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ISRO Mechanical Engineering Mathematics Questions, Answers and Explanation

ISRO Mechanical Engineering Mathematics Questions, Answers and Explanation

Linear Algebra

2017.2.36. A point P moves along the path y = x2 -4. What is the displacement when the point moves from x =1 to x = 3.
a) 8.24, @ tan-1 (4)
b) 2,  @ tan-1 (1)
c) 11.3, @ tan-1 (4)
d) 11.3,  @ tan-1 (1)

Answer
a) 8.24, @ tan-1 (4)

Explanation
when x = 1, y = 1-4 = -3
when x = 3, y = 9-4 = 5
distance between the 2 points = √ (22+82) = √ (68), which is just greater than 8. So other options are eliminated.

2017.2.47. A cubic B-spline curve requires minimum ......... control points
a) 3
b) 4
c) 5
d) 6

Answer
b) 4

Explanation
A line (linear curve) requires minimum 2 control points
A quadratic curve requires minimum 3 control points
In general a curve of degree n requires minimum n+1 control points

Derivatives

2017.2.80.  A point moves according to the relation x = 8t2 cos ω t , y = -4t3 sin ωt, where x and y are in meters and t in seconds. If angular velocity ω = 8 π rad/s, velocity vector of the point at t = 0.5s is given by
a) 8 + j4π
b) 6 + j20π
c) 8 - j4π
d) -20π -j6

Answer
c) 8 - j4π

Explanation
x = 8t2 cos ω t
y = -4t3 sin ωt
u = dx/dt = 16t cos ω t - 8ωt2 sin ω t
v = dy/dt = -12t2 sin ωt - -4ωt3 cos ωt
velocity vector = u + jv
 = 16t cos ω t - 8ωt2 sin ω t + j(-12t2 sin ωt - 4ωt3 cos ωt)
Substituting ω = 8 π rad/s and t = 0.5s,
velocity vector = 16*0.5 cos 8 π *0.5 - 8*8 π*0.52 sin 8 π *0.5 + j(-12*0.52 sin 8 π*0.5 -4*8 π*0.53 cos 8 π*0.5)
 = 8 cos 4 π  - 2 sin 4 π + j(-3 sin 4 π -4 πcos 4 π)
 = 8 - j4 π

Laplace Transforms

2006. Laplace transform of t2 + 2t + 3 is
a) - 2 ⁄ s3 - 2 ⁄ s2  - 3 ⁄ s
b) 2 ⁄ s3 + 2 ⁄ s2 - 3 ⁄ s
c) 2 ⁄ s3 + 2 ⁄ s2+3 ⁄ s
d) - 2 ⁄ s3 + 2 ⁄ s2 - 3 ⁄ s

2007. Laplace transform of sin3 3t is
a) 24 ⁄ (s2+4)(s2+36)
b) 1 ⁄ (s2+4)(s2+64)
c) 48 ⁄ (s2+4)(s2+36)
d) 64 ⁄ (s2+4)(s2+36)

2017_1. Laplace transform of t cos (at) is
a) s2+a2 ⁄ (s2-a2)
b) s ⁄ (s2-a2)
c) s2-a2 ⁄ (s2+a2)
d) s ⁄ (s2+a2)

2017.2.4 If f(t) = eat, its Laplace transform (for s > a) is given by
a) a⁄s2 + (s-a)
b) √π⁄2(s-a)
c) 1⁄(s-a)
d) Does not exist

Answer
c) 1⁄(s-a)

Explanation
If L{f(t)}=F(s), then L{eatf(t)}=F(s-a)
In the problem f(t) = 1, L(1) =  F(s) = 1/s
Therefore, L(eat1) = F(s-a) = 1/(s-a)

Probability

2017.2.23. A man draws 3 balls from a jug containing 5 white balls and 7 black balls. He gets Rs. 20 for each white ball and Rs. 10 for each black ball. What is his expectation?
a) Rs. 21.25
b) Rs 42.50
c) Rs 31.50
d) Rs 45.21

Answer
b) Rs 42.50

Explanation
3 balls can be drawn in the following ways
- 3 white balls, P = 5C3/11C3 = 10/220 = 2/44
- 2 white balls and 1 black ball, P = 5C2*7C1/11C3 = 70/220 = 14/44
- 1 white ball and 2 black balls, P = 5C1*7C2/11C3 = 125/220 = 21/44
- 3 black balls, P = 7C3/11C3 = 35/220 = 7/44
Expectation = sum of the product of probability and the money he gets for each combination
= (2/44)*60 + (14/44)*50 + (21/44)*40 + (7/44)*30 = 935/22 = 85/2 = 42.50
2018.77. In a party, each person shook hands with every other person present. The total number of hand shakes was 45. The number of people in the party are
a) 12
b) 10
c) 8
d) 6

Answer
b) 10

Explanation
First person can shake the hand with n-1 persons.
The second person can shake hand with n-2 persons, because the second person has already shaken hands with the first person.
Proceeding this way the last person has no one to shake hands with. That persons handshake has been accounted previously.
So if there are n persons there are (n-1)+(n-2)+(n-3)+.....+1 hand shakes = nC2 = (n-1)n/2 = Sum of first n-1 integers.
Here (n-1)n/2 = 45
(n-1)n = 90 = 9 x 10

So n = 10 


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