The correct answer is option 2  i.e. 17

a² + 2(b - 1) (c - 1) = 73

b² + 2(c-1)(a-1) = 75.5

c² + 2(a - 1) (b - 1) = 78.5

Adding all equations:

(a²+b²+c²) + 2 (ab+bc+ca) - 4 (a+b+c) = 227 - 6

(a+b+c)²  - 4 (a+b+c) - 221 = 0

a+b+c = 17 (a,b,c are positive)

According to the question;

⇒ cot 13° cot 27° cot 60° cot (90° - 27°) cot (90° - 13°)

⇒ cot 13° cot 27° cot 60° cot (90° - 27°) cot (90° - 13°)

⇒ cot 13° cot 27° cot 60° tan 27° tan 13°

⇒ cot 13° × cot 27° × cot 60° × 1/cot 27° × 1/cot 13°

⇒ cot 60° = 1/√3

The correct answer is Option 1 i.e. 56.25 km/h.

60 minutes = 1 hour and 90 minutes = 1.5 hours

First distance = 80 × 1 = 80 km

Second distance = 100 × 1.5 = 150 km

Let speed for the last segment = x km/h

(80 + 150 + 4x)/(1 + 1.5 + 4) = 70

⇒ (230 + 4x)/6.5 = 70

⇒ 230 + 4x = 455

⇒ 4x = 225

⇒ x = 225/4 = 56.25 km/h

The correct answer is option 2 i.e. 1

N = 50 log₁₀N

Log₁₀N = N/50

⇒ N = (10)^N/50

N^1/N = (10)^1/50 = (10)^2/100 = (100)^1/100

∴ N = 100.

The correct answer is Option 2 i.e. 10.

X = LCM(12, 18, 20, 27)K + 2

X = 540K + 2

X = 47 × 11K + 23K + 2

X is divisible by 47 if K = 4

X = 540 × 4 + 2 = 2162

HCF(2162, 1932) = 46 = y

Sum of digits of y = 4 + 6 = 10

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

Angles in the same segment are equal.

∠BDA = ∠BCA = 50°

The correct answer is option 4 i.e. 36.76%.

Volume of sphere (V) = (4/3)πr³

New radius (R) = 1.11r

New volume = (4/3)π(R)³ = (4/3)π(1.11r)³ = (4/3)πr³ × 1.367631 = 1.367631V

Increase = (1.367631V - V) = 0.367631V

Percentage increase = (0.367631V)/V × 100 ≈ 36.76%

The correct answer is Option 3 i.e. 4.

(23¹⁰ - 1024) = (23¹⁰ - 2¹⁰)

aⁿ - bⁿ is divisible by (a - b) and (a + b) if n = even

So, (23¹⁰ - 2¹⁰) is divisible by 21 and 25

Hence, it is not divisible by 4

tan 3x + cot(2x + π/3) = 0

⇒ tan 3x = -cot(2x + π/3)

⇒ tan 3x = tan[(2x + π/3) + π/2]

⇒ tan 3x = tan(2x + 5π/6)

For tan A = tan B, the general solution is A = nπ + B, where 'n' is an integer

So, 3x = nπ + (2x + 5π/6)

⇒ 3x - 2x = nπ + 5π/6

⇒ x = nπ + 5π/6