Quantitative Aptitude PYQs

Given:

Angle at center = 120°

Formula Used:

Angle at circumference = (1 ÷ 2) × Angle at center

Angle between tangent and chord = Angle in opposite arc

Calculation:

⇒ Angle at circumference = (1 ÷ 2) × 120 = 60°

⇒ Angle between tangent and chord = 60°

∴ The correct answer is 60°.

Given:

CSA : TSA = 7 : 10

Volume = 12672 cm³

Formula Used:

CSA of cylinder = 2 × π × r × h

TSA of cylinder = 2 × π × r × (h + r)

Volume = π × r² × h

Calculation:

⇒ (2 × π × r × h) ÷ [2 × π × r × (h + r)] = 7 ÷ 10

⇒ h ÷ (h + r) = 7 ÷ 10

⇒ 10h = 7h + 7r

⇒ 3h = 7r

⇒ h = (7 ÷ 3)r

⇒ Volume = π × r² × (7r ÷ 3)

⇒ 12672 = (7 ÷ 3) × π × r³

⇒ Using π = 22 ÷ 7

⇒ 12672 = (22 ÷ 3) × r³

⇒ r³ = (12672 × 3) ÷ 22 = 1728

⇒ r = 12

⇒ h = (7 ÷ 3) × 12 = 28

⇒ CSA = 2 × (22 ÷ 7) × 12 × 28

⇒ CSA = 2112 cm²

∴ The correct answer is 2112 cm².

Given:

PQ = QR + 5

PQ = PR + 2

Perimeter = 32 cm

Formula Used:

Perimeter of triangle = PQ + QR + PR

Calculation:

⇒ QR = PQ − 5

⇒ PR = PQ − 2

⇒ Perimeter = PQ + (PQ − 5) + (PQ − 2)

⇒ 32 = 3PQ − 7

⇒ 3PQ = 39

⇒ PQ = 13

⇒ PR = 13 − 2 = 11

∴ The correct answer is 11.

Given:

Rate = 5%

Time for A = 4 years

Time for B = 6 years

B's investment = A − 12000

Formula Used:

Simple Interest = (P × R × T) ÷ 100

Calculation:

⇒ (A × 5 × 4) ÷ 100 = [(A − 12000) × 5 × 6] ÷ 100

⇒ A × 20 = (A − 12000) × 30

⇒ 20A = 30A − 360000

⇒ 10A = 360000

⇒ A = 36000

∴ The correct answer is 36000.

Given:

P investment = 50000

Q investment = 60000

R investment = 70000

Total profit = 24000

Formula Used:

Profit share ∝ Investment

Individual share = (Individual investment ÷ Total investment) × Total profit

Calculation:

⇒ Total investment = 50000 + 60000 + 70000 = 180000

⇒ Q's share = (60000 ÷ 180000) × 24000

⇒ Q's share = (1 ÷ 3) × 24000

⇒ Q's share = 8000

∴ The correct answer is 8000.

Given:

Ratio = 5 : 8

New ratio = 11 : 17

Formula Used:

Numbers = 5x and 8x

Ratio = (a ÷ b)

Calculation:

⇒ (5x + 12) ÷ (8x + 12) = 11 ÷ 17

⇒ 17(5x + 12) = 11(8x + 12)

⇒ 85x + 204 = 88x + 132

⇒ 204 − 132 = 88x − 85x

⇒ 72 = 3x

⇒ x = 24

⇒ Numbers = 5x = 120 and 8x = 192

⇒ Sum = 120 + 192 = 312

∴ The correct answer is 312.

Given:

Number = 1812

Divisors = 7, 11, 14

Formula Used:

LCM of numbers

Required number = Next multiple of LCM − Given number

Calculation:

⇒ LCM of 7, 11, 14 = 154

⇒ 1812 ÷ 154 = 11 remainder 118

⇒ Required addition = 154 − 118 = 36

∴ The correct answer is 36.

Given:

Sum = 84

HCF = 12

LCM : HCF = 12 : 1

Formula Used:

Product of two numbers = HCF × LCM

Numbers = HCF × m and HCF × n where gcd(m, n) = 1

Calculation:

⇒ LCM = 12 × 12 = 144

⇒ Product = 12 × 144 = 1728

⇒ Let numbers = 12m and 12n

⇒ Sum = 12(m + n) = 84

⇒ m + n = 7

⇒ mn = (Product) ÷ 12² = 1728 ÷ 144 = 12

⇒ m, n = 3 and 4

⇒ Numbers = 12 × 3 = 36 and 12 × 4 = 48

∴ The correct answer is 36 and 48.

Shortcut Trick

If every number in a data set is increased, decreased, multiplied, or divided by a constant k, the average of the set changes in the same way.

New Average = Old Average − Decrease Value

New Average = 45 − 7 = 38

∴ The correct answer is 38.

Alternate Method

Given:

Total observations = 20

Original Average = 45

Constant decrease (k) = 7

⇒ Total Sum of 20 numbers = 20 × 45 = 900

⇒ Total reduction in sum = 20 × 7 = 140

⇒ New Total Sum = 900 − 140 = 760

⇒ New Average = New Sum ÷ Total numbers

⇒ New Average = 760 ÷ 20 = 38

∴ The correct answer is 38.

Additional Information

Arithmetic Mean Properties

If every value in a set is multiplied by a constant x, the average also becomes x times the original average.

Summation Formula

The Sum of all observations is calculated as Average × Number of items.

Effect of Addition

If each observation is increased by k, the new average is Original Average + k.

Shortcut Trick

Excess contribution by the new employee = 50,000 − 20,000 = ₹30,000.

This excess is distributed to increase the average of everyone by ₹500.

Total employees (after join) = 30,000 ÷ 500 = 60.

Original employees = 60 − 1 = 59.

∴ The correct answer is 59.

Alternate Method

Given: Original average = ₹20,000 | New salary = ₹50,000 | Increase in average = ₹500.

Formula Used: Total Sum = Average × Number of items.

Let n be the original number of employees.

⇒ Total salary initially = 20,000 × n

⇒ New total salary = 20,000n + 50,000

⇒ New average = 20,000 + 500 = 20,500

⇒ (20,000n + 50,000) ÷ (n + 1) = 20,500

⇒ 20,000n + 50,000 = 20,500(n + 1)

⇒ 20,000n + 50,000 = 20,500n + 20,500

⇒ 20,500n − 20,000n = 50,000 − 20,500

⇒ 500n = 29,500

⇒ n = 29,500 ÷ 500 = 59

∴ The correct answer is 59.

Additional Information

Inclusion Formula

Value of new person = Old Average + (Total count × Increase in Average).

Weighted Average

Combined Average = (n₁a₁ + n₂a₂) ÷ (n₁ + n₂), where n is count and a is average.

Deviation Method

Sum of deviations from the average is always zero. Net gain by new person = Net loss distributed among others.

Given:

B = 100

A is 50% more than B

C is 30% less than (A + B)

Formula Used:

Percentage increase = (Increase ÷ Original) × 100

Percentage decrease = (Decrease ÷ Original) × 100

Percentage = (Part ÷ Whole) × 100

Calculation:

⇒ A = 100 + 50 = 150

⇒ A + B = 150 + 100 = 250

⇒ C = 250 × (70 ÷ 100) = 175

⇒ Required % = (175 ÷ 150) × 100

⇒ = 116.67%

∴ The correct answer is 116.67%.

Given:

Distance₁ = 100 km

Speed₁ = 50 km/h

Distance₂ = 60 km

Speed reduced by 40%

Formula Used:

Time = Distance ÷ Speed

New speed = Original speed × (100 − reduction%) ÷ 100

Calculation:

⇒ Time₁ = 100 ÷ 50 = 2 hours

⇒ New speed = 50 × (60 ÷ 100) = 30 km/h

⇒ Time₂ = 60 ÷ 30 = 2 hours

⇒ Total time = 2 + 2 = 4 hours

∴ The correct answer is 4 hours.

Given:

Men = 18

Total days = 20

Work done in 10 days

Remaining men = 12

Formula Used:

Total work = Men × Days

Remaining work = Total work − Work done

Days = Remaining work ÷ Remaining men

Calculation:

⇒ Total work = 18 × 20 = 360

⇒ Work done in 10 days = 18 × 10 = 180

⇒ Remaining work = 360 − 180 = 180

⇒ Days required = 180 ÷ 12 = 15

∴ The correct answer is 15.

Given:

Cost of items = ₹2050

Discount rate = 26%

Formula Used:

Total savings = (Cost of items × Discount rate) ÷ 100

Calculation:

⇒ Total savings = (₹2050 × 26) ÷ 100

⇒ Total savings = ₹53300 ÷ 100

⇒ Total savings = ₹533

∴ The correct answer is ₹533.

Given:

Selling price after 25% discount = 7200

New discount = 30%

Formula Used:

Selling price = Marked price × (100 − discount%) ÷ 100

Calculation:

⇒ 7200 = MP × (75 ÷ 100)

⇒ MP = 7200 × (100 ÷ 75) = 9600

⇒ New selling price = 9600 × (70 ÷ 100)

⇒ New selling price = 6720

∴ The correct answer is 6720.

Given:

Container A = 20 litres, ratio = 3 : 2

Container B = 30 litres, ratio = 4 : 1

Formula Used:

Quantity = Total × (part ÷ sum of ratio)

Calculation:

⇒ Milk in A = 20 × (3 ÷ 5) = 12

⇒ Water in A = 20 × (2 ÷ 5) = 8

⇒ Milk in B = 30 × (4 ÷ 5) = 24

⇒ Water in B = 30 × (1 ÷ 5) = 6

⇒ Total milk = 12 + 24 = 36

⇒ Total water = 8 + 6 = 14

⇒ Ratio = 36 : 14

⇒ Simplified ratio = 18 : 7

∴ The correct answer is 18 : 7.

Given:

Numbers = 2^{1 ÷ 3}, 3^{1 ÷ 2}, 5^{1 ÷ 6}

Formula Used:

If a^m and bⁿ, compare by taking common power

Calculation:

⇒ LCM of 3, 2, 6 = 6

⇒ (2^{1 ÷ 3})⁶ = 2² = 4

⇒ (3^{1 ÷ 2})⁶ = 3³ = 27

⇒ (5^{1 ÷ 6})⁶ = 5

⇒ 27 > 5 > 4

∴ The correct answer is √3.

Shortcut Trick

Identify the slope m = −2 and the point (1, −3).

Use the Slope-Intercept form: y = mx + c.

Substitute the coordinates: −3 = (−2 × 1) + c.

⇒ −3 = −2 + c ⇒ c = −1.

The required equation is y = −2x − 1.

∴ The correct answer is y = − 2x − 1.

Alternate Method

Given:

Slope (m) = −2

Point (x₁, y₁) = (1, −3)

Formula Used:

The Point-Slope form of a linear equation is: (y − y₁) = m(x − x₁)

Calculations:

⇒ y − (−3) = −2(x − 1)

⇒ y + 3 = −2(x) − (−2 × 1)

⇒ y + 3 = −2x + 2

⇒ y = −2x + 2 − 3

⇒ y = −2x − 1

∴ The correct answer is y = − 2x − 1.

Additional Information

Slope-Intercept Form

Defined as y = mx + c, where m represents the slope and c represents the y-intercept (the value of y when x = 0).

Slope Formula

If two points (x₁, y₁) and (x₂, y₂) are known, the slope m is calculated as (y₂ − y₁) ÷ (x₂ − x₁).

Parallel and Perpendicular Lines

Parallel lines have equal slopes (m₁ = m₂). Perpendicular lines have slopes whose product is −1 (m₁ × m₂ = −1).

Shortcut Trick

In similar triangles, the ratio of any corresponding linear segments (such as sides, perimeters, medians, altitudes, or angle bisectors) is always equal.

Ratio of perimeters = 3 : 4

Ratio of corresponding medians = Ratio of perimeters

∴ Ratio of medians = 3 : 4

∴ The correct answer is 3:4.

Alternate Method

Given: ΔPQR ∼ ΔXYZ and Perimeter(ΔPQR) : Perimeter(ΔXYZ) = 3 : 4.

Formula Used: For two similar triangles, Ratio of Perimeters = Ratio of Sides = Ratio of Medians = Ratio of Altitudes.

Calculations:

⇒ Let the sides of ΔPQR be a₁, b₁, c₁ and sides of ΔXYZ be a₂, b₂, c₂.

⇒ Perimeter of ΔPQR / Perimeter of ΔXYZ = 3 / 4

⇒ By the property of similarity, the ratio of corresponding sides is equal to the ratio of perimeters.

⇒ Ratio of sides = 3 / 4

⇒ Corresponding medians of similar triangles are in the same ratio as their corresponding sides.

⇒ Length of median from P / Length of median from X = Ratio of sides = 3 / 4

∴ The correct answer is 3:4.

Additional Information

Area Ratio Property

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding linear dimensions (sides, medians, or perimeters). Area Ratio = (3/4)² = 9/16.

Median Definition

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.

Every triangle has exactly three medians, which intersect at the centroid.

Similarity Criteria

Triangles are similar if their corresponding angles are equal (AA) or if their corresponding sides are in the same proportion (SSS or SAS).

Given:

AD = 5 cm

DB = 7 cm

AE = 10 cm

Formula Used:

If a line is parallel to one side of a triangle, then it divides the other two sides proportionally

AD ÷ DB = AE ÷ EC

Calculation:

⇒ 5 ÷ 7 = 10 ÷ EC

⇒ 5 × EC = 7 × 10

⇒ 5EC = 70

⇒ EC = 70 ÷ 5

⇒ EC = 14 cm

∴ The correct answer is 14.