EC GATE 2014 PAPER 01 (Test 4)



Tag: ec gate 2014 paper 01
Q.1
The Taylor series expansion of 3 sin x + 2 cos x is
A.

2 + 3x – x2 – x3/2 + ….

B.

2 - 3x + x2 – x3/2 + ….

C.

2 + 3x + x2 + x3/2 + ….

D.

2 + 3x + x2 + x3/2 + ….

Answer : Option A
Explaination / Solution:

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


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Q.2
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

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


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Q.3
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,

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

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Q.4
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
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


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Q.5
Consider the state space model of a system, as given below
 
The system is
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

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 Cis
Rank (CM) = 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.

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Q.6
Let X be a real-valued random variable with E[X] and E[X2] denoting the mean values of X and X2 , respectively. The relation which always holds true is
A.

(E[X])2 > E[X2]

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.


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Q.7
Consider a random process   where the random phase  is uniformly distributed in the interval [0, 2π]. The auto-correlation E[X(t1) X(t2)] is
A. cos(2π(t1 + t2)) 
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

Where random phase ϕ is uniformly distributed in the interval  [0, 2π]. So, we obtain the probability density function as
fϕ(ϕ)= 1/2π
Therefore, the auto-correlation is given as


Using the trigonometric relation,


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