The correct answer is option 3 i.e. cosec x.

{sin x × (1 + cos x)}/(1 + cos x - cos² x - cos³ x)

⇒  {sin x × (1 + cos x)} / (1 + cos x - cos² x - cos³ x)

⇒ {sin x (1 + cos x)}/{(1 - cos² x) (1 + cos x)}

⇒ sin x/(1 - cos² x)

⇒ sin x/ sin²x [∵ 1 - cos²A = sin²A]

⇒ 1/sin x = cosec x [∵ 1/ sin A = cosec A]

The correct answer is option 2 i.e. 48.

Given:

b + 1/b = 4

Formula used:

(a + b)³ = a³ + b³ + 3ab(a + b) ---- (1)

Calculations:

Using equation (1), we get

⇒ (b + 1/b)³ = (4)³

⇒ b³ + 1/b³ + 3 × 4 = 64

⇒ b³ + 1/b³ = 64 - 12

⇒ b³ + 1/b³ = 52

Subtracting 4 from both sides we get

⇒ b³ + 1/b³ - 4 = 52 - 4

⇒ b³ + 1/b³ - 4 = 48

The correct answer is option 1 i.e. 4.

Concept used:

If x + 1/x = √3

x³ + 1/x³ = 3√3 - 3√3

x⁶ = -1 ---- (1)

Calculations:

⇒ x²⁴ + x³⁶ + x⁶ + 3

It can be written as

⇒ (x⁶)⁴ + (x⁶)⁶ + x⁶ + 3

⇒ (-1)⁴ + (-1)⁶ + -1 + 3

⇒ 1 + 1 - 1 + 3 = 4

The correct answer is option 1 i.e. 26.67%.

The cost price and selling price of apple are the same, B first incurs a loss while purchasing and then incurs a gain while selling.

Profit % = (Selling price- Cost price)/Cost price× 100

The S.P : C.P ratio is given by

Percentage profit earned = (95 - 75)/75 × 100 = 2000/75 = 26.67%

The correct answer is option 2 i.e. 10 : 3.

Given:

30% of x is equal to 5 times of 20% of y.

Calculations:

Hence, the value of x : y = 10 : 3

The correct answer is option 3 i.e. -660.

20% of 300 - {40 ÷ 1/(2)^-1}[5.64 + 0.36]²

By using the BODMAS rule:

= (20/100) × 300 - (40 ÷ 2)[6.00]²

= 60 - 20 [36]

= 60 - 720

= -660

The correct answer is Option 3 i.e. 1.

Let the number be 8

When 8 is divided by 7, the remainder is 1

Thus,

When the cube of the number is divided by 7

⇒ (8)³/7 or 512/7

The remainder will be 1

The correct answer is option 4 i.e. 21.

Average = Sum of all weights/Total number of people

⇒ Sum of all weights = 55 × 7

⇒ 385kg

When X is removed from the group the total weight becomes,

⇒ 385 - X

When Y is added,

⇒ 385 - X + Y

New average = 58 × 7

⇒ 406

⇒ 385 - X + Y = 406

⇒ X - Y = -21

Or |-21| = 21

The correct answer is option 1 i.e. 22.5 km/hr.

If the speed of the boat in still water  = x km/hr and, the speed of the stream = y km/hr

The downstream speed = (x + y)

The upstream speed = (x - y)

The distance covered in 4 hr upstream = 60 km

The distance covered in 1.5 hr downstream = 45 km

Speed = distance/time

The relative speed in upstream = (x - y)

⇒ (x - y) = 60/4 = 15 ....(I)

The relative speed downstream = (x + y)

⇒ (x + y) = 45/1.5 = 30 ....(II)

Adding (I) and (II) we get,

⇒ 2x = (30 + 15) = 45

⇒ x = 45/2 = 22.5 km/hr

So, the speed of the boat = 22.5 km/hr