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?

**A. ** The residue of

**B. **

**C. **

**D. ** (complex conjugate of z ) is an analytical function

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

**Explaination / Solution: **

No Explaination.

No Explaination.

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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

Given the conditional density function of R as

Decision error probability that receiver decides 0 for a transmitted bit 1 is

f_{R/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

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

**A. ** [p, q , r ] = [b, a, c]

**B. ** [p, q , r ] = [b, c, a]

**C. ** [p, q , r ] = [c, a, b]

**D. ** [p, q , r ] = [c, b, a]

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

**Explaination / Solution: **

Given relation is

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

then the relationship between the sequences [p, q , r ] and [a, b, c] is

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]^{T }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]

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The solution of the differential equation is

**A. ** (2 – t) e^{t}

**B. ** (1 + 2t) e^{-t}

**C. ** (2 + t) e-t

**D. ** (1 - 2t) et

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

**Explaination / Solution: **

We have the differential equation,

We have the differential equation,

Given equation is linear constant coefficient differential equation. Let

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Which one of the following graphs describes the function

**A. **

**B. **

**C. **

**D. **

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

**Explaination / Solution: **

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

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