Quantitative Aptitude PYQs

Given:

Chord AB = radius

PA and PB are tangents

Formula Used:

Chord = 2 × r × sin(θ ÷ 2)

Angle between tangents = 180° − central angle

Calculation:

⇒ r = 2 × r × sin(θ ÷ 2)

⇒ 1 = 2 × sin(θ ÷ 2)

⇒ sin(θ ÷ 2) = 1 ÷ 2

⇒ θ ÷ 2 = 30°

⇒ θ = 60°

⇒ ∠APB = 180 − 60 = 120°

∴ The correct answer is 120°.

Given:

Radius = 6 cm

Distance from center = 10 cm

Formula Used:

Length of tangent = √(distance² − radius²)

Calculation:

⇒ Length = √(10² − 6²)

⇒ Length = √(100 − 36)

⇒ Length = √64

⇒ Length = 8 cm

∴ The correct answer is 8 cm.

Shortcut Trick

Test the given points in the options to find the correct linear equation.

For the point (1, 2): y = 2(1) = 2. This satisfies the equation y = 2x.

For the point (3, 6): y = 2(3) = 6. This also satisfies the equation y = 2x.

Since both coordinates satisfy the relation, the equation must be y = 2x.

∴ The correct answer is y = 2x.

Alternate Method

Given: Point 1 (x₁, y₁) = (1, 2) and Point 2 (x₂, y₂) = (3, 6).

Formula Used:

Slope (m) = (y₂ − y₁) / (x₂ − x₁)

Equation of line: (y − y₁) = m(x − x₁)

⇒ Slope (m) = (6 − 2) / (3 − 1)

⇒ m = 4 / 2 = 2

⇒ Equation: (y − 2) = 2(x − 1)

⇒ y − 2 = 2x − 2

⇒ y = 2x

∴ The correct answer is y = 2x.

Additional Information

Slope-Intercept Form

The standard equation of a line is y = mx + c, where m is the slope and c is the y-intercept.

Intercept Form

If a line cuts the x-axis at 'a' and the y-axis at 'b', its equation is (x/a) + (y/b) = 1.

Parallel Lines

Two parallel lines have the same slope (m₁ = m₂).

Perpendicular Lines

The product of slopes of two perpendicular lines is −1 (m₁ × m₂ = −1).

Given:

Central angle = 60°

Radius = 7 cm

Formula Used:

Chord length = 2 × r × sin(θ ÷ 2)

Calculation:

⇒ Length = 2 × 7 × sin(60 ÷ 2)

⇒ Length = 14 × sin30

⇒ Length = 14 × (1 ÷ 2)

⇒ Length = 7 cm

∴ The correct answer is 7 cm.

Shortcut Trick

Old Selling Price (105%) = ₹4200

Unit value (1%) = 4200 ÷ 105 = 40

New Selling Price required (115%) = 40 × 115

Required SP = ₹4600

∴ The correct answer is ₹4600.

Alternate Method

Given:

First Selling Price (SP₁) = ₹4200

Profit Percentage (P₁) = 5%

Desired Profit Percentage (P₂) = 15%

Formula Used:

Cost Price (CP) = [SP × 100] ÷ [100 + Profit %]

New Selling Price = CP × [(100 + New Profit %) ÷ 100]

Calculations:

⇒ Cost Price = (4200 × 100) ÷ (100 + 5)

⇒ Cost Price = 420000 ÷ 105

⇒ Cost Price = ₹4000

⇒ New Selling Price = 4000 × (115 ÷ 100)

⇒ New Selling Price = 40 × 115

⇒ New Selling Price = ₹4600

∴ The correct answer is ₹4600.

Additional Information

Profit and Loss Calculation Base

Profit and Loss percentages are always calculated on the Cost Price (CP) of the article unless the question specifies calculation on Selling Price.

Direct SP Relation

When multiple selling prices are involved for the same article, the ratio of Selling Prices is equal to the ratio of their respective percentage values: SP₁ ÷ (100 + P₁) = SP₂ ÷ (100 + P₂).

Marked Price and Discount

Selling Price can also be calculated as Marked Price × (100 − Discount%) ÷ 100.

Given:

(a + b)² = 121

ab = 10

Formula Used:

(a + b)² = a² + b² + 2ab

Calculation:

⇒ (a + b)² = a² + b² + 2ab

⇒ 121 = a² + b² + 2(10)

⇒ 121 = a² + b² + 20

⇒ a² + b² = 121 - 20

⇒ a² + b² = 101

∴ The correct answer is a² + b² = 101.

Given:

A investment = 30000 for 12 months

B investment = 40000 for (12 - x) months

Profit ratio = 3 : 2

Formula Used:

Profit ∝ Investment × Time

Calculation:

⇒ A = 30000 × 12 = 360000

⇒ B = 40000 × (12 - x)

⇒ 360000 : 40000(12 - x) = 3 : 2

⇒ 360000 ÷ [40000(12 - x)] = 3 ÷ 2

⇒ 360000 × 2 = 3 × 40000(12 - x)

⇒ 720000 = 120000(12 - x)

⇒ 6 = 12 - x

⇒ x = 6

∴ The correct answer is 6.

Given:

sec²θ = 4

0° < θ < 90°

Formula Used:

sec²θ = 1 ÷ cos²θ

sin²θ = 1 − cos²θ

cosec(90 − θ) = secθ

Calculation:

⇒ 1 ÷ cos²θ = 4

⇒ cos²θ = 1 ÷ 4

⇒ sin²θ = 1 − (1 ÷ 4) = 3 ÷ 4

⇒ secθ = 2

⇒ Required value = (3 ÷ 4) + 2

⇒ = 3 ÷ 4 + 8 ÷ 4 = 11 ÷ 4

∴ The correct answer is 11 ÷ 4.

Given:

Amount = 39200

Rate = 12%

Time = 2 years

Formula Used:

Amount = Principal × (1 + R ÷ 100)ⁿ

Calculation:

⇒ 39200 = P × (1 + 12 ÷ 100)²

⇒ 39200 = P × (112 ÷ 100)²

⇒ 39200 = P × (12544 ÷ 10000)

⇒ P = 39200 × (10000 ÷ 12544)

⇒ P = 31250

∴ The correct answer is 31250.

Given:

∠ACD = 119°

∠BAC = (4 ÷ 3) × ∠ABC

Formula Used:

Exterior angle = Sum of opposite interior angles

Calculation:

⇒ ∠ACD = ∠BAC + ∠ABC

⇒ 119 = (4 ÷ 3 × ∠ABC) + ∠ABC

⇒ 119 = (4 ÷ 3 + 1) × ∠ABC

⇒ 119 = (7 ÷ 3) × ∠ABC

⇒ ∠ABC = 119 × (3 ÷ 7) = 51°

⇒ ∠BAC = (4 ÷ 3) × 51 = 68°

∴ The correct answer is 68°.

Given:

Quantity removed each time = 3 litres

Operation repeated = 3 times

Final ratio (wine : water) = 8 : 19

Formula Used:

Remaining wine = Initial × (1 − removed ÷ total)ⁿ

Calculation:

⇒ Let initial quantity = x

⇒ Remaining wine = x × (1 − 3 ÷ x)³

⇒ Wine fraction = 8 ÷ (8 + 19) = 8 ÷ 27

⇒ x × (1 − 3 ÷ x)³ = (8 ÷ 27) × x

⇒ (1 − 3 ÷ x)³ = 8 ÷ 27

⇒ 1 − 3 ÷ x = 2 ÷ 3

⇒ 3 ÷ x = 1 ÷ 3

⇒ x = 9

∴ The correct answer is 9 litres.

Given:

tanθ + cotθ = 2

θ is acute

Formula Used:

(a + b)³ = a³ + b³ + 3ab(a + b)

tanθ × cotθ = 1

Calculation:

⇒ (tanθ + cotθ)³ = tan³θ + cot³θ + 3 × tanθ × cotθ × (tanθ + cotθ)

⇒ 2³ = tan³θ + cot³θ + 3 × 1 × 2

⇒ 8 = tan³θ + cot³θ + 6

⇒ tan³θ + cot³θ = 2

∴ The correct answer is 2.

Given:

Speed of A = 35 m/s

Speed of B = 21 m/s

Track length = 3500 m

Formula Used:

Relative speed (same direction) = Difference of speeds

Time = Distance ÷ Relative speed

Calculation:

⇒ Relative speed = 35 − 21 = 14 m/s

⇒ Time = 3500 ÷ 14

⇒ Time = 250 seconds

∴ The correct answer is 250 seconds.

Given:

Expression = [(√5 + √2)² − (√5 − √2)²] ÷ (√5 × √2)

Formula Used:

(a + b)² − (a − b)² = 4ab

Calculation:

 [(√5 + √2)2 − (√5 − √2)2] ÷ (√5 × √2)

⇒ 4 × √5 × √2 ÷ (√5 × √2)

⇒ 4

∴ The correct answer is 4.

Given:

Parallel sides = 8.4 cm and 10.8 cm

Height = 7.5 cm

Formula Used:

Area of trapezium = (1 ÷ 2) × (sum of parallel sides) × height

Calculation:

⇒ Sum of parallel sides = 8.4 + 10.8 = 19.2

⇒ Area = (1 ÷ 2) × 19.2 × 7.5

⇒ Area = 9.6 × 7.5

⇒ Area = 72 cm²

∴ The correct answer is 72 cm².

Shortcut Trick

In a square, the Side = Diagonal ÷ √2.

Side = 2√34 ÷ √2 = 2√(34/2) = 2√17 cm.

Area of square = (Diagonal)² ÷ 2 = (2√34)² ÷ 2 = (4 × 34) ÷ 2 = 68 cm².

∴ The correct answer is 2√17 cm and 68 cm².

Alternate Method

Given:

Diagonal of the square (d) = 2√34 cm.

Formula Used:

1. Diagonal (d) = Side (a) × √2
2. Area (A) = Side² = a²

Calculations:

⇒ Let the side of the square be 'a'.

⇒ d = a × √2

⇒ 2√34 = a × √2

⇒ a = (2√34) / √2

⇒ a = 2 × √(34 / 2)

⇒ a = 2√17 cm

⇒ Area of square = a²

⇒ Area = (2√17)²

⇒ Area = 4 × 17

⇒ Area = 68 cm²

∴ The correct answer is 2√17 cm and 68 cm².

Additional Information

Relationship between Area and Diagonal

The Area of a square can be calculated directly from the diagonal (d) using the formula: Area = d² / 2.

Pythagoras Theorem in Square

Since each angle in a square is 90°, the diagonal forms a right-angled triangle where a² + a² = d², leading to d = a√2.

Perimeter of Square

The total boundary length or Perimeter is given by 4 × Side (4a).

Shortcut Trick

Assume P = 6 units.

Q = 2.5 × 6 = 15 units and S = 60% of 15 = 9 units.

Since R is the average of P, Q, R, S ⇒ 3R = P + Q + S = 6 + 15 + 9 = 30 units ⇒ R = 10 units.

Given P + R = 6 + 10 = 16 units = ₹4800 ⇒ 1 unit = ₹300.

Amount of R = 10 units × 300 = ₹3000.

∴ The correct answer is Rs. 3000.

Alternate Method

Given:

Q = 2.5P | S = Q − 40% of Q = 0.6Q | R = (P + Q + R + S) ÷ 4 | P + R = Rs. 4800

Calculations:

⇒ Expressing all in terms of P:

⇒ Q = 2.5P

⇒ S = 0.6 × 2.5P = 1.5P

⇒ R = (P + 2.5P + R + 1.5P) ÷ 4

⇒ 4R = 5P + R

⇒ 3R = 5P ⇒ P = (3 ÷ 5)R = 0.6R

⇒ Given P + R = 4800

⇒ 0.6R + R = 4800

⇒ 1.6R = 4800

⇒ R = 4800 ÷ 1.6 = 3000

∴ The correct answer is Rs. 3000.

Additional Information

Average Property

If one number in a set is equal to the average of all numbers in the set, then that number is also equal to the average of the remaining numbers.

Percentage Change

To find a value after an x% decrease, multiply the initial value by (1 − x/100). For a 40% decrease, multiplying factor is 0.6.

Ratio Balancing

Always relate all variables to a single common variable (like P or R) to simplify multi-variable linear equations quickly.

Given:

Selling price of 15 kg sugar = 450

Profit = Selling price of 3 kg sugar

Formula Used:

Profit percentage = (Profit ÷ Cost price) × 100

Cost price = Selling price - Profit

Calculation:

⇒ Selling price of 1 kg sugar = 450 ÷ 15

⇒ Selling price of 1 kg sugar = 30

⇒ Profit = Selling price of 3 kg sugar = 3 × 30

⇒ Profit = 90

⇒ Cost price = Selling price - Profit

⇒ Cost price = 450 - 90

⇒ Cost price = 360

⇒ Profit percentage = (Profit ÷ Cost price) × 100

⇒ Profit percentage = (90 ÷ 360) × 100

⇒ Profit percentage = 25%

∴ The correct answer is option 2.

Given:

Gain = 25%

Loss = 25%

Formula Used:

Net loss % = r² ÷ 100

Calculation:

⇒ r = 25

⇒ Net loss % = 25² ÷ 100

⇒ Net loss % = 625 ÷ 100

⇒ Net loss % = 6.25%

∴ The correct answer is 6.25%.

Given:

Initial ratio = 2 : 3 : 4

Increase = 50%, 40%, 30%

Formula Used:

New value = Original × (1 + increase/100)

Calculation:

⇒ 2 × 100 = 200

⇒ 3 × 100 = 300

⇒ 4 × 100 = 400

⇒ 200 × 150 ÷ 100 = 300

⇒ 300 × 140 ÷ 100 = 420

⇒ 400 × 130 ÷ 100 = 520

⇒ Ratio = 300 : 420 : 520

⇒ Divide by 20

⇒ Ratio = 15 : 21 : 26

∴ The correct answer is 15 : 21 : 26.