The Taylor series expansion of 3 sin x + 2 cos x is

**A. **
**B. **
**C. **
**D. **
**Answer : ****Option A**

**Explaination / Solution: **

Given the function

2 + 3x – x^{2} – x^{3}/2
+ ….

2 - 3x + x^{2} – x^{3}/2
+ ….

2 + 3x + x^{2} + x^{3}/2
+ ….

2 + 3x + x^{2} + x^{3}/2
+ ….

Given the function

f(x) = 3 sin x + 2 cos x

Now, we have the Taylor’s expansion for the trigonometric function as

Substituting it in equation (1), we get

Workspace

Report

For a function g (t), it is given that for any real value 𝜔. If is

**A. ** 0

**B. ** -j

**C. ** -(j/2)

**D. ** j/2

**Answer : ****Option B**

**Explaination / Solution: **

Given the relations

Given the relations

The Fourier transformation of f (t) is defined as

Now, from equation (2), we have

where u (t) is unit step function. Taking Fourier transform both the sides, we have

Workspace

Report

Let The Region of Convergence (ROC) of the z -transform of x[n].

**A. ** is |z| > (1/9)

**B. ** is |z| < (1/3)

**C. ** is (1/3) > |z| > (1/9)

**D. ** does not exist

**Answer : ****Option C**

**Explaination / Solution: **

Given the discrete signal,

Given the discrete signal,

So, the z -transform of signal is obtained as

The above series converges, if

Combining the two inequalities, we get

(1/9) < |z| < (1/3)

This is the ROC of the z -transform

Workspace

Report

A system is described by the following differential equation, where u(t) is the input to the system and y(t) is the output of the system
y(t) + 5y(t) = u(t)
when y(t) = 1 and u(t) is a unit step function, y(t) is

**A. ** 0.2 + 0.8e^{-5t}

**B. ** 0.2 - 0.2e^{-5t}

**C. ** 0.8 + 0.2e^{-5t}

**D. ** 0.8 - 0.8e^{-5t}

**Answer : ****Option A**

**Explaination / Solution: **

Given the differential equation of the system

Given the differential equation of the system

y(t) + 5y(t) = u(t)

Applying Laplace transform both the sides,

We obtain the constants A and B as

Substituting there values in equation (1), we get

Taking inverse Laplace transform, we get

Workspace

Report

Consider the state space model of a system, as given below

**A. ** controllable and observable

**B. ** uncontrollable and observable

**C. ** uncontrollable and unobservable

**D. ** controllable and unobservable

**Answer : ****Option B**

**Explaination / Solution: **

Given the state-space model of system

The system is

Given the state-space model of system

In standard form, we define the state space model as

[X] = A[X] + Bu

y = C[X] + Du

Comparing it to the given space model, we have the matrix

So, we obtain the controllability matrix as

Therefore, the rank of matrix C_{M }is

Rank (C_{M}) = 2 < 3 (order of system)

Hence, the system is uncontrollable

Again, we obtain the observability matrix as

Therefore, the rank of observability matrix is

Rank (OM) = 3 = order of system

Hence, the system is observable.

Workspace

Report

Let X be a real-valued random variable with E[X] and E[X^{2}] denoting the
mean values of X and X2 , respectively. The relation which always holds true is

**A. **
**B. ** E[X2] ≥ (E[X])2

**C. ** E[X2] = (E[X])2

**D. ** E[X2] > (E[X])2

**Answer : ****Option B**

**Explaination / Solution: **

No Explaination.

(E[X])^{2} > E[X^{2}]

No Explaination.

Workspace

Report

Consider a random process where the random
phase is uniformly distributed in the interval [0, 2π]. The auto-correlation E[X(t_{1}) X(t_{2})] is

**A. ** cos(2π(t_{1} + t_{2}))

**B. ** sin(2π(t1 - t2))

**C. ** sin(2π(t1 + t2))

**D. ** cos(2π(t1 - t2))

**Answer : ****Option D**

**Explaination / Solution: **

We have the random process

We have the random process

Where random phase ϕ is uniformly distributed in the interval [0, 2π]. So, we
obtain the probability density function as

f_{ϕ}(ϕ)= 1/2π

Therefore, the auto-correlation is given as

Using the trigonometric relation,

Workspace

Report