Explanation

Make b the same in both ratios:  

a : b = 4 : 5 = 12 : 15

b : c = 3 : 2 = 15 : 10

So a : b : c = 12 : 15 : 10

Therefore a : c = 12 : 10 = 6 : 5

Last verified on 07-05-2026

Explanation

Mean = (3 + 5 + 7 + 9 + 11) / 5 = 35 / 5 = 7  

Last verified on 07-05-2026

Explanation

• From a point outside a circle, exactly two tangents can be drawn.
• These tangents touch the circle at two distinct points.
• Both tangents are equal in length from the external point.

Last verified on 07-06-2026

Explanation

This is a Pythagorean trig identity:  

sec²θ = 1 + tan²θ

So sec²θ - tan²θ = 1

Last verified on 07-05-2026

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