Question 1 |
For matrices of same dimension M, N and scalar c, which one of these properties DOES NOT ALWAYS hold ?
(M^{T})^{T}=M | |
(cM)^{T}=c(M)^{T} | |
(M+N)^{T}=M^{T}+N^{T} | |
MN=NM |
Question 1 Explanation:
Matrix multiplication is not commutative.
Question 2 |
In a housing society, half of the families have a single child per family, while the
remaining half have two children per family. The probability that a child picked
at random, has a sibling is _____
1.5 | |
0.67 | |
2 | |
3 |
Question 2 Explanation:
Required probability
=\frac{\frac{1}{2} \times \frac{2}{3}}{\frac{1}{2} \times \frac{1}{3}+\frac{1}{2} \times \frac{2}{3}}=\frac{2}{3}=0.666
=\frac{\frac{1}{2} \times \frac{2}{3}}{\frac{1}{2} \times \frac{1}{3}+\frac{1}{2} \times \frac{2}{3}}=\frac{2}{3}=0.666
Question 3 |
C is a closed path in the z -plane by |z| = 3. The value of the integral \oint_{c}(\frac{z^{2}-z+4j}{z+2j})dz is
-4\pi (1+ j2 ) | |
4\pi (3- j2 ) | |
-4\pi (3+ j2 ) | |
4\pi (1- j2) |
Question 3 Explanation:
\oint_{c} \frac{z^{2}-z+4 j}{z+2 j}
Pole =z=-2 j
which is inside of |z|=3
From Cauchy integral formula
\begin{aligned} \oint \frac{z^{2}-z+4 j}{z+2 j} &=2 \pi i\left[\lim _{z \rightarrow-2 j} z^{2}-z+4 j\right] \\ &=2 \pi j[-4+2 j+4 j] \\ &=2 \pi j[-4+6 j] \\ &=-4 \pi[2 j+3] \end{aligned}
Question 4 |
A real (4x4) matrix A satisfies the equation A_{2} = I, where I is the (4x4)
identity matrix. The positive eigen value of A is _____.
1 | |
2 | |
3 | |
4 |
Question 4 Explanation:
\begin{aligned} \text{since, }A^{2}=l, \text{eig}\left(A^{2}\right)&=\text{eig}(I)=1 \\ \Rightarrow \quad \text{eig}(A)^{2}&=1\\ \Rightarrow \quad \text{eig}(A)&=\pm 1 \end{aligned}
Therefore, the positive eigen value of A is +1.
Therefore, the positive eigen value of A is +1.
Question 5 |
Let X_{1} , X_{2}, \; and \; X_{3} be independent and identically distributed random variables with the uniform distribution on [0,1]. The probability P{X_{1} is the largest} is
0.5 | |
0.33 | |
0.25 | |
0.75 |
Question 5 Explanation:
If multiple independent random variables are uniformly distributed in the same interval then each random variable will have equal chances to be largest and to be lowest.
P\left(X_{1} \text { is the largest) }=\frac{1}{3}\right.
P\left(X_{1} \text { is the largest) }=\frac{1}{3}\right.
There are 5 questions to complete.