Search Previous Year Questions

Given:

Height of cone = 8 cm

Radius of cone = 2 cm

Formula used:

Volume of sphere = $\frac{4}{{3}}$πR³

Volume of cone = $\frac{1}{{3}}$πr²h

Calculation:

According to the question

The volume of a sphere = Volume of a cone

⇒  $\frac{4}{{3}}$πR³ = $\frac{1}{{3}}$πr2h

⇒ $\frac{4}{{3}}$ × π × R³ = $\frac{1}{{3}}$ × π × 2 × 2 × 8

⇒ R³ = 8

⇒ R = 2 cm

∴ The radius of the sphere is 2 cm.

Given:

(81)-1 × 3-5 × 39 × (64)5/6 ×$(\sqrt[3]{3})^6=?$

Calculation:

Let the unknown number be x.

(81)-1 × 3-5 × 39 × (64)5/6 × $(\sqrt[3]{3})^6$ = x

⇒ (3⁴)-1 × 3-5 × 39 × (2⁶)5/6 × (3^1/3)⁶ = x

⇒ 3^-4× 3^-5 ×  39 × 2^(6 × 5/6) × 3 ^6/3 = x

⇒ 3-4× 3-5 ×  39 × 2(6 × 5/6) × 3² = x

⇒ 3^{(- 4 - 5 + 9 + 2)} × 2⁵ = x

⇒ 3² × 32 = x 

⇒ x = 288

∴ The correct answer is option 2 i.e. 288

Shortcut Trick

Volume of material in a hemispherical bowl = 2/3 × π × (R³ − r³)

Inner radius (r) = 5 cm, Outer radius (R) = 5 + 0.25 = 5.25 cm

Volume = 2/3 × 22/7 × (5.25³ − 5³) ≈ 2.095 × (144.703 − 125) ≈ 2.095 × 19.703

Volume ≈ 41.28 cm³

∴ The correct answer is 41.28 cm³.

Alternate Method

Given:

Inner radius (r) = 5 cm

Thickness (t) = 0.25 cm

Outer radius (R) = Inner radius + Thickness = 5 + 0.25 = 5.25 cm

Formula Used:

Volume of material = $\frac{2}{3}$ × π × (R³ − r³)

Calculations:

⇒ Volume = 2/3 × 22/7 × (5.25³ − 5³)

⇒ Volume = 44/21 × (144.703125 − 125)

⇒ Volume = 44/21 × 19.703125

⇒ Volume = 866.9375 / 21

⇒ Volume ≈ 41.2827... cm³

∴ The correct answer is 41.28 cm³.

Additional Information

Volume of Hemisphere

The volume of a solid hemisphere is given by V = 2/3 × π × r³.

Surface Area of Hemispherical Bowl

Inner Curved Surface Area = 2πr²; Outer Curved Surface Area = 2πR².

Total Surface Area of Bowl

TSA = Outer CSA + Inner CSA + Area of Ring = 2πR² + 2πr² + π(R² − r²).

Shortcut Trick

Radius (r) = 42 cm.

Curved Surface Area (CSA) of a hemisphere = 2πr².

Half of CSA = (1/2) × 2πr² = πr².

Calculation: (22/7) × 42 × 42 = 22 × 6 × 42 = 132 × 42 = 5544 cm².

∴ The correct answer is 5544 cm².

Alternate Method

Given:

Radius (r) of the hemisphere = 42 cm.

Value of π = 22/7.

Formula Used:

Curved Surface Area (CSA) of a hemisphere = 2πr²

Required Value = (1/2) × CSA = πr²

Calculations:

⇒ Required Area = π × r × r

⇒ Required Area = (22/7) × 42 × 42

⇒ Required Area = 22 × (42 ÷ 7) × 42

⇒ Required Area = 22 × 6 × 42

⇒ Required Area = 132 × 42

⇒ Required Area = 5544 cm²

∴ The correct answer is 5544 cm².

Additional Information

Total Surface Area of Hemisphere

The Total Surface Area (TSA) includes the curved part and the flat circular base: TSA = 3πr².

Volume of Hemisphere

The Volume of a hemisphere is exactly half the volume of a sphere: V = (2/3)πr³.

Sphere Properties

For a complete Sphere, the Surface Area is 4πr² and the Volume is (4/3)πr³.

Interest rate for first 2 years = 8%

Interest rate for 5/8 year = 5/8 × 8 = 5%

P = 12,000

As we know,

A = P (1 + r/100)³

⇒ A = 12,000 × (108/100) × (108/100) × (105/100)

⇒ A = 12,000 × (27/25) × (27/25) × (21/20)

⇒ A = 14697

∴ CI = 14697 – 12000 = 2697

Short Trick:

12500 unit = 12000

15309 – 12500 = 2809 unit

∴ 2809 unit = 12000/12500 × 2809 = 2697

Shortcut Trick

We know that, Profit Ratio = Ratio of Equivalent Capital (Investment × Time).

Profit of P / Profit of Q = (P's Capital × P's Time) / (Q's Capital × Q's Time)

3 / 5 = (4 × 21) / (7 × Q's Time)

3 / 5 = 12 / Q's Time ⇒ Q's Time = (12 × 5) / 3 = 20

∴ The correct answer is 20 months.

Alternate Method

Given:

Investment ratio of P : Q = 4 : 7

P's investment period = 21 months

Profit ratio of P : Q = 3 : 5

Formula Used:

Profit Ratio = (Capital of A × Time of A) : (Capital of B × Time of B)

Calculation:

Let the investments of P and Q be 4x and 7x respectively.

Let Q's investment period be t months.

⇒ Profit of P / Profit of Q = (4x × 21) / (7x × t)

⇒ 3 / 5 = 84x / 7xt

⇒ 3 / 5 = 12 / t

⇒ 3 × t = 12 × 5

⇒ 3t = 60

⇒ t = 20

∴ The correct answer is 20 months.

Additional Information

Basic Principle of Partnership

In a partnership, the profit is distributed in proportion to the product of capital invested and the time period of investment. Hence, Profit = Capital × Time.

Active and Sleeping Partners

An active partner actively manages the business and usually receives a fixed salary or a specific percentage of profit for their effort, while the remaining profit is distributed according to capital investments.

Joining Mid-Way

If a business runs for a specific period (e.g., 12 months) and a partner joins after 'x' months, their capital is considered invested only for the remaining time, i.e., (12 − x) months.

Given:

P : Q : R = 3 : 7 : 12

Q - P = Rs. X

R - Q = Rs. 30000

Solution:

Let, k = constant

R - Q = 12k - 7k = 30000

⇒k = 30000/5 = 6000

Q - P = 7k - 3k = 4k = X

X = 4 x 6000 = 24000

Total money = 3k + 7k + 12k = 22k = 22 x 6000 = Rs. 132000

Hence, the correct option is 3.

Shortcut Trick

Ratio of Father to Son Today = 3 : 1 (Difference = 2 units)

Ratio after 5 years = 2.5 : 1 = 5 : 2 (Difference = 3 units)

To equate difference, multiply Today by 3 and Future by 2:

Today = 9 : 3 | After 5 years = 10 : 4

Growth of 1 unit = 5 years. Son age today = 3 units = 3 × 5 = 15 years.

∴ The correct option is B.

Alternate Method

Given:

Let Son age today = S years

Father age today = 3S years

After 5 years: Son = S + 5 | Father = 3S + 5

Present F = 3S S = S After 5 Yrs F + 5 = 2.5(S+5)

• 5 years

Ratio change: 3:1 → 2.5:1

Calculation:

⇒ Father_future = 2.5 × Son_future

⇒ 3S + 5 = 2.5(S + 5)

⇒ 3S + 5 = 2.5S + 12.5

⇒ 3S − 2.5S = 12.5 − 5

⇒ 0.5S = 7.5

⇒ S = 7.5 ÷ 0.5

⇒ S = 15 years

∴ The correct option is B.

Additional Information

Constant Age Difference

The difference between the ages of two persons always remains constant regardless of the number of years passed.

Ratio Balancing Method

If A:B is the current ratio and C:D is the ratio after n years, equate (B−A) and (D−C) by multiplying with suitable constants to find the value of 1 unit.

Ages n years ago

If current age is X, then age n years ago was (X − n) and age n years hence will be (X + n).

Given:

A number when increased by 50% gives 2430.

Formula used:

If a number is increased by x%, the new value = original number × (1 + x/100).

Calculation:

Let the number be N.

According to the problem:

⇒ N × (1 + 50/100) = 2430

⇒ N × 1.5 = 2430

⇒ N = 2430 ÷ 1.5

⇒ N = 1620

∴ The correct answer is option (1).

Given:

SI = Rs. 862.50

Rate = 7(2/3)%

Time = 2(1/2) years

Formula Used:

SI = PRT/100

Where, P = Principal, R = Rate, T = Time(in years)

Calculation:

According to the question,

862.50 = $\frac{x\times7\frac{2}{3}\times2\frac{1}{2}}{100}$

⇒ 862.50 = $\frac{x\times23\times5}{3\times2\times100}$

⇒ x = $\frac{862.5\times3\times2}{23\times5}$ = 4500

∴ The value of x is Rs. 4500.