The correct answer is option 1 i.e. 480 cm².

The perpendicular line from the vertex to the unequal side bisects the side

Applying Pythagoras' theorem in the triangle,

⇒ AD² = √(34² - 30²)

⇒ AD² = √(256) = 16 cm

Unequal side of the triangle = (16 × 2) = 32 cm

Area of a triangle = (1/2 × b × h)

⇒ (1/2) × 32 × 30

⇒ 480 cm²

The correct answer is option 4 i.e. Rs 20328.

Area of circular park = πr²

⇒ 5544 = πr²

⇒ r² = 5544/π = 1764 (where  π = 22/7)

⇒ r = √1764 = 42 m

The path length is 7 m (given)

Inner circle area (without including path area)

Radius of inner circle = (42 - 7) m = 35 m

Area₂ = 22/7 × 35 × 35 = 3850 m²

Total area of path = Area₁ - Area₂

= (5544 - 3850) m²

= 1694 m²

Total cost for paving = 1694 × 12 

= Rs 20328

The correct answer is Option 1 i.e. 2592π cm².

let the radius of the cylinder = 4x

Let the height of the cylinder  = 5x

Now 5x – 4x = 6, x = 6

∴ radius  = 4 × 6 = 24cm

Height  = 5 × 6 = 30 cm

Total surface area of cylinder  = 2π r(r + h)

=2 × π × 24 (24 + 30)

=π × 48 × 54 = 2592π cm²

The correct answer is Option 4 i.e. (1 – cos 8x)/8.

1 + cos 4x = 2 cos² 2x [∵ 1 + cos 2A = 2 cos² A]

2 sin² x cos² x (1 + cos 4x) = 2 sin² x cos² x × 2 cos² 2x

⇒ 4 sin² x cos² × cos² 2x

⇒ (2 sin x cos x)² × cos² 2x

⇒ sin² 2x × cos² 2x [∵ 2 sin A cos A = sin 2A]

⇒ (2 sin²2x cos²2x)²/4

⇒ sin² 4x/4

⇒ (1 – cos 8x)/8 [∵ sin2 A = (1 – cos 2A)/2]

The correct answer is option 3 i.e. 50.

H.C.F. of the two numbers = 50

L.C.M. of the two numbers = 300

Let the two numbers be a and b.

LCM × HCF = Product of numbers

300 × 50 = 15000

ab = 15000

a + b = 250

(a - b)² = (a + b)² - 4ab

⇒ (a - b)² = (250)² - 4 × 15000

⇒ (a - b)² = 62500 - 60000 = 2500

a - b = 50

The correct answer is option 3 i.e. 16.

Let the H.C.F. of those two numbers be 'k'

The first number would be = N1k

The second number would be = N2k

⇒ N1k + N2k = 128

⇒ k(N1 + N2) = 128   (1)

⇒ N1N2k = 240

⇒ N1N2k = 24 × 3 × 5   (2)

By (1) and (2)

⇒ k = 16

The correct answer is option 1 i.e. 14.

Given:

x + 1/(x - 4) = 8

Formula used:

a² + b² + 2ab = (a + b)² ---- (1)

Calculations:

⇒ x + 1/(x - 4) = 8

Subtracting 4 from both sides, we get

⇒ (x - 4) + 1/(x - 4) = 8 - 4

⇒ (x - 4) + 1/(x - 4) = 4

Using equation (1), we get

⇒ (x - 4)² + 1/(x - 4)² + 2 = 4²

⇒ (x - 4)² + 1/(x - 4)² = 16 - 2 = 14

The correct answer is option 3 i.e. 2756.

Given:

a = 53 and b = 13

Formula used:

(a + b)² - (a - b)² = 4ab ---- (1)

Calculations:

Using equation (1), we get

⇒ (a + b)² - (a - b)² = 4ab

⇒ 4 × 53 × 13 = 2756