Quantitative Aptitude PYQs

Shortcut Trick

The fastest way to convert radians to degrees is to substitute π with 180°.

⇒ Angle = 2 × 180° / 3

⇒ Angle = 360° / 3 = 120°

∴ The correct answer is 120°.

Alternate Method

Given:

Angle in radians = 2π / 3

Formula Used:

Angle in Degrees = Angle in Radians × (180° / π)

120° 2π/3 Radians 0°

Calculations:

⇒ Degrees = (2π / 3) × (180° / π)

⇒ Degrees = (2 / 3) × 180°

⇒ Degrees = 2 × 60°

⇒ Degrees = 120°

∴ The correct answer is 120°.

Additional Information

Radians to Degrees Conversion

To convert any angle from radians to degrees, multiply the value by 180° / π.

Degrees to Radians Conversion

To convert any angle from degrees to radians, multiply the value by π / 180°.

Common Benchmark Angles

π = 180°, π/2 = 90°, π/3 = 60°, π/4 = 45°, and π/6 = 30°.

Shortcut Trick

Given A + B = 90°, this means A and B are complementary angles.

For any two complementary angles, tan B = cot A.

We need to find the value of tan A × tan B.

Substitute tan B with cot A: tan A × cot A.

Since tangent and cotangent are reciprocals, their product is always exactly 1, regardless of the value of tan A.

∴ The correct answer is 1.

Alternate Method

Given:

A + B = 90°

tan A = 2

 A B C A B 2 (Opposite to A) 1 (Adjacent to A)

Calculations:

⇒ In a right-angled triangle, if one acute angle is A, the other must be B = 90° − A.

⇒ The tangent of an angle is defined as Perpendicular ÷ Base.

⇒ For angle A, tan A = 2 ÷ 1, so Opposite Side = 2 and Adjacent Side = 1.

⇒ For angle B, the reference sides reverse: Opposite Side = 1 and Adjacent Side = 2.

⇒ Therefore, tan B = Opposite ÷ Adjacent = 1 ÷ 2.

⇒ Now, calculating the required product: tan A × tan B

⇒ 2 × (1 ÷ 2) = 1

∴ The correct answer is 1.

Additional Information

Complementary Angles

Two angles are complementary if their sum is exactly 90°. In any right-angled triangle, the two acute angles are inherently complementary because the sum of all angles is 180° and the right angle occupies 90°.

Trigonometric Co-function Identities

The value of a trigonometric ratio for an angle is exactly equal to the co-function of its complement. Key properties include: sin(90° − θ) = cos θ, and tan(90° − θ) = cot θ.

Reciprocal Identity

Tangent and cotangent are reciprocal functions. Thus, their direct product evaluates to unity: tan θ × cot θ = 1 for any defined angle.

Shortcut Trick

The orthocenter is the point of intersection of the three altitudes of a triangle.

In a right-angled triangle, the two perpendicular legs containing the right angle act as altitudes for each other.

Since these two legs naturally intersect at the vertex where the right angle is formed, the orthocenter must be exactly at this vertex.

∴ The correct answer is the vertex of the right angle.

Alternate Method

Given:

A right-angled triangle.

Concept Used:

The Orthocenter is the common intersection point of the three altitudes of a triangle. An altitude is a perpendicular line segment drawn from a vertex to the line containing the opposite side.

        A B (Orthocenter) C D  Altitude BD

Calculations:

⇒ Consider a right-angled triangle ABC, with the right angle at vertex B (90°).

⇒ The altitude drawn from vertex A to the opposite side BC is the side AB itself, because AB ⊥ BC.

⇒ The altitude drawn from vertex C to the opposite side AB is the side CB itself, because CB ⊥ AB.

⇒ The altitude drawn from vertex B to the hypotenuse AC is the line segment BD.

⇒ We can observe that all three altitudes (AB, CB, and BD) intersect precisely at a single point, which is vertex B.

⇒ Therefore, the orthocenter of the right-angled triangle lies exactly at the vertex containing the right angle.

∴ The correct answer is the vertex of the right angle.

Additional Information

Orthocenter Position by Triangle Type

In an acute triangle, the orthocenter lies strictly inside the triangle. In an obtuse triangle, it lies completely outside the triangle. In a right-angled triangle, it lies exactly at the vertex with the 90° angle.

Circumcenter Position

The circumcenter (intersection of perpendicular bisectors) of a right-angled triangle always lies exactly at the midpoint of the hypotenuse.

Centroid & Incenter

The centroid (intersection of medians) and the incenter (intersection of angle bisectors) always lie strictly inside the triangle, regardless of whether it is acute, obtuse, or right-angled.

Euler Line Property

In any triangle, the Orthocenter, Centroid, and Circumcenter are collinear. The straight line passing through these points is called the Euler line. The centroid divides the distance from the orthocenter to the circumcenter internally in a 2:1 ratio.

Shortcut Trick

In a circle, the radius, the perpendicular distance to the chord, and half the chord always form a right-angled triangle.

We can directly use the standard Pythagorean triplet (5, 12, 13).

Since the hypotenuse (radius) is 13 cm and one side is 5 cm, the other side (half the chord) must be 12 cm.

Length of the full chord = 2 × 12 cm = 24 cm.

∴ The correct answer is 24 cm.

Alternate Method

Given:

Radius of the circle (OA) = 13 cm

Perpendicular distance from the center to the chord (OM) = 5 cm

Formula Used:

According to Pythagoras theorem: Hypotenuse² = Base² + Perpendicular²

The perpendicular drawn from the center of a circle to a chord bisects the chord.

   O A B M 13 cm 5 cm x Chord AB = 2 × AM = 24 cm

Calculations:

In right-angled triangle OMA:

⇒ OA² = OM² + AM²

⇒ 13² = 5² + AM²

⇒ 169 = 25 + AM²

⇒ AM² = 169 − 25

⇒ AM² = 144

⇒ AM = √144 = 12 cm

Since the perpendicular from the center bisects the chord, full chord AB = 2 × AM

⇒ Length of chord AB = 2 × 12 = 24 cm

∴ The correct answer is 24 cm.

Additional Information

Perpendicular from Center to Chord

A perpendicular drawn from the center of a circle to any chord bisects the chord into two equal halves.

Common Pythagorean Triplets

Memorizing integer sets that satisfy a² + b² = c² saves critical calculation time. Important triplets include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25) along with their multiples.

Intersecting Chords Theorem

If two chords AB and CD intersect internally at point P, then the products of their segments are always equal: PA × PB = PC × PD.

Angle in a Semicircle

The angle subtended by a diameter at any point on the circumference of a circle is exactly 90°.

Shortcut Trick

The radius, tangent, and the distance from the center form a right-angled triangle.

In this triangle, the distance from the center is the hypotenuse.

We know the standard Pythagorean triplet (8, 15, 17).

Since the radius is 8 cm and the hypotenuse is 17 cm, the tangent must be 15 cm.

∴ The correct answer is 15 cm.

Alternate Method

Given:

Radius of the circle (OT) = 8 cm

Distance from the center to external point P (OP) = 17 cm

Formula Used:

In a circle, the tangent is always perpendicular to the radius at the point of tangency.

Using Pythagoras theorem in right-angled ΔOTP: OP² = OT² + PT²

   O P T 17 cm 8 cm ? (Tangent)

Calculations:

⇒ In right-angled ΔOTP, OP is the hypotenuse.

⇒ 17² = 8² + PT²

⇒ 289 = 64 + PT²

⇒ PT² = 289 − 64

⇒ PT² = 225

⇒ PT = √225 = 15 cm

∴ The correct answer is 15 cm.

Additional Information

Tangent Perpendicularity Property

The radius of a circle drawn to the point of tangency is always perpendicular (90°) to the tangent line.

Equal Tangents Theorem

If two tangent segments are drawn to a single circle from an external point, then those segments are strictly equal in length.

Secant-Tangent Theorem

If a tangent segment (PT) and a secant segment (PAB) are drawn to a circle from an exterior point, then PT² = PA × PB.

Important Pythagorean Triplets

Memorizing common triplets speeds up geometry calculations: (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), and (9, 40, 41).

Shortcut Trick

The effective Compound Interest rate for 3 years at r% is calculated as: 3r . 3r² . r³

For r = 4%, Effective Rate = 3(4) . 3(16) . (64) = 12.4864%

Total Amount = Principal + Interest = 100% + 12.4864% = 112.4864%

Amount = 112.4864% of 6000 = 6749.184 ≈ 6749.18

∴ The correct answer is Rs. 6749.18.

Alternate Method

Given:

⇒ Principal (P) = Rs. 6000

⇒ Rate (R) = 4% per annum

⇒ Time (T) = 3 years

Formula Used:

⇒ Amount (A) = P(1 + R/100)^T

  Year 0 Year 1 Year 2 Year 3  Rs. 6000  Amount?       × 1.04  × 1.04  × 1.04 Amount = 6000 × (1.04)³

Calculations:

⇒ A = 6000 × (1 + 4/100)³

⇒ A = 6000 × (1 + 1/25)³

⇒ A = 6000 × (26/25)³

⇒ A = 6000 × (17576 / 15625)

⇒ A = 120 × (17576 / 3125)

⇒ A = 24 × (17576 / 625)

⇒ A = 421824 / 625

⇒ Multiply numerator and denominator by 16 for easier division:

⇒ A = (421824 × 16) / (625 × 16)

⇒ A = 6749184 / 10000

⇒ A = 6749.184 ≈ 6749.18

∴ The correct answer is Rs. 6749.18.

Additional Information

Effective Rate of CI

For 2 years, effective CI = 2r + (r² ÷ 100)%. For 3 years, effective CI = 3r + (3r² ÷ 100) + (r³ ÷ 10000)%.

Compound Interest (Half-Yearly)

When interest is compounded half-yearly, Amount = P(1 + R/200)^2T, meaning the rate is halved and time periods are doubled.

Difference Between CI and SI

For 2 years: Difference = P × (R ÷ 100)². For 3 years: Difference = P × (R ÷ 100)² × (3 + R ÷ 100).

Shortcut Trick

Whole numbers are simply the set of natural numbers (counting numbers) along with zero.

Set of Whole Numbers (W) = {0, 1, 2, 3, 4, ...}

This means they include zero and all positive integers, but no fractions or negative numbers.

∴ The correct answer is They include zero and positive integers.

Alternate Method

Checking all the given statements:

⇒ Option 1: They include fractions. (False, fractions like 1/2 or 3/4 are Rational Numbers, not Whole Numbers)

⇒ Option 2: They include negative numbers. (False, negative numbers like −1 or −5 belong to Integers, not Whole Numbers)

⇒ Option 4: They start from 1. (False, numbers starting from 1 are called Natural Numbers)

⇒ Option 3: They include zero and positive integers. (True)

Derivation:

⇒ The set of Whole Numbers is mathematically denoted by W.

⇒ W = {0, 1, 2, 3, 4, ...}

⇒ The numbers {1, 2, 3, 4, ...} are positive integers.

⇒ Therefore, Whole Numbers = {0} + Positive Integers.

∴ The correct answer is They include zero and positive integers.

Additional Information

Natural Numbers (N)

These are the counting numbers starting from 1. N = {1, 2, 3, 4, ...}. They do not include zero, negative numbers, or fractions.

Integers (Z)

This set includes all whole numbers and their negative counterparts. Z = {..., −3, −2, −1, 0, 1, 2, 3, ...}.

Rational Numbers (Q)

Any number that can be expressed in the form of p/q, where p and q are integers and q is not equal to zero. It includes fractions, terminating decimals, and repeating decimals.

Real Numbers (R)

The set of all rational and irrational numbers. They represent all possible continuous points on the number line.

Shortcut Trick

When total time (T) to travel a distance (D) back and forth with speeds S₁ and S₂ is given, use the direct formula:

D = T × (S₁ × S₂) ÷ (S₁ + S₂)

D = 5 × (3.8 × 2.6) ÷ (3.8 + 2.6)

D = 5 × 9.88 ÷ 6.4 = 7.71875 ≈ 7.72 km

∴ The correct answer is 7.72 km.

Alternate Method

Given:

Speed going to school (v₁) = 3.8 km/hr

Speed returning from school (v₂) = 2.6 km/hr

Total time taken (T) = 5 hours

Formula Used:

Time = Distance ÷ Speed

Total Time = (Distance ÷ v₁) + (Distance ÷ v₂)

House   School   Distance = D    3.8 km/hr   2.6 km/hr   Total Time = 5 hours

Calculations:

⇒ Let the distance between his house and school be D km.

⇒ Total time = Time taken to go + Time taken to return

⇒ (D ÷ 3.8) + (D ÷ 2.6) = 5

⇒ D × [(1 ÷ 3.8) + (1 ÷ 2.6)] = 5

⇒ D × [(2.6 + 3.8) ÷ (3.8 × 2.6)] = 5

⇒ D × (6.4 ÷ 9.88) = 5

⇒ D = (5 × 9.88) ÷ 6.4

⇒ D = 49.4 ÷ 6.4

⇒ D = 7.71875 km

⇒ Rounding to two decimal places, we get D ≈ 7.72 km

∴ The correct answer is 7.72 km.

Additional Information

Average Speed for Equal Distances

When a person travels equal distances at two different speeds S₁ and S₂, the average speed for the complete journey is calculated using the formula: Average Speed = (2 × S₁ × S₂) ÷ (S₁ + S₂).

Distance from Difference in Time

If instead of total time, the difference in time (ΔT) taken to cover a distance D at speeds S₁ and S₂ is given, the distance is: D = ΔT × (S₁ × S₂) ÷ |S₁ − S₂|.

Relative Speed

When two entities travel in opposite directions towards each other, their relative speed is the sum of their individual speeds (S₁ + S₂). If they travel in the same direction, it becomes the difference of their speeds (S₁ − S₂).

Shortcut Trick

Outer radius (R) = 20 ÷ 2 = 10 cm, Inner radius (r) = 10 − 1 = 9 cm.

Volume of spherical shell = (4/3) × π × (R³ − r³)

Volume = (4/3) × 3.14 × (10³ − 9³) = (4/3) × 3.14 × (1000 − 729)

Volume = (4/3) × 3.14 × 271 = 3403.76 ÷ 3 ≈ 1134.6 cm³

∴ The correct answer is 1134.6 cm³.

Alternate Method

Given:

Outer diameter of the spherical shell = 20 cm

Thickness of the shell = 1 cm

Value of π = 3.14

 r = 9 cm  R = 10 cm  Thickness = 1 cm Cross-section of the Spherical Shell

Formula Used:

Volume of a spherical shell = (4/3) × π × (R³ − r³)

Where R = Outer radius and r = Inner radius

Calculations:

⇒ Outer radius (R) = Outer diameter ÷ 2 = 20 ÷ 2 = 10 cm

⇒ Inner radius (r) = Outer radius − Thickness = 10 − 1 = 9 cm

⇒ Volume = (4/3) × 3.14 × (10³ − 9³)

⇒ Volume = (4/3) × 3.14 × (1000 − 729)

⇒ Volume = (4/3) × 3.14 × 271

⇒ Volume = 3403.76 ÷ 3

⇒ Volume = 1134.586... ≈ 1134.6 cm³

∴ The correct answer is 1134.6 cm³.

Additional Information

Volume of a Solid Sphere

The volume of a solid sphere of radius r is given by V = (4/3) × π × r³.

Surface Area of a Sphere

The total surface area of a solid sphere of radius r is given by A = 4 × π × r².

Volume of a Hemisphere

The volume of a solid hemisphere of radius r is exactly half of a full sphere: V = (2/3) × π × r³.

Total Surface Area of a Hemispherical Shell

For a hemispherical shell with outer radius R and inner radius r, the total surface area = 3 × π × R² + π × r².

Shortcut Trick

Rise in water level = (Total Volume of submerged objects) ÷ (Base Area of the container)

Rise in level = [10 × (4/3) × π × 5³] ÷ (100 × 50)

Rise in level = (5000π / 3) ÷ 5000

Rise in level = π/3 cm

∴ The correct answer is π/3 cm.

Alternate Method

Given:

⇒ Aquarium length (L) = 100 cm, width (W) = 50 cm

⇒ Number of metal spheres (n) = 10

⇒ Diameter of each sphere = 10 cm ⇒ Radius (r) = 5 cm

Formula Used:

⇒ Volume of a sphere = (4/3) × π × r³

⇒ Displaced Volume = Base Area of container × Rise in fluid level

          20 cm   Rise: h Base Area = 100 × 50 = 5000 cm²  10 Spheres r = 5 cm Volume Displaced = Total Volume of Spheres

Calculations:

⇒ Base area of the aquarium = 100 × 50 = 5000 cm²

⇒ Volume of one sphere = (4/3) × π × 5³ = 500π / 3 cm³

⇒ Total volume of 10 spheres = 10 × (500π / 3) = 5000π / 3 cm³

⇒ Let the rise in water level be h cm.

⇒ Volume of water displaced = Total volume of the fully submerged spheres

⇒ Base Area × h = 5000π / 3

⇒ 5000 × h = 5000π / 3

⇒ h = π / 3 cm

∴ The correct answer is π/3 cm.

Additional Information

Archimedes' Principle (Submerged Volume)

When solid objects are completely submerged in a liquid, the volume of liquid displaced is exactly equal to the total volume of the submerged objects.

Volume of a Sphere

The volume of a sphere is given by V = (4÷3) × π × r³, where r is the radius of the sphere.

Rise in Liquid Level

The rise in the fluid level (h) in any container with a uniform vertical cross-section can be calculated directly by the relation: h = (Total Volume of submerged objects) ÷ (Base Area).

Shortcut Trick

Effective Capital = Capital × Time

A's effective capital = (6 × 3) + (5 × 6) = 18 + 30 = 48

B's effective capital = (4 × 3) + (5 × 6) = 12 + 30 = 42

C's effective capital = 5 × 9 = 45

Ratio = 48 : 42 : 45 = 16 : 14 : 15

∴ The correct answer is 16:14:15.

Alternate Method

Given:

Initial investments: A = Rs. 6,00,000, B = Rs. 4,00,000, C = Rs. 5,00,000

After 3 months: A withdrew Rs. 1,00,000; B added Rs. 1,00,000

Total time of partnership = 9 months

Formula Used:

Ratio of Profit = Ratio of Effective Capital = (C₁ × T₁) : (C₂ × T₂) : (C₃ × T₃)

A 6L × 3m 5L × 6m B 4L × 3m 5L × 6m C 5L × 9m 0 3 Months 9 Months

Calculations:

⇒ A's investment for first 3 months = Rs. 6,00,000

⇒ A's investment for remaining 6 months = 6,00,000 − 1,00,000 = Rs. 5,00,000

⇒ A's effective capital = (6,00,000 × 3) + (5,00,000 × 6) = 18,00,000 + 30,00,000 = Rs. 48,00,000

 

⇒ B's investment for first 3 months = Rs. 4,00,000

⇒ B's investment for remaining 6 months = 4,00,000 + 1,00,000 = Rs. 5,00,000

⇒ B's effective capital = (4,00,000 × 3) + (5,00,000 × 6) = 12,00,000 + 30,00,000 = Rs. 42,00,000

 

⇒ C's investment for entire 9 months = Rs. 5,00,000

⇒ C's effective capital = 5,00,000 × 9 = Rs. 45,00,000

 

⇒ Ratio of their effective capitals = 48,00,000 : 42,00,000 : 45,00,000

⇒ Canceling 1,00,000 from all terms, Ratio = 48 : 42 : 45

⇒ Dividing by 3, Ratio = 16 : 14 : 15

∴ The correct answer is 16:14:15.

Additional Information

Simple Partnership

When all partners invest their capital for the exact same period, profits are distributed strictly in the ratio of their original investments.

Compound Partnership

When partners invest different amounts for different time periods, profits are distributed in the ratio of their effective capitals (calculated as Capital × Time).

Active and Sleeping Partners

An active partner receives a fixed salary or a certain percentage of total profit for managing the business operations, while the remaining profit is shared among all partners according to their effective capital ratios.

Shortcut Trick

When a value is first decreased by x% and then increased by x%, the net change is always a decrease of (x² ÷ 100)%.

Given the net decrease is 10%:

⇒ x² ÷ 100 = 10

⇒ x² = 1000

⇒ x = √1000 ≈ 31.62

∴ The correct answer is 31.6.

Alternate Method

Given:

Initial decrease in price = x%

Subsequent increase in price = x%

Final price is 10% less than the original price.

Formula Used:

Net percentage change for successive changes a% and b% = (a + b + a × b ÷ 100)%

Calculation:

Original Price 100 − x% Intermediate Decreased + x% Final Price 90 Net Decrease: 10%

⇒ Let the original price be 100.

⇒ Since the final price is 10% less, Final Price = 100 − 10 = 90.

⇒ Using the successive percentage formula where a = −x and b = +x:

⇒ Net Change = −x + x + (−x × x) ÷ 100

⇒ Net Change = −x² ÷ 100

⇒ The negative sign confirms an overall decrease.

⇒ Equating the net decrease to the given 10%:

⇒ x² ÷ 100 = 10

⇒ x² = 1000

⇒ x = √1000 ≈ 31.62

∴ The correct answer is 31.6.

Additional Information

Successive Percentage Change

When a value undergoes two successive percentage changes of a% and b%, the net percentage change is given by the formula: Net Change = (a + b + a × b ÷ 100)%. Use a positive sign for an increase and a negative sign for a decrease.

Equal Increase and Decrease

If a quantity is increased by x% and subsequently decreased by the same x%, or vice versa, the final value is always less than the original. The net percentage decrease is always exactly (x² ÷ 100)%.

Base Value Concept

Any percentage calculation is computed on a reference base value. When multiple percentage changes occur sequentially, the intermediate value immediately becomes the new base for the next percentage calculation.

Shortcut Trick

Number of tiles along the length = 12 ÷ 1.5 = 8

Number of tiles along the breadth = 9 ÷ 1.5 = 6

Total number of tiles required = 8 × 6 = 48

∴ The correct answer is 48 tiles.

Alternate Method

Given:

Length of the floor (L) = 12 m

Breadth of the floor (B) = 9 m

Side of one square tile (a) = 1.5 m

Formula Used:

Number of tiles = Total Area of floor ÷ Area of one tile

Number of tiles = (L × B) ÷ a²

12 m 9 m Tile: 1.5 m × 1.5 m

Calculation:

⇒ Area of the floor = 12 × 9 = 108 m²

⇒ Area of one square tile = 1.5 × 1.5 = 2.25 m²

⇒ Number of tiles = 108 ÷ 2.25

⇒ Number of tiles = 10800 ÷ 225 = 48

∴ The correct answer is 48 tiles.

Additional Information

Area of a Rectangle

The area of a rectangle is the product of its length and breadth: Area = L × B. Its perimeter is defined as 2 × (L + B).

Area of a Square

The area of a square is the square of its side length: Area = a². Its perimeter is defined as 4 × a.

Tiling Concept

When covering a larger flat area with smaller uniform regular shapes (like tiles), the total number required is always the total area divided by the area of one unit shape.

Shortcut Trick

Total area of the sign = 5 × 4 = 20 m²

Area of cut-out = 2 × 1 = 2 m²

Exposed area to be painted = 20 − 2 = 18 m²

Total painting cost = 18 × 180 = ₹3,240

∴ The correct answer is ₹3,240.

Alternate Method

5 m 4 m 2 m 1 m Cut out Exposed Area

Given:

Dimensions of the full store sign = 5 m × 4 m

Dimensions of the cut-out section = 2 m × 1 m

Rate of painting = ₹180 per m²

Formula Used:

Area of a Rectangle = Length × Width

Calculations:

⇒ Area of the entire store sign = 5 × 4 = 20 m²

⇒ Area of the rectangular cut-out = 2 × 1 = 2 m²

⇒ Exposed Area = Area of store sign − Area of cut-out

⇒ Exposed Area = 20 − 2 = 18 m²

⇒ Cost of painting = Exposed Area × Rate of painting

⇒ Cost of painting = 18 × 180 = 3240

∴ The correct answer is ₹3,240.

Additional Information

Area of a Rectangle

The space occupied by a 2D rectangle is found by the formula: Area = Length × Width.

Area of Composite Figures

The area of a composite shape having a missing portion (cut-out) is determined by calculating the total area of the primary shape and subtracting the exact area of the removed portion.

Total Cost Calculation

The total cost of painting, paving or fencing a specific region is obtained by multiplying the required area (or perimeter) by the designated unit rate. (Cost = Total Area × Unit Rate).

Shortcut Trick

Area of a hollow regular hexagon = (3√3 ÷ 2) × (Outer Side² − Inner Side²)

⇒ Area = (3 × 1.732 ÷ 2) × (6² − 3²)

⇒ Area = 2.598 × (36 − 9)

⇒ Area = 2.598 × 27 = 70.146 ≈ 70 cm²

∴ The correct answer is 70 cm².

Alternate Method

Given:

Outer side of the hexagonal nut (R) = 6 cm

Inner hollow side of the hexagonal nut (r) = 3 cm

   6 cm    3 cm Cross-section of Hexagonal Nut

Formula Used:

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

Calculations:

⇒ Area of the outer regular hexagon = (3√3 ÷ 2) × 6²

⇒ Area of outer hexagon = (3√3 ÷ 2) × 36 = 54√3 cm²

⇒ Area of the inner hollow regular hexagon = (3√3 ÷ 2) × 3²

⇒ Area of inner hexagon = (3√3 ÷ 2) × 9 = 13.5√3 cm²

⇒ Cross-sectional area of the material = Area of outer hexagon − Area of inner hexagon

⇒ Material Area = 54√3 − 13.5√3 = 40.5√3 cm²

⇒ Taking √3 ≈ 1.732, Area = 40.5 × 1.732 = 70.146 cm²

⇒ Rounding to the nearest whole number, Area ≈ 70 cm²

∴ The correct answer is 70 cm².

Additional Information

Area of a Regular Hexagon

A regular hexagon can be completely divided into 6 equilateral triangles. The total area is exactly 6 × (√3 ÷ 4) × a² = (3√3 ÷ 2) × a², where 'a' is the length of its side.

Inradius of a Regular Hexagon

The radius of the inscribed circle (also called the apothem) is the perpendicular distance from the geometric center to any flat side, given by r = (√3 ÷ 2) × a.

Circumradius of a Regular Hexagon

The radius of the circumscribed circle passing through all six outer vertices is exactly equal to the side length of the regular hexagon, therefore R = a.

Shortcut Trick

Reinvesting the profit is equivalent to calculating Compound Interest.

Effective interest rate for 2 successive years at 10% = 10 + 10 + (10 × 10) ÷ 100 = 21%.

Total value after 2 years will be 121% of the initial investment.

Total Value = (121 ÷ 100) × 5000 = 121 × 50 = 6050.

∴ The correct answer is Rs. 6050.

Alternate Method

Given:

Initial Investment (P) = Rs. 5000

Profit Rate (R) = 10% per year

Time (n) = 2 years

Since the earned profit is reinvested at the same rate, this acts as Compound Interest.

Formula Used:

Total Amount (A) = P × (1 + R ÷ 100)ⁿ

    ₹5000 Initial   ₹5500 End Year 1   ₹6050 End Year 2   +10%  +10% Total Value after 2 Years = ₹6050

Calculations:

⇒ A = 5000 × (1 + 10 ÷ 100)²

⇒ A = 5000 × (1 + 1 ÷ 10)²

⇒ A = 5000 × (11 ÷ 10)²

⇒ A = 5000 × (121 ÷ 100)

⇒ A = 50 × 121

⇒ A = 6050

∴ The correct answer is Rs. 6050.

Additional Information

Effective Percentage Rate

The effective net percentage change for two successive changes of x% and y% is calculated using the formula: Net Change = x + y + (x × y) ÷ 100.

Concept of Reinvestment

When profit or interest is reinvested along with the principal to earn further returns, it acts as Compound Interest because you are generating "interest on interest".

SI vs CI Difference

The exact difference between Compound Interest and Simple Interest for 2 years on a principal P at rate R is given by the formula: Difference = P × (R ÷ 100)².

Shortcut Trick

In this problem, the Selling Price (SP) is given directly as Rs. 120 and the profit is 20%.

The information about the 25% discount on the marked price is extraneous and not required to find the Cost Price (CP).

We can calculate the CP directly using the SP and Profit percentage:

CP = (SP × 100) ÷ (100 + Profit%)

CP = (120 × 100) ÷ 120 = 100

∴ The correct answer is Rs. 100.

Alternate Method

Given:

Selling Price (SP) = Rs. 120

Discount = 25%

Profit = 20%

Formulas Used:

MP = (SP × 100) ÷ (100 − Discount%)

CP ÷ MP = (100 − Discount%) ÷ (100 + Profit%)

       CP = ?   SP = 120   MP = 160   +20% Profit   -25% Discount  CP × 1.20 = 120 → CP = 100

Calculations:

⇒ First, let's determine the Marked Price (MP) for reference:

⇒ MP = (120 × 100) ÷ (100 − 25)

⇒ MP = 12000 ÷ 75 = Rs. 160

⇒ Now, applying the direct ratio formula between CP and MP:

⇒ CP ÷ 160 = (100 − 25) ÷ (100 + 20)

⇒ CP ÷ 160 = 75 ÷ 120

⇒ CP ÷ 160 = 5 ÷ 8

⇒ CP = (160 × 5) ÷ 8

⇒ CP = 800 ÷ 8 = 100

∴ The correct answer is Rs. 100.

Additional Information

Marked Price & Discount

Marked Price (MP) is the price printed on the label. Discount is always calculated on the MP. Formula: SP = MP × (100 − D%) ÷ 100.

Direct CP and MP Relationship

When profit percentage and discount percentage are known, the ratio of Cost Price to Marked Price is given directly by: CP ÷ MP = (100 − Discount%) ÷ (100 + Profit%).

Extraneous Data in Competitive Exams

In many competitive exams, extra data (like the 25% discount in this problem) is provided to consume your time. Always identify the minimum variables needed to reach the final value before calculating blindly.

Shortcut Trick

Salary of the person who left = Initial Average + (Decrease in Average × Remaining Persons)

Decrease in average = 60,000 − 58,000 = 2,000

Remaining engineers = 5 − 1 = 4

Salary = 60,000 + (2,000 × 4) = 60,000 + 8,000 = 68,000

∴ The correct answer is Rs. 68,000.

Alternate Method

Given:

Initial number of engineers = 5

Initial average salary = Rs. 60,000

New average salary = Rs. 58,000

Formula Used:

Sum of observations = Average × Number of observations

Total Salary of 5 Engineers = 3,00,000 4 Engineers = 2,32,000 Left = ? Salary of Left Engineer = 3,00,000 − 2,32,000 = 68,000

Calculations:

⇒ Total salary of 5 engineers initially = 5 × 60,000 = 3,00,000

⇒ Total salary of remaining 4 engineers = 4 × 58,000 = 2,32,000

⇒ Salary of the engineer who left = Total salary of 5 engineers − Total salary of 4 engineers

⇒ Salary = 3,00,000 − 2,32,000 = 68,000

∴ The correct answer is Rs. 68,000.

Additional Information

Included or Excluded Observation Trick

When a person joins a group, the new value = Old Average + (Increase in Average × New Total Members). When a person leaves a group, the value of the left person = Old Average + (Decrease in Average × Remaining Members).

Combined Average

If group 1 has size n₁ with average A₁, and group 2 has size n₂ with average A₂, the combined average is calculated as (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

Net Deviation Method

Instead of multiplying large numbers, you can select an assumed average. Calculate the net deviation of all values from this assumed average, and find the true average by adding (Total Deviation ÷ Number of items) to the assumed average.

Shortcut Trick

The Lateral Surface Area (LSA) of any right prism depends only on its base perimeter and height.

LSA = Perimeter of the base × Height of the prism

LSA = 60 × 8 = 480 cm²

The side ratio 5:12:13 is extra data and is not required to find the lateral area.

∴ The correct answer is 480 cm².

Alternate Method

Given:

Base side ratio = 5 : 12 : 13

Base perimeter = 60 cm

Prism height (h) = 8 cm

Formula Used:

LSA = (Side₁ × h) + (Side₂ × h) + (Side₃ × h)

Base of the Triangular Prism 10 cm 24 cm 26 cm Base Perimeter = 60 cm Prism Height = 8 cm

Calculations:

⇒ Let base sides be 5x, 12x, and 13x

⇒ Perimeter = 5x + 12x + 13x = 30x

⇒ 30x = 60 ⇒ x = 2

⇒ Base sides = 10 cm, 24 cm, 26 cm

⇒ Lateral Surface Area = Area of the 3 rectangular side faces

⇒ LSA = (10 × 8) + (24 × 8) + (26 × 8)

⇒ LSA = 80 + 192 + 208 = 480 cm²

∴ The correct answer is 480 cm².

Additional Information

Total Surface Area of a Prism

The total surface area includes the lateral surface and the two identical bases: TSA = LSA + 2 × (Area of Base).

Volume of a Prism

The volume of any right prism is given by the product of the base area and the height: Volume = Area of Base × Height.

Pythagorean Triples

The base side ratio 5:12:13 forms a standard Pythagorean triple since 5² + 12² = 13². Other standard triples include 3:4:5, 8:15:17, and 7:24:25.

Shortcut Trick

Using the average deviation method:

Removed Number = Old Average + (Remaining terms × Decrease in average)

⇒ Removed Number = 63 + (8 × 3)

⇒ Removed Number = 63 + 24 = 87

∴ The correct answer is 87.

Alternate Method

Given:

Average of 9 numbers = 63

Average of remaining 8 numbers = 60

Total Sum of 9 numbers = 9 × 63 = 567 Sum of 8 numbers = 8 × 60 = 480 87 Removed Removed Number = 567 − 480 = 87

Formula Used:

Sum of observations = Average × Number of observations

Calculations:

⇒ Sum of 9 numbers = 9 × 63 = 567

⇒ Sum of 8 numbers = 8 × 60 = 480

⇒ Removed Number = Sum of 9 numbers − Sum of 8 numbers

⇒ Removed Number = 567 − 480 = 87

∴ The correct answer is 87.

Additional Information

Combined Average

When two groups of sizes n₁ and n₂ have averages A₁ and A₂, the combined average is given by (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Replacement in Averages

If a new value replaces an old one and the average increases: New Value = Old Value + (Increase in Average × Total elements).

Addition of a New Term

When a new term is added and the average increases: New Term = Old Average + (Increase in Average × New number of terms).