Quantitative Aptitude PYQs

Given:

x + y + z = 0

x² + y² + z² = 32

Formula used:

(x + y + z)² = x² + y² + z² + 2(xy + yz + zx)

Calculation:

⇒ (0)² = 32 + 2(xy + yz + zx)

⇒ 0 = 32 + 2(xy + yz + zx)

⇒ 2(xy + yz + zx) = -32

⇒ xy + yz + zx = -16

The value of xy + yz + zx is -16.

Given:

The ratio of capitals of A, B, and C = 2 : 3 : 4

Total time = 2 years = 24 months

After 6 months, A increases his capital by 50%

Total profit = Rs. 90,000

Step 1: Assume the initial capitals

Let the initial capitals of A, B, and C be:

A = 2x, B = 3x, C = 4x

Step 2: Calculate time-wise investment of A

For the first 6 months, A invests 2x

After increasing by 50%, A’s new capital = 2x + (50% of 2x) = 3x

So, for the remaining 18 months, A invests 3x

Time-weighted investment of A:

= (2x × 6) + (3x × 18)

= 12x + 54x

= 66x

Step 3: Calculate time-wise investment of B and C

B invests 3x for 24 months:

= 3x × 24 = 72x

C invests 4x for 24 months:

= 4x × 24 = 96x

Step 4: Find the ratio of profits

Ratio of profits = 66x : 72x : 96x

= 11 : 12 : 16

Step 5: Calculate A’s share of profit

Sum of ratio parts = 11 + 12 + 16 = 39

A’s profit share:

= (11 / 39) × 90,000

= Rs. 25,384.62 (approx.)

Therefore, A receives Rs. 25,384.62 as profit.

Shortcut Trick

Identify the expression as the Trigonometric Compound Angle Identity: sin(A + B) = sin A cos B + cos A sin B.

Using Pythagorean triplets (5, 12, 13): if cos A = 12/13, then sin A = 5/13.

Similarly, if sin B = 5/13, then cos B = 12/13.

Substitute values: (5/13 × 12/13) + (12/13 × 5/13) = 60/169 + 60/169 = 120/169.

∴ The correct answer is 120/169.

Alternate Method

Given: cos A = 12/13 and sin B = 5/13

Formula: sin² θ + cos² θ = 1

⇒ sin A = √(1 − cos² A) = √(1 − (12/13)²)

⇒ sin A = √(1 − 144/169) = √(25/169) = 5/13

⇒ cos B = √(1 − sin² B) = √(1 − (5/13)²)

⇒ cos B = √(1 − 25/169) = √(144/169) = 12/13

⇒ Required expression = sin A cos B + cos A sin B

⇒ Required expression = (5/13 × 12/13) + (12/13 × 5/13)

⇒ Required expression = 60/169 + 60/169 = 120/169

∴ The correct answer is 120/169.

Additional Information

Compound Angle Formulae

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

Fundamental Identity

The core relation is sin²θ + cos²θ = 1, used to find one ratio if the other is known.

Common Pythagorean Triplets

Triplets like (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25) help solve trigonometry problems faster.

Given:

Cost price of 10 apples = ₹120

Cost price of 1 apple = ₹120 ÷ 10 = ₹12

Selling price of 8 apples = ₹100

Selling price of 1 apple = ₹100 ÷ 8 = ₹12.5

Formula used:

Profit/Loss Percentage = (Selling Price - Cost Price) ÷ Cost Price × 100

Calculations:

Cost Price of 1 apple = ₹12

Selling Price of 1 apple = ₹12.5

Profit = ₹12.5 - ₹12 = ₹0.5

⇒ Profit Percentage = (₹0.5 ÷ ₹12) × 100

⇒ Profit Percentage = 4.17%

∴ The correct answer is option (1).

Given:

A + B = ₹9000

(4/11) × A = (3/5) × B

Formula used:

To solve, use the equation derived from the given ratios.

Calculation:

(4/11) × A = (3/5) × B

⇒ B = (4/11) × A ÷ (3/5)

⇒ B = (4 × 5) ÷ (11 × 3) × A

⇒ B = (20/33) × A

We know A + B = ₹9000

⇒ A + (20/33) × A = ₹9000

⇒ A × (1 + 20/33) = ₹9000

⇒ A × (33/33 + 20/33) = ₹9000

⇒ A × (53/33) = ₹9000

⇒ A = ₹9000 × (33/53)

⇒ A = ₹5603.8

∴ The correct answer is option (1).

Given:

An amount doubles in 5 years with compound interest.

Find the time required for the amount to grow to 8 times its original value.

Formula used:

A = P(1 + r)^t

Where:

• A = Final amount
• P = Principal amount
• r = Rate of interest (constant)
• t = Time in years

Calculation:

Let the amount double in 5 years. So:

2 = (1 + r)⁵

⇒ (1 + r) = 2^1/5

To find the time for the amount to grow to 8 times:

8 = (1 + r)^t

⇒ 8 = (2^1/5)^t

⇒ 8 = 2^t/5

⇒ t/5 = 3 (as 8 = 2³)

⇒ t = 3 × 5

⇒ t = 15

∴ The correct answer is option (3).

Given:

The ratio of speeds of bike and car = 4 : 5

Distance traveled by bike = 160 km

Time taken by bike = 4 hours

Time taken by car = 5 hours

Formula used:

Speed = Distance ÷ Time

Distance = Speed × Time

Calculations:

Speed of bike = 160 ÷ 4 = 40 km/h

Speed of car = (Speed of bike) × (Ratio of car to bike)

⇒ Speed of car = 40 × (5 ÷ 4) = 50 km/h

Distance traveled by car in 5 hours = Speed × Time

⇒ Distance = 50 × 5 = 250 km

∴ The correct answer is option (2) 250 km.

Given:

Number of students (n) = 30

Initial average marks = 55

Marks of the replaced student = 40

Marks of the new student = 70

Formula used:

New average = Total marks of all students/Number of students

Calculation:

Total marks of all students = Initial average × Number of students

⇒ Total marks = 55 × 30 = 1650

After replacement:

New total marks = Total marks - Marks of replaced student + Marks of new student

⇒ New total marks = 1650 - 40 + 70 = 1680

New average = New total marks/Number of students

⇒ New average = 1680 ÷ 30

⇒ New average = 56

∴ The correct answer is option (1).

Given:

Required number leaves remainder 4 when divided by 9, 12 and 15

Concept:

If a number leaves remainder 4 in each case, then (Number − 4) is divisible by 9, 12 and 15

Calculation:

Find LCM of 9, 12 and 15

9 = 3 × 3

12 = 2 × 2 × 3

15 = 3 × 5

LCM = 2 × 2 × 3 × 3 × 5

⇒ LCM = 180

Smallest number = 180 + 4

⇒ 184

∴ The required smallest number = 184

Shortcut Trick

For two successive percentage increases of x% and y%, use the formula:

Net % Increase = x + y + (x × y / 100)%

Given: x = 20 and y = 10.

Net % Increase = 20 + 10 + (20 × 10 / 100) = 30 + 2 = 32%

∴ The correct answer is 32%.

Alternate Method

Given:

First increase = 20%

Second increase = 10%

Formula: Final Value = Initial Value × (1 + r₁/100) × (1 + r₂/100)

⇒ Let the initial price of the shirt be 100.

⇒ Price after the first increase of 20% = 100 + (20/100 × 100) = 100 + 20 = 120.

⇒ Price after the second increase of 10% = 120 + (10/100 × 120) = 120 + 12 = 132.

⇒ Net increase in price = 132 − 100 = 32.

⇒ Net percentage increase = (32 / 100) × 100 = 32%.

∴ The correct answer is 32%.

Additional Information

Successive Percentage Change

When a value is changed by a% and then by b%, the net change is (a + b + ab/100)%. Use a positive sign for increase and a negative sign for decrease.

Fraction Method

Convert percentages to fractions: 20% = 1/5 (ratio 5:6) and 10% = 1/10 (ratio 10:11). Net ratio = (5×10) : (6×11) = 50 : 66. Increase = 16/50 × 100 = 32%.

Successive Discount

If two successive discounts of d₁% and d₂% are given, the equivalent single discount is (d₁ + d₂ − d₁d₂/100)%.

Given:

A can complete the task in 10 days.

B can complete the task in 15 days.

A works alone for 4 days, then B joins A.

Formula used:

Work completed = (Work rate of A × Days worked by A) + (Work rate of A and B together × Remaining days).

Calculation:

Work rate of A = 1/10 (since A completes 1 task in 10 days).

Work rate of B = 1/15 (since B completes 1 task in 15 days).

Work done by A in 4 days = (1/10) × 4 = 4/10 = 2/5.

Remaining work = 1 - 2/5 = 3/5.

Combined work rate of A and B = (1/10) + (1/15) = (3 + 2)/30 = 5/30 = 1/6.

Time taken to complete remaining work = Remaining work ÷ Combined work rate.

⇒ Time = (3/5) ÷ (1/6).

⇒ Time = (3/5) × (6/1) = 18/5 = 3.6 days.

∴ The correct answer is option (1): 3.6 days.

Given:

Marked Price (M.P.) = ₹5,000

Discount = 20%

Profit = 25%

Cost Price (C.P.) = ?

Formula used:

Selling Price (S.P.) = Marked Price × (1 - Discount/100)

S.P. = Cost Price × (1 + Profit/100)

Calculation:

S.P. = 5000 × (1 - 20/100)

⇒ S.P. = 5000 × 0.8

⇒ S.P. = ₹4,000

Now, S.P. = C.P. × (1 + Profit/100)

⇒ 4000 = C.P. × (1 + 25/100)

⇒ 4000 = C.P. × 1.25

⇒ C.P. = 4000 / 1.25

⇒ C.P. = ₹3,200

∴ The original cost price of the fan is ₹3,200. The correct answer is option (3).

Shortcut Trick

Simple Interest (SI) for 2 years (6 − 4) = 7,680 − 6,720 = ₹960

SI for 1 year = 960 ÷ 2 = ₹480

Principal (P) = Amount in 4 years − SI for 4 years = 6,720 − (480 × 4) = ₹4,800

Annual Rate (R) = (SI for 1 year ÷ Principal) × 100 = (480 ÷ 4,800) × 100 = 10%

∴ The correct answer is 10%.

Alternate Method

Given:

Amount after 4 years (A₄) = ₹6,720

Amount after 6 years (A₆) = ₹7,680

Formula Used:

Amount = Principal (P) + Simple Interest (SI)

SI = (P × R × T) ÷ 100

Calculations:

⇒ SI for 2 years = A₆ − A₄ = 7,680 − 6,720 = ₹960

⇒ SI for 1 year = 960 ÷ 2 = ₹480

⇒ SI for 4 years = 480 × 4 = ₹1,920

⇒ Principal (P) = A₄ − SI₄ = 6,720 − 1,920 = ₹4,800

⇒ Rate (R) = (SI for 1 year × 100) ÷ P

⇒ R = (480 × 100) ÷ 4,800

⇒ R = 48,000 ÷ 4,800 = 10%

∴ The correct answer is 10%.

Additional Information

Simple Interest Constant Property

In Simple Interest, the interest earned remains the same every year as it is always calculated on the original Principal.

Finding Rate of Interest

The annual rate can be found using R = (SI for 1 year × 100) ÷ P, where P is the initial sum.

Amount and Principal Relationship

Amount (A) = P × [1 + (R × T)/100]. To find P when two amounts are given, subtract the respective SI from the total amount.

Let the given expression be:

x = √(12 + √(12 + √(12 + … )))

Observe the repeating pattern

The expression inside the square root repeats infinitely, so:

x = √(12 + x)

Square both sides

x² = 12 + x

Rearrange the equation

x² − x − 12 = 0

Solve the quadratic equation

(x − 4)(x + 3) = 0

So, x = 4 or x = −3

Reject the negative value

Since the square root of a number cannot be negative, x = −3 is not acceptable.

Final Answer:

The value of √(12 + √(12 + √(12 + … ))) is 4.

Shortcut Trick

New Acid Percentage = Initial Percentage × (1 − Replacement Volume ÷ Total Volume)

New Acid Percentage = 90% × (1 − 8 ÷ 80)

New Acid Percentage = 90% × (1 − 1/10)

New Acid Percentage = 90% × 0.9 = 81%

∴ The correct answer is 81%.

Alternate Method

Given:

Total volume of acid solution = 80 litres

Initial concentration of pure acid = 90%

Volume removed and replaced with pure water = 8 litres

Formula Used:

Final Quantity = Initial Quantity × (1 − x/V)

Calculations:

⇒ Initial amount of pure acid = 90% of 80 = (90 ÷ 100) × 80 = 72 litres

⇒ Amount of pure acid in 8 litres removed = 90% of 8 = (90 ÷ 100) × 8 = 7.2 litres

⇒ Pure acid remaining in the drum = 72 − 7.2 = 64.8 litres

⇒ Total volume of solution after adding pure water = (80 − 8) + 8 = 80 litres

⇒ New percentage of pure acid = (Amount of remaining acid ÷ Total volume) × 100

⇒ New percentage = (64.8 ÷ 80) × 100 = 648 ÷ 8 = 81%

∴ The correct answer is 81%.

Additional Information

Successive Replacement Formula

If 'x' volume is replaced 'n' times from a total volume 'V', the final quantity of the original liquid is Q_f = Q_i × (1 − x/V)ⁿ.

Dilution Concept

When pure water is added to a mixture, the quantity of the solute (acid) remains constant, but its percentage decreases as the total solution volume increases.

Concentration Ratio

The ratio of the final concentration to the initial concentration is equal to the ratio of the volume of acid remaining to the initial volume of acid.

Given:

Length of the chord = 16 cm

Perpendicular distance from the center to the chord = 6 cm

Formula used:

Radius (r) = √((Length of the chord/2)² + (Perpendicular distance)²)

Calculation:

r = √((16/2)² + 6²)

⇒ r = √(8² + 6²)

⇒ r = √(64 + 36)

⇒ r = √100

⇒ r = 10 cm

∴ The correct answer is option (1).

Given:

Radius of original sphere (R) = 3 cm

Radius of smaller sphere (r) = 1 cm

Formula used:

Volume of a sphere = (4/3) × π × r³

Number of smaller spheres = Volume of original sphere / Volume of one smaller sphere

Calculation:

Volume of original sphere = (4/3) × π × (3)³

⇒ Volume of original sphere = (4/3) × π × 27

⇒ Volume of original sphere = 36π

Volume of one smaller sphere = (4/3) × π × (1)³

⇒ Volume of one smaller sphere = (4/3) × π × 1

⇒ Volume of one smaller sphere = (4/3)π

Number of smaller spheres = (36π) / ((4/3)π)

⇒ Number of smaller spheres = 36 × (3/4)

⇒ Number of smaller spheres = 27

∴ The correct answer is option (3).

Given:

A solid hemispherical block is melted and recast into a cylinder of the same radius.

Radius of hemisphere = r

Formula used:

Volume of hemisphere = (2/3)πr³

Volume of cylinder = πr²h

Since the volume of the hemisphere is equal to the volume of the cylinder:

(2/3)πr³ = πr²h

Calculation:

(2/3)πr³ = πr²h

⇒ h = (2/3)r

∴ The height of the cylinder is (2/3) of the radius of the hemisphere.

The correct answer is Option (1).

Given:

(3.5)² + √1.44 - (2.5 ÷ 0.5)

Formula used:

Basic arithmetic operations:

• Square: (a)² = a × a
• Square root: √a = number whose square equals a
• Division: a ÷ b = quotient of a and b

Calculation:

(3.5)² + √1.44 - (2.5 ÷ 0.5)

⇒ (3.5 × 3.5) + √1.44 - (2.5 ÷ 0.5)

⇒ 12.25 + 1.2 - 5

⇒ 12.25 + 1.2 = 13.45

⇒ 13.45 - 5 = 8.45

∴ The correct answer is option (2).

Given Expression:

1 / (√5 − 2) + 1 / (√5 + 2)

Calculation:

Taking the sum:

= [(√5 + 2) + (√5 − 2)] / [(√5 − 2)(√5 + 2)]

= (2√5) / (5 − 4)

= 2√5 / 1

= 2√5

The simplified value of the given expression is 2√5.