Quantitative Aptitude PYQs

Given:

Printed price of TV = 10,000

First discount = 10%

Final selling price = 8,100

Formula Used:

Price after first discount = Printed price × (1 - First discount ÷ 100)

Second discount = [(Price after first discount - Final selling price) ÷ Price after first discount] × 100

Calculation:

⇒ Price after first discount = 10,000 × (1 - 10 ÷ 100)

⇒ Price after first discount = 10,000 × 0.9

⇒ Price after first discount = 9,000

⇒ Second discount = [(9,000 - 8,100) ÷ 9,000] × 100

⇒ Second discount = [900 ÷ 9,000] × 100

⇒ Second discount = 0.1 × 100

⇒ Second discount = 10%

∴ The correct answer is 10%.

Given:

Time = 5 years

Final amount = 13 ÷ 8 of principal

Formula Used:

Simple Interest = P × R × T ÷ 100

Amount = Principal + Simple Interest

Calculation:

⇒ Let Principal = P

⇒ Amount = 13P ÷ 8

⇒ Simple Interest = Amount − Principal

⇒ Simple Interest = 13P ÷ 8 − P

⇒ Simple Interest = 13P ÷ 8 − 8P ÷ 8

⇒ Simple Interest = 5P ÷ 8

⇒ P × R × 5 ÷ 100 = 5P ÷ 8

⇒ R × 5 ÷ 100 = 5 ÷ 8

⇒ R = 5 × 100 ÷ (8 × 5)

⇒ R = 100 ÷ 8

⇒ R = 12.5

∴ The correct answer is 12.5%.

Shortcut Trick

Apply the algebraic identity for the difference of two squares: p² − q² = (p + q)(p − q).

Substitute the given values into the formula: 72 = 12 × (p − q).

Divide the difference of squares by the sum: p − q = 72 ÷ 12 = 6.

∴ The correct answer is 6.

Alternate Method

Given:

The sum of two variables: p + q = 12

The difference of their squares: p² − q² = 72

Formula Used:

The Difference of Squares identity is given by:

a² − b² = (a + b)(a − b)

Calculations:

⇒ p² − q² = (p + q)(p − q)

⇒ 72 = 12 × (p − q)

⇒ p − q = 72 ÷ 12

⇒ p − q = 6

∴ The correct answer is 6.

Additional Information

Difference of Squares Identity

The formula a² − b² = (a + b)(a − b) is used to factorize the difference of two perfect squares.

Finding Individual Variables

Once (p + q) and (p − q) are known, you can find p by adding the equations and q by subtracting them.

Square of a Sum

Related identity: (a + b)² = a² + 2ab + b², which represents the expansion of the square of a binomial.

Given:

Number = 6561

Formula Used:

Square root of a number = √(Number)

Calculation:

⇒ √6561

⇒ 81

∴ The correct answer is 81.

Given:

Total mixture = 90 L

Milk : Water = 8 : 7

Mixture removed = 15 L

Pure milk added = 15 L

Formula Used:

Quantity of component = Total mixture × Ratio part ÷ Total ratio

Remaining quantity after removal = Initial quantity − Removed quantity

Calculation:

⇒ Total ratio = 8 + 7 = 15

⇒ Initial milk = 90 × 8 ÷ 15

⇒ Initial milk = 48

⇒ Initial water = 90 × 7 ÷ 15

⇒ Initial water = 42

⇒ Milk removed = 48 × 15 ÷ 90

⇒ Milk removed = 8

⇒ Water removed = 42 × 15 ÷ 90

⇒ Water removed = 7

⇒ Remaining milk = 48 − 8

⇒ Remaining milk = 40

⇒ Remaining water = 42 − 7

⇒ Remaining water = 35

⇒ New milk after adding pure milk = 40 + 15

⇒ New milk = 55

⇒ Water = 35

⇒ Ratio milk : water = 55 : 35

⇒ Ratio milk : water = 11 : 7

∴ The correct answer is 11 : 7.

Given:

Diameter of sphere = 28 cm

Radius of sphere = 28 ÷ 2 = 14 cm

Length of block = 44 cm

Breadth of block = 14 cm

π = 22 ÷ 7

Formula Used:

Volume of sphere = 4 ÷ 3 × π × r³

Volume of rectangular block = Length × Breadth × Height

Calculation:

⇒ Volume of sphere = 4 ÷ 3 × 22 ÷ 7 × 14³

⇒ Volume of sphere = 4 ÷ 3 × 22 ÷ 7 × 2744

⇒ Volume of sphere = 4 ÷ 3 × 22 × 392

⇒ Volume of sphere = 4 × 8624 ÷ 3

⇒ Volume of sphere = 34496 ÷ 3

⇒ Volume of block = 44 × 14 × Height

⇒ Volume of block = 616 × Height

⇒ 616 × Height = 34496 ÷ 3

⇒ Height = 34496 ÷ (3 × 616)

⇒ Height = 34496 ÷ 1848

⇒ Height = 56 ÷ 3

⇒ Height = 18.67 cm

∴ The correct answer is 18.67 cm.

Given:

Radius of hemisphere = 21 cm

Diameter of cylinder = 7 cm

Radius of cylinder = 7 ÷ 2 = 3.5 cm

Height of cylinder = 9 cm

π = 22 ÷ 7

Formula Used:

Volume of hemisphere = 2 ÷ 3 × π × r³

Volume of cylinder = π × r² × h

Calculation:

⇒ Volume of hemisphere = 2 ÷ 3 × 22 ÷ 7 × 21³

⇒ Volume of hemisphere = 2 ÷ 3 × 22 ÷ 7 × 9261

⇒ Volume of hemisphere = 2 ÷ 3 × 22 × 1323

⇒ Volume of hemisphere = 58212 ÷ 3

⇒ Volume of hemisphere = 19404

⇒ Volume of cylinder = 22 ÷ 7 × 3.5² × 9

⇒ Volume of cylinder = 22 ÷ 7 × 12.25 × 9

⇒ Volume of cylinder = 346.5

⇒ Number of bottles = 19404 ÷ 346.5

⇒ Number of bottles = 56

∴ The correct answer is 56.

Given:

Expression = \( \frac{1}{2} \left( \frac{3}{4} + \frac{1}{4} \right) \div \frac{1}{3} \)

Formula Used:

\( a \div b = a \times \frac{1}{b} \)

Calculation:

⇒ \( \frac{1}{2} \left( \frac{3}{4} + \frac{1}{4} \right) \div \frac{1}{3} \)

⇒ \( \frac{1}{2} \left( \frac{4}{4} \right) \div \frac{1}{3} \)

⇒ \( \frac{1}{2} \times 1 \div \frac{1}{3} \)

⇒ \( \frac{1}{2} \times 3 \)

⇒ \( \frac{3}{2} \)

∴ The correct answer is \( \frac{3}{2} \).

Given:

\( 9 \times 3^{4} = 3^{x} \)

Formula Used:

\( a^{m} \times a^{n} = a^{m+n} \)

Calculation:

⇒ \( 9 \times 3^{4} = 3^{x} \)

⇒ \( 3^{2} \times 3^{4} = 3^{x} \)

⇒ \( 3^{2+4} = 3^{x} \)

⇒ \( 3^{6} = 3^{x} \)

⇒ \( x = 6 \)

∴ The correct answer is \( 6 \).

Given:

Diameter of wheel = 63 cm

Number of revolutions = 1000

π = 22 ÷ 7

Formula Used:

Circumference of circle = π × Diameter

Distance travelled = Circumference × Number of revolutions

Calculation:

⇒ Circumference = 22 ÷ 7 × 63

⇒ Circumference = 198 cm

⇒ Distance = 198 × 1000

⇒ Distance = 198000 cm

⇒ Distance = 198000 ÷ 100

⇒ Distance = 1980 m / 1000 = 1.98 km

∴ The correct answer is 1.98 Km.

Shortcut Trick

The slope of the given line y = 3x + 4 is m₁ = 3.

For a perpendicular line, the slope m₂ = −1/m₁ = −1/3.

Substitute the point (2, 5) into the equation y = mx + c: 5 = (−1/3)×2 + c.

Solving for c: c = 5 + 2/3 = 17/3.

The required equation is y = −x/3 + 17/3.

∴ The correct answer is y = -(x)/(3) + (17/3).

Alternate Method

Given:

Equation of the given line: y = 3x + 4

Coordinates of the point: (x₁, y₁) = (2, 5)

Formula Used:

Slope-intercept form: y = mx + c

Condition for perpendicular lines: m₁ × m₂ = −1

Point-slope form: y − y₁ = m(x − x₁)

Calculations:

⇒ Comparing y = 3x + 4 with y = m₁x + c, we get m₁ = 3.

⇒ Since the required line is perpendicular to the given line:

⇒ m₂ = −1 / m₁

⇒ m₂ = −1 / 3

⇒ Using the point-slope form for point (2, 5) and slope −1/3:

⇒ y − 5 = (−1/3) × (x − 2)

⇒ 3(y − 5) = −1(x − 2)

⇒ 3y − 15 = −x + 2

⇒ 3y = −x + 17

⇒ y = −x/3 + 17/3

∴ The correct answer is y = -(x)/(3) + (17/3).

Additional Information

Slope of Parallel Lines

If two lines are parallel, their slopes are equal (m₁ = m₂).

Slope Calculation from Coordinates

The slope m between two points (x₁, y₁) and (x₂, y₂) is given by m = (y₂ − y₁) ÷ (x₂ − x₁).

Standard Form of a Line

The standard form is Ax + By + C = 0, where the slope is −A/B.

Given:

x = 4

y = 3

z = 2

Formula Used:

a² + b² + c² + 2ab − 2bc − 2ca = (a + b − c)²

Calculation:

⇒ 4x² + y² + 25z² + 4xy − 10yz − 20zx

⇒ (2x)² + y² + (5z)² + 2(2x)(y) − 2(y)(5z) − 2(2x)(5z)

⇒ (2x + y − 5z)²

⇒ (2×4 + 3 − 5×2)²

⇒ (8 + 3 − 10)²

⇒ 1²

⇒ 1

∴ The correct answer is 1.

Shortcut Trick

Convert sec(x) and tan(x) into sin(x) and cos(x) forms mentally.

Since both sec(x) = 1/cos(x) and tan(x) = sin(x)/cos(x) share the same denominator cos(x), it cancels out from both the numerator and denominator of the main fraction.

Resulting expression = (1 − sin(x)) / (1 + sin(x)).

∴ The correct answer is ((1 − sin(x))) / ((1 + sin(x))).

Alternate Method

Given: (sec(x) − tan(x)) / (sec(x) + tan(x))

Formula Used: sec(x) = 1/cos(x) and tan(x) = sin(x)/cos(x)

Calculations:

⇒ [(1/cos(x)) − (sin(x)/cos(x))] / [(1/cos(x)) + (sin(x)/cos(x))]

⇒ [ (1 − sin(x)) / cos(x) ] / [ (1 + sin(x)) / cos(x) ]

⇒ [ (1 − sin(x)) / cos(x) ] × [ cos(x) / (1 + sin(x)) ]

⇒ (1 − sin(x)) / (1 + sin(x))

This expression matches the simplified form given in Option 2.

∴ The correct answer is ((1 − sin(x))) / ((1 + sin(x))).

Additional Information

Pythagorean Identity

One of the most used identities is sec²(x) − tan²(x) = 1, which implies (sec(x) − tan(x)) = 1 / (sec(x) + tan(x)).

Reciprocal Identities

sec(x) = 1/cos(x), cosec(x) = 1/sin(x), and cot(x) = 1/tan(x) = cos(x)/sin(x) are essential for simplifying rational trigonometric expressions.

Alternative Rationalization

Multiplying both numerator and denominator by (sec(x) − tan(x)) transforms the expression into (sec(x) − tan(x))², which can further be expanded or simplified based on the requirements.

Shortcut Trick

Identify the Pythagorean triplet for the given sin value.

Since sin 3A = 3/5 (Perpendicular/Hypotenuse), the sides of the triangle are 3, 4, 5.

The Base of this right-angled triangle is 4.

Therefore, cos 3A = Base / Hypotenuse = 4/5.

∴ The correct answer is 4/5.

Alternate Method

Given: sin 3A = 3/5

Formula Used: sin²θ + cos²θ = 1

⇒ cos²(3A) = 1 − sin²(3A)

⇒ cos²(3A) = 1 − (3/5)²

⇒ cos²(3A) = 1 − 9/25

⇒ cos²(3A) = (25 − 9) / 25

⇒ cos²(3A) = 16/25

⇒ cos 3A = √(16/25)

⇒ cos 3A = 4/5

∴ The correct answer is 4/5.

Additional Information

Trigonometric Identities

The fundamental identity sin²θ + cos²θ = 1 relates the sine and cosine of the same angle.

Triple Angle Formulas

For any angle A: sin 3A = 3sin A − 4sin³A and cos 3A = 4cos³A − 3cos A.

Pythagorean Triplets

Common triplets like (3, 4, 5), (5, 12, 13), and (8, 15, 17) satisfy the condition a² + b² = c² and are very useful in trigonometry.

Given:

tan A = 2

Formula Used:

tan A = Opposite side ÷ Adjacent side

sin A = Opposite side ÷ Hypotenuse

cos A = Adjacent side ÷ Hypotenuse

2 sin A cos A = 2 × sin A × cos A

Calculation:

⇒ tan A = 2 ÷ 1

⇒ Opposite side = 2

⇒ Adjacent side = 1

⇒ Hypotenuse = √(2² + 1²)

⇒ Hypotenuse = √(4 + 1)

⇒ Hypotenuse = √5

⇒ sin A = 2 ÷ √5

⇒ cos A = 1 ÷ √5

⇒ 2 sin A cos A = 2 × (2 ÷ √5) × (1 ÷ √5)

⇒ 2 sin A cos A = 4 ÷ 5

∴ The correct answer is 4 ÷ 5.

Given:

A capital = 80000

B capital = 60000

C capital = 40000

B increases capital by 25% after 6 months

C decreases capital by 10% after 6 months

Total profit = 120000

Formula Used:

Profit share ∝ Capital × Time

Calculation:

⇒ A investment for 12 months = 80000 × 12

⇒ A = 960000

⇒ B investment for first 6 months = 60000 × 6

⇒ B investment for next 6 months = (60000 × 125 ÷ 100) × 6

⇒ B = 360000 + 450000

⇒ B = 810000

⇒ C investment for first 6 months = 40000 × 6

⇒ C investment for next 6 months = (40000 × 90 ÷ 100) × 6

⇒ C = 240000 + 216000

⇒ C = 456000

⇒ Ratio A : B : C = 960000 : 810000 : 456000

⇒ Ratio = 320 : 270 : 152

⇒ Total ratio = 320 + 270 + 152

⇒ Total ratio = 742

⇒ C share = 120000 × 152 ÷ 742

⇒ C share = 18240000 ÷ 742

⇒ C share ≈ 24582.21

∴ The correct answer is ₹24582.

Given:

sin A = 7 ÷ 25

cos B = 24 ÷ 25

Formula Used:

sin²θ + cos²θ = 1

sin A cos B + cos A sin B = sin(A + B)

Calculation:

⇒ cos A = √(1 − (7 ÷ 25)²)

⇒ cos A = √(1 − 49 ÷ 625)

⇒ cos A = √(576 ÷ 625)

⇒ cos A = 24 ÷ 25

⇒ sin B = √(1 − (24 ÷ 25)²)

⇒ sin B = √(1 − 576 ÷ 625)

⇒ sin B = √(49 ÷ 625)

⇒ sin B = 7 ÷ 25

⇒ sin A cos B + cos A sin B

⇒ (7 ÷ 25 × 24 ÷ 25) + (24 ÷ 25 × 7 ÷ 25)

⇒ 168 ÷ 625 + 168 ÷ 625

⇒ 336 ÷ 625

∴ The correct answer is 336 ÷ 625.

Given:

First quantity = 50 kg

Cost price of first variety = 50 per kg

Second quantity = 30 kg

Cost price of second variety = 60 per kg

Selling price = 55 per kg

Formula Used:

Total cost price = Sum of (Quantity × Cost price)

Total selling price = Total quantity × Selling price per kg

Profit or Loss = Total selling price − Total cost price

Calculation:

⇒ Cost of first variety = 50 × 50

⇒ Cost = 2500

⇒ Cost of second variety = 30 × 60

⇒ Cost = 1800

⇒ Total cost price = 2500 + 1800

⇒ Total cost price = 4300

⇒ Total quantity = 50 + 30

⇒ Total quantity = 80 kg

⇒ Total selling price = 80 × 55

⇒ Total selling price = 4400

⇒ Profit = 4400 − 4300

⇒ Profit = 100

∴ The correct answer is Profit of ₹100.

Shortcut Trick

Assume the third number is 100.

The first number (20% less than 100) = 80.

The second number (30% more than 100) = 130.

The required ratio = 80 : 130 = 8 : 13.

∴ The correct answer is 8:13.

Alternate Method

Given:

Let the third number be x.

First number = x − (20/100) × x

Second number = x + (30/100) × x

Calculations:

⇒ First number = 0.8x

⇒ Second number = 1.3x

⇒ Ratio = First number : Second number

⇒ Ratio = 0.8x : 1.3x

⇒ Ratio = 8 : 13

∴ The correct answer is 8:13.

Additional Information

Percentage to Fraction Conversion

To simplify calculations, convert percentages to fractions: 20% = 1/5 and 30% = 3/10. If the third number is 1, the first is 1 − 1/5 = 4/5 and the second is 1 + 3/10 = 13/10.

Base Value Selection

In problems where two values are compared to a common third value, always assume the third value to be 100 or a multiple of the denominators to avoid decimal calculations.

Ratio Properties

The ratio a : b is the same as ka : kb for any non-zero constant k. Multiplying 0.8 : 1.3 by 10 removes the decimal to give 8 : 13.

Shortcut Trick

Net change in total sum = (4 × 5) + (4 × (−3)) = 20 − 12 = +8

Net change in average = Total change ÷ Total numbers = 8 ÷ 8 = +1

New Average = Old Average + Change = 45 + 1 = 46

∴ The correct answer is 46.

Alternate Method

Given:

Total numbers = 8

Original Average = 45

Formula Used:

Total Sum = Average × Total Numbers

New Average = New Total Sum ÷ Total Numbers

Calculations:

⇒ Original Sum of 8 numbers = 8 × 45 = 360

⇒ Increase in first 4 numbers = 4 × 5 = 20

⇒ Decrease in last 4 numbers = 4 × 3 = 12

⇒ New Total Sum = 360 + 20 − 12 = 368

⇒ New Average = 368 ÷ 8 = 46

∴ The correct answer is 46.

Additional Information

Basic Average Property

If a constant 'k' is added to or subtracted from every number in a set, the new average increases or decreases by 'k'.

Weighted Average

When two or more groups are combined, the new average is calculated as A_avg = (n₁a₁ + n₂a₂) ÷ (n₁ + n₂).

Sum Property

The total sum of observations is always the product of the average and the total number of observations (Sum = Avg × n).