Explanation

(a + bi)(a - bi) is the product of conjugates.  

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

Since i² = -1:  

= a² - b²(-1) = a² + b²  

Given (a + bi)(a - bi) = 61,  

so a² + b² = 61.

Last verified on 24-05-2026

Explanation

Given:  

a/b = 9/10  

b/c = 5/6  

Multiply both equations:  

(a/b) × (b/c) = (9/10) × (5/6)  

a/c = 45/60  

a/c = 3/4  

So a:c = 3:4.

Last verified on 23-05-2026

Explanation

Mean = (Sum of numbers) / (Total numbers)  

= (10 + 12 + 14) / 3  

= 36 / 3  

= 12  

Last verified on 23-05-2026

Explanation

• A secant is a line that cuts a circle and intersects it at exactly 2 distinct points.
• If it touches at 1 point, it’s a tangent.

Last verified on 23-05-2026

Explanation

cotθ = 1/tanθ  

So:  

cotθ × tanθ = (1/tanθ) × tanθ = 1  

Last verified on 23-05-2026

Explanation

Given:  

cos θ = 9/41  

Use the formula:  

sin²θ + cos²θ = 1  

Step 1: Put cos θ value  

sin²θ + (9/41)² = 1  

sin²θ + 81/1681 = 1  

Step 2: Find sin²θ  

sin²θ = 1 - 81/1681  

sin²θ = 1681/1681 - 81/1681  

sin²θ = 1600/1681  

Step 3: Take square root  

sin θ = ±√(1600/1681)  

sin θ = ±40/41  

Last verified on 23-05-2026

Explanation

Area = base × height  

Let original base = b, height = h  

Original area = b × h  

New base = b + 50% of b = b + 0.5b = 1.5b  

Height stays same = h  

New area = 1.5b × h = 1.5 × (b × h)  

So the area becomes 1.5 times the original, or increases by 50%.

Last verified on 23-05-2026

Explanation

Given:  

a² + b² = 100  

a = 6  

Step 1: Put a = 6 in the equation  

6² + b² = 100  

36 + b² = 100  

Step 2: Solve for b²  

b² = 100 - 36

b² = 64  

Step 3: Take square root  

b = ±√64  

b = ±8  

So b = 8 or b = -8

Last verified on 23-05-2026

Explanation

(9x + 4) / [x(x + 4)] = A/x + B/(x + 4)

Multiply both sides by x(x + 4):

9x + 4 = A(x + 4) + B x

9x + 4 = A x + 4A + B x

9x + 4 = (A + B)x + 4A

Compare both sides:

4A = 4 => A = 1

A + B = 9 => 1 + B = 9 => B = 8

Therefore:

(9x + 4) / [x(x + 4)] = 1/x + 8/(x + 4)

Last verified on 23-05-2026

Explanation

f(x) = x² - 3

Find f(4):  

f(4) = 4² - 3 = 16 - 3 = 13

Find f(2):  

f(2) = 2² - 3 = 4 - 3 = 1

Now f(4) - f(2) = 13 - 1 = 12

Last verified on 23-05-2026