Engineering Mathematics - Online Test
Q1. You are given three coins: one has heads on both faces, the second has tails on both faces, and the third has a head on one face and a tail on the other. You choose a coin at random and toss it, and it comes up heads. The probability that the other face is tails is
Answer : Option B
Explaination / Solution:
No Explaination.
Q2. For matrices of same dimension M , N and scalar c, which one of these properties DOES NOT ALWAYS hold ?
Answer : Option D
Explaination / Solution:
Let the matrices
i.e. the property holds always
i.e. property does not hold always.
Q3. C is a closed path in the z -plane by |z| = 3 The value of the integral 
is
is
Answer : Option C
Explaination / Solution:
Integral,
So, we have the singularity
z j + 2 = 0
z =- 2j
Since, z = -2j lies inside |z| = 3. Therefore, using cauchy’s integral, we get
Q4. The 4-point Discrete Fourier Transform (DFT) of a discrete time sequence {1,0,2,3} is
Answer : Option D
Explaination / Solution:
Q5. The Taylor series expansion of 3 sin x + 2 cos x is
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
Q6. For a function g (t), it is given that 
for any real value 𝜔. If 
is
for any real value 𝜔. If is
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
Q7. Let 
The Region of Convergence (ROC) of the z -transform of x[n].
The Region of Convergence (ROC) of the z -transform of x[n].
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
Q8. 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
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
Q9. Consider the state space model of a system, as given below
The system is
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 CM is
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.
Q10. A discrete random variable X takes values from 1 to 5 with probabilities as shown in the table. A student calculates the mean X as 3.5 and her teacher calculates the variance of X as 1.5. Which of the following statements is true ?
Answer : Option B
Explaination / Solution:
