and y is the vertical coordinate with the origin at the centre of the cable.

The expression for the total length of the cable is (GATE-10[CE])

L L/2 L/2 L/2

∫a) 1+64 h2x2 dx ∫b) 2 1+64 h3x2 dx ∫ ∫c) 1+64 h2x2 dx d) 2 1+64 h2x2 dx
L4 L4 L4 L4
0 0 00

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ENGINEERING MATHEMATICS

sin 2 x (GATE-10[CE])
80. The Lim 3 is c) 3 d) ∞

x→0 x 2

a) 2 b) 1
3

81. Given a function f(x, y)= 4x2 + 6y2 - 8x - 4y + 8, the optimal values of f(x, y) is (GATE-10[CE])

a) a minimum equal to 10 10
3 b) a maximum equal to 3
d) a maximum equal to 8
c) a minimum equal to 8
3 3

82. What should be the value of λ such that the function defined below is continuous at (GATE-11[CE])
π

x= ?
2

λ cos x , if x ≠ π
π 2
{ f(x)=
2 -x

π
1 , if x = 2

a) 0 b) 2π π
c) 1 d) 2

83. What is the value of the definite integral (GATE-11[CE])

a d) 2a
(GATE-12[CE])
∫ √x dx ? d) 1 + sin2x
√x +√a - x
0

a) 0 b) a c) a
2

84. The infinite series

1+x+ x2 + x3 + x4 + ..... corresponds to
2! 3! 4!

a) sec x b) ex c) cos x

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MECHANICAL ENGINEERING

π/6 c) 0 (GATE-13[CE])
c) 1 d) 8
∫85 The solution for cos4 3θ sin3 6θdθ is :
0 3
1
a) 0 b) 15 (GATE-14-CE-Set 1)
d) ∞
Lim x + sin x equal to
86. x→0 x

a) - ∞ b) 0

87. With reference to the conventional Cartesian (x, y) coordinate system, the vertices of a triangle have

the following coordinates : (GATE-14-CE-Set 1)

(x ,y ) = (1, 0) ; (x , y ) = (2, 2) ; and (x , y ) = (4, 3). The area of the triangle is equal to
11 22 33

a) 3 b) 3 c) 4 d) 5
24 52

88. The expression lim a→0 xa - 1 is equal to (GATE-14-CE-Set 2)
a d) ∞

a) log x b) 0 c) x log x

π/2

∫89. Given i = √ - 1, the value of the definite integral, I = cos x + i sin x dx is :
0 cos x - i sin x

(GATE-15-CE-Set 2)

a) 1 b) -1 c) i d) -i

Lim 1 is equal to (GATE-15-CE-Set 2)
90. x→0 1+ x c) 1 d) e2

a) e-2 b) e

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ENGINEERING MATHEMATICS

91. What is the value of Lim cos x - sin x (GATE-07[PI])
x→π/4 x- π/4 d) Limit does not exist

a) √2 b) 0 c) -√2

92. For function f(x, y) = x2 - y2 defined on R2, the point (0, 0) is (GATE-07[PI])

a) a local minimum b) Neither a local minimum (nor) a local maximum

c) a local maximum d) Both a local minimum and a local maximum

π/2 (GATE-08[PI])

∫93. The value of the intergral (x cos x) dx is
−π/2

a) 0 b) π - 2 c) π d) π + 2

94. The value of expression Lim sin (x) is (GATE-08[PI])
x→0 exx

1 c) 1 1
a) 0 b) 2 c) dx + dy d) 1 + e

95. The total derivative of the function ‘xy’ is (GATE-09[PI])
d) dx dy
a) x dy + y dx b) x dx + y dy

96. If (x) = sin |x| then the value of df at x = - π is (GATE-10[PI])
dx 4

a) 0 b) 1 c) - 1 d) 1
√2 √2 (GATE-10[PI])

∫97. 1 α -x2
√2π
The integral e 2 dx is equal to

−α

a) 1 b) 1 c) 1 d) ∞
2 √2

98. The area enclosed between the straight line y = x and the parabola y = x2 in the x- y plane is

a) 1/6 b) 1/4

c) 1/3 d) 1/2 (GATE-12[ME, PI])

99. Lim 1 - cos x is (GATE-12[ME, PI])
x2 c) 1 d) 2
x→0

a) 1/4 b) 1/2

100. At x = x, the function f(x) = x3 + 1 has c) a singlularity (GATE-12[ME, PI])
a) a mximum value b) a minimum value d) a point of inflection

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MECHANICAL ENGINEERING

101. The function f(x) = x2 = x+x+x+x+ ....... x times, is defined (GATE-15-PI)

a) at all real values of x b) only at positive integer values of x

c) only at negative integer values of x d) only at rational values of x

102. The value of Lim x2 - xy is (GATE-15-PI)
(x,y)→(0,0) √x - √y
c) 1 d) ∞
a) 0 b) 1
2 (GATE-15-PI)
b) concave down for all values of x
103. The curve y = x4 is d) concave up only for negative values of x
a) concave up for all values of x
c) concave up only for positive values of x

**********************

“A person who aims at nothing is sure to hit it”

2. CALCULUS (ANS.) :
1 - a, 2 - 1, 3 - sol, 4 - b, 5-a, 6 -b, 7 - b, 8 - b, 9 - a, 10 - c, 11 - b, 12 - d, 13 - d, 14 - d,
15 - a, 16 - a, 17 - b, 18 - d, 19 - c, 20 - b, 21 - d, 22 - d, 23 - c, 24 - a, 25 - b, 26 - d, 27
- b, 28 - b, 29 - 0.333, 30 - c, 31 - 1.8614, 32 - 2, 33 - a, 34 - b, 35 - a, 36 - sol., 37 - c,
38 - a, 39 - b, 40 - c, 41 - c, 42 - b, 43 - b, 44 - b, 45 - a, 46 - b, 47 - a, 48 - b, 49 - c, 50
- a, 51 - a, 52 - a, 53 - b, 54 - d, 55 - c, 56 - d, 57 - c, 58 - a, 59 - d, 60 - d, 61 - a, 62 -
b, 63 - d, 64 - a, 65 - d, 66 - b, 67 - d, 68 - c, 69 - c, 70 - b, 71 - a, 72 - b, 73 - c, 74 - a,
75 - b, 76 - a, 77 - a, 78 - a, 79 - d, 80-a, 81 - a, 82 - c, 83 - b, 84 - b, 85 - b, 86 - c, 87 -
a, 88 - a, 89 - c, 90 - d, 91- c, 92 - b, 93 - a, 94 - c, 95 - a, 96 - c, 97 - c, 98 - a, 99 - b,
100 - d, 101 - b, 102 - a, 103 - a.

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ENGINEERING MATHEMATICS

3 VECTOR CALCULUS

1. If V is a differentiable vector function and f is sufficiently differentiable scalar function then curl

( f V) = ______ (GATE-95[ME])

a) (grad f) x V + (f curl V) b) 0

c) f curl V d) (grad f) x V

2. The expression curl (grad f) where f is a scalar function is (GATE-96[ME])
a) Equal to ∆2 f
a) Equal to div (grad f)

c) Ascalar of zero magnitude d) a vector of zer o magnitude

3. Stokes theorem connects (GATE-05[ME[)
a) a line integral and a surface intergral b) a surface intergral and a volume integral
c) a line integral and a volume integral d) gradient of a function and its surface integral

4. The area of a triagnel formed by the tips vectors a, b and c is (GATE-07[ME])

a) 1 (a - b) (a - c) b) 1 (a - b) x (a - c)
2 2

c) 1 (a x b x c) d) 1 (a x b) c
2 2

5. The divergence of the vector field (x- y)i + (y -x )j + (x+y+z)k is (GATE-08[ME])
d) 3
a) 0 b) 1 c) 2

6. The directional derivative of the scalar function f(x, y, z) = x2 + 2y2 + z at the point P = (1, 1, 2) in the

direction of the vector a = 3i - 4j is (GATE-08[ME])

a) -4 b) -2 c) -1 d) 1

7. The divergence of the vector field 3xz^i + 2xy^j - yz2^k at a point (1, 1, 1) is equal to

(GATE-09[ME])

a) 7 b) 4 c) 3 d) 0

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MECHANICAL ENGINEERING

8. The following surface integral is to be evaluated over a sphere for the given steady velocity vector field
F = xi+yj+zk defined with respect to a Cartesian coordinate system having i, j and k as unit base

∫∫vectors. 1 (F.n)dA
s 4

Where S is the sphere, x2+y2+z2 = 1 and n is the outward unit normal vector to the sphere. The value

of the surface integral is c) 3π /4 (GATE-2013[ME])
a) π b) 2π d) 4π

∫9. The integral (ydx - xdy) is evaluated along the circle x2 + y2 = 1 traversed in counter
4
c

clockwise direction. The interal is equal to (GATE-14-ME-Set 1)

a) 0 b) - π π d) π
4 c) - 4

2 (GATE-14-ME-Set 2)

10. Curl of vector → = x2z2^i - 2xy2z^j + 2y2z3^k is

F

a) (4yz3 + 2xy2)^i + 2x2z^j - 2y2z^k b) (4yz3 + 2xy2)^i - 2x2z^j - 2y2z^k

c) 2xz2^i - 4xyz^j + 6y2z2^k d) 2xz2^i + 4xyz^j + 6y2z2^k

11. Divergence of the vector field x2z^i + xy^j - yz2^k at (1, -1, 1) is (GATE-14-ME-Set 3)
d) 6
a) 0 b) 3 c) 5

12. Let φ be an arbitrary smooth real valued scalar function and → be an arbitrary smooth vector valued

V

function in a three-dimensional space. Which one of the following is an identify ?

→ ∆ → → (GATE-15-ME-Set 3)

a) Curl (φV)→= (φDiv V) b) DCuivrVl (φ=V→0) →
d)
c) Div Curl V = 0 = φDiv V

∫13. The value of [(3x - 8y2)dx + (4y-6xy)dy], (where C is the region bounded by x = 0, y = 0 and
c

x + y = 1) is _______________ (GATE-15-ME-Set 3)

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ENGINEERING MATHEMATICS

14. The velocity field on an incompressible flow is given by (GATE-15-ME-Set 1)

V = (a1x + z2y + a3z)i + (b1x + b2y + b3z)j + c1x + c2y + c3z) k, (GATE-15-ME-Set 2)
where a1 = 2 and c3 = -4. The value of b2 is _____________ d) 3i - 6k
(GATE-15-ME-Set 2)
15. Curl of vector V(x,y,z) = 2x2i + 3z2j + y3k at x = y = z = 1 is

a) -3i b) 3i c) 3i - 4j

∫∫16. The surface integral 1 (9xi - 3yj).n dS
π

s

over the sphere given by x2 + y2 + z2 = 9 is

17. If the linear velocity V is given by V = x2 y i + xyz j - yz2 k then the angular velocity W at the point

(1, 1, -1) is _____ (GATE-93)

18. The→dire→ctional derivative of f(x, y) = 2x2 + 3y2 + z2 at point P(2, 1, 3) in the direction of the vector
a = i - 2k is (GATE-14)

a) 4 / √5 b) -4 /√ 5 c) √5/4 d) - √5/4

19. The derivative of f(x, y) at point (1, 2) in the direction of vector i + j is 2√2 and in the direction of the

vector -2j is -3. Then the derivative of f(x, y) in direction -i - 2j is (GATE-95)

a) 2 /√2 + 3/2 b) -7/√ 5 c) 2 /√2 - 3/2 d) 1 /√5

20. The directional derivative of the function f(x, y, z) = x + y at the point P(1, 1, 0) along the direction

→→ b) √ 2 c) -√2 (GATE - 96)
d) 2
i + j is

a) 1 /√2

21. For the function φ = ax2 y - y3 to represent ∆2 φshoud be equal to zero. In that case, the value of

‘a’ has to be (GATE-99)

a) -1 b) 1 c) -3 d) 3

22. The directional derivative of the following function at (1, 2) in the direction of (4i + 3i) is :

f(x, y) = x2+y2 (GATE-02)

a) 4 / 5 b) 4 c) 2 / 5 d) 1

23. The vector field F = xi - y j (where i and j are unit vectors) is (GATE-03)

a) divergence free, but not irrotational b) irrtational, but not divergence free

c) devergence free and irrotational d) neither divergence free nor irrotational

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MECHANICAL ENGINEERING

∫24. Value of the integral xydy - y2dx, where , c is the square cut from the first quadrant by the line

x = 1 and y =1 will be (Use Green’s theorem to change the time integral into double integral)

a) 1/2 b) 1 c) 3/2 d) 5/3 (GATE-05)

∫25. The line integral V.dr of the vector function V(r) = 2xyzi + x2 zj + x2 yk from the origin to the point

P(1, 1, 1) (GATE-05)
a) is 1 b) is zero
c) is -1 d) cannot be determined without specifying the path

26. The directional derivative of f(x, y, z) = 2x2 + 3y2 + z2 at the point p(2, 1, 3) in the direction of the

vector a = i - 2k is ___________ (GATE-06[CE])
d) 1.000
a) - 2.785 b) -2.145 c) -1.789

27. The velocity vector is given as v = 5xyi + 2y2j + 3yz2k. The divergence of this velocity vector at

(1, 1, 1) is (GATE-07[CE])

a) 9 b) 10 c) 14 d) 15

28. For a scalar function f(x, y, z) = x2 + 3y2 + 2z2, the gradient at the point P(1, 2, -1) is

a) 2i + 6j + 4k b) 2i + 12j - 4k c) 2i + 12j + 4k d) √56 (GATE-09[CE])

29. For a scalar function f(x, y, z) = x2 + 3y2 + 2z2, the directional derivative at the point P(1, 2, -1) in the

direction of a vector i - j -+2k is (GATE-09[CE])
d) 18
a) -18 b) -3√6 c) 3√6

30. If a and b are two arbitrary vectors with magnitudes a and and b respectively, | a x b | 2 will be equal

to (GATE-11[CE])

a) a2b2 - (a.b)2 b) ab - a.b c) a2b2 - (a.b)2 d) ab - a.b

31. For the parallelogram OPQR shown in the sketch. OP = a^i + b^j and OR = c^i + d^j . The area of the

parallelogram is Q (GATE-12[CE])

a) ad - bc R >
b) ac + bd

c) ad + bc

d) ab - cd >P

S

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ENGINEERING MATHEMATICS

32. A particle moves along a curve whose parametric euqations are : x = t3 + 2t, y = -3e-2t and z = 2 sin
(5t), where x, y and z show variations of the distance covered by the particle (in cm) with time t(in s).
The magnitude of the acceleration of the particle (in cm/s2) at t = 0 is ___ (GATE-14-CE-Set 1)

33. The directional derivative of the field u(x,y,z) = x2 - 3yz in the direction of the vector (^i +^j - 2^k) at

point (2, -1, 4) is (GATE-15-CE-Set 1)

34. Which one of the following is not associated with vector calculus ? (GATE-05[PI])

a) Stoke’s theorem b) Gauss Divergence theorem

c) Green’s theorem d) Kennedy’s theorem

35. The angle (in degrees) between two planar vectorsa = √3 i + 1 j and b= √3 i + 1 j
2 2 2 2

a) 30 b) 60 d) 90 d) 120 (GATE-07[PI])

36. If r is the positon vector of any point on a closed surface S that encloses the volume V then

∫∫ (r . ds) is equal tos (GATE-08[PI])
d) 3V
1 b) V c) 2V
a) 2 V

37. The line integral of the vector function F = 2x^i + x2 ^j along the x - axis from x = 1 to x = 2 is
a) 0 b) 2.33 c) 3 d) 5.33 (GATE-09[PI])

38. If A ( 0, 4, 3) , B(0, 0, 0) and C(3, 0, 4) are there points defined in x, y , z coordinate system, then

which one of the following vectors is perpendicular to both the vectors AB and BC

(GATE-11[PI])

a) 16i + 9j - 12K b) 16i - 9j + 12K c) 16i - 9j - 12K d) 16i + 9j + 12K

∫P1

39. The line integral (ydx + xdy) from P1(x1, y1) to P2(x2, y2) along the semi circle P1P2 shown in the
figure is P2 (GATE-11[PI])

^ P2(x2, y2)

a) x2y2 - x1y1 P1(x1, y1)
b) (y22 - y21) + (x22 - x21)
c) (x2 - x1) (y2 - y1) ^
d) (y2 - y1)2 + (x2 - x1)2

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MECHANICAL ENGINEERING

40. If T(x, yx z) = x2 + y2 + 2z2 defines the temperature at any location (x, y, z) then the magnitude of the

temperature gradient at point P(1, 1, 1) is ---------- (GATE-11[PI])
d) √6
a) 2√6 b) 4 c) 24

41. For the spherical surface x2 + y2 + z2 = 1, the unit outward normal vector at the point

1 , 1 ,0 is given by (GATE-12[ME, PI])
√2 √2

a) 1 ^i + 1 ^j b) 1 ^i - 1 ^j
√2 √2 √2 √2

c)^k d) 1 ^i + 1 ^j + 1 ^k
√3 √3 √3

42. Directional derivative of φ = 2xz - y2 at the point (1, 3, 2) becomes maximum in the direction of

(GATE-14-PI-Set 1)

a) 4i+2j-3k b) 4i-6j+2k c) 2i-6j+2k d) 4i-6j-2k

43. If φ = 2x3y2z4 then ∆2 φ is (GATE-14-PI-Set 1)
a) 12xy2z4 + 4x2z2 + 20x3y2z3 b) 2x2y2z + 4x3z4 + 24x3y2z2
c) 12xy2z4 + 4x2z3 + 24x3y2z2 d) 4xy2z + 4x2z4 + 24x3y2z2

*****************

“The successful warrior is the average man, with laser-link focus”

3. VECTOR CALCULUS (ANS.) :
1 - a, 2 - d, 3 - a, 4 - b, 5 - d, 6 - b, 7 - c, 8 - a, 9 - c, 10 - a, 11 - c, 12 - c, 13 - 1.666,
14 - 2, 15 - a, 16 - 216, 17 - sol, 18 - b, 19 - b, 20 - b, 21 - d, 22 - b, 23 - c, 24 - c, 25 -
a, 26 - c, 27 - d, 28 - b, 29 - b, 30 - a, 31 - a, 32 - 12, 33 - 5.714, 34 - d, 35 - d, 36 - d,
37 - c, 38 - a, 39 - a, 40 - a, 41 - a, 42 - b, 43 - c.

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ENGINEERING MATHEMATICS

4 DIFFERENTIALEQUATIONS

1. The differential equation y11 + y = 0 is subjected to the conditions y(0) = 0, y(λ) = 0. In order that the

equation has non-trivial solutions, the general value of λ is (GATE-93[ME])

d2y dy (GATE-93[ME])
2. The differential dx2 + dx + sin y = 0 is d) of degree two

a) linear b) non-linear c) homogeneous

3. For the defferential euation dy + 5y = 0with y(0) = 1, the general solution is (GATE-94[ME])
dt

a) e5t b) e-5t c) 5 e-5t d) e√-5t

d2y dy (GATE-94[ME])
4. Solve for y if dt2 + 2 dt + y = 0 with y(0) = 1 and y1(0) = 2.

5. If H(x, y) is homogeenous function of degree n then x ∂H + y ∂H = nH
∂x ∂y

a) True b) False (GATE-94[ME])

6. The solution to the differential equation f11(x) + 4f1(x) + 4f(x) = 0 (GATE-95[ME])

a) f1(x) = e-2x b) f1(x) = e2x , f2(x) = e-2x
c) f1(x) = e-2x , f2(x) = xe-2x d) f1(x) = e-2x , f2(x) = e-x

dy
7. A differential equation of the form dx = f(x, y) is homogeneous if the function f(x, y) depends

only on the ratio y or x (GATE-95[ME])
xy

a) True b) False

8. The particular solution for the differential equation d2y dy
dt2 + 2 dx + 2y = 5 Cos x is

a) 0.5 Cos x + 1.5 Sin x b) 1.5 Cos x + 0.5 Sin x (GATE-96[ME])
c) 1.5 Sin x d) 0.5 Cos x

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MECHANICAL ENGINEERING

9. The solution of the differential equation dy + y2 = 0 is (GATE-03[ME])
dx

1 x3
a) y = x + c b) y = - 3 + c

c) c ex d) unsolvable as equations is non - linear

dy 2 lnx (GATE-05[ME])
10. If x2 dx + 2xy = x and y(1) = 0 then what is y(e) ? d) 1

a) e b) 1 1 e2
c) e

d2y dy
11. The complete solution of the ordinary differential equation dx2 + P dx + q y = 0

is y = C1 e-x + C2 e-3x then P and q are (GATE-05[ME])

a) P = 3, q = 3 b) P =3, q = 4 c) P = 4, q =3 d) P = 4, q = 4

12. Which of the following is a solution of the differential equation d2y dy +(q +1)y = 0
dx + P dx

Where p = 4, q = 3

a) e-3x b) xe-x c) x e-2x d) x2 e-2x

13. The dy + 2xy = e-x2 with y(0) = 1 is (GATE-06[ME])
solution of the differential equation dx

a) (1 + x) ex2 b) (1 + x) e-x2 c) (1 - x) ex2 d) (1 - x) e-x2

d2y dy (GATE-06[ME])
14. For dx2 + 4 dx + 3y = 3e2x, the particular integral is d) c1e-x + c2e-3x

a) 1 e2x b) 1 e2x c) 3e2x
15 5

15. The solution of dy = y2with initial value y(0) = 1 is bounded in the interval is (GATE-07[ME])
dx

a) -∞ < x < ∞ b) -∞ < x < 1 c) x < 1, x > 1 d) -2 < x < 2

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ENGINEERING MATHEMATICS

16. Given that x + 3x = 0 and x(0) = 1, x(0) = 1, what is x(1) ______ (GATE-08([ME])
d) 0.4096
a) -0.99 b) -0.16 c) 0.16

17. It is given that y’’+2y’+y = 0, y(0) = 0 & y(1) = 0. What is Y(0.5) (GATE-08([ME])
d) 1.13
a) 0 b) 0.37 c) 0.62

dy 6 (GATE-09([ME])
18. The solution of x dx + y = x4 with condition y(1) = 5 x5

x4 1 b) y = 4x4 + 4 c) y = x4 + 1 d) y = + 1
a) y = 5 + x 5 5x 5 5

19. Consider the differential equationdy = (1 + y2)xT. he general solution with constant ‘C’ is
dx (GATE-11[ME])

a) y = tan x2 + C b) y = tan2 x2 + C c) y = tan2 x + C d) y = tan x2 + C
2 2 2 2

20. The solution to the differential equatiodn2u - k du = 0 where ‘k’ is a constant, subjected to the
dx2 dx

boundary conditions u(0) = 0 and u(L) = U, is (GATE-13[ME])

a) u = U x b) u = U 1 - ekx c) u = U 1 - e-kx d) u = U 1 + ekx
L e - ekL e - e-kL e + ekL

21. The one dimensional heat conduction partial differential equation ∂T = ∂2T is
∂t ∂x2

a) parabolic b) hyperbolic c) elliptic d) mixed (GATE-96[ME])

∂2φ ∂2φ ∂φ ∂φ (GATE-07[ME])
22. The partial differential equation ∂x2 + ∂y2 + ∂x + ∂y = 0 has

a) degree 1 and order 2 b) degree 1 and order 1
c) degree 2 and order 1 d) degree 2 and order 2

23. Left f = yx. What is ∂2f at x = 2, y = 1 ? (GATE-08[ME])
∂x∂y d) 1

a) 0 b) ln2 c) 1 ln2
(GATE-10[ME])
24. The blasius equationdd3ηf3 + f d2f = 0 is a
2 dη2

a) 2nd order non-linear ordinary differential equation
b) 3rd order non-linear ordinary differential equation
c) 3rd order linear ordinary differential equation
d) mixed order non-linear ordinary differential equation

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MECHANICAL ENGINEERING

25. The partial differential equation ∂u +u ∂u = ∂2u is a (GATE-13[ME])
∂t ∂x ∂x2

a) Linear equation of order 2 b) Non-linear equation of order 1
c) Liner equation of order 1 d) non-linear equation of order 2

26. The matrix form of the linear systemdx = 3x - 5y dy
dt anddt = 4x + 8y is

(GATE-14-ME-Set 1)

a) d x 3 -5 x b) d x 3 8 x c) d x 4 -5 x d) d x 4 8 x
dt y =4 8 y dt y =4 -5 y dt y =3 8 y dt y =3 -5 y

d2y dy
27. If y = f(x) is the solution of dx2 = 0with the boundary conditions y = 5 at x = 0, and dx = 2

at x = 10, f(15) = ____ (GATE-14-ME-Set 1)

28. The general solution of the differential equatiodny = cos (x+ y) , with c as a constant , is
dx

a) y + sin (x + y) = x + c b) tan x+y =y+c (GATE-14-ME-Set 2)
2

c) cos x+y =x+c d) tan x+y =x+c
2 2

29. Consider two solutions x(t) = x1(t) and x(t) = x2(t) of the differential equation

d2x(t) such that (GATE-14-ME-Set 3)
dt2 + x (t) = 0, t > 0

x1(0) = 1, dx1(t) =0
dt
t = 0

x2(0) = 0, dx2(t) =1
dt
t = 0

The wronskian w(t) = x1(t) x2(t) at t = π/2 is
dx1 dx2(t)
dt dt

a) 1 b) -1 c) o d) π/2

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ENGINEERING MATHEMATICS

30. The solution of the initial value problem dy (GATE-14-ME-Set 4)
dx = -2xy; y(0) = 2 is

a) 1 + e-x2 b) 2e-x2 c) 1 + ex2 d) 2ex2

31. Find the solution of d2y = ywhich passed through origin and the point ln2, 3
dx2 4

(GATE-15-ME-Set 1)

a) y= 1 ex - e-x b) 1 (ex + e-x) c) 1 (ex - e-x) d) 1 ex - e-x
2 2 2 2

32. Consider the following differential euqatiodny = -5y ; initial condition ; y = 2 at t = 0.
dt

The value of y at t = 3 is (GATE-15-ME-Set 2)
d) -15e2
a) -5e-10 b) 2e-10 c) 2e-15

33. The necessary & sufficient condition for the differential equation of the form (GATE-94)
M(x,y) dx + N(x,y) dy = 0 to be exact is

a) M = N b) ∂M = ∂N c) ∂M = ∂N ∂2M ∂2N
∂x ∂y ∂y ∂x d) ∂x2 = ∂y2

34. The differential equationd4y +p d2y + ky = 0 is (GATE-94)
dx4 dx2

a) Linear of fourth order b) Non - Linear of foruth order
c) Non - Homogeneous d) Linear and foruth degree

35. The differential equation y11 + (x3 Sin x)5 y1 + y = Cos x3 is (GATE-95)

a) homogeneous b) non-linear

c) 2nd order linear d) non-homogeneous with constant coefficients

36. The solution of a differential equation y11 + 3y1 + 2y = 0 is of the form (GATE-95)

a) c1ex + c2e2x b) c1e-x + c2e3x c) c1e-x + c2e-2x d) c1e-2x + c22-x

37. Solve d4v +4λ4v = 1 + x + x2 (GATE-96)
dx4

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MECHANICAL ENGINEERING

38. For the differential equation f(x, y) dy + g(x,y) = 0 to be exact is (GATE-97[CE])
dx ∂2f ∂2g

∂f ∂g b) ∂f = ∂g c) f = g d) ∂x2 = ∂y2
a) ∂y = ∂x ∂x ∂y

39. The differential equation dy + py = Q,is a linear equation of first order only if, (GATE-97[CE])
dx

a) P is a constant but Q is a function of y b) P and Q are functions of y (or) constants
c) P is a function of y but Q is a constant d) P and Q are fucntions of x (or) constants

d4v (GATE-98[CE])
40. Solve dx4 - y = 15 Cos 2x (GATE-98)

41. The general solution of the differential equation x2 d2y -x dy + y = 0 is
dx2 dx

a) Ax + Bx2 (A, B are constants) b) Ax + B logx (A, B are constants)
c) Ax + Bx2logs (A, B are constatns) d) Ax + Bxlogx (A, B are constants)

42. The radial displacement in a rotating disc is governed by the differential equation

d2u + 1 du - u =8
dx2 x dx x2

where us is the displacement and x is the radius. If y = 0 at x = 0 and u = 2 at x = 1, calculate the

displacement at x = 1 (GATE-98)
2

43. The euation d2y + (x2 + 4x) dy + y = x8 - 8 is a (GATE-99)
dx2 dx

a) partial differential equation b) non-linear differential equation
c) non-homogeneous differential equation d) ordinary differential equation

dy (GATE-99[CE])
44. If Cc is a constant, then the solution of dx = 1 + y2 is d) y = ex + c

a) y = sin (x + c) b) y = cos (x + c) c) y = tan (x + c)

45. Find the solution of differential d2y + λ2y = Cos (wt + k) with initial conditoins
equationdt2

y(0) = 0, dy(0) = 0. Here λ, w and k are constants. Use either the method of undetermined
dt

cofficients (or) the operator D = d based method. (GATE-2000)
dt

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ENGINEERING MATHEMATICS

46. The solution for the following differential equation with boundary conditions y(0) =2 and y1(1) = -3 is

d2y (GATE-01[CE])
where dx2 = 3x - 2
b) y = 3x3 - x2 - 5x + 2
a) y = x3 - x2 = 3x - 2 2
3 2
d) y = 3x3 - x2 + 5x + 3
c) y = x3 - x2 5x + 2 22
22

47. Solve the different equation d2y (GATE-2001)
dx2 + y = x with the following conditions

(i) at x = 0, y = 1 (ii) at x = 0, y1 = 1

48. Biotransformation of an organic comound having concentration (x) can be modeled using an ordinary

differential equation dx + kx2 0 ,where k is the reaction rate constant. If x = a at t = 0 then solution
dt

of the equation is (GATE-04[CE])

a) x = ac-kt b) 1 1 +kt c) x = a ( 1 -e-kt) d) x = a + kt
x= a

49. Transformation to linear form by substituing v = y1-n of the equation (GATE-05[CE])

dy
dt + p(t) y = q(t) yn , n > 0 will be

a) dv + (1 - n)pv = (1 - n)q b) dv + (1 + n)pv = (1 + n)q
dt dt

c) dv d) dy
dt + (1 + n)pv = (1 - n)q dt + (1 + n)pv = (1 + n)q

50. The solution d2y dy + 17y = 0 ; y (0)= 1, dy
dx2 + 2 dx
dx x= π = 0 in the range 0 < x < π
4

π/ 4 is given by (GATE-05[CE])

1 sin 4x ] 1
a) e-x [ cos 4x + 4 b) ex [ cos 4x - sin 4x ]
4

1 sin x ] 1
c) e-4x [ cos 4x - 4 d) e-4x [ cos 4x - sin 4x ]
4

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MECHANICAL ENGINEERING

51. The solution of the differential equation x2 dy + 2xy - x + 1 = 0 given that at x = 1, y = 0 is
dx (GATE-06[CE])

a) 1 - 1 + 1 b) 1 - 1 - 1 c) 1 + 1 + 1 d) - 1 + 1 + 1
2 x 2x2 2 x 2x2 2 x 2x2 2 x 2x2

52. The degree of the differential equation d2x (GATE-07[CE])
dt2 + 2x3 = 0 is c) 3

a) 0 b) 1 b) 2

53. The solution for the differential equationdy = x2 y with the condition that y = 1 at x = 0 is
dx b)
x2 (GATE-07[CE])
1 In(y) = 2 + 4

a) y = e 2x

c) In(y) = x2 X3
2
d) y = e 3

54. A body originally at 600 cools down to 40 in 15 minutes when kept in air at a temperature of 250c.

What will be the temperature of the body at the and of 30 minutes ? (GATE-07[CE])

a) 35.20C b) 31.50C c) 28.70C d) 150C

55. Solution of the differential equation 3y dy + 2x = 0 represents a family of
a) ellipses dx
b) circles (GATE-09[CE])

c) parabolas d) hyperbolas

56. The order and degree of a differential equationd3y + 4 dy 3 are respectively
+ y2 = 0
dx3 dx (GATE-10[CE])

a) 3 and 2 b) 2 and 3 c) 3 and 3 d) 3 and 1

57. The solution to the ordinary differential equation d2y dy - 6y = 0 is (GATE-10[CE])
dx2 + dx

a) y = C1e3x + C2e-2x b) y = C1e3x + C2e2x c) y = C1e-3x + C2e2x d) y = C1e-3x + C2e-2x

dy y
58. The solution of the differential equation dx + x = x with the condition that y = 1 at x = 1 is

(GATE-11[CE])

a) y= 2 x b) y = x1 c) y = 2x 2 x2
3x2 + 3 2 + 2x 3 +3 d) y = 3 + 3

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ENGINEERING MATHEMATICS

59. Consider the differential equation x2 d2y +x dy - 4y =0 with the boundary conditions of
dx2 dx

y(o) = 0 and y(1) = 1. The complete solution of the differential equation is (GATE-12[ME, PI])

a) x2 b) sin πx c) ex sin πx d) e-x sin πx
2 2 2

60. The solution of the ordinary differential equation dy 2y = 0 for the boudary condition, y = 5 at
dx (GATE-12[CE])
x = 1 is

a) y = e-2x b) y = 2e-2x c) y = 10.95e-2x d) y = 36.95c-2x

∂ 2φ ∂ 2φ
61. The number of boundary conditions required to solve the differential equation + = 0 is
∂x2 ∂y2

a) 2 b) 0 c) 4 d) 1 (GATE-01[CE])

62. The partial differential euqation that can be formed from z = ax + by + ab has the form

p= ∂z , q = ∂z (GATE-10[CE])
∂x ∂y

a) z = px + qy b) z = py + qx c) z = px + qy + pq d) z = qy + pq

63. The intergrating factor for the differential equation (GATE-14-CE-Set 2)

dP is
dt + k2 P = k1L0e-k1t

a) e-k1t b) e-k2t c) ek1t d) ek2t

64. Water is flowing at a steady rate through a homogeneous and satured horizontal soil strip of 10m
length. The strip is being subjected to a constant water head (H) of 5m at the beginning and 1m at the

end. If the governing equation of flow in the soil strip is d2H =0 (where x is the distance
dx2

along the soil strip), the value of H (in m) at the middle of the strip is ______ (GATE-14CE- Set 1)

65. Consider the following differential equation x(y dx +x dy)cos y = y(x dy - y dx)sin y .
x x

Which of the following is the solution of the above euqation ( C is an arbitrary constant)

(GATE-15-CE- Set 1)

a) x cos y =C b) x sin y c) xy cos y =C d) xysin y
yx y x =C x x =C

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MECHANICAL ENGINEERING

66. Consider the following second order linear differential equation d2y = -12x2 + 24x - 20
dx2

The boundary conditions are : at x = 0, y = 5 and at a x= 2, y = 21. The value of y at a x = 1 is.

(GATE-15-CE- Set 2)

67. The differential equation 1+ dy 2 3 2 (GATE-05[PI])
dx = C2
d2y
dx2 is of

a) 2nd order and 3rd degree b) 3rd order and 2nd degree
c) 2nd order and 2nd degre d) 3rd order and 3rd degree

d2y dy
68. The homogeneous part of the differential equation dx2 + P dx + qy = r

(p, q, r are constants) has real distinct roots if (GATE-09[PI])

a) p2 - 4q > 0 b) p2 - 4q < 0 c) p2 - 4q = 0 d) p2 - 4q = r

69. The solution of the differential equation d2y = 0 with boundary conditions
dx2
(GATE-09[PI])
(i) dy = 1 at x = 0 (ii) dy = 1 at x = 1 is
dx dx

a) y = 1
b) y = x
c) y = x + c where c is an arbitrary constant
d) y = C1x + C2 where C1, C2 are arbitary constants

70. The solution of the differential equation dy - y2 = 1 satisfying the condition y(0) = 1 is
dx

(GATE-10[PI])

a) y = ex2 b) y = √x c) y = cot(x + π ) d) y = tan(x + π )
4 4

71. Which one of the following differential equations has a solution given by the function

π (GATE-10[PI])
y = 5 sin 3x + 3

a) dx - 5 cos(3x) = 0 b) dy + 5 (cos3x) = 0
dy 3 dx 3

c) d2y + 9y = 0 d) d2y - 9y = 0
d2x dx2

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ENGINEERING MATHEMATICS

72. The solution of the differential equation d2y +6 dy + 9y = 9x + 6
dx2 dx

with C1 and C2 as constants is (GATE-11[PI])

a) y = (C1x + C2)e-3x b) y = C1e3x + C2 e-3x
d) y = (C1x + C2)e3x + x
c) y = (C x + C2)e-3x + x
1

73. The solution to 6yy1 - 25x = 0 represents a (GATE-15-PI)
a) family of circles b) family of ellipses c) family of parabolas d) family of hyperbolas

74. The solution to x2y11 + xy1 - y = 0 is c) y = C1 x + C2 (GATE-15-PI)
a) y = C1x2 + C2x-3 b) y = C1 + C2x-2 x d) y = C1x + C2x4

**********************

“The world belongs to the enthusiast who keeps cool”

4. DIFFERENTIAL EQUATIONS (ANS.) :
1 - sol., 2- b, 3 - b, 4 - sol., 5 - a, 6 - c, 7 - a, 8 - a, 9 - a, 10 - d, 11 - c, 12 - c, 13 - b,
14 - b, 15 - c, 16 - d, 17 - a, 18 - a, 19 - d, 20 - b, 21 - a, 22 - a, 23 - c, 24 - b, 25 - d, 26 - a,
27 - 35, 28 - d, 29 - a, 30 - b, 31 - c, 32 - c, 33 - c, 34 - a, 35 - c, 36 - c, 37 - sol, 38 - b, 39 -
d, 40 - sol, 41 - d, 42 - sol. ,43 - c, 44 - c, 45 - 0, 46 - c, 47 - sol, 48 - b, 49 - a,
50 - a, 51 - a, 52 - b, 53 - d, 54 - b, 55 - a, 56 - a, 57 - c, 58 - d, 59 - a, 60 - d, 61 - c, 62 - c,
63 - d, 64 - 3, 65 - c, 66 - 18, 67 - c, 68 - a, 69 - c, 70 -d, 71 - c, 72 - c, 73 - d, 74 - c.

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MECHANICAL ENGINEERING

5 TRANSFORM THEORY

1. The laplace trnasform of the periodic function f(t) described by the curve below

i.e. f(t) = sin t, if (2n -1) π < t < 2np(n = 1, 2, 3 ...........) (GATE-93[ME])

0 other wise

2. If f(t) is a finite and continuous Function for t > 0 the laplace transformation is given by

∞

∫F = e-stf(t)dt, then for f(t) = cons h mt,
0
(GATE-94[ME])
the laplace transformation is _______

3. The inverse Laplace transform of (s +9)/(s2 +6x + 13) is (GATE-1995)

a) cos 2t + 9 sin 2t b) e-3t cos2t - 3 e-3t sin 2t

c) e-3t sin2t + 3 e-3t cos 2t d) e-3t cos2t + 3 e-3t sin 2t

4. Using Laplace transform, solve the initial value problem 9y11 - 6y1 + y = 0 y(0) = 3 and y1(0) = 1,

where prime denotes derivative with respect to t. (GATE-96)

5. Solve the initial value problem d2y - 4 dy + 3y = 0 y = 3 and dy = 7 at x = 0
dx2 dx dt

using the laplace transform technique ? (GATE-97-[ME])

6. The Laplace Transform of the unit step function ua(t), defined as (GATE-98)
d) se-as - 1
ua(t) = 0 for t < a is

= 1 for t > a ,

a) e-as /s b) se-as c) s - u(0)

7. (s +1)-2 is the Laplace transform of c) e-2t (GATE-98)
a) t2 b) t3 d) te-t

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