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.