# GATE EC 2014 SET-1

 Question 1
For matrices of same dimension M, N and scalar c, which one of these properties DOES NOT ALWAYS hold ?
 A $(M^{T})^{T}=M$ B $(cM)^{T}=c(M)^{T}$ C $(M+N)^{T}=M^{T}+N^{T}$ D MN=NM
Engineering Mathematics   Linear Algebra
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 _____
 A 1.5 B 0.67 C 2 D 3
Engineering Mathematics   Probability and Statistics
Question 2 Explanation:
Required probability
$=\frac{\frac{1}{2} \times \frac{2}{3}}{\frac{1}{2} \times \frac{1}{3}+\frac{1}{2} \times \frac{2}{3}}=\frac{2}{3}=0.666$

 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
 A $-4\pi (1+ j2 )$ B $4\pi (3- j2 )$ C $-4\pi (3+ j2 )$ D $4\pi (1- j2)$
Engineering Mathematics   Complex Analysis
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 _____.
 A 1 B 2 C 3 D 4
Engineering Mathematics   Linear Algebra
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.
 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
 A 0.5 B 0.33 C 0.25 D 0.75
Communication Systems   Random Processes
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.
$P\left(X_{1} \text { is the largest) }=\frac{1}{3}\right.$

There are 5 questions to complete.