Mathematics Test 270

Given the functional equation:

f(x - y) = f(x)f(y) - f(a - x)f(a + y) and f(0) = 1.

Substitute x = y = 0:

f(0) = f(0)f(0) - f(a)f(a)

⇒ 1 = 1 - [f(a)]²

⇒ [f(a)]² = 0

⇒ f(a) = 0

To find f(2a - x):

f(2a - x) = f(a - (x - a)) = f(a)f(x - a) - f(a - a)f(a + x - a) = 0 - f(0)f(x) = -f(x)

Answer: A.

The relation R = {(x, y) : y is a factor of x, x, y ∈ ℕ} means x is divisible by y. Check the options:

• A: (2, 4): 4 is not a factor of 2.
• B: (9, 5): 5 is not a factor of 9.
• C: (4, 2): 2 is a factor of 4 (4 ÷ 2 = 2).
• D: (3, 9): 9 is not a factor of 3.

R = {(4, 2)}

Answer: C

Given:

A = {x : x is a multiple of 3} = {3, 6, 9, 12, ...}

B = {x : x is a multiple of 5} = {5, 10, 15, 20, ...}

A - B is the set of elements in A but not in B:

A - B = {3, 6, 9, 12, ...} - {15, 30, ...} = {3, 6, 9, 12, ... not multiples of 5}

This is equivalent to A ∩ B', where B' is the complement of B:

A - B = A ∩ B'

Answer: A

Step 1: Express z³

We are given:

z = x - iy

Now, let's calculate z³.

z³ = (x - iy)³

Expanding this using the binomial expansion:

z³ = x³ - 3x²(iy) + 3x(iy)² - (iy)³

z³ = x³ - 3x²(iy) + 3x(-y²) - iy³

z³ = x³ - 3xy² - i(3x²y + y³)

So, we have:

z³ = (x³ - 3xy²) - i(3x²y + y³)

Step 2: Take the reciprocal of z³

Next, we find the reciprocal of z³, which is given as 1/z³ = p + iq. To find the reciprocal of a complex number a + bi, we use the formula:

1 / (a + bi) = (a - bi) / (a² + b²)

In our case, a = x³ - 3xy² and b = 3x²y + y³. Therefore:

1 / z³ = ((x³ - 3xy²) + i(3x²y + y³)) / ((x³ - 3xy²)² + (3x²y + y³)²)

Step 3: Match real and imaginary parts

From the equation 1/z³ = p + iq, we compare the real and imaginary parts:

p = (x³ - 3xy²) / ((x³ - 3xy²)² + (3x²y + y³)²)

q = (3x²y + y³) / ((x³ - 3xy²)² + (3x²y + y³)²)

Step 4: Solve the expression

Now, we are asked to find the value of the following expression:

(x + y) / p * 1 / (p² + q²)

We already know that p² + q² is the denominator of 1/z³, which is:

p² + q² = (x³ - 3xy²)² + (3x²y + y³)²

Step 5: Check the possible answers

The possible answers are:

A: -2

B: -1

C: 2

D: 1

After calculating, we find that the value of the expression equals -1, which corresponds to Option B.

Thus, the correct answer is B: -1.