Engineering Mathematics Test - 2 - PDF Flipbook
Engineering Mathematics Test - 2
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GATE
EEE
Engineering
Mathematics
Test-02Solutions
ENGINEERING MATHEMATICS
1. The rank of ( × ) matrix (m < n) cannot be more than
a) m + n
b) n
c) mn
d) None
Answer: (a)
Solution:
We know that, ( × ) ≤ { , }
But it is given that m < n
∴ ( × ) ≤
ℎ ( × ) cannot be more than ‘m’.
0 0 −3
2. The rank of matrix �9 3 5 �is
31 1
a) 0
b) 1
c) 2
d) 3
Answer: (c)
Solution:
00 −3
= �9 3 5�
1
31
1 ↔ 3
1
31 1
∽ �9 3 5 �
0 0 −3
2 ⟶ 2 − 3 1
31 1
∽ �0 0 2 �
0 0 −3
3 ⟶ 2 3 + 3 2
311
∽ �0 0 2�
000
∴ ( ) = 2
14 9
3. The value of the following determinant �4 9 16�
9 16 25
a) 8
b) 12
c) -12
d) -8
Answer: (d)
Solution:
14 9
| | = �4 9 16�
9 16 25
= 1(225 − 256) − 4(100 − 144) + 9(64 − 81)
= −8
2
1 −1 0
4. The inverse of the matrix = �1 1 1� is
001
101
a) �0 0 0�
011
0 11
b) �−1 −1 1�
01
1
2 −2
2 2 −2�
c) �−2 22
0
1 1 −1
222
d) �−1 1 −1�
222
001
Answer: (d)
Solution:
1 −1 0
Given = �1 1 1�
001
⇒ | | = 1(1 + 1) = 2 ≠ 0
1 1 −1
( ) = �−1 1 −1�
002
1 1 − 1
2 2
∴ −1 ( ) 2 1 1�
= | | = �− 1 2
2 − 2
0
0 1
3
5. Let AX = B be a system of linear equations where A is an m×n
matrix B is an n×1 column matrix which if the following is
false?
a) The system has a solution, if ρ(A) = ρ(A/B)
b) If m = n and B is a non-zero vector then the system has a
unique solution.
c) If m < n and B is a zero vector then the system has infinitely
many solutions.
d) The system will have a trivial solution when m = n, B is the
zero vector and rank of A is n.
Answer: (b)
Solution:
Given that × ×1 = ×1
⇒ × ×1 = ×1( = )
In this case, the given system may (or) may not have unique
solution.
If A is singular then unique solution does not exist. And if A is
non- singular then unique solution exists.
∴Option b is wrong statement
6. Consider the following statements
I. S1: The sum of two singular matrices may be singular.
II. S2: The sum of two non-singular matrices may be non-
singular.
Which of the following statements is true?
a) S1 & S2 are both true
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b) S1 & S2 are both false
c) S1 is true & S2 is false
d) S1 is false & S2 is true
Answer: (a)
Solution:
S1 is true, = �01 00� = �00 10�
Where | | = 0, | | = 0
⇒ + = �01 10� ⇒ | + | = 0
S2 is true, = �10 32� = �21 31�
Where | | ≠ 0, | | ≠ 0
⇒ + = �22 63� ⇒ | + | ≠ 0
7. Eigen values of a matrix = �23 23� are 5 and 1.
What are the eigen values of the matrix S2 = SS?
a) 1 and 25
b) 6, 4
c) 5, 1
d) 2, 10
Answer: (a)
Solution:
Eigen values of a matrix ‘S’ are 1 and 5. We know that, the
eigen values of matrix 52 are 12 and 52 i.e. 1 & 25.
5
8. The curve given by the equation x2 + y2 = 3axy is
a) Symmetrical about x-axis
b) Symmetrical about y-axis
c) Symmetrical about the line y = x
d) Tangential to x = y = a/3
Answer: (c)
Solution:
f(x, y) = f(y, x) ⇒ curve is symmetric about the line y = x
9. The continuous function f(x, y) is said to have saddle point at (a,
b) if
a) fx (a, b) = fy (a, b) = 0
2 - fxxfyy < 0 at (a, b)
b) fx (a, b) = fy (a, b) = 0,
2 - fxxfyy > 0 at (a, b)
c) fx (a, b) = fy (a, b) = 0,
fxx and fyy < 0 at (a, b)
d) fx (a, b) = fy (a, b) = 0,
2 - fxxfyy = 0 at (a, b)
Answer: (b)
Solution:
Condition for no maxima and minima at (a, b)
10. The value of the integral is = ∫0 �4 2
a) + 1
8 4
b) − 1
8 4
6
c) − − 1
8 4
d) − + 1
8 4
Answer: (a)
Solution:
= ∫0 �4 �1+co2s 2 � = � 2 + 4 2 � 0 �4
11. The minimum value of function y = x2 in the interval [1, 5] is
a) 0
b) 1
c) 25
d) Undefined
Answer: (b)
Solution:
y = x2 in [1, 5]
y is minimum when x is minimum in [1, 5]
∴y is minimum at x = 1
∴minimum value of y = (1)2 = 1
12. Consider the function f (x) = x2 - x - 2. The maximum value of
f(x) in the closed interval [-4, 4] is
a) 18
b) 10
c) -2.25
d) indeterminate
Answer: (a)
7
Solution:
f (x) = x2 – x – 2 in [-4, 4]
f'(x) = 2x – 1 = 0 x = ½ is a stationary point
f "(x) = 2 > 0 ⇒ f(x) has a minimum at x = 1/2
∴The greatest value lies at extreme points here f(-4) = 18, f(4) =
10
∴f(x) has a Maximum at x = - 4
Maximum value f(-4) = 18
13. Which one of the following is Not associated with vector
calculus?
a) Stoke's theorem
b) Gauss Divergence theorem
c) Green’s theorem
d) Kennedy’s theorem
Answer: (d)
Solution:
Kennedy’s theorem
14. Divergence of the vector field v (x, y, z) = - (x cosxy + y) ̅ +
(y cosxy) ̅ + [(sinz2) + x2 + y2] � is
a) 2z cosz2
b) sin xy + 2z cosz2
c) x sin xy – cos z
d) none of these
Answer: (a)
8
Solution:
v (x, y, z) = - (x cosxy + y) ̅ + (y cosxy) ̅ + [sinz2 + x2+ y2] �
� = [−( cos + )] + [ cos ] +
[ 2 + 2 + 2] = 2 cos 2
15. If ̅ is the position vector of any point on a closed surface S
that encloses the volume V then ∬ ( ̅. � � � )is equal to
a) 1
2
b) V
c) 2V
d) 3V
Answer: (d)
Solution:
As ‘S’ is a closed surface, using Gauss-Divergence theorem,
∬ ̅. ̅ = ∭ (∇. ̅) = ∭ 3 = 3
16. The velocity vector is given as ̅ = 5 ̅ + 2 2 ̅ + 3 2 � .
The divergence of this velocity vector at (1, 1, 1) is
a) 9
b) 10
c) 14
d) 15
Answer: (d)
Solution:
� = 5 ̅ + 2 2 ̅ + 3 2 �
� = 5 + 4 + 6
9
At (1, 1, 1) div � = 5 + 4 + 6 = 15
17. The angle (in degrees) between two planar vectors � = √3 +
2
1 � = −√3 + 1 is
2 2 2
a) 30
b) 60
c) 90
d) 120
Answer: (d)
Solution:
� = √3 + 1 ,
2 2
� = − √3 + 1
2 2
cos = � . �
| � |� � �
= −43+41 = − 1
�43+41.�43+41 2
⇒ = 120
18. In a manufacturing plant, the probability of making a defective
bolt is 0.1. The mean and standard deviation of defective bolts
in a total of 900 bolts are respectively
a) 90 and 9
b) 9 and 90
c) 81 and 9
d) 9 and 81
10
Answer: (a)
Solution:
= 0.1, = 900, = 1 − = 0.9
Mean = np = 90
S.D = = � = 9
19. A bag contains 10 blue marbles, 20 black marbles and 30 red
marbles. A marble is drawn from the bag, its color recorded and
it is put back in the bag. This process is repeated 3 times. The
probability that no two of the marbles drawn have the same
color is
a) 1
36
b) 1
6
c) 1
4
d) 1
3
Answer: (b)
Solution:
3 balls of different colors can be drawn in 6 ways
Required probability = 6. �6100 20 6300� = 1
60 6
20. Two dice are thrown simultaneously. The probability that the
sum of numbers on both exceeds 8 is
a) 4
36
b) 7
36
11
c) 9
36
d) 10
36
Answer: (d)
Solution:
( ) = 6 × 6
= 36
= �(3,6), (4,5), (5,4), (6,3), (4,6), (5,5),�
(6,4), (5,6), (6,5), (6,6)
Required probability = ( )/ ( )10/36
21. A single die is thrown two times. What is the probability that
the sum is neither 8 nor 9?
a) 1
9
b) 5
36
c) 1
4
d) 3
4
Answer: (d)
Solution:
Let E be the event of getting the sum 8 or 9
⇒ n(E) = 9
( ) = 9 = 1
36 4
Required probability = 1 – p(E) = 1 − 1 = 3
4 4
12
22. A fair dice is rolled twice. The probability that an odd number
will follow an even number is
a) 1
2
b) 1
6
c) 1
3
d) 1
4
Answer: (d)
Solution:
Probability of getting an odd number when a fair dice is rolled
= 3 = 1
6 2
Probability of getting an even number when a fair dice is rolled
= 3 = 1
6 2
∴ Required probability = 1 × 1 = 1
2 2 4
23. The general solution of the differential equation (D2 – 4D + 4)
y = 0 is of the form
(given D = and C1, C2 are constants)
a) C1e2x
b) C1e2x + C2e-2x
c) C1e2x + C2e2x
d) C1e2x + C2xe2x
Answer: (d)
Solution:
Given (D2 – 4D + 4) y = 0
13
The Auxiliary equation is
D2 – 4D + 4 = 0 ⇒ (D – 2)2 = 0
⇒ D = 2, 2
∴ = ( 1 + 2 ) 2
24. The differential equation �1 + � �2�3 = 2 � 2 2 �2is of
a) 2nd order and 3rd degree
b) 3rd order and 2nd degree
c) 2nd order and 2nd degree
d) 3rd order and 3rd degree
Answer: (c)
Solution:
Given �1 + � �2�3 = 2 � 2 2 �2
According to the definition of order and degree of a differential
equation, order = 2 and degree = 2
25. The complete solution of the ordinary differential equation
2 + + = 0 is C1e-x + C2e-3x then P and q are
2
a) P = 3, q = 3
b) P = 3, q = 4
c) P = 4, q = 3
d) P = 4, q = 4
Answer: (c)
Solution:
Given 2 + + = 0 ….. (1) and its
2
14
Solution is y = C1e-x + C2e-3x ……. (2)
From (1), we have
⇒ ( 2 + + ) = 0 ….. (3)
∴ ( + 1)( + 3) = 0 (∵ (2))
⇒ 2 + 4 + 3 = 0 …… (4)
Comparing (3) and (4), we have
P = 4 and q = 3
26. A solution of the differential equation 2 − 5 + 6 = 0 is
2
given by
a) y = e2x + e-3x
b) y = e2x + e3x
c) y = e-2x + e3x
d) y = e-2x + e-3x
Answer: (b)
Solution:
Given ( 2 − 5 + 6) = 0
⇒ 2 − 5 + 6 = 0
⇒ = 2, 3
∴ = 1 2 + 2 3
If we choose 1 = 1 and 2 = 1, we get
= 2 + 3
27. If F(s) = L{f(t)} = 2( +1) then the initial and final values of
2+4 +7
f(t) are respectively
a) 0, 2
15
b) 2, 0
c) 0, 2
7
d) 2 , 0
7
Answer: (b)
Solution:
L →im0 ( ) = L →im∞ ( )
= L →im∞ 2 2+ = 2
2+4 +7
L →im∞ ( ) = L →im0 ( ) = 2 2+ = 0
2+4 +7
28. The Laplace transform of eαt cosαt is equal to
a) −
( − )2+ 2
b) +
( + )2+ 2
c) 1
( − )2
d) None
Answer: (a)
Solution:
{cos } = = ( )
2+ 2
{ cos } = ( − ) = −
( − )2+ 2
29. If f(z) = Co + C1 z-1 then ∮| |=1 1+ ( ) is given
a) 2 C1
b) 2 (1+Co)
c) 2 jC1
16
d) 2 j(1+Co)
Answer: (d)
Solution:
= ∫ 1+ ( )
Where f(z) = C0 + C1z-1 = C0 + 1
= ∫ 1+ 0+( 1 / )
= ∫ + 0 + 1 , | | = 1
2
Here z = 0 lies insides C
By Cauchy Integral formula = 2 1 (0)
1!
Where ( ) = 0 + 0 + 1
& 1( ) = 1 + 0 + 0
= 2 (0 + 1 + 0)
= 2 ( 0 + 1)
30. Matching exercise choose the correct one out of the
alternatives A, B, C, D
Group-I
P. 2nd order differential equations
Q. Nonlinear algebraic equations
R. linear algebraic equations
S. numerical integration
Group-II
1. Runge - Kutta method
2. Newton - Raphson method
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3. Gauss elimination
4. Simpson’s Rule
a) P - 3, Q - 2, R - 4, S - 1
b) P - 2, Q - 4, R - 3, S - 1
c) P - 1, Q - 2, R - 3, S - 4
d) P - 1, Q - 3, R - 2, S - 4
Answer: (c)
Solution:
P - 1, Q - 2, R - 3, S – 4
(1) Simpson’s Rule is one of the numerical integration
technique (method).
(2) Gauss-elimination method is used to solve only system of
linear algebraic equations.
(3) Runge –Kutta method is used to solve the ordinary
differential equations.
(4) Newton-Raphson method is used to solve the linear and
non-linear algebraic equations.
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