Quantitative Aptitude PYQs

Given:

∠AOC = 80

∠BOD = 60

Formula Used:

Angle between intersecting chords = 1 ÷ 2 × (sum of intercepted arcs)

Calculation:

⇒ ∠APC = 1 ÷ 2 × (80 + 60)

⇒ = 1 ÷ 2 × 140

⇒ = 70

∴ The correct answer is 70°.

Given:

Central angle = 120

Chord length = 10√3

Formula Used:

Chord length = 2 × r × sin(θ ÷ 2)

Calculation:

⇒ 10√3 = 2 × r × sin(120 ÷ 2)

⇒ 10√3 = 2 × r × sin60

⇒ sin60 = √3 ÷ 2

⇒ 10√3 = 2 × r × (√3 ÷ 2)

⇒ 10√3 = r × √3

⇒ r = 10

∴ The correct answer is 10 cm.

Shortcut Trick

From volume of hemisphere: 2/3 π r³ = 144π ⇒ r³ = 216 ⇒ r = 6 cm.

For conical part: r = 6, h = 8 ⇒ slant height l = 10 cm (using 6-8-10 Pythagorean triplet).

Total Surface Area = CSA of hemisphere + CSA of cone = 2πr² + πrl.

TSA = 2π(6)² + π(6)(10) = 72π + 60π = 132π cm².

∴ The correct answer is 132 π cm².

Alternate Method

Given: Volume of hemisphere = 144π cm³, Height of cone (h) = 8 cm.

Formula: Volume of hemisphere = 2/3 π r³ | Slant height l = √(r² + h²) | TSA = 2π r² + π r l.

⇒ 2/3 × π × r³ = 144 × π

⇒ r³ = (144 × 3) ÷ 2

⇒ r³ = 72 × 3 = 216

⇒ r = √³216 = 6 cm.

⇒ Slant height (l) = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.

⇒ Total Surface Area = Curved Surface Area of hemisphere + Curved Surface Area of cone

⇒ TSA = 2 × π × 6² + π × 6 × 10

⇒ TSA = 2 × π × 36 + 60π

⇒ TSA = 72π + 60π = 132π cm².

∴ The correct answer is 132 π cm².

Additional Information

Volume of Combined Solids

Total volume is the sum of volumes: V_total = V_cone + V_hemisphere = 1/3 πr²h + 2/3 πr³.

Surface Area of Combined Solids

When two solids are joined at identical bases, the common base area is hidden; only the external curved areas are added.

Cone Slant Height

Slant height l is the distance from the apex to any point on the base circumference, calculated as l = √(r² + h²).

Surface Area of Sphere

Total Surface Area of a full sphere is 4πr², hence for a hemisphere, the curved area is 2πr².

Shortcut Trick

Identify that a and b are conjugate pairs of the form (x + y) and (x − y).

Use the direct algebraic identity: a² + b² = (x + y)² + (x − y)² = 2(x² + y²)

Here, x = √7 and y = √5.

a² + b² = 2[(√7)² + (√5)²] = 2(7 + 5) = 2 × 12 = 24

∴ The correct answer is 24.

Alternate Method

Given: a = √7 + √5 and b = √7 − √5

Formula Used: (x + y)² = x² + y² + 2xy and (x − y)² = x² + y² − 2xy

⇒ a² = (√7 + √5)² = (√7)² + (√5)² + 2(√7 × √5)

⇒ a² = 7 + 5 + 2√35 = 12 + 2√35

⇒ b² = (√7 − √5)² = (√7)² + (√5)² − 2(√7 × √5)

⇒ b² = 7 + 5 − 2√35 = 12 − 2√35

⇒ a² + b² = (12 + 2√35) + (12 − 2√35)

⇒ a² + b² = 12 + 12 = 24

Therefore, the value of the expression is 24, which corresponds to option (3).

∴ The correct answer is 24.

Additional Information

Algebraic Identities for Conjugates

For any two numbers x and y: (x + y)² + (x − y)² = 2(x² + y²) and (x + y)² − (x − y)² = 4xy.

Product of Conjugate Surds

The product of conjugate surds (√m + √n) and (√m − √n) always results in a rational number: (√m + √n)(√m − √n) = m − n.

Sum of Conjugate Surds

Adding conjugates (√m + √n) + (√m − √n) eliminates the second irrational term, resulting in 2√m.

Given:

Point 1: (1, 2)

Point 2: (3, 6)

Formula Used:

Slope (m) = (y₂ - y₁) ÷ (x₂ - x₁)

Calculation:

⇒ y₂ = 6, y₁ = 2, x₂ = 3, x₁ = 1

⇒ Slope (m) = (6 - 2) ÷ (3 - 1)

⇒ Slope (m) = 4 ÷ 2

⇒ Slope (m) = 2

∴ The correct answer is 2.

Shortcut Trick

Rational numbers and Irrational numbers are mutually exclusive categories by definition.

A Rational number can be written as a fraction, while an Irrational number cannot.

Since no number can satisfy both properties simultaneously, their intersection is always empty.

∴ The correct answer is Empty.

Alternate Method

Given: Two sets of numbers − Rational (Q) and Irrational (Q^c).

Property: Real Numbers (R) = Rational Numbers (Q) ∪ Irrational Numbers (Q^c).

Calculations:

⇒ Let Q = {p/q : p, q are integers and q ≠ 0}

⇒ Let Q' = {x : x is a real number and x is not rational}

⇒ Intersection (Q ∩ Q') represents elements present in both sets.

⇒ Since Q and Q' are disjoint sets by definition, Q ∩ Q' = ∅

∴ The correct answer is Empty.

Additional Information

Real Number System

The set of Real Numbers (R) is the union of Rational (Q) and Irrational (Q') numbers; mathematically, R = Q ∪ Q'.

Rational Numbers

These include integers, terminating decimals (e.g., 0.25), and non-terminating repeating decimals (e.g., 0.333...).

Irrational Numbers

These consist of non-terminating and non-repeating decimals, such as π and √2.

Disjoint Sets

Two sets are disjoint if they have no common elements, meaning their intersection is the Null set or Empty set (∅).

Concept Used:

A real number is a number that can be located on a number line and does not involve the square root of a negative number.

Calculation:

⇒ √7 is a real number because the square root of a positive number is real.

⇒ √-7 is not a real number because the square root of a negative number is imaginary.

⇒ 1/√7 is a real number because it involves only the division of real numbers.

⇒ -√7 is a real number because it is the negative of a real square root.

∴ The correct answer is option 2.

Given:

The number of rational numbers between any two rational numbers is:

Formula Used:

Rational numbers are numbers that can be expressed as a ratio of integers (p/q).

Between any two rational numbers, there exists infinitely many other rational numbers.

Calculation:

⇒ Let two rational numbers be a/b and c/d.

⇒ Calculate the average: (a/b + c/d) ÷ 2 = (ad + bc)/(2bd).

⇒ This average is also a rational number and lies between a/b and c/d.

⇒ Repeat the process to find infinitely many rational numbers between the two rational numbers.

∴ The correct answer is Infinite.

Given:

PS = 18

QT = 7.5

Medians intersect at right angle

Formula Used:

Median property: PX = 2 ÷ 3 × PS

Median property: QX = 2 ÷ 3 × QT

If medians are perpendicular: PQ² = PX² + QX²

Calculation:

⇒ PX = 2 ÷ 3 × 18 = 12

⇒ QX = 2 ÷ 3 × 7.5 = 5

⇒ PQ² = 12² + 5²

⇒ PQ² = 144 + 25 = 169

⇒ PQ = √169 = 13

∴ The correct answer is 13 cm.

Given:

8 men can complete the work in 14 days.

12 men need to complete the same work.

Formula Used:

Work = Men × Days

Men × Days (initial) = Men × Days (final)

Calculation:

⇒ Work by 8 men in 14 days = Work by 12 men in x days

⇒ 8 × 14 = 12 × x

⇒ 112 = 12x

⇒ x = 112 ÷ 12

⇒ x = 9.33 days

∴ The correct answer is 9.33 days.

Given:

A does 60% work in 18 days

Remaining work = 40%

A + B complete remaining work in 4 days

Formula Used:

Work Rate = Work ÷ Time

Calculation:

⇒ A’s rate = 60 ÷ 18 = 10 ÷ 3 % per day

⇒ (A + B) rate = 40 ÷ 4 = 10 % per day

⇒ B’s rate = 10 - (10 ÷ 3)

⇒ B’s rate = (30 ÷ 3 - 10 ÷ 3) = 20 ÷ 3 % per day

⇒ Time taken by B = 100 ÷ (20 ÷ 3)

⇒ Time = 100 × (3 ÷ 20)

⇒ Time = 15 days

∴ The correct answer is 15 days.

Concept Used:

The diameter of a circle is the longest chord.

Calculation:

⇒ By definition, the diameter passes through the center and is the longest chord in a circle.

∴ The correct answer is Diameter.

Shortcut Trick

The angle between the tangents drawn from an external point and the angle subtended by the chord at the center are supplementary (sum to 180°).

⇒ Angle between tangents + Angle at center = 180°

⇒ Angle between tangents + 60° = 180°

⇒ Angle between tangents = 180° − 60° = 120°

∴ The correct answer is 120°.

Alternate Method

Given:

⇒ Angle subtended by chord AB at the center (O) = ∠AOB = 60°.

⇒ Tangents are drawn at points A and B, meeting at an external point P.

Formula Used:

⇒ The radius is perpendicular to the tangent at the point of contact (∠OAP = ∠OBP = 90°).

⇒ Sum of internal angles of a quadrilateral = 360°.

Calculations:

⇒ In quadrilateral OAPB:

⇒ ∠OAP = 90° (Radius ⊥ Tangent)

⇒ ∠OBP = 90° (Radius ⊥ Tangent)

⇒ ∠AOB = 60° (Given)

⇒ ∠AOB + ∠OAP + ∠APB + ∠OBP = 360°

⇒ 60° + 90° + ∠APB + 90° = 360°

⇒ 240° + ∠APB = 360°

⇒ ∠APB = 360° − 240° = 120°

∴ The correct answer is 120°.

Additional Information

Tangent-Radius Property

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Length of Tangents

The lengths of tangents drawn from an external point to a circle are always equal (PA = PB).

Angle Bisector Property

The line joining the external point to the center of the circle bisects the angle between the two tangents (∠APO = ∠BPO).

Cyclic Nature

Quadrilateral OAPB is always cyclic because the sum of opposite angles (∠OAP + ∠OBP = 180°) is constant.

Given:

Sides = 6, 8, 10

Formula Used:

For right triangle: Radius of circumcircle = Hypotenuse ÷ 2

Calculation:

⇒ 6² + 8² = 36 + 64 = 100

⇒ √100 = 10

⇒ Radius = 10 ÷ 2

⇒ Radius = 5 cm

∴ The correct answer is 5 cm.

Given:

Circumference = 132

Height of cone = 12

π = 22 ÷ 7

Formula Used:

Circumference = 2 × π × r

Volume of cone = (1 ÷ 3) × π × r² × h

Calculation:

⇒ 132 = 2 × (22 ÷ 7) × r

⇒ 132 = (44 ÷ 7) × r

⇒ r = 132 × (7 ÷ 44)

⇒ r = 21

⇒ Volume = (1 ÷ 3) × (22 ÷ 7) × 21² × 12

⇒ Volume = (1 ÷ 3) × (22 ÷ 7) × 441 × 12

⇒ Volume = (1 ÷ 3) × 22 × 63 × 12

⇒ Volume = (1 ÷ 3) × 16632

⇒ Volume = 5544

∴ The correct answer is 5544.

Given:

AB - BC = 4

AB - AC = 2

Perimeter = 36

Calculation:

Let AB = c, BC = a, AC = b

Semiperimeter s = (a + b + c) ÷ 2

AP = AR = s - a

BP = s - b

⇒ c - a = 4

⇒ c - b = 2

⇒ a + b + c = 36

⇒ a = c - 4, b = c - 2

⇒ (c - 4) + (c - 2) + c = 36

⇒ 3c - 6 = 36

⇒ 3c = 42

⇒ c = 14

⇒ a = 10, b = 12

⇒ s = (10 + 12 + 14) ÷ 2 = 18

⇒ PB = s - b = 18 - 12 = 6

⇒ AR = s - a = 18 - 10 = 8

⇒ PB + AR = 6 + 8 = 14

∴ The correct answer is 14.

Given:

40% at 7.5%

60% at 11.25%

Time = 2.5 years

Total interest = 6825

Formula Used:

Simple Interest = (Principal × Rate × Time) ÷ 100

Total interest = Sum of individual interests

Calculation:

⇒ Let total principal = P

⇒ Interest = (0.4P × 7.5 × 2.5 ÷ 100) + (0.6P × 11.25 × 2.5 ÷ 100)

⇒ Interest = (0.4P × 18.75 ÷ 100) + (0.6P × 28.125 ÷ 100)

⇒ Interest = 0.075P + 0.16875P

⇒ Total interest = 0.24375P

⇒ 0.24375P = 6825

⇒ P = 6825 ÷ 0.24375

⇒ P = 28000

∴ The correct answer is 28000.

Given:

Deepak saving = 7500

Karan saving = 7500

Deepak spends = 75%

Karan spends = 80%

Sanjay income = (1/3) of (Deepak income + Karan income)

Sanjay spends = 7/9 of income

Formula Used:

Income = Saving ÷ Saving percentage

Saving = Income × Saving fraction

Calculation:

⇒ Deepak saving % = 100 - 75 = 25%

⇒ Deepak income = 7500 ÷ 0.25 = 30000

⇒ Karan saving % = 100 - 80 = 20%

⇒ Karan income = 7500 ÷ 0.20 = 37500

⇒ Total income = 30000 + 37500 = 67500

⇒ Sanjay income = 67500 ÷ 3 = 22500

⇒ Sanjay saving fraction = 1 - 7/9 = 2/9

⇒ Sanjay saving = 22500 × 2 ÷ 9

⇒ Sanjay saving = 5000

∴ The correct answer is 5000.

Given:

Capital ratio between A and B = 5 : 7

A's investment = Rs 50,000

Formula Used:

B's investment = (B's ratio ÷ A's ratio) × A's investment

Calculation:

⇒ B's investment = (7 ÷ 5) × 50,000

⇒ B's investment = 1.4 × 50,000

⇒ B's investment = Rs 70,000

∴ The correct answer is Rs 70,000.

Given:

First student's score = 80% of the test's total score

Second student's score = 90% of the test's total score

Formula Used:

Ratio = (First student's score) ÷ (Second student's score)

Percentage conversion: x% = x ÷ 100

Calculation:

⇒ First student's score = 80 ÷ 100

⇒ Second student's score = 90 ÷ 100

⇒ Ratio = (80 ÷ 100) ÷ (90 ÷ 100)

⇒ Ratio = 80 ÷ 90

⇒ Ratio = 8 ÷ 9

⇒ Ratio = 8 : 9

∴ The correct answer is 8 : 9.