Search Previous Year Questions

Given:

The difference between the circumference and the diameter of a circle = 105 cm.

Formula used:

Circumference of a circle = 2πr

Diameter of a circle = 2r

Calculation:

Let the diameter of the circle be D cm and radius be r cm.

⇒ D = 2r

Circumference - Diameter = 105 cm

⇒ 2πr - 2r = 105

⇒ 2 × $\dfrac{22}{7}$ ×  r - 2r = 105 

⇒ $\dfrac{44\text{r}}{7}$ - 2r = 105

⇒ $\dfrac{44\text{r} - 14\text{r}}{7}$ = 105

⇒ $\dfrac{30\text{r}}{7}$ = 105

⇒ 30r = 105 × 7 = 735

⇒ r = $\dfrac{735}{30}$ = 24.5 cm

Diameter (D) = 2r = 2 ×  24.5 = 49 cm

∴ The diameter of the circle is $49$ cm.

Given:

Radius and height ratio = 8:15

Volume of the cone = 1005$\frac{5}7$ cm³

Formula Used:

Volume of a cone = (1/3) × π × r² × h

Slant height (l) = √(r² + h²)

Calculation:

Let radius be 8k and height be 15k.

Volume = 1005$\frac{5}7$ = 1005 + 5/7 = 7040/7 cm³

Volume of the cone = (1/3) × π × (8k)² × (15k)

⇒ 7040/7 = (1/3) × π × 64k² × 15k

⇒ 7040/7 = (1/3) × π × 960k³

⇒ 7040/7 = 320πk³

⇒ k³ = 1

⇒ k = 1

Therefore, radius r = 8k = 8 × 1 = 8 cm

Height h = 15k = 15 × 1 = 15 cm

Slant height (l) = √(r² + h²)

⇒ l = √(8² + 15²)

⇒ l = √(64 + 225)

⇒ l = √289

⇒ l = 17 cm

The slant height of the cone is 17 cm.

Given:

Time to fill a cistern = 10 hours

Time to fill a cistern when there is a leak in it = 10 + 2 = 12 hours

Calculation:

When there is a no-leak,

Pipe's 1-hour work = $1\over10$

When there is a leak,

Pipe's 1-hour work = $1\over12$

Now, a leak can empty a cistern in 'x' hour then,

⇒ $1\over10$ - $1\over12$ = $1\over x$

⇒ $1\over60$ = $1\over x$

⇒ x = 60 hours

Therefore, a cistern will empty in 60 hours.

Hence, the correct answer is 60 hours.

Shortcut Trick

Let the Cost Price (CP) be 100.

Marked Price (MP) = 100 + 20% of 100 = 120.

Selling Price (SP) = 120 − 10% of 120 = 120 − 12 = 108.

Profit Percentage = SP − CP = 108 − 100 = 8%.

Alternatively, use Successive Percentage formula: a + b + (a × b) ÷ 100.

Net change = 20 − 10 + (20 × −10) ÷ 100 = 10 − 2 = 8%.

∴ The correct answer is 8%.

Alternate Method

Given: Markup% = 20%, Discount% = 10%.

Formulas: SP = CP × (100 + Markup%) ÷ 100 × (100 − Discount%) ÷ 100 | Profit% = (SP − CP) ÷ CP × 100

⇒ Let Cost Price = 100 units.

⇒ Marked Price = 100 × (120 ÷ 100) = 120 units.

⇒ Selling Price = 120 × (90 ÷ 100) = 108 units.

⇒ Profit = SP − CP = 108 − 100 = 8 units.

⇒ Profit Percentage = (8 ÷ 100) × 100 = 8%.

∴ The correct answer is 8%.

Additional Information

Marked Price and Discount

The Marked Price (MP) is the price listed on the product. Discount is always calculated on the MP to arrive at the Selling Price (SP).

Successive Percentage Change

When a value is increased by x% and then decreased by y%, the net percentage change is given by: [x − y − (xy ÷ 100)]%.

Calculation:

The given expression is:

${{5^3}\ ×\ {2^4}\ ×{\ 25^{-2}}}\over{{125} \ ×\ {5^{-4}}}$

⇒ ${{5^3}×{2^4}×{5^{4}}}\over{{125}×{25^{2}}}$

⇒ ${{5^7}×{2^4}}\over{{5^3}×{5^{4}}}$

Since, 5³ = 125 and 5⁴ = 625

⇒ ${{5^7}×{2^4}}\over{{5^7}}$

⇒ 2⁴

⇒ 16

∴ the value of the expression is 16.

Shortcut Trick

Difference in percentage = 3% − (−1%) = 4%

This 4% difference is equal to the numerical difference of 122.

⇒ 4% → 122

⇒ 1% → 122 ÷ 4 = 30.5

⇒ 100% → 30.5 × 100 = 3050

∴ The correct answer is 3050.

Alternate Method

Given:

Let the original population be P.

Increased population = P + 3% of P = 1.03P

Decreased population = P − 1% of P = 0.99P

Calculations:

According to the question, the difference between these two states is 122.

⇒ 1.03P − 0.99P = 122

⇒ 0.04P = 122

⇒ P = 122 ÷ 0.04

⇒ P = 12200 ÷ 4

⇒ P = 3050

∴ The correct answer is 3050.

Additional Information

Percentage Change

To find the net percentage change between an increase of x% and a decrease of y%, the total gap is (x + y)% if they are on opposite sides of the base value.

Base Value Concept

In population or price problems, the original value is always considered as 100% unless specified otherwise.

Successive Percentage

If a value changes by a% and then b%, the effective change is [a + b + (ab/100)]%.

Shortcut Trick

Let the numbers be 10x, 15x, and 13x.

Given: 10% of 10x = 66 ⇒ 1x = 66.

Required value: 50% of (15x − 13x) = 50% of 2x = 1x.

Since 1x = 66, the required value is 66.

∴ The correct answer is 66.

Alternate Method

Given: Ratio = 10 : 15 : 13 | 10% of 1^st number = 66

Calculations:

⇒ Let the numbers be 10x, 15x, and 13x.

⇒ 10% of 10x = (10/100) × 10x = 1x

⇒ x = 66

⇒ Difference between 2^nd and 3^rd number = 15x − 13x = 2x

⇒ 50% of the difference = (50/100) × 2x = 1x

⇒ Since x = 66, the value is 66.

∴ The correct answer is 66.

Additional Information

Ratio and Proportion

If a : b : c is the ratio, then the numbers can be expressed as ax, bx, and cx where x is a common multiplier.

Percentage Calculation

To find p% of a number N, use the formula (p/100) × N. Note that 50% is equivalent to half (1/2) of the value.

Base Value Logic

When the percentage and the ratio part result in 1 unit of 'x', that value directly represents the scaling factor for all other calculations.

Given:

Distance covered by car = 30 km

Speed of car = 60 km/h

Distance covered by cycle = 20 km

Total journey time = 3 hours

Formula used:

Time = Distance ÷ Speed

Total time = Time by car + Time by cycle

Calculation:

Time by car = Distance covered by car ÷ Speed of car

⇒ Time by car = 30 ÷ 60 = 0.5 hours

Total time = Time by car + Time by cycle

⇒ 3 = 0.5 + (Distance by cycle ÷ Speed of cycle)

⇒ 3 - 0.5 = (20 / Speed of cycle)

⇒ 2.5 = (20 / Speed of cycle)

⇒ Speed of cycle = 20 / 2.5 = 8 km/h

∴ The correct answer is option (3).

Given:

The curved surface area of a cylinder is three-fifth of its total surface area.

Formula Used:

Curved Surface Area (CSA) of a cylinder = 2πrh

Total Surface Area (TSA) of a cylinder = 2πrh + 2πr²

CSA = (3/5) TSA

Calculation:

Let the radius of the cylinder be r and the height be h.

Given, CSA = (3/5) TSA

⇒ 2πrh = (3/5)(2πrh + 2πr²)

⇒ 2πrh = (6πrh + 6πr²)/5

⇒ 10πrh = 6πrh + 6πr²

⇒ 10rh = 6rh + 6r²

⇒ 4rh = 6r²

⇒ h/r = 6/4

⇒ h/r = 3/2

The ratio between the height and radius of the cylinder is 3:2.

Shortcut Trick

In a set of consecutive numbers, the Average is always the middle term.

Average = Total Sum ÷ Number of terms = 260 ÷ 5 = 52

For any set of numbers in Arithmetic Progression, the sum of the 1^st and last terms is equal to twice the average.

Sum of first and last = 2 × Average = 2 × 52 = 104

∴ The correct answer is 104.

Alternate Method

Given: Sum of 5 consecutive numbers = 260.

Formula Used: Let the five consecutive numbers be x, (x + 1), (x + 2), (x + 3), and (x + 4).

Calculations:

⇒ Sum = x + (x + 1) + (x + 2) + (x + 3) + (x + 4) = 260

⇒ 5x + 10 = 260

⇒ 5x = 260 − 10

⇒ 5x = 250

⇒ x = 50

⇒ First number = x = 50

⇒ Last number = x + 4 = 50 + 4 = 54

⇒ Sum of first and last numbers = 50 + 54 = 104

∴ The correct answer is 104.

Additional Information

Arithmetic Progression (AP)

Consecutive numbers form an Arithmetic Progression with a common difference of 1. The sum of such a series is given by S = n ÷ 2 × (a + l), where a is the first term and l is the last term.

Middle Term Property

For an odd number of terms (n), the Average is exactly equal to the middle term. For a set of 5 numbers, the 3^rd number is the average.

Equidistant Sum Rule

In any AP, the sum of terms equidistant from the beginning and the end is constant. Thus, First + Last = Second + Second Last = 2 × Middle term.