Engineering Mathematics - Online Test
Q1. 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
Answer : Option B
Explaination / Solution:
No Explaination.
Q2. 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
where the random
phase is uniformly distributed in the interval [0, 2π]. The auto-correlation E[X(t1) X(t2)] is
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,
Q3.
The minimum eigen value of the following matrix is
Answer : Option A
Explaination / Solution:
For, a given matrix [A] the eigen value is calculated as 
where 
gives the eigen values of matrix. Here, the minimum eigen value among the given options is 
We check the characteristic equation of matrix for this eigen value
where gives the eigen values of matrix. Here, the minimum eigen value among the given options is We check the characteristic equation of matrix for this eigen value= 3(60 - 49h- 5(25 - 14)+ 2)35 - 24h
= 33 - 55 + 22
= 0
Hence, it satisfied the characteristic equation and so, the minimum eigen value is
Q4. A polynomial 
with all coefficients positive has
with all coefficients positive has
Answer : Option D
Explaination / Solution:
Given, the polynomial 
Since, all the coefficients are positive so, the roots of equation is given by
f(x) = 0
It will have at least one pole in right hand plane as there will be least one sign change from (a1) to (a0) in the Routh matrix 1st column. Also, there will be a corresponding pole in left hand plane
i.e.; at least one positive root (in R.H.P)
and at least one negative root (in L.H.P)
Rest of the roots will be either on imaginary axis or in L.H.P
Q5. The solution of the differential equation 
is
is
Answer : Option C
Explaination / Solution:
lny = kx + A
Since y(0) = c thus lnc = A
lny = kx + lnc
Iny = Inekx + Inc
y = cekx
Q6. The differential equation 100 
describes a system with an
in
put x (t) and an output y(t). The system, which is initially relaxed, is excited by
a unit step input. The output y(t) can be represented by the waveform
describes a system with an
in
put x (t) and an output y(t). The system, which is initially relaxed, is excited by
a unit step input. The output y(t) can be represented by the waveform
Answer : Option A
Explaination / Solution:
100 
Applying Laplace transform we get
Roots are s = 1/10, 1/10 which lie on Right side of s plane thus unstable
Q7. The trigonometric Fourier series of an even function does not have the
Answer : Option C
Explaination / Solution:
For an even function Fourier series contains dc term and cosine term (even and odd harmonics).
Q8. The value of the integral 
where c is the circle |z| = 1 is given
by
where c is the circle |z| = 1 is given
by
Answer : Option A
Explaination / Solution:
C R Integrals is 
where C is circle |z| = 1
where C is circle |z| = 1So poles are outside the unit circle.
Q9. Let 
, and h(t) is a filter matched to g(t). if g(t) is applied as input to h(t) then the Fourier transform of the output is
, and h(t) is a filter matched to g(t). if g(t) is applied as input to h(t) then the Fourier transform of the output is
Answer : Option D
Explaination / Solution:
The matched filter is characterized by a frequency response that is given as
Q10.
Let U and V be two independent zero mean Gaussain random variables of variances 1/4 and 1/9 respectively. The probability
Answer : Option B
Explaination / Solution:
Given, random variables U and V with mean zero and variances 1/4 and 1/9
The distribution is shown in the figure below
We can express the distribution in standard form by assuming
The distribution is shown in the figure below
We can express the distribution in standard form by assuming
