Engineering Mathematics - Online Test

Q1. The rank of the matrix M =  is
Answer : Option C
Explaination / Solution:



Q2. A periodic signal x(t) has a trigonometric Fourier series expansion  
If  we can conclude that
Answer : Option A
Explaination / Solution:

If x(t) = -x(-t) the given periodic signal is odd symmetric. For an odd symmetric signal an = 0 for all n.
If x(t) = -x(t- π/𝜔0),  π/𝜔T0/2 where T0 is fundamental period then the given condition satisfies half-wave symmetry.
For half-wave symmetrical signal all coefficients an and bn are zero for even value of n.

Q3. Which one of the following is the general solution of the first order differential equation  where x, y are real?

Answer : Option D
Explaination / Solution:




Q4. Let  for real x. From among the following, choose the Taylor series approximation of f(x) around x = 0, which included all powers of x less than or equal to 3.
Answer : Option C
Explaination / Solution:



Q5. Suppose A and B are two independent events with probabilities P(A) ≠ 0 and P(B) ≠ 0. Let  and   be their complements. Which one of the following statements is FALSE?
Answer : Option C
Explaination / Solution:

A and B are two independent events with probabilities,  P(A) ≠ 0 and P(B) ≠ 0
Now, we check the given options.
                                             True
 True
              False
                                               True

Q6. Let z = x + iy be a complex variable. Consider that contour integration is performed along the unit circle in anticlockwise direction. Which one of the following statements is NOT TRUE?
Answer : Option D
Explaination / Solution:
No Explaination.


Q7.
A source emits bit 0 with probability 1/3 and bit 1 with probability 2/3. The emitted bits are communicated to the receiver. The receiver decides for either 0 or 1 based on the received value R. It is given that the conditional density functions of R are as

The minimum decision error probability is
Answer : Option D
Explaination / Solution:

Given the conditional density function of R as


Decision error probability that receiver decides 0 for a transmitted bit 1 is
fR/1 (r = 0) = 1/6
Again, the decision error probability that receiver decides 1 for a transmitted bit 0 is
fR/0 (r = 1) = 1/4
Hence, the minimum decision error probability is
fR/1 (r = 0) = 1/6


Q8. Two sequences [a, b, c] and [A, B, C ] are related as.

If another sequence [p, q , r ] is derived as,

then the relationship between the sequences [p, q , r ] and [a, b, c] is
Answer : Option C
Explaination / Solution:

Given relation is

Comparing it with the DFT concept of taking fourier transform by matrix form. We may calculate that here we are taking the 3 order DFT of [a b c]whose
transformed output is [A B C]T . So,

Again, we consider the equation (1),

Since, the relation between cube roots of unity is given as

So, we solve the matrix equation as

Again, we consider the equation (2),

In above equation, we apply elementary row operation as


Hence, we can conclude that
[p q r] =[c a b]


Q9. The solution of the differential equation  is
Answer : Option B
Explaination / Solution:

We have the differential equation,

Given equation is linear constant coefficient differential equation. Let


Q10. Which one of the following graphs describes the function 
Answer : Option B
Explaination / Solution:


This condition is satisfied by the graph shown in option (B).