Sample Questions: GATE Mechanical Engineering Mathematics Set1

1. A Coin is tossed 4 times. What is the probability of getting heads exactly 3 times?

(A) 1/4
(B) 3/8
(C) 1/2
(D) 3/4
Ans: Option A : when coin is tossed getting a Head H or Tail T has pobability = 1/2;
If the coin was tossed 4 times to get the Head H exactly 3 times there are four different ways
1: THHH : 1/2 * 1/2 * 1/2 * 1/2 = 1/16
2: HTHH : 1/2 * 1/2 * 1/2 * 1/2 = 1/16
3: HHTH : 1/2 * 1/2 * 1/2 * 1/2 = 1/16
4: HHHT : 1/2 * 1/2 * 1/2 * 1/2 = 1/16
as there are 4 different ways this result can be obtained hence the actual probability is
Ans = 1/16 + 1/16 + 1/16 + 1/16 = 1/4

2. The divergence of the vector field (x-y)i + (y-x)j + (x+y+z)k is
(A) 0
(B) 1
(C) 2
(D) 3

Ans: D : the divergence is given by:
$\operatorname{div}\,\mathbf{F} = \nabla\cdot\mathbf{F} =\frac{\partial U}{\partial x} +\frac{\partial V}{\partial y} +\frac{\partial W}{\partial z }.$

3. If 3 coins are tossed simultaniously, the probability of getting atleast 1 Head is
(A) 1/8
(B) 3/8
(C) 1/2
(D) 7/8
Ans: D:
traditional solution:
The possible outcome when 3 coins are tossed are
TTT, HTT, THT, TTH, HHT, HTH, THH, HHH
We Need atleast 1 Head, there are 7 combinations out of 8 in which user will get atleast 1 Head
Hence answer is 7/8.

Quick Solution:
When single coin is tossed the probability of getting Head is 1/2.
When more than 1 coins are tossed and only one Head is expected then it is likely to occur more easily than when single coin is tossed. Hence the probability of getting atleast 1 Head when 3 coins are tossed is more than 1/2.
There is only one option out of four having value more than 1/2 is Option "D" i.e 7/8

4. For the given matrix find the value of ‘x’ if Transpose of the matrix is equal to Inverse of the Matrix.