Explanation

Given y = 18 when x = 3:

18 = k(3²)

18 = 9k

k = 2

Now for x = 5:

y = 2(5²)

y = 2 × 25

y = 50

Last verified on 15-05-2026

Explanation

Find f(3):

f(3) = 3² − 2(3)

= 9 − 6

= 3

Find f(1):

f(1) = 1² − 2(1)

= 1 − 2

= −1

Now add:

f(3) + f(1) = 3 + (−1) = 2

Last verified on 15-05-2026

Explanation

(x - 3)/2 = (x + 1)/4

Multiply both sides by 4 to remove denominators:

4 × (x - 3)/2 = 4 × (x + 1)/4

2(x - 3) = x + 1

Expand:

2x - 6 = x + 1

Bring x terms to one side, numbers to other:

2x - x = 1 + 6

x = 7

Last verified on 15-05-2026

Explanation

Given: x² - 7x + 10 = 0

Step 1: Find original roots

x² - 7x + 10 = (x - 2)(x - 5) = 0

So roots are 2 and 5

Step 2: Multiply roots by 3

New roots = 2 × 3 = 6, 5 × 3 = 15

Step 3: Form new equation

If roots are α and β, equation is x² - (α+β)x + αβ = 0

Here α+β = 6 + 15 = 21

     αβ = 6 × 15 = 90

Therefore new equation = x² - 21x + 90 = 0

Last verified on 15-05-2026

Explanation

Let the two factors be (x - a) and (x - b)

Then:

1. (x - a)(x - b) = x^2 - (a + b)x + ab

2. Compare with x^2 - px + 48:

   ab = 48

   a + b = p

3. Given: a - b = 2 => a = b + 2

Substitute:

(b + 2)b = 48

b^2 + 2b - 48 = 0

(b + 8)(b - 6) = 0

So b = 6 or b = -8

If b = 6, then a = 8, p = a + b = 14

If b = -8, then a = -6, p = a + b = -14

Last verified on 15-05-2026

Explanation

(a + bi)(a − bi) = a² + b²

Given:

(a + bi)(a − bi) = 41

Therefore:

a² + b² = 41

Last verified on 15-05-2026

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