Explanation

Using trigonometric identity:

tan²θ + 1 = sec²θ

So, tan²θ + 1 = sec²θ.

Last verified on 14-05-2026

Explanation

Given cosθ = 5/13

sinθ = √(1 − 25/169)

= √(144/169)

= 12/13

Last verified on 14-05-2026

Explanation

(7x + 5)/[x(x + 1)] = A/x + B/(x + 1)

7x + 5 = A(x + 1) + B(x)

Put x = 0: 5 = A(1) => A = 5

Put x = -1: -2 = B(-1) => B = 2

(7x + 5)/[x(x + 1)] = 5/x + 2/(x + 1)

Last verified on 13-05-2026

Explanation

f(2) = 2² + 1 = 4 + 1 = 5  

f(3) = 3² + 1 = 9 + 1 = 10  

f(2) + f(3) = 5 + 10 = 15  

Last verified on 13-05-2026

Explanation

Since y ∝ x, we have y = kx for some constant k.

When x = 4, y = 16:  

16 = k × 4 → k = 4  

Now for x = 6:  

y = 4 × 6 = 24  

Last verified 13-05-2026

Explanation

4x - 3x = 7 + 5  

x = 12

Last verified on 13-05-2026

Explanation

Roots of x² - 5x + 4 = 0:  

x² - 5x + 4 = (x - 1)(x - 4) → roots are 1 and 4  

Increase each by 1:  

New roots = 1 + 1 = 2, and 4 + 1 = 5  

New equation with roots 2 and 5:  

(x - 2)(x - 5) = x² - 7x + 10

Last verified on 13-05-2026

Explanation

Let the roots be r and r + 2, since they differ by 2.

For x² - px + 35 = 0:

Product of roots = r(r + 2) = 35  

Sum of roots = r + (r + 2) = 2r + 2 = p  

Solve r(r + 2) = 35:  

r² + 2r - 35 = 0  

(r + 7)(r - 5) = 0  

So r = 5 or r = -7  

Case 1: r = 5 → roots are 5 and 7 → p = 5 + 7 = 12  

Case 2: r = -7 → roots are -7 and -5 → p = -7 + (-5) = -12  

So p = 12 or -12

Last verified on 13-05-2026

Explanation

(a + bi)(a - bi) is a standard identity:  

(a + bi)(a - bi) = a² - (bi)² = a² - b²i² = a² - b²(-1) = a² + b²

So a² + b² = 37 directly.

Last verified on 13-05-2026

Explanation

(2x + y) + (x − y) = 11 + 1

3x = 12

x = 4

Last verified on 13-05-2026