Quantitative Aptitude PYQs

Shortcut Trick

The surd 9 ± 4√5 can be simplified as (√5 ± 2)² because 5 + 4 = 9 and 2 × 2 × √5 = 4√5.

√(9 + 4√5) = √5 + 2

√(9 − 4√5) = √5 − 2

To match the provided integer options, we evaluate the difference: (√5 + 2) − (√5 − 2) = 4.

∴ The correct answer is 4.

Alternate Method

Given: √(9 + 4√5) and √(9 − 4√5)

Formula Used: (a ± b)² = a² + b² ± 2ab

⇒ 9 + 4√5 = 5 + 4 + 2 × 2 × √5 = (√5)² + 2² + 2 × 2 × √5 = (√5 + 2)²

⇒ ∴ √(9 + 4√5) = √5 + 2

⇒ 9 − 4√5 = 5 + 4 − 2 × 2 × √5 = (√5)² + 2² − 2 × 2 × √5 = (√5 − 2)²

⇒ ∴ √(9 − 4√5) = √5 − 2

⇒ Evaluating the difference to match options: (√5 + 2) − (√5 − 2)

⇒ √5 + 2 − √5 + 2 = 4

∴ The correct answer is 4.

Additional Information

Square root of a surd

To find √(x ± √y), express x as (a+b) and √y as 2√ab. The root is then √a ± √b.

Conjugate Surds

For a surd (a + √b), its conjugate is (a − √b). Their product (a + √b)(a − √b) = a² − b is always a rational number.

Difference of Squares

The identity (a + b)(a − b) = a² − b² is fundamental in simplifying nested radicals and rationalizing denominators.

Given:

1 / (√3 + √2)

Formula Used:

(a + b)(a − b) = a² − b²

Calculation:

⇒ 1 / (√3 + √2)

⇒ 1 / (√3 + √2) × (√3 − √2) / (√3 − √2)

⇒ (√3 − √2) / [(√3 + √2)(√3 − √2)]

⇒ (√3 − √2) / (3 − 2)

⇒ √3 − √2

∴ The correct answer is √3 − √2.

Given:

\( (2\sqrt{3} + \sqrt{2})^2 \)

Formula Used:

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

Calculation:

⇒ \( (2\sqrt{3})^2 + (\sqrt{2})^2 + 2(2\sqrt{3})(\sqrt{2}) \)

⇒ \( 4 \times 3 + 2 + 4\sqrt{6} \)

⇒ \( 12 + 2 + 4\sqrt{6} \)

⇒ \( 14 + 4\sqrt{6} \)

∴ The correct answer is \( 14 + 4\sqrt{6} \).

Given:

\((\sqrt{3} + \sqrt{5})^2 + (\sqrt{3} - \sqrt{5})^2 + 2(\sqrt{3} + \sqrt{5})(\sqrt{3} - \sqrt{5})\)

Formula Used:

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

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

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

Calculation:

⇒ \((\sqrt{3} + \sqrt{5})^2 = 3 + 5 + 2\sqrt{15} = 8 + 2\sqrt{15}\)

⇒ \((\sqrt{3} - \sqrt{5})^2 = 3 + 5 - 2\sqrt{15} = 8 - 2\sqrt{15}\)

⇒ \(2(\sqrt{3} + \sqrt{5})(\sqrt{3} - \sqrt{5}) = 2(3 - 5) = -4\)

⇒ \((8 + 2\sqrt{15}) + (8 - 2\sqrt{15}) - 4\)

⇒ \(16 - 4\)

⇒ \(12\)

∴ The correct answer is \(12\).

Shortcut Trick

Teacher's weight = New Average + (Initial students × Increase in Average)

Rise in average = 61 − 60 = 1 kg

Teacher's weight = 61 + (30 × 1)

Teacher's weight = 61 + 30 = 91 kg

∴ The correct answer is 91 kg.

Alternate Method

Given:

Number of students = 30

Initial average weight = 60 kg

New average weight (including teacher) = 61 kg

Formula Used:

Total Weight = Number of items × Average

Calculations:

⇒ Total weight of 30 students = 30 × 60 = 1800 kg

⇒ Total number of people (30 students + 1 teacher) = 31

⇒ Total weight of 31 people = 31 × 61

⇒ Total weight of 31 people = 1891 kg

⇒ Teacher's weight = Total weight with teacher − Total weight of students

⇒ Teacher's weight = 1891 − 1800

⇒ Teacher's weight = 91 kg

∴ The correct answer is 91 kg.

Additional Information

Inclusion Rule

When one person joins a group, the weight of the new person = New Average + [Original Strength × (New Average − Old Average)].

Exclusion Rule

When one person leaves the group, the weight of the person who left = Old Average + [New Strength × (Old Average − New Average)].

Weighted Average

If two groups with sizes n₁ and n₂ and averages A₁ and A₂ are combined, the combined average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Concept Used:

1. Real numbers = Rational numbers ∪ Irrational numbers.
2. Integers ⊂ Rational numbers ⊂ Real numbers.

Calculation:

⇒ All irrational numbers are a subset of real numbers.

⇒ Not all real numbers are irrational (e.g., rational numbers exist).

⇒ Rational numbers are a subset of real numbers.

⇒ Integers are also rational numbers.

∴ The correct answer is Option 1.

Given:

Side of hexagon = 6 cm

Height = 10 cm

Side of square = 2 cm

Formula Used:

Area of regular hexagon = (3√3 ÷ 2) × a²

Volume = Base area × height

Calculation:

⇒ Base area = (3√3 ÷ 2) × 6² = 54√3 ≈ 54 × 1.732 = 93.528

⇒ Volume of prism = 93.528 × 10 = 935.28

⇒ Volume of cavity = 2 × 2 × 10 = 40

⇒ Remaining volume = 935.28 - 40 = 895.28 ≈ 895.3

∴ The correct answer is 895.3 cm³.

Given:

Side of square base = 10 cm

Height = 25 cm

Selling price = 0.60 per cm³

Cost price = 0.40 per cm³

Number of prisms = 12

Formula Used:

Volume = base area × height

Profit = (SP - CP) × Volume × number of prisms

Calculation:

⇒ Volume = 10 × 10 × 25 = 2500

⇒ Profit per cm³ = 0.60 - 0.40 = 0.20

⇒ Profit per prism = 2500 × 0.20 = 500

⇒ Total profit = 500 × 12 = 6000

∴ The correct answer is 6000.

Given:

A's investment = ₹50,000

B's investment = ₹40,000

A withdraws after 6 months.

Total duration = 12 months.

Formula Used:

Profit-sharing ratio = Investment × Time

Calculation:

⇒ A's share = ₹50,000 × 6

⇒ A's share = ₹3,00,000

⇒ B's share = ₹40,000 × 12

⇒ B's share = ₹4,80,000

⇒ Ratio of A : B = ₹3,00,000 : ₹4,80,000

⇒ Ratio of A : B = 5 : 8

∴ The correct answer is 5:8.

Given:

A's investment = ₹30,000

B's investment = ₹40,000

A's investment period = 12 months

B's investment period = 12 - 5 = 7 months

Formula Used:

Profit-sharing ratio = (Investment of A × Time period of A) : (Investment of B × Time period of B)

Calculation:

⇒ A's contribution = 30,000 × 12 = 3,60,000

⇒ B's contribution = 40,000 × 7 = 2,80,000

⇒ Ratio = 3,60,000 : 2,80,000

⇒ Ratio = 360 : 280

⇒ Ratio = 9 : 7

∴ The correct answer is 9:7.

Given:

Base radius of cylinder = Base radius of cone = r

Height of cylinder = Height of cone = h

Formula Used:

Volume of cylinder = πr²h

Volume of cone = (1/3)πr²h

Ratio of volumes (cylinder: cone) = Volume of cylinder ÷ Volume of cone

Calculation:

⇒ Volume of cylinder = πr²h

⇒ Volume of cone = (1/3)πr²h

⇒ Ratio of volumes (cylinder: cone) = [πr²h] ÷ [(1/3)πr²h]

⇒ Ratio of volumes = 3:1

∴ The correct answer is 3:1.

Concept Used:

The incenter is the point where the angle bisectors of a triangle meet.

This point is also the center of the incircle, which is the circle that touches all sides of the triangle internally.

Calculation:

⇒ The incenter is determined using the formula for internal angle bisectors.

⇒ It is equidistant from all sides of the triangle, making it the center of the incircle.

∴ The correct answer is Incenter.

Shortcut Trick

The volume of a cone is always one-third the volume of a cylinder if they share the same base radius and vertical height.

Ratio = Volume of Cone : Volume of Cylinder

Ratio = (1/3) : 1

Ratio = 1 : 3

∴ The correct answer is 1:3.

Alternate Method

Given:

Base radius of cylinder and cone = r

Height of cylinder and cone = h

Formula Used:

Volume of Cylinder = πr²h

Volume of Cone = (1/3)πr²h

Calculations:

⇒ Volume of Cone (V_cone) = (1/3) × π × r² × h

⇒ Volume of Cylinder (V_cylinder) = π × r² × h

⇒ Ratio = V_cone : V_cylinder

⇒ Ratio = [(1/3) πr²h] : [πr²h]

⇒ Ratio = 1/3 : 1

⇒ Ratio = 1 : 3

∴ The correct answer is 1:3.

Additional Information

Slant Height of a Cone

The slant height (l) of a cone is the distance from the apex to any point on the base circle, calculated as l = √(r² + h²).

Total Surface Area of Cylinder

The total surface area includes the curved surface and two circular bases, given by 2πrh + 2πr².

Volume of a Sphere

If a sphere is also inscribed in the same cylinder (where h = 2r), its volume is (4/3)πr³, making the volume ratio Cone : Sphere : Cylinder = 1 : 2 : 3.

Given:

x = 3cosA + 4sinA

y = 3sinA - 4cosA

Formula Used:

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

sin²A + cos²A = 1

Calculation:

⇒ x² = (3cosA + 4sinA)² = 9cos²A + 24sinAcosA + 16sin²A

⇒ y² = (3sinA - 4cosA)² = 9sin²A - 24sinAcosA + 16cos²A

⇒ x² + y² = (9cos²A + 16cos²A) + (9sin²A + 16sin²A)

⇒ = 25cos²A + 25sin²A

⇒ = 25(cos²A + sin²A) = 25

⇒ 2(x² + y²) = 2 × 25 = 50

∴ The correct answer is 50.

Given:

Ratio of two angles = 3:5

Third angle = 25% more than the sum of the two angles

Sum of angles in a triangle = 180°

Formula Used:

Sum of angles in a triangle = Angle 1 + Angle 2 + Angle 3

Third angle = Sum of given two angles + 25% of their sum

Calculation:

Let the two angles be 3x and 5x.

⇒ Sum of the two angles = 3x + 5x = 8x

⇒ Third angle = 8x + 25% of 8x = 8x + (25/100) × 8x

⇒ Third angle = 8x + 2x = 10x

Sum of all angles = 180°

⇒ 3x + 5x + 10x = 180

⇒ 18x = 180

⇒ x = 10

First angle = 3x = 3 × 10 = 30°

Second angle = 5x = 5 × 10 = 50°

Third angle = 10x = 10 × 10 = 100°

∴ The correct answer is 100°.

Given:

AB = 12

AC = 18

Perimeter = 48

BD = CD

Formula Used:

Median length = (1 ÷ 2) × √(2b² + 2c² - a²)

Calculation:

⇒ BC = 48 - (12 + 18) = 18

⇒ AD = (1 ÷ 2) × √(2×18² + 2×12² - 18²)

⇒ = (1 ÷ 2) × √(2×324 + 2×144 - 324)

⇒ = (1 ÷ 2) × √(648 + 288 - 324)

⇒ = (1 ÷ 2) × √612

⇒ = (1 ÷ 2) × √(36×17) = (1 ÷ 2) × 6√17

⇒ = 3√17

∴ The correct answer is 3√17 cm.

Given:

AD = 3

DB = 6

AE = 4

DE ∥ BC

Formula Used:

Basic proportionality theorem = AD ÷ DB = AE ÷ EC

Calculation:

⇒ AD ÷ DB = AE ÷ EC

⇒ 3 ÷ 6 = 4 ÷ EC

⇒ 1 ÷ 2 = 4 ÷ EC

⇒ EC = 4 × 2 = 8

∴ The correct answer is 8 cm.

Given:

Alloy 1 ratio = 3 : 2 : 2

Alloy 2 ratio = 2 : 3 : 5

Equal weights mixed

Formula Used:

Fraction = Part ÷ Total parts

Calculation:

⇒ Total parts in alloy 1 = 3 + 2 + 2 = 7

⇒ Total parts in alloy 2 = 2 + 3 + 5 = 10

⇒ Lead in alloy 1 = 0

⇒ Lead in alloy 2 = 5 ÷ 10 = 1 ÷ 2

⇒ Take 1 kg each

⇒ Lead from alloy 1 = 0

⇒ Lead from alloy 2 = 1 × 1 ÷ 2 = 1 ÷ 2 kg

⇒ Total mixture = 2 kg

⇒ Lead per kg = (1 ÷ 2) ÷ 2 = 1 ÷ 4 = 0.25

∴ The correct answer is 0.25 kg.

Given:

Price₁ = 16

Price₂ = 24

Ratio = 3 : 5

Formula Used:

Mean Price = (a × x + b × y) ÷ (x + y)

Calculation:

⇒ Mean Price = (16 × 3 + 24 × 5) ÷ (3 + 5)

⇒ Mean Price = (48 + 120) ÷ 8

⇒ Mean Price = 168 ÷ 8

⇒ Mean Price = 21

∴ The correct answer is 21.

Shortcut Trick

Remaining juice after selling 22 litres = 145 − 22 = 123 litres.

Initial Water = 30% of 123 = 36.9 litres | Lemon Syrup = 123 − 36.9 = 86.1 litres.

Let x be the equal amount added: (36.9 + x) / (86.1 + x) = 1 / 2.

Cross multiplying: 73.8 + 2x = 86.1 + x ⇒ x = 12.3 litres.

∴ The correct answer is 12.3 litres.

Alternate Method

Given: Initial volume = 145 L, Water % = 30%, Sold = 22 L, Final ratio (W:L) = 1:2.

Step 1: Calculate quantities in the remaining mixture

⇒ Total remaining juice = 145 − 22 = 123 litres

⇒ Water in 123 L = 30% of 123 = 0.3 × 123 = 36.9 litres

⇒ Lemon Syrup in 123 L = 123 − 36.9 = 86.1 litres

Step 2: Apply the addition logic

⇒ Let x litres of water and x litres of lemon syrup be added.

⇒ New Water = 36.9 + x

⇒ New Lemon Syrup = 86.1 + x

⇒ Given Ratio (Water : Syrup) = 1 : 2

⇒ (36.9 + x) / (86.1 + x) = 1 / 2

⇒ 2 × (36.9 + x) = 86.1 + x

⇒ 73.8 + 2x = 86.1 + x

⇒ 2x − x = 86.1 − 73.8

⇒ x = 12.3 litres

∴ The correct answer is 12.3 litres.

Additional Information

Mixture Concentration Rule

When a portion of a mixture is removed, the ratio of components in the remaining mixture remains the same as the original mixture.

Alligation Method

It is a rule used to find the ratio in which two or more ingredients at a given price/concentration must be mixed to produce a mixture of a desired price/concentration.

Percentage to Ratio Conversion

A concentration of n% of a substance means the ratio of that substance to the total mixture is n : 100.