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

M^{4k+1} | |

M^{4k+2} | |

M^{4k+3} | |

M^{4k} |

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}

\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 _______

0.5 | |

1 | |

2 | |

3 |

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}

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}.

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}.

P is true, Q is false, R is false | |

P is false, Q is true, R is true | |

P is false, Q is true, R is false | |

P is true, Q is false, R is true |

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

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?

The solutions have neither maxima nor minima anywhere except at the boundaries. | |

The solutions are not separable in the coordinates. | |

The solutions are not continuous | |

The solutions are not dependent on the boundary conditions |

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) ?

Suppose F(x)=\int_{-5}^{x}f(y)dy . Which one of the following is a graph of F(x) ?

A | |

B | |

C | |

D |

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.

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.

Question 6 |

Which one of the following is an eigen function of the class of all continuous-time, linear, timeinvariant systems (u(t) denotes the unit-step function)?

e^{j\omega_{0}t }u(t) | |

cos(\omega _{0}t) | |

e^{j\omega_{0}t } | |

sin(\omega _{0}t) |

Question 6 Explanation:

If the input to the system is eigen signal output is also the same eigen signal.

Question 7 |

A continuous-time function x(t) is periodic with period T. The function is sampled uniformly with a sampling period T_{s}. In which one of the following cases is the sampled signal periodic?

T=\sqrt{2}T_{s} | |

T=1.2T_{s} | |

Always | |

Never |

Question 7 Explanation:

A signal is said to be periodic if \frac{T}{T_{s}} is a rational
number.

Here, T=1.2 T_{s}

\Rightarrow \frac{T}{T_{s}}=\frac{6}{5} \quad Which is a rational number

Here, T=1.2 T_{s}

\Rightarrow \frac{T}{T_{s}}=\frac{6}{5} \quad Which is a rational number

Question 8 |

Consider the sequence x[n]=a^{n}u[n]+b^{n}u[n] , where u[n] denotes the unit-step sequence and 0 \lt |a| \lt |b| \lt 1. The region of convergence (ROC) of the z-transform of x[n] is

|z|\gt|a| | |

|z|\gt|b| | |

|z|\lt|a| | |

|a|\lt|z|\lt|b| |

Question 8 Explanation:

\begin{array}{l} \text { Given, } x[n]=a^{n} u[n]+b^{n} u[n] \\ \text { Also given, } \quad \begin{aligned} 0 &<|a|<|b|<1 \\ & A O C=(|z|>|a|) \text { and }(|z|>|b|) \\ & R O C=|z|>|b| \end{aligned} \end{array}

Question 9 |

Consider a two-port network with the transmission matrix: T=\begin{pmatrix} A & B\\ C& D \end{pmatrix}. If the network is reciprocal, then

T^{-1}=T | |

T^{2}=T | |

Determinant (T) = 0 | |

Determinant (T) = 1 |

Question 9 Explanation:

For reciprocal network AD-BC= 1

|T|=1

|T|=1

Question 10 |

A continuous-time sinusoid of frequency 33 Hz is multiplied with a periodic Dirac impulse train of frequency 46 Hz. The resulting signal is passed through an ideal analog low-pass filter with a cutoff frequency of 23 Hz. The fundamental frequency (in Hz) of the output is _________

5 | |

12 | |

18 | |

24 |

Question 10 Explanation:

If x(t) is a message signal and y(t) is a sampled
signal, then y(t) is related to x(t) as

y(t)=x(t) \sum_{n=-\infty}^{\infty} \delta\left(t-n T_{s}\right)

r(f)=f_{s} \sum_{n=--\infty}^{\infty} X\left(f-n f_{s}\right)

Spectrum of X(f) and Y(f) are as shown

Cut off frequency of LPF = 23 Hz

Hence, frequency at the output is 13 Hz

y(t)=x(t) \sum_{n=-\infty}^{\infty} \delta\left(t-n T_{s}\right)

r(f)=f_{s} \sum_{n=--\infty}^{\infty} X\left(f-n f_{s}\right)

Spectrum of X(f) and Y(f) are as shown

Cut off frequency of LPF = 23 Hz

Hence, frequency at the output is 13 Hz

There are 10 questions to complete.