Explanation

(a + bi)(a - bi) = a² - (bi)² = a² - b²i² = a² + b²  

Given: a² + b² = 13  

Last verified on 07-05-2026

Explanation

sinθ = cosθ → tanθ = 1  

θ = 45° + 180°k  

Principal value: θ = 45°

Last verified on 06-05-2026

Explanation

tan θ = 1 when θ = 45° + 180°k  

Principal value: θ = 45°  

Last verified on 06-05-2026

Explanation

Use (a - b)² = a² - 2ab + b²  

5² = 169 - 2ab  

25 = 169 - 2ab  

2ab = 169 - 25 = 144  

ab = 72  

Last verified on 06-05-2026

Explanation

x² + 2x = x(x + 2)  

Set (3x + 5) / [x(x + 2)] = A/x + B/(x + 2)  

3x + 5 = A(x + 2) + Bx = (A + B)x + 2A  

Equate coefficients:  

2A = 5 → A = 5/2  

A + B = 3 → 5/2 + B = 3 → B = 1/2  

So: 5/(2x) + 1/[2(x + 2)]  

Last verified on 06-05-2026

Explanation

f(x) = x² - 4  

f(-x) = (-x)² - 4 = x² - 4  

f(-x) - f(x) = (x² - 4) - (x² - 4) = 0  

Last verified on 06-05-2026

Explanation

Inverse variation: y = k/x  

When x = 2, y = 12  

12 = k/2 → k = 24  

Now y = 24/x  

When x = 8: y = 24/8 = 3  

Last verified on 06-05-2026

Explanation

3(x - 2) = 2(x + 3) + 1  

3x - 6 = 2x + 6 + 1  

3x - 6 = 2x + 7  

3x - 2x = 7 + 6  

x = 13  

Last verified on 06-05-2026

Explanation

(x + m)(x + n) = x² + (m + n)x + mn  

So p = m + n = 10  

q = mn = 21  

p + q = 10 + 21 = 31  

Last verified on 06-05-2026

Explanation

(a + bi)² = a² - b² + 2abi  

Purely real means imaginary part = 0  

So 2ab = 0  

Given a ≠ 0, then b = 0

Last verified on 06-05-2026