The correct answer is Option 4 i.e. 70.2.

Given,

The number of students = 50 

Average marks = 70

Total marks of 50 students = 50 × 70 = 3500

Incorrect marks of 3 students = 68 + 65 + 73 = 206

Correct marks of 3 students = 70 + 62 + 84 = 216

Marks required to add = 216 - 206 = 10

New total marks = 3500 + 10 = 3510 

∴ Correct average = 3510/50 = 70.2

The correct answer is Option 4 i.e. 100%.

x% of half of 84 = x/100 × 1/2 × 84 = 0.42x

y% of one fourth of 48 = y/100 × 1/4 × 48 = 0.12y

⇒ 0.42x = 7 × 0.12y

⇒ 6x = 12y

⇒ x/y = 2/1

Required percentage = (2 - 1)/1 × 100

⇒ 100%

The correct answer is Option 2 i.e. 1.

⇒ 1/(tan A tan B) + 1/(tan B tan C) + 1/(tan C tan A) = (tan A + tan B + tan C)/(tan A tan B tan C)

⇒ {tan A + tan B + tan (π – A – B)}/(tan A tan B tan C) [∵ Sum of angles in triangle is π]

⇒ {tan A + tan B – tan (A + B)}/(tan A tan B tan C)

[We know that ⇒ tan A + tan B = tan (A + B)/(1 – tan A tan B)]

⇒ {tan (A + B)/(1 – tan A tan B) – tan (A + B)}/(tan A tan B tan C)

⇒ - tan (A + B) (tan A tan B)/(tan A tan B tan C)

⇒ (tan A tan B tan C)/(tan A tan B tan C) = 1 [∵ tan (A + B) = tan (π – C) = - tan C]

The correct answer is option 2 i.e. 33.6.

Let the length of the other diagonal be x cm

The Area of rhombus = 1/2 × d₁d₂

⇒ 1176 = 1/2 × 56 × x

⇒ 1176 = 28x

⇒ x = 42 cm

Its diagonals bisect each other at 90°. So by using the Pythagoras theorem

Side² = Sum of squares of lengths of semi-diagonals

⇒ Side² = (56/2)² + (42/2)²

⇒ Side² = 1225

⇒ Side = 35 cm

Area = Side × Height

⇒ 1176 = 35 × Height

⇒ Height = 33.6 cm

The correct answer is option 2 i.e. c³ - a³ - b³.

(x + y)³ = x³ + y³ + 3xy(x + y)

If a + b = c ..........(I)

a + b - c = 0

So, (a + b - c)³ = 0

Cubing eq. (I) On both sides we get 

(a + b)³ = c³

⇒ a³ + b³ + 3ab(a + b) = c³

⇒ 3abc = c³ - a³ - b³

So,

(a + b - c)³ + 3abc

= 0 + c³ - a³ - b³ 

= c³ - a³ - b³

The correct answer is Option 3 i.e. 230 m.

Speed of faster train = 58 km/hr

Speed of slower train = 49 km/hr

Relative speed (same direction) = 58 - 49 = 9 km/hr = 9 × (5/18) = 2.5 m/s

Time = 3 minutes = 3 × 60 = 180 seconds.

Distance travelled = Sum of lengths of both trains = Relative speed × time

Let length of trains be x and x + 10.

⇒ x + x + 10 = 2.5 × 180 

⇒ 2x + 10 = 450

⇒ 2x = 440

⇒ x = 220

Length of longer train = x + 10 = 220 + 10 = 230 m

The correct answer is option 3 i.e. 88.5%.

Total number of youth addicted to drugs in states A and E

= 12 + 15 = 27 lakhs

And

Total number of youth inclined towards Yoga in states C and D

= 18 + 12.5 = 30.5 lakhs

Hence,

Required percentage = [27/30.5] × 100 = 88.5% (Approx.)

The correct answer is option 1 i.e. 24 cm.

TS × TR = TP × TQ

⇒ 16 × 12 = TP × 8

⇒ TP = (16 × 12)/8

⇒ TP = (2 × 12) cm = 24 cm

Hence, the length of TP is 24 cm

The correct answer is option 1 i.e. Rs. 2500.

Let the amount lent at an interest rate of 8% be Rs. x

Amount lent at 10% interest rate = Rs. (5000 - x)

Simple Interest = (P × R × T)/100

Where, P = Principal , R = Rate of interest, T = time period in years

Simple Interest from amount lent at 8% = x × 8 × 1/100 = 0.08x

Simple Interest from amount lent at 10% = (5000 - x) × 10 × 1/100 = 500 - 0.1x

According to the question -

⇒ 0.08x + 500 - 0.1x = 450

⇒ 0.02x = 50

⇒ x = 50/0.02

⇒ x = Rs. 2500

The correct answer is option 1 i.e. 160.

Total apples = 450

⇒ Number of apples with Sita = [450/(27 + 11 + 7)] × 27

⇒ 450 × 27/45

⇒ 270

⇒ Number of apples with Reeta = [450/(27 + 11 + 7)] × 11

⇒ 450 × 11/45

⇒ 110

Required number of apples = 270 - 110 = 160