Quantitative Aptitude PYQs

Shortcut Trick

Multiply the speed in kmph directly by (5 ÷ 18) to convert it to m/s, then multiply by time to find the distance.

Distance = Speed × Time

Distance = 110 × (5 ÷ 18) × 100 = (550 ÷ 18) × 100

Distance = 27500 ÷ 9 = 3055.555... m

∴ The correct answer is 3055.56 m.

Alternate Method

Given:

Speed of train = 110 kmph

Time = 100 seconds

 Train   110 kmph t = 100 sec    Distance = ?

Formula Used:

Distance = Speed × Time

To convert kmph to m/s, multiply by 5 ÷ 18.

Calculations:

⇒ Speed in m/s = 110 × (5 ÷ 18)

⇒ Speed = 550 ÷ 18 = 275 ÷ 9 m/s

⇒ Distance = Speed × Time

⇒ Distance = (275 ÷ 9) × 100

⇒ Distance = 27500 ÷ 9

⇒ Distance ≈ 3055.56 m

∴ The correct answer is 3055.56 m.

Additional Information

Unit Conversion

To convert km/hr to m/s, multiply the speed value by 5 ÷ 18. To convert m/s to km/hr, multiply the speed value by 18 ÷ 5.

Relative Speed

When two objects move in the same direction, their relative speed is |S₁ − S₂|. When they move in opposite directions, their relative speed is S₁ + S₂.

Average Speed

Average Speed is calculated as Total Distance ÷ Total Time. If equal distances are covered at two different speeds a and b, Average Speed = 2ab ÷ (a + b).

Shortcut Trick

Profit ratio is directly proportional to the product of capital and time for each partner.

Profit Ratio = (Capital Ratio of A) × (Time Ratio of A) : (Capital Ratio of B) × (Time Ratio of B)

Profit Ratio = (3 × 2) : (2 × 3)

Profit Ratio = 6 : 6 = 1 : 1

∴ The correct answer is 1:1.

Alternate Method

Partner A Capital = 3 units Time = 2 units Partner B Capital = 2 units Time = 3 units Profit Ratio (C × T) 3 × 2 : 2 × 3 = 1 : 1

Given:

Ratio of capital of A and B = 3 : 2

Ratio of time for which they invested = 2 : 3

Formula Used:

Profit = Capital × Time

Calculations:

⇒ Let the capital of A and B be 3x and 2x respectively.

⇒ Let the time period for A and B be 2y and 3y respectively.

⇒ Profit of A = Capital of A × Time of A

⇒ Profit of A = 3x × 2y = 6xy

⇒ Profit of B = Capital of B × Time of B

⇒ Profit of B = 2x × 3y = 6xy

⇒ Profit Ratio = Profit of A : Profit of B

⇒ Profit Ratio = 6xy : 6xy

⇒ Profit Ratio = 1 : 1

∴ The correct answer is 1:1.

Additional Information

Working and Sleeping Partner

A working partner actively manages the business and usually receives a set salary or fixed percentage of the overall profit. The remaining profit is distributed among all partners (working and sleeping) in the ratio of their effective investments.

Mid-way Investment or Withdrawal

If a partner adds or withdraws capital mid-way through the term, the effective investment is calculated by splitting the time periods: Effective Capital = (Initial Capital × Time₁) + (New Capital × Time₂).

Three Partners Rule

If three partners A, B, and C invest capitals C₁, C₂, and C₃ for times T₁, T₂, and T₃ respectively, their profit sharing ratio will strictly follow: C₁×T₁ : C₂×T₂ : C₃×T₃.

Shortcut Trick

Total Volume = (1/3) × a² × h = (1/3) × 100 × 12 = 400 cm³

Filled Volume (90%) = 0.9 × 400 = 360 cm³

Total Cost = 360 × 0.50 = ₹180

∴ The correct answer is ₹180.

Alternate Method

Given:

Base side (a) = 10 cm

Height (h) = 12 cm

Volume filled = 90%

Cost of chocolate = ₹0.50 per cm³

Formula Used:

Volume of a Pyramid = (1/3) × Base Area × Height

    a = 10 cm h = 12 cm

Calculations:

⇒ Base Area of the square box = 10 × 10 = 100 cm²

⇒ Total Volume of the pyramid box = (1/3) × 100 × 12

⇒ Total Volume = 400 cm³

⇒ Volume of chocolate to be filled = 90% of 400

⇒ Filled Volume = (90 ÷ 100) × 400 = 360 cm³

⇒ Total cost of chocolate = 360 × 0.50

⇒ Total cost = ₹180

∴ The correct answer is ₹180.

Additional Information

Volume of a Pyramid

The volume of any pyramid is given by V = (1/3) × Base Area × Height. This applies irrespective of whether the base is a triangle, square, or any other polygon.

Surface Area of a Regular Pyramid

Lateral Surface Area = (1/2) × Base Perimeter × Slant Height. The Total Surface Area is the sum of the Lateral Surface Area and the Base Area.

Slant Height of a Square Pyramid

For a regular right pyramid with a square base of side a and vertical height h, the slant height l is given by l = √(h² + (a/2)²).

Shortcut Trick

For a right trapezoidal field, Area = 1/2 × (Sum of parallel sides) × Height.

⇒ Area = 1/2 × (30 + 50) × 20

⇒ Area = 1/2 × 80 × 20

⇒ Area = 40 × 20 = 800 m²

∴ The correct answer is 800 m².

Alternate Method

50 m 30 m 20 m Area of Right Trapezium = 800 m²

Given:

Parallel side 1 (a) = 30 m

Parallel side 2 (b) = 50 m

Height (h) = 20 m (as one non-parallel side is perpendicular to the base)

Formula Used:

Area of a trapezium = 1/2 × (a + b) × h

Calculations:

⇒ Area = 1/2 × (30 + 50) × 20

⇒ Area = 1/2 × 80 × 20

⇒ Area = 40 × 20

⇒ Area = 800 m²

∴ The correct answer is 800 m².

Additional Information

Area of a Trapezium

The formula for the area of any trapezium is 1/2 × (Sum of parallel sides) × (Distance between them).

Right Trapezium

A right trapezium is a special type of trapezium in which one non-parallel side is perpendicular to the parallel bases, directly acting as its height.

Perimeter of Right Trapezium

The perimeter is the sum of all sides. For this trapezium, the length of the slant side can be found using the Pythagoras theorem: √((50 − 30)² + 20²) = √(400 + 400) = 20√2 m.

Shortcut Trick

For a regular hexagon inscribed in a circle, its side length is equal to the radius of the circle.

Area of a regular hexagon = 6 × (Area of equilateral triangle of side R)

Area = 6 × (√3 ÷ 4) × R²

Area = 1.5 × √3 × 21² = 1.5 × 1.732 × 441

Area ≈ 1145.7 cm²

∴ The correct answer is 1145 cm².

Alternate Method

      O 21 cm 21 cm Regular Hexagon Inscribed in a Circle

Given:

Radius of the circle, R = 21 cm

Formula Used:

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

where 'a' is the side length of the regular hexagon.

For a regular hexagon inscribed in a circle, Side (a) = Radius of the circumcircle (R)

Calculation:

⇒ Side of the hexagon (a) = 21 cm

⇒ Area = (3√3 ÷ 2) × (21)²

⇒ Area = (3 × 1.732 ÷ 2) × 441

⇒ Area = 1.5 × 1.732 × 441

⇒ Area = 2.598 × 441

⇒ Area = 1145.718 cm²

⇒ Taking the closest approximate integer, Area ≈ 1145 cm²

∴ The correct answer is 1145 cm².

Additional Information

Equilateral Triangles in Hexagon

A regular hexagon is composed of 6 identical equilateral triangles. The area is simply 6 times the area of one equilateral triangle: 6 × (√3 ÷ 4) × a².

Circumradius vs Inradius

The circumradius (R) is the distance from the center to a vertex and is equal to the side length (a). The inradius (r) is the perpendicular distance from the center to a side and equals (√3 ÷ 2) × a.

Interior and Exterior Angles

Every interior angle of a regular hexagon is 120°, while every exterior angle is 60°.

Shortcut Trick

Let the initial population of males and females be 300 and 200 respectively (to maintain the 3:2 ratio).

New male population = 300 × 110% = 330

New female population = 200 × 125% = 250

New ratio = 330 : 250 = 33 : 25

∴ The correct answer is 33:25.

Alternate Method

Given:

Initial ratio of males to females = 3 : 2

Increase in male population = 10%

Increase in female population = 25%

  Initial Base (300 : 200) Final Population  Male: 300  +10%  Male: 330  Fem: 200  +25%  Fem: 250  Final Ratio → 330 : 250 = 33 : 25

Calculations:

⇒ Let the number of males = 3x and females = 2x.

⇒ New male population = 3x × (100% + 10%) = 3x × 1.10 = 3.3x

⇒ New female population = 2x × (100% + 25%) = 2x × 1.25 = 2.5x

⇒ New Ratio = 3.3x : 2.5x

⇒ Multiply both terms by 10 to remove decimals ⇒ 33x : 25x = 33 : 25

∴ The correct answer is 33:25.

Additional Information

Percentage Increase

When a value increases by R%, the new value is obtained directly by multiplying the original value by (100 + R)/100.

Properties of Ratio

A ratio remains unchanged when both its antecedent (first term) and consequent (second term) are multiplied or divided by the same non-zero number.

Base Assumption Technique

In ratio and percentage problems, assuming base values as multiples of 100 (e.g., 300 and 200 for a 3:2 ratio) perfectly avoids fractional calculations and speeds up the entire solution.

Shortcut Trick

Area of a triangle = ½ × a × b × sin(θ)

Given sides a = 8 cm, b = 10 cm and angle θ = 120°

Area = ½ × 8 × 10 × sin(120°)

Area = 40 × √3 / 2 = 20√3 cm²

∴ The correct answer is 20√3 cm².

Alternate Method

Given:

⇒ Two sides of a triangle, a = 8 cm and b = 10 cm

⇒ Included angle between the sides, θ = 120°

Formula Used:

⇒ Area of triangle = ½ × a × b × sin(θ)

120° A B C 10 cm 8 cm

Calculations:

⇒ sin(120°) = sin(180° − 60°) = sin(60°) = √3 / 2

⇒ Substituting the values in the area formula:

⇒ Area = ½ × 8 × 10 × (√3 / 2)

⇒ Area = 4 × 10 × (√3 / 2)

⇒ Area = 40 × √3 / 2

⇒ Area = 20√3 cm²

∴ The correct answer is 20√3 cm².

Additional Information

Area Using Base and Height

When the base and perpendicular height of a triangle are given: Area = ½ × base × height.

Heron's Formula

When all three sides (a, b, c) of a triangle are known: Area = √[s × (s − a) × (s − b) × (s − c)], where s = (a + b + c) / 2 is the semi-perimeter.

Sine Rule Formula for Area

The area of any triangle can be calculated using two sides and the included angle: Area = ½ × a × b × sin(C) = ½ × b × c × sin(A) = ½ × c × a × sin(B).

Shortcut Trick

Assume Initial Income = 100 ⇒ Expenditure = 80, Savings = 20.

New Income = 125 (25% increase), New Expenditure = 80 (unchanged) ⇒ New Savings = 45.

Increase in Savings = 45 − 20 = 25.

Percentage Increase = (25 ÷ 20) × 100 = 125%.

∴ The correct answer is 125%.

Alternate Method

Given:

Initial expenditure = 80% of income.

Income increases by 25%, while expenditure remains constant.

Formula Used:

Income = Expenditure + Savings

Percentage Increase = (Increase ÷ Original Value) × 100

Original Income: 100 Expenditure: 80 Sav: 20 New Income: 125 (+25%) Expenditure: 80 Savings: 45 +25 Increase

Calculations:

⇒ Let the initial income be 100x.

⇒ Initial expenditure = 80x

⇒ Initial savings = 100x − 80x = 20x

⇒ New income = 100x + 25% of 100x = 125x

⇒ New expenditure = 80x (since it remains the same)

⇒ New savings = 125x − 80x = 45x

⇒ Increase in savings = 45x − 20x = 25x

⇒ Percentage increase in savings = (25x ÷ 20x) × 100

⇒ Percentage increase in savings = 1.25 × 100 = 125%

∴ The correct answer is 125%.

Additional Information

Percentage Change

Percentage change is calculated as the ratio of the net change in a value to its original value, multiplied by 100. Formula: % Change = (New Value − Old Value) ÷ Old Value × 100.

Base of Percentage

Whenever comparing two quantities or calculating an increase/decrease, the original (initial) quantity always acts as the base value in the denominator.

Income, Expenditure & Savings

The fundamental relation governing personal finance in aptitude questions is always: Income = Expenditure + Savings. Any change in income must reflect as a combined change in expenditure and savings.

Shortcut Trick

Efficiency ratio of Sameer to Ajay = 100 : 160 = 5 : 8

Time ratio of Sameer to Ajay = 8 : 5 (since time is inversely proportional to efficiency)

Given, 8 parts → 16 days

⇒ 1 part → 2 days

Ajay's time (5 parts) = 5 × 2 = 10 days.

∴ The correct answer is 10 days.

Alternate Method

Given:

Time taken by Sameer to complete the work = 16 days.

Ajay's efficiency is 60% more than Sameer's efficiency.

Formula Used:

Total Work = Efficiency × Time

Calculations:

⇒ Let the efficiency of Sameer be 100 units/day.

⇒ Efficiency of Ajay = 100 + 60% of 100 = 160 units/day.

⇒ Ratio of their efficiencies (Sameer : Ajay) = 100 : 160 = 5 : 8.

⇒ Let Sameer's efficiency be 5 units/day and Ajay's efficiency be 8 units/day.

⇒ Total Work = Sameer's efficiency × Sameer's time = 5 × 16 = 80 units.

⇒ Time taken by Ajay = Total Work ÷ Ajay's efficiency

⇒ Time taken by Ajay = 80 ÷ 8 = 10 days.

∴ The correct answer is 10 days.

Additional Information

Efficiency and Time Relationship

Efficiency is inversely proportional to time. If A is twice as efficient as B, A will take half the time B takes to complete the same amount of work.

Combined Work Concept

If person A takes x days and person B takes y days, their combined efficiency is (1/x + 1/y). Thus, the time taken to complete the work together is (x × y)/(x + y) days.

Work and Wages

Wages are distributed in direct proportion to the actual amount of work done by each individual. If they work for the exact same number of days, wages are divided directly in the ratio of their efficiencies.

Shortcut Trick

Volume of Prism = Base Area × Prism Height

⇒ V = [1/2 × (Sum of parallel sides) × Base height] × Prism height

⇒ V = [1/2 × (10 + 6) × 4] × 15

⇒ V = [1/2 × 16 × 4] × 15

⇒ V = 32 × 15 = 480

∴ The correct answer is 480 cm³.

Alternate Method

Given:

Parallel sides of the trapezium base (a, b) = 10 cm and 6 cm

Distance between parallel sides or height of base (h) = 4 cm

Height of the prism (H) = 15 cm

6 cm 10 cm 4 cm Trapezium Base Area

Prism H = 15 cm

Formula Used:

Area of Trapezium = 1/2 × (Sum of parallel sides) × distance between them

Volume of a Prism = Area of Base × Height of Prism

Calculations:

⇒ Area of Trapezium Base = 1/2 × (10 + 6) × 4

⇒ Area of Trapezium Base = 1/2 × 16 × 4

⇒ Area of Trapezium Base = 8 × 4 = 32 cm²

⇒ Volume of Prism = Area of Base × Prism Height

⇒ Volume of Prism = 32 × 15

⇒ Volume of Prism = 480 cm³

∴ The correct answer is 480 cm³.

Additional Information

Volume of a Prism

The volume of any right prism is calculated by multiplying its base area by its height. Formula: V = Base Area × Height.

Total Surface Area of a Prism

The total surface area includes the lateral surface faces and the two parallel bases. Formula: TSA = (Perimeter of Base × Height) + 2 × Base Area.

Volume of a Pyramid

A pyramid tapers to an apex, making its volume exactly one-third of a prism with the same base and height. Formula: V = 1/3 × Base Area × Height.

Shortcut Trick

Using the Net Deviation Method from the overall average (61):

Deviation of first 6 numbers = 6 × (57 − 61) = 6 × (−4) = −24

Deviation of last 6 numbers = 6 × (65 − 61) = 6 × (+4) = +24

6^th number = Overall Average + Net Deviation = 61 + (−24 + 24) = 61

∴ The correct answer is 61.

Alternate Method

Given:

Total numbers = 11, Overall average = 61

Average of first 6 numbers = 57

Average of last 6 numbers = 65

11 Numbers (Average = 61, Sum = 671) First 6 (Sum = 342) Last 6 (Sum = 390) 6^th 6^th Number = (342 + 390) − 671 = 61

Formula Used:

Sum = Average × Number of values

6^th number = (Sum of first 6) + (Sum of last 6) − (Total Sum of 11 numbers)

Calculations:

⇒ Total sum of 11 numbers = 11 × 61 = 671

⇒ Sum of first 6 numbers = 6 × 57 = 342

⇒ Sum of last 6 numbers = 6 × 65 = 390

⇒ 6^th number = 342 + 390 − 671

⇒ 6^th number = 732 − 671 = 61

∴ The correct answer is 61.

Additional Information

Basic Average Formula

The average is equal to the sum of all observations divided by the total number of observations: Average = Sum ÷ Count.

Net Deviation Method

The algebraic sum of deviations of all individual values from their exact average is zero. This principle is used to find an unknown value rapidly by balancing positive and negative deviations.

Overlapping Groups

When a series is divided into two groups that overlap by exactly one element, the overlapping element is calculated by: Overlapping Element = (Sum of Group 1 + Sum of Group 2) − Total Sum.

Shortcut Trick

The total volume of 60 liters is not needed to find the average cost when the mixing ratio is given.

Average Cost = (Ratio of A × Cost of A + Ratio of B × Cost of B) ÷ (Ratio of A + Ratio of B)

Average Cost = (2 × 45 + 3 × 30) ÷ (2 + 3)

Average Cost = (90 + 90) ÷ 5 = 180 ÷ 5 = 36

∴ The correct answer is Rs. 36.

Alternate Method

Given:

Total volume = 60 liters

Ratio of Liquid 'A' to Liquid 'B' = 2 : 3

Cost of Liquid 'A' = Rs. 45/liter | Cost of Liquid 'B' = Rs. 30/liter

Liquid A 45 Liquid B 30 Mean x x − 30 45 − x

Calculations:

By the rule of alligation, the ratio of Liquid 'A' to Liquid 'B' is derived by cross-subtracting the mean price from the respective individual prices:

⇒ (Mean Cost − Cost of B) ÷ (Cost of A − Mean Cost) = Ratio of A ÷ Ratio of B

⇒ (x − 30) ÷ (45 − x) = 2 ÷ 3

⇒ 3 × (x − 30) = 2 × (45 − x)

⇒ 3x − 90 = 90 − 2x

⇒ 3x + 2x = 90 + 90

⇒ 5x = 180

⇒ x = 36

∴ The correct answer is Rs. 36.

Additional Information

Rule of Alligation

It is used to find the ratio in which two ingredients at given prices must be mixed to produce a mixture at a desired mean price. The ratio is derived as: Quantity of Cheaper ÷ Quantity of Dearer = (Price of Dearer − Mean Price) ÷ (Mean Price − Price of Cheaper).

Weighted Average Method

When components with quantities n₁ and n₂ have individual averages A₁ and A₂, the net average of the entire mixture can be calculated using the direct combined formula: Combined Average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Irrelevant Data in Ratio Questions

Whenever a question asks for a rate (such as the average cost per liter) and provides the relative ratio of the mixing components, the absolute total quantity (e.g., 60 liters) does not affect the final rate and can be ignored to speed up calculations in exams.

Shortcut Trick

Given, sin A + cos A = 1

This condition is satisfied when A = 0° or A = 90°.

Let A = 0°, then sin 0° = 0 and cos 0° = 1.

Now put A = 0° in the given expression: sin⁴ A + cos⁴ A

⇒ 0⁴ + 1⁴ = 1

∴ The correct answer is 1.

Alternate Method

Given:

sin A + cos A = 1

Formula Used:

(a + b)² = a² + b² + 2ab

sin² θ + cos² θ = 1

a⁴ + b⁴ = (a² + b²)² − 2a²b²

Calculations:

⇒ sin A + cos A = 1

Squaring both sides, we get:

⇒ (sin A + cos A)² = 1²

⇒ sin² A + cos² A + 2 sin A cos A = 1

Since sin² A + cos² A = 1:

⇒ 1 + 2 sin A cos A = 1

⇒ 2 sin A cos A = 1 − 1

⇒ 2 sin A cos A = 0

⇒ sin A cos A = 0

Now, we need to find the value of sin⁴ A + cos⁴ A.

⇒ sin⁴ A + cos⁴ A = (sin² A)² + (cos² A)²

Applying the algebraic formula for a⁴ + b⁴:

⇒ (sin² A + cos² A)² − 2 sin² A cos² A

⇒ (1)² − 2 (sin A cos A)²

Substituting the value of sin A cos A = 0:

⇒ 1 − 2 × (0)²

⇒ 1 − 0 = 1

∴ The correct answer is 1.

Additional Information

Trigonometric Identities

The three fundamental Pythagorean trigonometric identities are: sin² θ + cos² θ = 1, 1 + tan² θ = sec² θ, and 1 + cot² θ = cosec² θ.

Algebraic Identities in Trigonometry

Expressions like sin⁴ θ + cos⁴ θ can be simplified using a⁴ + b⁴ = (a² + b²)² − 2a²b² to get 1 − 2 sin² θ cos² θ.

Similarly for Power of 6

sin⁶ θ + cos⁶ θ can be easily evaluated using a³ + b³ = (a + b)(a² + b² − ab) or (a + b)³ − 3ab(a + b) to get the direct result 1 − 3 sin² θ cos² θ.

Shortcut Trick

Using the basic trigonometric quotient identity:

⇒ tanA = sinA ÷ cosA

⇒ tanA = 0.6 ÷ 0.8

⇒ tanA = 6 ÷ 8 = 0.75

∴ The correct answer is 0.75.

Alternate Method

Given:

sinA = 0.6

cosA = 0.8

A B C Base (AB) = 0.8 Perp. (BC) = 0.6 Hypotenuse = 1

Formula Used:

In a right-angled triangle:

⇒ sinA = Perpendicular ÷ Hypotenuse

⇒ cosA = Base ÷ Hypotenuse

⇒ tanA = Perpendicular ÷ Base

Calculation:

Let the Hypotenuse of the right-angled triangle be 1 unit.

⇒ Perpendicular = 0.6 units

⇒ Base = 0.8 units

Now, substituting the values to find tanA:

⇒ tanA = 0.6 ÷ 0.8

⇒ tanA = 6 ÷ 8

⇒ tanA = 3 ÷ 4 = 0.75

∴ The correct answer is 0.75.

Additional Information

Pythagorean Identity

The fundamental identity states that sin²A + cos²A = 1. (Here, 0.6² + 0.8² = 0.36 + 0.64 = 1, confirming the values are valid).

Reciprocal Identities

The reciprocal trigonometric ratios are: cosecA = 1 ÷ sinA, secA = 1 ÷ cosA, and cotA = 1 ÷ tanA.

Quotient Identity

The tangent and cotangent can be expressed as: tanA = sinA ÷ cosA and cotA = cosA ÷ sinA.

Shortcut Trick

From the equation y = mx + 1, we know the y-intercept is 1, meaning the line passes through (0, 1).

We are given another point (2, 5) on the line.

Slope m = (y₂ − y₁) ÷ (x₂ − x₁)

m = (5 − 1) ÷ (2 − 0) = 4 ÷ 2 = 2.

∴ The correct answer is 2.

Alternate Method

Given:

Equation of the straight line: y = mx + 1

Point on the line: (x, y) = (2, 5)

Method Used:

Since the line passes through the point (2, 5), the coordinates x = 2 and y = 5 must satisfy the given equation.

x y 0 (0, 1) (2, 5) 2 5 y = mx + 1

Calculations:

⇒ y = mx + 1

⇒ Substitute x = 2 and y = 5 into the equation.

⇒ 5 = m × 2 + 1

⇒ 5 − 1 = 2m

⇒ 4 = 2m

⇒ m = 4 ÷ 2

⇒ m = 2

∴ The correct answer is 2.

Additional Information

Slope-Intercept Form

The equation y = mx + c represents a straight line, where m is the slope of the line and c is the y-intercept.

Slope Formula

The slope m of a line passing through two specific points (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁) ÷ (x₂ − x₁).

Line Intercepts

The x-intercept is the point where the line crosses the x-axis (set y = 0), and the y-intercept is where it crosses the y-axis (set x = 0).

Shortcut Trick

We know the complementary angle trigonometric identity: cot(90° − A) = tan A.

Given sin A = 4/5, the perpendicular is 4 and the hypotenuse is 5.

This forms a standard 3-4-5 Pythagorean triplet, so the base is 3.

Therefore, tan A = Perpendicular ÷ Base = 4/3.

∴ The correct answer is 4/3.

Alternate Method

Given:

sin A = 4/5

A is an acute angle.

Formula Used:

sin A = Perpendicular ÷ Hypotenuse

Base = √(Hypotenuse² − Perpendicular²)

cot(90° − A) = tan A = Perpendicular ÷ Base

A Base = 3 Perp = 4 Hypotenuse = 5 A B C

Calculations:

⇒ Let Perpendicular = 4x and Hypotenuse = 5x.

⇒ By Pythagoras theorem: Base² + Perpendicular² = Hypotenuse²

⇒ Base² + (4x)² = (5x)²

⇒ Base² + 16x² = 25x²

⇒ Base² = 9x²

⇒ Base = 3x

⇒ cot(90° − A) = tan A = Perpendicular ÷ Base

⇒ tan A = 4x ÷ 3x = 4/3

∴ The correct answer is 4/3.

Additional Information

Complementary Angles in Trigonometry

For any acute angle A: sin(90° − A) = cos A, cos(90° − A) = sin A, and tan(90° − A) = cot A.

Pythagorean Triplets

A set of three positive integers a, b, and c that satisfy a² + b² = c². Common triplets include (3, 4, 5), (5, 12, 13), and (8, 15, 17).

Trigonometric Ratios

In a right-angled triangle, sin θ = Perpendicular ÷ Hypotenuse, cos θ = Base ÷ Hypotenuse, and tan θ = Perpendicular ÷ Base.

Shortcut Trick

For any regular polygon, the sum of all exterior angles is always 360°.

Number of sides (n) = 360° ÷ Exterior angle

⇒ n = 360° ÷ 20°

⇒ n = 18

∴ The correct answer is 18.

Alternate Method

Given:

Measure of an exterior angle = 20°

Formula Used:

Number of sides of a regular polygon (n) = 360° ÷ Measure of each exterior angle

    20°  160° Vertex Side Side Exterior Angle = 360° ÷ n

Calculations:

⇒ n = 360° ÷ 20°

⇒ n = 18

∴ The correct answer is 18.

Additional Information

Interior Angle Formula

The measure of each interior angle of a regular polygon with n sides is given by: [(n − 2) × 180°] ÷ n. Also, Interior angle + Exterior angle = 180°.

Sum of Interior Angles

The sum of all interior angles of a polygon with n sides is (n − 2) × 180°.

Number of Diagonals

The total number of diagonals in an n-sided polygon can be calculated using the formula: [n(n − 3)] ÷ 2.

Shortcut Trick

In a triangle, the coordinates of the centroid are the average of the coordinates of its vertices.

Sum of x-coordinates = 3 × x-coordinate of Centroid.

⇒ x + 5 + 2 = 3 × 4 ⇒ x + 7 = 12 ⇒ x = 5

Sum of y-coordinates = 3 × y-coordinate of Centroid.

⇒ y + (−1) + 6 = 3 × 2 ⇒ y + 5 = 6 ⇒ y = 1

∴ The correct answer is (5, 1).

Alternate Method

  P(x, y) Q(5, −1) R(2, 6) G(4, 2)

Given:

Vertices of the triangle: P(x, y), Q(5, −1), and R(2, 6)

Centroid of the triangle: G(4, 2)

Formula Used:

The coordinates of the Centroid G(X, Y) of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) are given by:

G(X, Y) = [ (x₁ + x₂ + x₃) ÷ 3 , (y₁ + y₂ + y₃) ÷ 3 ]

Calculations:

⇒ Comparing the x-coordinate of the centroid:

⇒ 4 = (x + 5 + 2) ÷ 3

⇒ 12 = x + 7

⇒ x = 12 − 7

⇒ x = 5

⇒ Comparing the y-coordinate of the centroid:

⇒ 2 = (y − 1 + 6) ÷ 3

⇒ 6 = y + 5

⇒ y = 6 − 5

⇒ y = 1

⇒ The coordinates of vertex P are (5, 1).

∴ The correct answer is (5, 1).

Additional Information

Centroid of a Triangle

The centroid is the point of intersection of the medians of a triangle. It divides each median in a 2:1 ratio. Formula: G(X, Y) = [(x₁ + x₂ + x₃) ÷ 3, (y₁ + y₂ + y₃) ÷ 3].

Incenter of a Triangle

The incenter is the point of intersection of the internal angle bisectors. Coordinates: I(X, Y) = [(ax₁ + bx₂ + cx₃) ÷ (a + b + c), (ay₁ + by₂ + cy₃) ÷ (a + b + c)], where a, b, and c are side lengths.

Circumcenter of a Triangle

The circumcenter is the point where the perpendicular bisectors of the sides intersect. It is equidistant from all three vertices of the triangle.

Orthocenter of a Triangle

The orthocenter is the point where all three altitudes of the triangle intersect. For a right-angled triangle, the orthocenter lies at the vertex containing the right angle.

Shortcut Trick

Square the given value of x directly using the identity (a + b)² = a² + b² + 2ab.

x² = (√2 + 1)² = 2 + 1 + 2√2 = 3 + 2√2.

Subtract 2 from the result to get the required expression directly.

x² − 2 = 3 + 2√2 − 2 = 1 + 2√2.

∴ The correct answer is 1 + 2√2.

Alternate Method

Given:

x = √2 + 1

Formula Used:

(a + b)² = a² + b² + 2ab

Calculations:

⇒ x² = (√2 + 1)²

⇒ x² = (√2)² + (1)² + 2 × √2 × 1

⇒ x² = 2 + 1 + 2√2

⇒ x² = 3 + 2√2

Now, substitute the value of x² into the required expression x² − 2:

⇒ x² − 2 = (3 + 2√2) − 2

⇒ x² − 2 = 1 + 2√2

∴ The correct answer is 1 + 2√2.

Additional Information

Difference of Squares

The algebraic identity (a + b)(a − b) = a² − b² is particularly useful for rationalizing denominators involving surds or square roots.

Square of a Difference

Similar to the sum of squares, the formula for subtraction is (a − b)² = a² + b² − 2ab.

Cube of a Sum

The standard identity for the cube of a sum is expressed as (a + b)³ = a³ + b³ + 3ab(a + b).

Shortcut Trick

A median of a triangle divides the triangle into two smaller triangles of exactly equal area.

Since AD is the median to side BC, it bisects the base into two equal segments (BD = CD).

Both △ABD and △ACD share the same perpendicular height from vertex A.

Therefore, their areas are identical.

∴ The correct answer is 1:1.

Alternate Method

     A B C D M      Base (x) Base (x) Area 1 Area 2 h

Given:

⇒ In △ABC, AD is the median to the side BC.

Formula Used:

⇒ Area of a triangle = 1/2 × Base × Height

Calculations:

⇒ Since AD is the median, it bisects the base BC into two equal halves.

⇒ Therefore, length of BD = length of CD = x (let).

⇒ Let the perpendicular height of the triangle from vertex A to side BC be h.

⇒ Area of △ABD = 1/2 × BD × h = 1/2 × x × h

⇒ Area of △ACD = 1/2 × CD × h = 1/2 × x × h

⇒ Ratio of Area of △ABD to Area of △ACD = (1/2 × x × h) ÷ (1/2 × x × h)

⇒ Ratio = 1:1

∴ The correct answer is 1:1.

Additional Information

Centroid of a Triangle

The intersection point of all three medians is called the centroid. It divides each median in the ratio of 2:1.

Area Division by Medians

The three medians of a triangle divide the triangle into six smaller triangles of exactly equal area.

Apollonius's Theorem

It connects the length of a median to the sides of the triangle: AB² + AC² = 2 × (AD² + BD²).