Question 1

If z = cos θ + i sin θ, then the value of z^n + 1/z^n is:

Accepted Answer

Step 1: By De Moivre's theorem, z^n = cos(nθ) + i sin(nθ).
Step 2: Also, 1/z^n = z^(−n) = cos(nθ) − i sin(nθ)  [since 1/z = cos θ − i sin θ when |z|=1].
Step 3: Adding: z^n + 1/z^n = [cos(nθ) + i sin(nθ)] + [cos(nθ) − i sin(nθ)] = 2 cos(nθ).
Rule used: De Moivre's theorem and the property that for |z|=1, 1/z = conjugate of z.

Question 2

Let A = {1, 2, 3, 4, 5} and B = {2, 4, 6, 8}. A relation R is defined from A to B such that R = {(a, b) : a ∈ A, b ∈ B, and a divides b}. Which of the following statements is correct?

Accepted Answer

We need to find the relation R where (a, b) ∈ R if and only if a divides b (written as a | b).

Step 1: Check divisibility for each element of A with elements of B.

For a = 1: 1 divides all elements of B → (1,2), (1,4), (1,6), (1,8)
For a = 2: 2 divides 2, 4, 6, 8 → (2,2), (2,4), (2,6), (2,8)
For a = 3: 3 divides 6 only → (3,6)
For a = 4: 4 divides 4, 8 → (4,4), (4,8)
For a = 5: 5 divides none of {2,4,6,8} → no pairs

Step 2: Write the complete relation.
R = {(1,2), (1,4), (1,6), (1,8), (2,2), (2,4), (2,6), (2,8), (3,6), (4,4), (4,8)}

Step 3: Count the number of ordered pairs.
Total ordered pairs in R = 11

Step 4: Verify domain and range.
Domain of R = {1, 2, 3, 4} (all elements of A that have at least one relation to B)
Range of R = {2, 4, 6, 8} (all elements of B that are related to some element of A)

Note: The domain excludes 5 because 5 does not divide any element in B.
The range includes all elements of B because: 1|2, 2|4, 3|6, 4|8.

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