Explanation

• A proper noun is the name of a specific place, person, or thing.
• “Karachi” refers to a particular city.
• Proper nouns always start with a capital letter.

• Common noun: city, boy, river (general names)
• Abstract noun: happiness, beauty, honesty (feelings/ideas)

Last verified on 04-05-2026

Explanation

Make b the same in both ratios  

a:b = 3:4  

b:c = 2:5 → multiply by 2 → b:c = 4:10  

Now a:b:c = 3:4:10  

So a:c = 3:10

Last verified on 04-05-2026

Explanation

• In geometry, a tangent is a line that touches a circle at only one point.
• At the point of contact, it forms a right angle (90°).
• The tangent is always perpendicular to the radius of the circle.

Last verified on 04-05-2026

Explanation

Identity: 1 + tan²θ = sec²θ

Last verified on 04-05-2026

Explanation

For similar triangles: area ratio = (side ratio)²  

(side ratio)² = 9/16  

side ratio = √9/16

side ratio = 3/4

Last verified on 04-05-2026

Explanation

Right triangle: a² + b² = c²  

5² + b² = 13²  

25 + b² = 169  

b² = 169 - 25  

b² = 144  

b = 12  

Last verified on 04-05-2026

Explanation

Third side rule: difference < x < sum  

10 - 7 < x < 10 + 7  

3 < x < 17

16 → 3 < 16 < 17, possible

Last verified on 04-05-2026

Explanation

5/[x(x + 2)] = A/x + B/(x + 2)  

Multiply both sides by x(x + 2)  

5 = A(x + 2) + Bx  

Put x = 0  

5 = A(0 + 2) + B×0  

5 = 2A  

A = 5/2  

Put x = -2  

5 = A(-2 + 2) + B(-2)  

5 = -2B  

B = -5/2  

So 5/[x(x + 2)] = 5/(2x) - 5/[2(x + 2)]

Last verified on 04-05-2026

Explanation

Let y = 2x + 1  

Swap x and y  

x = 2y + 1  

Solve for y  

x - 1 = 2y  

y = (x - 1) / 2  

f⁻¹(x) = (x - 1) / 2

Last verified on 04-05-2026

Explanation

y = k × x²  

Put y = 36, x = 3  

36 = k × 3²  

36 = k × 9  

k = 4

Now y = 4 × x²  

Put x = 5  

y = 4 × 5²  

y = 4 × 25

y = 100

Last verified on 04-05-2026