# GATE EC 2016 SET-1

 Question 1
Let $M^{4}=I$, (where $I$ denotes the identity matrix) and $M \neq I$, $M^{2} \neq I$ and $M^{3} \neq I$. Then, for any natural number $k,M^{-1}$ equals
 A $M^{4k+1}$ B $M^{4k+2}$ C $M^{4k+3}$ D $M^{4k}$
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
Given that $M^{4}=I$ or $M^{4 k}=I$ or $M^{4(k+1)}=I$
\begin{aligned} \therefore \quad M^{-1} \times I & =M^{4(k+1)} \times M^{-1} \\ \therefore \quad M^{-1} & =M^{4 k+3}\end{aligned}
 Question 2
The second moment of a Poisson-distributed random variable is 2. The mean of the random variable is _______
 A 0.5 B 1 C 2 D 3
Engineering Mathematics   Probability and Statistics
Question 2 Explanation:
In Poisson distribution,
Mean = First moment $=\lambda$
secondmoment $=\lambda^{2}+\lambda$
Given, that second moment is 2
$\begin{array}{r} \lambda^{2}+\lambda=2 \\ \lambda^{2}+\lambda-2=0 \\ (\lambda+2)(\lambda-1)=0 \\ \lambda=1 \end{array}$

 Question 3
Given the following statements about a function $f:\mathbb{R}\rightarrow \mathbb{R}$, select the right option:
P: If f(x) is continuous at x=$x_{0}$, then it is also differentiable at x =$x_{0}$
Q: If f(x) is continuous at x = $x_{0}$, then it may not be differentiable at x= $x_{0}$.
R: If f(x) is differentiable at x= $x_{0}$, then it is also continuous at x= $x_{0}$.
 A P is true, Q is false, R is false B P is false, Q is true, R is true C P is false, Q is true, R is false D P is true, Q is false, R is true
Engineering Mathematics   Calculus
Question 3 Explanation:
P: If f(x) is continuous at $x=x_{0},$ then it is also differentiable at $x=x_{0}$
Q: If f(x) is continuous at $x=x_{0},$ then it may or may not be derivable at $x=x_{0}$
R: If f(x) is differentiable at $x=x_{0}$, then it is also continuous at $x=x_{0}$
P is false
Q is true
R is true
Option (B) is correct
 Question 4
Which one of the following is a property of the solutions to the Laplace equation: $\bigtriangledown ^{2} f= 0$?
 A The solutions have neither maxima nor minima anywhere except at the boundaries. B The solutions are not separable in the coordinates. C The solutions are not continuous D The solutions are not dependent on the boundary conditions
Signals and Systems   Laplace Transform
 Question 5
Consider the plot of $f(x)$ versus $x$ as shown below. Suppose F(x)=$\int_{-5}^{x}f(y)dy$ . Which one of the following is a graph of F(x) ? A A B B C C D D
Engineering Mathematics   Calculus
Question 5 Explanation:
$F^{\prime}(x)=f(x)$ which is density function
$F^{\prime}(x)=f(x) \lt 0$ when $x \lt 0$
$\therefore \quad F(x)$ is decreasing for $x \lt 0$
$F^{\prime}(x)=f(x) \gt 0$
when $x\gt 0$
$\therefore \quad F(x)$ is increasing for $x\gt 0$.

There are 5 questions to complete.