Explanation

sinθ = 5/13 → right triangle with opposite = 5, hypotenuse = 13  

Adjacent = √(13² - 5²) = √(169 - 25) = √144 = 12  

tanθ = opposite/adjacent = 5/12  

Last verified on 07-05-2026

Explanation

• If two triangles have equal heights, their areas depend only on their bases.
• Area of triangle = ½ × base × height.
• With same height, the ratio of areas equals the ratio of their bases.

Last verified on 07-05-2026

Explanation

Use (a + b)² = a² + 2ab + b²  

7² = 25 + 2ab  

49 = 25 + 2ab  

2ab = 49 - 25 = 24  

ab = 12  

Last verified on 07-05-2026

Explanation

Triangle rule: sum of any two sides must be > the third side.

7, 8, 9: 7+8 > 9, 7+9 > 8, 8+9 > 7 ✓

6, 7, 8: 6+7 > 8, 6+8 > 7, 7+8 > 6 ✓

5, 5, 6: 5+5 > 6, 5+6 > 5 ✓

5, 6, 11: 5+6 = 11, which is not > 11 ✗

Last verified on 07-06-2026

Explanation

Use partial fractions: (4x + 6)/(x(x + 3)) = A/x + B/(x + 3)  

4x + 6 = A(x + 3) + Bx = (A + B)x + 3A  

Equate:  

3A = 6 → A = 2  

A + B = 4 → 2 + B = 4 → B = 2  

So (4x + 6)/(x(x + 3)) = 2/x + 2/(x + 3)

Last verified on 07-05-2026

Explanation

f(1 - x) = (1 - x)² + 3(1 - x) = 1 - 2x + x² + 3 - 3x = x² - 5x + 4  

f(x) = x² + 3x  

f(1 - x) - f(x) = (x² - 5x + 4) - (x² + 3x) = -8x + 4 = 4(1 - 2x)  

Last verified on 07-05-2026

Explanation

y ∝ 1/x → y = k/x  

When x = 3, y = 15:  

15 = k/3 → k = 45  

So y = 45/x  

When y = 5:  

5 = 45/x → x = 45/5 = 9  

Last verified on 07-05-2026

Explanation

(x + 2)/(x - 2) = 2, x ≠ 2

x + 2 = 2(x - 2)  

x + 2 = 2x - 4  

2 + 4 = 2x - x  

6 = x  

Last verified on 07-05-2026

Explanation

Original: x² - 4x + 3 = 0  

Roots: x = 1, x = 3

Square the roots: 1² = 1, 3² = 9

New equation with roots 1 and 9:  

(x - 1)(x - 9) = x² - 10x + 9 = 0  

Last verified on 07-05-2026

Explanation

Let roots be α and β. For x² - px + 20:  

Sum = α + β = p  

Product = αβ = 20  

Given: |α - β| = 1  

Use (α + β)² = (α - β)² + 4αβ  

p² = 1² + 4(20) = 1 + 80 = 81  

p = ±9  

Last verified on 07-05-2026