Quantitative Aptitude PYQs

Shortcut Trick

For two-digit numbers ab and cd, the property (ab)² + (cd)² = (ba)² + (dc)² holds true if a² + c² = b² + d².

In the given expression 14² + 87²: a = 1, b = 4, c = 8, d = 7.

Check the property: 1² + 8² = 1 + 64 = 65 and 4² + 7² = 16 + 49 = 65.

Since the sums are equal, the reversal of digits will yield the same value: 14 → 41 and 87 → 78.

∴ The correct answer is 41² + 78².

Alternate Method

Given:

Expression = 14² + 87²

Calculations:

⇒ 14² = 196

⇒ 87² = (90 − 3)² = 8100 − 540 + 9 = 7569

⇒ Value = 196 + 7569 = 7765

Evaluating Option (2):

⇒ 41² + 78²

⇒ 41² = 1681

⇒ 78² = (80 − 2)² = 6400 − 320 + 4 = 6084

⇒ Value = 1681 + 6084 = 7765

⇒ Since both sums are equal to 7765, the expressions are identical.

∴ The correct answer is 41² + 78².

Additional Information

Digit Reversal Square Property

A sum of two squares equals the sum of the squares of their reversed digits only when the sum of the squares of the first digits equals the sum of the squares of the second digits.

Algebraic Identity for Squaring

Use (a ± b)² = a² ± 2ab + b² for quick calculation of squares near base values like 10, 50, or 100.

Numbers ending in 1

To square a number ending in 1 (like 41), use (x+1)² = x² + 2x + 1. For example, 41² = 40² + 2(40) + 1 = 1600 + 80 + 1 = 1681.

Given:

Total alloy = 40 kg

Initial Zinc percentage = 10%

Required Zinc percentage = 20%

Formula Used:

Quantity of element = Total mixture × Percentage ÷ 100

Calculation:

⇒ Initial Zinc = 40 × 10 ÷ 100

⇒ Initial Zinc = 4

⇒ Let Zinc added = x

⇒ New total mixture = 40 + x

⇒ Total Zinc = 4 + x

⇒ \( \frac{4 + x}{40 + x} = \frac{20}{100} \)

⇒ \( \frac{4 + x}{40 + x} = \frac{1}{5} \)

⇒ \( 5(4 + x) = 40 + x \)

⇒ \( 20 + 5x = 40 + x \)

⇒ \( 5x - x = 40 - 20 \)

⇒ \( 4x = 20 \)

⇒ \( x = 5 \)

∴ The correct answer is 5 kg.

Given:

Inner radius (r₁) = 12 cm

Outer radius (r₂) = 13 cm

Formula Used:

Volume of a hemisphere = (2/3)πr³

Volume of the shell = Volume of outer hemisphere - Volume of inner hemisphere

Calculation:

⇒ Volume of outer hemisphere = (2/3)π(13)³

⇒ Volume of outer hemisphere = (2/3)π × 2197

⇒ Volume of outer hemisphere = (4394/3)π

⇒ Volume of inner hemisphere = (2/3)π(12)³

⇒ Volume of inner hemisphere = (2/3)π × 1728

⇒ Volume of inner hemisphere = (3456/3)π

⇒ Volume of the shell = Volume of outer hemisphere - Volume of inner hemisphere

⇒ Volume of the shell = (4394/3)π - (3456/3)π

⇒ Volume of the shell = (938/3)π

∴ The correct answer is (938/3)π .

Given:

Initial filled part of tank = \( \frac{3}{5} \)

Final filled part of tank = \( \frac{7}{10} \)

Water added = 20

Formula Used:

Added water = Total capacity × (Final fraction − Initial fraction)

Calculation:

⇒ Increase in fraction = \( \frac{7}{10} - \frac{3}{5} \)

⇒ Increase in fraction = \( \frac{7}{10} - \frac{6}{10} \)

⇒ Increase in fraction = \( \frac{1}{10} \)

⇒ \( \frac{1}{10} \) of capacity = 20

⇒ Capacity = 20 × 10

⇒ Capacity = 200

∴ The correct answer is 200 liters.

Given:

Coordinates of vertex A = (2, 3)

Coordinates of vertex B = (4, 5)

Coordinates of vertex C = (6, 7)

Formula Used:

Centroid (G) = ((x₁ + x₂ + x₃) ÷ 3, (y₁ + y₂ + y₃) ÷ 3)

Calculation:

⇒ x-coordinate of centroid = (2 + 4 + 6) ÷ 3

⇒ x-coordinate of centroid = 12 ÷ 3

⇒ x-coordinate of centroid = 4

⇒ y-coordinate of centroid = (3 + 5 + 7) ÷ 3

⇒ y-coordinate of centroid = 15 ÷ 3

⇒ y-coordinate of centroid = 5

⇒ Centroid (G) = (4, 5)

∴ The correct answer is (4, 5).

Given:

Ratio of the perimeters of two similar triangles = a : b

Formula Used:

Ratio of areas of two similar triangles = (Ratio of their perimeters)²

Calculation:

⇒ Ratio of areas = (a : b)²

⇒ Ratio of areas = a² : b²

∴ The correct answer is a² : b².

Given:

\( \frac{4}{\sqrt{7} + \sqrt{5}} \)

Formula Used:

\( (a+b)(a-b) = a^2 - b^2 \)

Calculation:

⇒ \( \frac{4}{\sqrt{7} + \sqrt{5}} \)

⇒ \( \frac{4}{\sqrt{7} + \sqrt{5}} \times \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} - \sqrt{5}} \)

⇒ \( \frac{4(\sqrt{7} - \sqrt{5})}{(\sqrt{7} + \sqrt{5})(\sqrt{7} - \sqrt{5})} \)

⇒ \( \frac{4(\sqrt{7} - \sqrt{5})}{7 - 5} \)

⇒ \( \frac{4(\sqrt{7} - \sqrt{5})}{2} \)

⇒ \( 2(\sqrt{7} - \sqrt{5}) \)

∴ The correct answer is \( 2(\sqrt{7} - \sqrt{5}) \).

Given:

\( \dfrac{(x+y)^3+(y+z)^3+(z+x)^3-3(x+y)(y+z)(z+x)}{x^3+y^3+z^3-3xyz} \)

Formula Used:

\( a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca) \)

Calculation:

⇒ \( (x+y)^3+(y+z)^3+(z+x)^3-3(x+y)(y+z)(z+x) \)

⇒ \( =[(x+y)+(y+z)+(z+x)]{(x+y)^2+(y+z)^2+(z+x)^2-(x+y)(y+z)-(y+z)(z+x)-(z+x)(x+y)} \)

⇒ \( =2(x+y+z){(x+y)^2+(y+z)^2+(z+x)^2-(x+y)(y+z)-(y+z)(z+x)-(z+x)(x+y)} \)

⇒ \( =(2x+2y+2z)(x^2+y^2+z^2-xy-yz-zx) \)

⇒ \( x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx) \)

⇒ \( \dfrac{(2x+2y+2z)(x^2+y^2+z^2-xy-yz-zx)}{(x+y+z)(x^2+y^2+z^2-xy-yz-zx)} \)

⇒ \( \dfrac{2(x+y+z)}{(x+y+z)} \)

⇒ \( 2 \)

∴ The correct answer is \( 2 \).

Shortcut Trick

Apply the algebraic identity a² − b² = (a + b)(a − b) to the expression.

sin⁴(x) − cos⁴(x) = (sin²(x) + cos²(x)) × (sin²(x) − cos²(x)).

Using the Pythagorean identity, sin²(x) + cos²(x) = 1.

The expression simplifies directly to 1 × (sin²(x) − cos²(x)), which is sin²(x) − cos2(x).

∴ The correct answer is sin²(x) − cos2(x).

Alternate Method

Given: sin⁴(x) − cos⁴(x)

Formula Used: a² − b² = (a − b)(a + b) and sin²(x) + cos²(x) = 1

⇒ sin⁴(x) − cos⁴(x) = (sin²(x))² − (cos²(x))²

⇒ (sin²(x) + cos²(x)) × (sin²(x) − cos²(x))

⇒ 1 × (sin²(x) − cos²(x))

⇒ sin²(x) − cos²(x)

Comparing the result with the given choices, the expression is represented as:

⇒ sin²(x) − cos2(x)

∴ The correct answer is sin²(x) − cos2(x).

Additional Information

Pythagorean Identities

The most fundamental identity is sin²(x) + cos²(x) = 1. Other variations include 1 + tan²(x) = sec²(x) and 1 + cot²(x) = cosec²(x).

Double Angle Formulas

The formula for cos(2x) can be written in three ways: cos²(x) − sin²(x), 2cos²(x) − 1, or 1 − 2sin²(x).

Algebraic Decomposition

Higher-order powers like a⁴ − b⁴ are simplified by treating them as (a²)² − (b²)², allowing the use of the difference of squares method.

Given:

Point 1: (1, 3)

Point 2: (4, 9)

Slope-intercept form: y = mx + c

Formula Used:

Slope (m) = (y₂ - y₁) ÷ (x₂ - x₁)

To find c: y = mx + c

Calculation:

⇒ Slope (m) = (9 - 3) ÷ (4 - 1)

⇒ Slope (m) = 6 ÷ 3

⇒ Slope (m) = 2

Using point (1, 3):

⇒ 3 = 2(1) + c

⇒ 3 = 2 + c

⇒ c = 3 - 2

⇒ c = 1

Equation of the line: y = 2x + 1

∴ The correct answer is y = 2x + 1.

Given:

Profit = 15%

Reduced cost price = 10% less

Reduced selling price = ₹18 less

New profit = 25%

Formula Used:

Selling Price = Cost Price × (100 + Profit%) ÷ 100

Reduced Cost Price = Original Cost Price × (100 − Reduction%) ÷ 100

Calculation:

⇒ Let Cost Price = x

⇒ Selling Price = x × (100 + 15) ÷ 100

⇒ Selling Price = 1.15x

⇒ Reduced Cost Price = x × (100 − 10) ÷ 100

⇒ Reduced Cost Price = 0.9x

⇒ New Selling Price = 0.9x × (100 + 25) ÷ 100

⇒ New Selling Price = 1.125x

⇒ Reduced Selling Price = Original Selling Price − 18

⇒ 1.125x = 1.15x − 18

⇒ 1.15x − 1.125x = 18

⇒ 0.025x = 18

⇒ x = 18 ÷ 0.025

⇒ x = 720

∴ The correct answer is ₹720.

Given:

3 × P = 4 × Q = 7 × R

Total Profit = 3150

Formula Used:

If a = b = c, then each quantity can be written equal to a common constant

Profit share ∝ Capital

Calculation:

⇒ 3P = 4Q = 7R = k

⇒ P = k ÷ 3

⇒ Q = k ÷ 4

⇒ R = k ÷ 7

⇒ Ratio P : Q : R = 1 ÷ 3 : 1 ÷ 4 : 1 ÷ 7

⇒ LCM of 3, 4, 7 = 84

⇒ P : Q : R = 28 : 21 : 12

⇒ Total ratio = 28 + 21 + 12 = 61

⇒ R's share = 3150 × 12 ÷ 61

⇒ R's share = 37800 ÷ 61

⇒ R's share = 619.67

∴ The correct answer is ₹619.67.

Shortcut Trick

The expression sin A cos B + cos A sin B follows the compound angle identity sin(A + B).

Given sin A = 8/17 and cos B = 15/17, notice they belong to the Pythagorean triplet (8, 15, 17).

This implies cos A = 15/17 and sin B = 8/17 as they share the same hypotenuse 17.

Substituting values: (8/17 × 15/17) + (15/17 × 8/17) = 120/289 + 120/289 = 240/289.

∴ The correct answer is 240/289.

Alternate Method

Given: sin A = 8/17 and cos B = 15/17

Formula Used: sin(A + B) = sin A cos B + cos A sin B | cos²θ = 1 − sin²θ

⇒ First, find cos A: cos A = √(1 − sin² A) = √(1 − (8/17)²)

⇒ cos A = √(1 − 64/289) = √(225/289) = 15/17

⇒ Next, find sin B: sin B = √(1 − cos² B) = √(1 − (15/17)²)

⇒ sin B = √(1 − 225/289) = √(64/289) = 8/17

⇒ Required value = (8/17 × 15/17) + (15/17 × 8/17)

⇒ Value = 120/289 + 120/289 = 240/289

∴ The correct answer is 240/289.

Additional Information

Trigonometric Identities for Sum/Difference

sin(A ± B) = sin A cos B ± cos A sin B and cos(A ± B) = cos A cos B ∓ sin A sin B.

Pythagorean Triplets

Sets of three positive integers a, b, and c such that a² + b² = c². Common examples include (3, 4, 5), (5, 12, 13), and (8, 15, 17).

Range of Sine and Cosine

For any real angle θ, −1 ≤ sin θ ≤ 1 and −1 ≤ cos θ ≤ 1.

Given:

Apples are 25% more than oranges

Formula Used:

Percentage increase = Original value × (100 + Increase%) ÷ 100

Calculation:

⇒ Let number of oranges = 100

⇒ Apples = 100 × (100 + 25) ÷ 100

⇒ Apples = 125

⇒ Ratio Apples : Oranges = 125 : 100

⇒ Ratio Apples : Oranges = 5 : 4

∴ The correct answer is 5 : 4.

Given:

Amount becomes 4 times in 6 years

Required amount = 16 times

Formula Used:

A = P × (1 + r)ⁿ

Calculation:

⇒ 4P = P × (1 + r)⁶

⇒ (1 + r)⁶ = 4

⇒ 16P = P × (1 + r)^t

⇒ (1 + r)^t = 16

⇒ 16 = 4²

⇒ (1 + r)^t = ((1 + r)⁶)²

⇒ (1 + r)^t = (1 + r)¹²

⇒ t = 12

∴ The correct answer is 12 years.

Given:

Remainder when divided by 12 = 5

Remainder when divided by 15 = 8

Remainder when divided by 20 = 13

Formula Used:

If N leaves remainder r when divided by d, then N − r is divisible by d

Required number = LCM of divisors + common remainder

Calculation:

⇒ N − 5 divisible by 12

⇒ N − 8 divisible by 15

⇒ N − 13 divisible by 20

⇒ 12 − 5 = 7

⇒ 15 − 8 = 7

⇒ 20 − 13 = 7

⇒ N − 5 = N − 8 = N − 13 = common multiple − 7

⇒ Required number = LCM(12, 15, 20) − 7

⇒ LCM(12, 15, 20)

⇒ 12 = 2 × 2 × 3

⇒ 15 = 3 × 5

⇒ 20 = 2 × 2 × 5

⇒ LCM = 2 × 2 × 3 × 5

⇒ LCM = 60

⇒ Required number = 60 − 7

⇒ Required number = 53

∴ The correct answer is 53.

Given:

Average of 14 consecutive even numbers = 107

Number of terms = 14

Formula Used:

Average = (Sum of observations) ÷ (Number of observations)

Average of consecutive even numbers = (First term + Last term) ÷ 2

Calculation:

⇒ Average = 107

⇒ (First term + Last term) ÷ 2 = 107

⇒ First term + Last term = 214

⇒ Last term = First term + 2 × (14 − 1)

⇒ Last term = First term + 26

⇒ First term + First term + 26 = 214

⇒ 2 × First term + 26 = 214

⇒ 2 × First term = 214 − 26

⇒ 2 × First term = 188

⇒ First term = 188 ÷ 2

⇒ First term = 94

∴ The correct answer is 94.

Given:

Increase in first year = 25%

Decrease in second year = 20%

Formula Used:

Final value = Initial value × (100 + Increase%) ÷ 100 × (100 − Decrease%) ÷ 100

Percentage change = (Final − Initial) ÷ Initial × 100

Calculation:

⇒ Let initial population = 100

⇒ After first year = 100 × (100 + 25) ÷ 100

⇒ After first year = 100 × 125 ÷ 100

⇒ After first year = 125

⇒ After second year = 125 × (100 − 20) ÷ 100

⇒ After second year = 125 × 80 ÷ 100

⇒ After second year = 100

⇒ Percentage change = (100 − 100) ÷ 100 × 100

⇒ Percentage change = 0

∴ The correct answer is 0%.

Given:

Time to cross a man = 12 s

Time to cross platform = 20 s

Length of platform = 150 m

Formula Used:

Speed = Distance ÷ Time

Distance = Speed × Time

Calculation:

⇒ Let length of train = x

⇒ Speed = x ÷ 12

⇒ Distance while crossing platform = x + 150

⇒ Speed = (x + 150) ÷ 20

⇒ x ÷ 12 = (x + 150) ÷ 20

⇒ 20x = 12 × (x + 150)

⇒ 20x = 12x + 1800

⇒ 20x − 12x = 1800

⇒ 8x = 1800

⇒ x = 1800 ÷ 8

⇒ x = 225

∴ The correct answer is 225 m.

Given:

A takes 12 days to complete the job.

B takes 36 days to complete the job.

Formula Used:

Work done in 1 day = 1 ÷ Time taken to complete the job.

Total work = 1 (entire job).

Time taken when working together = 1 ÷ (Work done by A in 1 day + Work done by B in 1 day).

Calculation:

⇒ Work done by A in 1 day = 1 ÷ 12 = 1/12.

⇒ Work done by B in 1 day = 1 ÷ 36 = 1/36.

⇒ Total work done in 1 day when A and B work together = 1/12 + 1/36.

⇒ Total work done in 1 day = (3 + 1) ÷ 36 = 4 ÷ 36 = 1/9.

⇒ Time taken when working together = 1 ÷ (1/9) = 9 days.

∴ The correct answer is 9 days.