Quantitative Aptitude PYQs

Shortcut Trick

Using the deviation method to find the average easily:

Let the assumed average be 220.

Deviations of Worker C's daily wages (220, 230, 215, 225) from 220 are: 0, +10, −5, +5.

Net deviation = 0 + 10 − 5 + 5 = +10

Average deviation = +10 ÷ 4 = +2.5

Actual average = 220 + 2.5 = 222.5

∴ The correct answer is 222.5.

Alternate Method

Given:

Daily earnings of Worker C extracted from the table:

Day 1 = 220, Day 2 = 230, Day 3 = 215, Day 4 = 225

Total number of days = 4

Formula Used:

Average = (Sum of all observations) ÷ (Total number of observations)

Calculations:

⇒ Total earnings of Worker C over 4 days = 220 + 230 + 215 + 225

⇒ Total earnings of Worker C = 890

⇒ Average daily earning = 890 ÷ 4

⇒ Average daily earning = 222.5

∴ The correct answer is 222.5.

Additional Information

Arithmetic Mean (Average)

It is the simplest and most widely used measure of central tendency. The formula is simply the sum of elements ÷ number of elements.

Deviation Method for Averages

Assume a base average value. Find the deviation (difference) of each number from this base. The actual average is the assumed base plus the net average of these deviations. This greatly reduces calculation time for large numbers.

Data Interpretation (Tables)

Tabular data presents numerical figures in rows and columns. Always read the row and column headers carefully to extract the precise data points needed before initiating any calculation.

Shortcut Trick

Net increase in population = 25250 − 25000 = 250.

Overall percentage change = (250 ÷ 25000) × 100 = +1%.

Using the Alligation method on the percentage change of Males (−5%) and Females (+10%):

Ratio of Males to Females = (10 − 1) : (1 − (−5)) = 9 : 6 = 3 : 2.

Present number of literate males = (3 ÷ 5) × 25000 = 15000.

∴ The correct answer is 15,000 males.

Alternate Method

Given:

Total current literate population = 25,000

New total literate population = 25,250

Net increase = 25,250 − 25,000 = 250

Total: 25,000Males (M)Females (F)−5%+10%Net Change: +250

Calculations:

Let the current number of literate males be M and females be F.

⇒ M + F = 25,000 --- (Equation 1)

The number of males decreases by 5% and females increases by 10%, resulting in a net increase of 250.

⇒ 10% of F − 5% of M = 250

⇒ (10 ÷ 100) × F − (5 ÷ 100) × M = 250

Multiplying the entire equation by 100:

⇒ 10F − 5M = 25,000

Dividing by 5 to simplify:

⇒ 2F − M = 5,000 --- (Equation 2)

Adding Equation 1 and Equation 2:

⇒ (M + F) + (2F − M) = 25,000 + 5,000

⇒ 3F = 30,000

⇒ F = 10,000

Substituting the value of F back into Equation 1:

⇒ M + 10,000 = 25,000

⇒ M = 15,000

∴ The correct answer is 15,000 males.

Additional Information

Alligation Rule

It is a powerful technique used to quickly find the ratio in which two components are combined to achieve a given overall change. The shortcut structure dictates: Ratio = (Higher % − Net %) : (Net % − Lower %).

Percentage Shift Method (Alternate Mental Trick)

If both groups decreased by 5%, the total decrease would be 5% of 25,000 = 1,250. The actual change is an increase of 250, creating a numerical gap of 1,500. This gap is entirely filled by the 15% difference in the females' group (from −5% to +10%). Hence, 15% of Females = 1,500 ⇒ Females = 10,000.

Linear Equations in Two Variables

When solving questions with two interconnected parts (like males and females in a total population), setting up two equations (Sum of parts = Total, and Sum of their changes = Net change) reliably yields an exact algebraic solution without getting confused by ratios.

Shortcut Trick

Let the price of standard flour = ₹1/kg and premium flour = ₹2/kg.

Original cost = 10 × 2 + x × 1 = 20 + x

New cost = x × 2 + 10 × 1 = 2x + 10

As cost decreased by 25%, New Cost = 75% of Original Cost ⇒ 3/4

⇒ (2x + 10) / (20 + x) = 3/4 ⇒ 8x + 40 = 60 + 3x ⇒ 5x = 20 ⇒ x = 4 kg.

∴ The correct answer is 5:2.

Alternate Method

Original Order Prem: 10 kg @ 2P Std: x kg @ 1P Cost = 20P + xP

New Order Prem: x kg @ 2P Std: 10 kg @ 1P Cost = 2xP + 10P

−25% Cost

Given:

⇒ Premium flour quantity = 10 kg

⇒ Let standard flour quantity = x kg

⇒ Let standard flour price = P per kg, then premium flour price = 2P per kg

Calculations:

⇒ Original Total Cost = (10 × 2P) + (x × P) = P(20 + x)

⇒ After interchange, New Total Cost = (x × 2P) + (10 × P) = P(2x + 10)

⇒ The new cost is 25% less than the original cost.

⇒ New Cost = (100% − 25%) of Original Cost = 75% of Original Cost

⇒ P(2x + 10) = 0.75 × P(20 + x)

⇒ 2x + 10 = 0.75(20) + 0.75x

⇒ 2x + 10 = 15 + 0.75x

⇒ 2x − 0.75x = 15 − 10

⇒ 1.25x = 5

⇒ x = 5 ÷ 1.25 = 4

⇒ The original quantity of standard flour is 4 kg.

⇒ Ratio of premium to standard flour = 10 : 4 = 5 : 2

∴ The correct answer is 5:2.

Additional Information

Cost-Quantity Relationship

The total cost or value of a commodity is the product of its unit price and the total quantity purchased: Total Cost = Price per unit × Quantity.

Percentage Change

A decrease of R% in a value means the new value becomes (100 − R)% of the original value. For example, a 25% decrease results in a multiplier of 0.75 or 3/4.

Ratio and Proportion

A ratio compares two quantities of the same unit, representing how many times one value contains or is contained within the other. Ratios should always be expressed in their simplest form.

Shortcut Trick

Using the combined ratio method: Amount₁ ÷ (Quantity₁ × SP₁%) = Amount₂ ÷ (Quantity₂ × SP₂%)

⇒ 3000 ÷ (15 × 125%) = 2816 ÷ (N × 110%)

⇒ 200 ÷ 125 = 2816 ÷ 110N

⇒ 8 ÷ 5 = 25.6 ÷ N ⇒ N = 25.6 × 5 ÷ 8 = 16

∴ The correct answer is 16 toys.

Alternate Method

Given:

Selling price of 15 toys = ₹3000

Initial profit percentage = 25%

Required profit percentage = 10%

Target selling amount = ₹2816

CP: ₹160

SP&sub1;: ₹200

SP&sub2;: ₹176

+25% Profit

+10% Profit

Calculations:

⇒ Selling price of 1 toy (SP₁) = 3000 ÷ 15 = ₹200

⇒ Cost price of 1 toy (CP) = SP₁ × [100 ÷ (100 + P₁%)]

⇒ CP = 200 × (100 ÷ 125) = ₹160

⇒ New selling price of 1 toy for 10% profit (SP₂) = CP × [100 + P₂%] ÷ 100

⇒ SP₂ = 160 × (110 ÷ 100) = ₹176

⇒ Number of toys to sell = Target amount ÷ SP₂

⇒ Number of toys = 2816 ÷ 176 = 16

∴ The correct answer is 16 toys.

Additional Information

Cost Price Formula

Cost Price = Selling Price × [100 ÷ (100 + Profit %)]. It acts as the fundamental base for calculating any subsequent profit or loss margins.

Successive Changes in Selling Price

When the SP varies due to a different profit or loss percentage on the same object, use the direct ratio: SP₁ ÷ (100 + P₁) = SP₂ ÷ (100 + P₂).

Unitary Method for Quantities

Total Value = Price per unit × Total Quantity. Applying this helps compare different transaction scenarios independently of the absolute Cost Price.

Shortcut Trick

CP of Brand A = 150 × (100 / 75) = ₹200/kg

CP of Brand B = 200 × (100 / 80) = ₹250/kg

For an equal mixture (1 kg each), Total CP = 200 + 250 = ₹450

Total SP for 2 kg = 175 × 2 = ₹350

Loss % = ((450 − 350) / 450) × 100 = (100 / 450) × 100 = 22.22%

∴ The correct answer is Loss of 22.22%.

Alternate Method

Given:

SP of Brand A = ₹150/kg, Loss = 25%

SP of Brand B = ₹200/kg, Loss = 20%

Mixture proportion = 1 : 1

SP of the mixture = ₹175/kg

Formula Used:

CP = SP × [100 / (100 − Loss %)]

Loss % = (Loss / Total CP) × 100

Brand A (1 kg) CP = ₹200

Brand B (1 kg) CP = ₹250

Mixture (2 kg) Total CP = ₹450

Total SP = ₹350 Overall Loss = ₹100

Calculations:

⇒ CP of Brand A = 150 × [100 / (100 − 25)]

⇒ CP of Brand A = 150 × (100 / 75) = ₹200/kg

⇒ CP of Brand B = 200 × [100 / (100 − 20)]

⇒ CP of Brand B = 200 × (100 / 80) = ₹250/kg

⇒ Assuming he mixes exactly 1 kg of each brand, Total Quantity = 2 kg

⇒ Total CP for 2 kg = 200 + 250 = ₹450

⇒ Total SP for 2 kg = 175 × 2 = ₹350

⇒ Overall Loss = Total CP − Total SP = 450 − 350 = ₹100

⇒ Overall Loss % = (Loss / Total CP) × 100

⇒ Overall Loss % = (100 / 450) × 100 = (2 / 9) × 100 ≈ 22.22%

∴ The correct answer is Loss of 22.22%.

Additional Information

Weighted Average Cost Price

When mixing two quantities in ratio n₁ : n₂, the average CP = (n₁ × CP₁ + n₂ × CP₂) / (n₁ + n₂).

Alligation Method

The ratio of quantities mixed is inversely proportional to the difference between their unit prices and the mean price: (Quantity 1) / (Quantity 2) = (CP₂ − Mean Price) / (Mean Price − CP₁).

Profit and Loss on Overall Transaction

To find the exact profit or loss percentage for mixed articles, always calculate the total Selling Price (SP) of the final mixture and compare it against the total Cost Price (CP) of the combined ingredients.

Shortcut Trick

Total work = LCM of 12, 15 and 20 = 60 units.

Combined efficiencies: (A + B) = 60 ÷ 12 = 5, (B + C) = 60 ÷ 15 = 4, (C + A) = 60 ÷ 20 = 3.

Total efficiency of 2(A + B + C) = 5 + 4 + 3 = 12 ⇒ Efficiency of (A + B + C) = 6.

Efficiency of A = Total Efficiency − Efficiency of (B + C) = 6 − 4 = 2 units/day.

Time taken by A alone = Total work ÷ Efficiency of A = 60 ÷ 2 = 30 days.

∴ The correct answer is 30 days.

Alternate Method

Given:

Time taken by A and B together = 12 days

Time taken by B and C together = 15 days

Time taken by C and A together = 20 days

Formula Used:

Total Work = Efficiency × Time

Calculations:

⇒ Adding all the combined efficiencies: (A + B) + (B + C) + (C + A) = 5 + 4 + 3

⇒ 2(A + B + C) = 12

⇒ Efficiency of A + B + C = 12 ÷ 2 = 6 units/day

⇒ Efficiency of A = (Efficiency of A + B + C) − (Efficiency of B + C)

⇒ Efficiency of A = 6 − 4 = 2 units/day

⇒ Time taken by A alone = Total Work ÷ Efficiency of A

⇒ Time taken by A alone = 60 ÷ 2 = 30 days

∴ The correct answer is 30 days.

Additional Information

Pipes & Cisterns

This concept works exactly like Time & Work. Inlet pipes have positive efficiency (filling) and outlet pipes have negative efficiency (emptying).

Work & Wages

Wages are always distributed in the direct ratio of the total work done by each person. If the working time is equal for all, wages are distributed in the ratio of their efficiencies.

Alternate Day Work

When people work on alternate days, calculate the combined work done in one complete cycle (e.g., 2 days for 2 people) and use it to find the number of cycles required to complete the total work.

Shortcut Trick

Direct Replacement Formula: Remaining Spirit = Initial Quantity × (1 − Replaced Quantity / Initial Quantity)ⁿ

⇒ Remaining Quantity = 64 × (1 − 8 / 64)³

⇒ Remaining Quantity = 64 × (7 / 8)³ = 64 × (343 / 512)

⇒ Remaining Quantity = 343 / 8 = 42.875 ≈ 42.88 liters

∴ The correct answer is 42.88 liters.

Alternate Method

Repeated Replacement Process

Initial 64L Pure Spirit

• 8L Mixture

• 8L Water (Repeated 3 Times)

Final 42.88L Spirit

Given:

Initial volume of pure spirit (V) = 64 liters

Volume of mixture replaced with water each time (x) = 8 liters

Total number of operations (n) = 3

Formula Used:

Final Quantity of Spirit = V × (1 − x / V)ⁿ

Calculations:

⇒ Final Quantity = 64 × (1 − 8 / 64)³

⇒ Final Quantity = 64 × (1 − 1 / 8)³

⇒ Final Quantity = 64 × (7 / 8)³

⇒ Final Quantity = 64 × (343 / 512)

⇒ Final Quantity = 343 / 8

⇒ Final Quantity = 42.875 ≈ 42.88 liters

∴ The correct answer is 42.88 liters.

Additional Information

Repeated Replacement Formula

If a container holds V units of pure liquid, and x units are replaced by water n times, the final quantity is given by V × (1 − x/V)ⁿ.

Successive Replacement

If varying amounts a, b, and c are replaced successively, the remaining pure liquid is V × (1 − a/V) × (1 − b/V) × (1 − c/V).

Rule of Alligation

Used to find the ratio in which two ingredients of different prices or concentration are mixed. The ratio is calculated as (Dearer Quantity − Mean Quantity) : (Mean Quantity − Cheaper Quantity).

Shortcut Trick

When a mixture is sold at the cost price of the pure liquid, the profit earned is entirely due to the free liquid (water) added.

Ratio of Water : Pure Liquid = Profit % : 100%

Ratio of Water : Pure Orange Juice = 20 : 100

Ratio of Water : Pure Orange Juice = 1 : 5

∴ The correct answer is 1 : 5.

Alternate Method

Given:

Profit earned = 20%

Selling price of the mixture = Cost price of pure orange juice

Formula Used:

Quantity of Cheaper / Quantity of Dearer = (Price of Dearer − Mean Price) / (Mean Price − Price of Cheaper)

Mean Price (CP of Mixture) = (SP × 100) ÷ (100 + Profit %)

Calculation:

Water CP = 0

Juice CP = 100

Mean = 500/6

100 − 500/6 100/6

500/6 − 0 500/6

Ratio = 1 : 5

⇒ Let the cost price (CP) of pure orange juice be ₹100 per unit.

⇒ Since the mixture is sold at the CP of pure juice, Selling Price (SP) of the mixture = ₹100.

⇒ The given profit is 20%, so CP of the mixture = (SP × 100) ÷ (100 + 20)

⇒ CP of the mixture = (100 × 100) ÷ 120 = 500/6.

⇒ The cost price of water is ₹0 per unit.

⇒ Applying the rule of alligation as shown in the diagram above:

⇒ Quantity of water = 100 − 500/6 = 100/6

⇒ Quantity of pure juice = 500/6 − 0 = 500/6

⇒ Ratio of water to pure juice = 100/6 : 500/6

⇒ Ratio = 1 : 5

∴ The correct answer is 1 : 5.

Additional Information

Rule of Alligation

It states that if two ingredients at given prices are mixed to produce a mixture at a given mean price, the ratio of their quantities is inversely proportional to the differences of their prices from the mean price. Ratio = (Price of Dearer − Mean Price) : (Mean Price − Price of Cheaper).

Dishonest Dealer Concept

When a dealer professes to sell his goods at cost price but uses a false weight or adds impurities, the profit percentage is calculated as: Profit % = (Error ÷ True Value) × 100. This is conceptually identical to adding free water to pure goods.

Replacement Formula in Mixtures

If from a vessel containing 'x' units of pure liquid, 'y' units are removed and replaced with water, and this operation is repeated 'n' times, the quantity of pure liquid left = x × (1 − y/x)ⁿ.

Shortcut Trick

Let the total distance be assumed as 100 km.

First part of the distance = 40 km, covered at 20 km/h.

Time taken = 40 ÷ 20 = 2 hours.

Remaining distance = 60 km, covered at 30 km/h.

Time taken = 60 ÷ 30 = 2 hours.

Average Speed = Total Distance ÷ Total Time = 100 ÷ (2 + 2) = 25 km/h.

∴ The correct answer is 25 km/h.

Alternate Method

Given:

First part of distance (D₁) = 40% of total distance

Speed for first part (S₁) = 20 km/h

Remaining distance (D₂) = 60% of total distance

Speed for remaining part (S₂) = 30 km/h

Formula Used:

Average Speed = Total Distance ÷ Total Time

Time = Distance ÷ Speed

40% Distance 60% Distance

20 km/h 30 km/h

Total Distance Journey Average Speed = ?

Calculations:

⇒ Let the total distance be D km.

⇒ Time taken for the first part = 0.4D ÷ 20 = 4D ÷ 200 = D ÷ 50 hours

⇒ Time taken for the remaining part = 0.6D ÷ 30 = 6D ÷ 300 = D ÷ 50 hours

⇒ Total Time taken = (D ÷ 50) + (D ÷ 50) = 2D ÷ 50 = D ÷ 25 hours

⇒ Average Speed = Total Distance ÷ Total Time

⇒ Average Speed = D ÷ (D ÷ 25)

⇒ Average Speed = 25 km/h

∴ The correct answer is 25 km/h.

Additional Information

Average Speed for Equal Distances

When a certain distance is covered at speed 'x' and an equal distance at speed 'y', the Average Speed = (2xy) ÷ (x + y).

Average Speed for Equal Time Intervals

When travelling for equal time periods at speeds 'x' and 'y', the Average Speed = (x + y) ÷ 2.

Relative Speed

If two bodies move in the same direction, relative speed = S₁ − S₂. If they move in opposite directions, relative speed = S₁ + S₂.

Shortcut Trick

Relative speed (opposite directions) = 5 + 7 = 12 km/h.

Time for first meeting = Distance ÷ Relative speed = 1.2 ÷ 12 = 0.1 hours = 6 minutes.

Time for 5^th meeting = 5 × 6 = 30 minutes.

Meeting Time = 6:00 AM + 30 minutes = 6:30 AM.

∴ The correct answer is 6:30 AM.

Alternate Method

Start Point (6:00 AM)

A: 5 km/h

B: 7 km/h

Perimeter = 1.2 km Opposite Directions

Given:

Perimeter of track (Distance) = 1.2 km

Speed of A (S₁) = 5 km/h

Speed of B (S₂) = 7 km/h

Formula Used:

Time = Distance ÷ Relative Speed

Relative Speed (Opposite Directions) = S₁ + S₂

Calculations:

⇒ Relative Speed = 5 + 7 = 12 km/h

⇒ Time for 1^st meeting = 1.2 ÷ 12 = 0.1 hours

⇒ Converting hours into minutes: 0.1 × 60 = 6 minutes

⇒ Time taken to meet for the 5^th time = 5 × 6 = 30 minutes

⇒ Meeting Time = 6:00 AM + 30 minutes = 6:30 AM

∴ The correct answer is 6:30 AM.

Additional Information

Relative Speed in Same Direction

When two bodies move in the same direction, their relative speed is the difference between their speeds: |S₁ − S₂|.

Meeting on a Circular Track

Time taken to meet for the first time anywhere on the track = Track Length ÷ Relative Speed.

Meeting at Starting Point

Time taken to meet for the first time exactly at the starting point = LCM of individual times taken by each person to complete one full round.

Shortcut Trick

A central angle of 90° means the sector is exactly a quarter of the full circle.

Area of a full circle = π × r²

⇒ Full Area = (22/7) × 7 × 7 = 154 cm²

Area of the sector = 154 ÷ 4 = 38.5 cm²

∴ The correct answer is 38.5 cm².

Alternate Method

r = 7 cm 90° Area of Sector

Given:

Radius of the circle (r) = 7 cm

Central angle (θ) = 90°

Formula Used:

Area of a sector = (θ ÷ 360°) × π × r²

Calculations:

⇒ Area = (90° ÷ 360°) × (22/7) × 7²

⇒ Area = (1/4) × (22/7) × 49

⇒ Area = (1/4) × 154

⇒ Area = 38.5 cm²

∴ The correct answer is 38.5 cm².

Additional Information

Length of an Arc

The length of the arc bounding the sector is given by: l = (θ ÷ 360°) × 2πr.

Perimeter of a Sector

The perimeter of a sector includes the arc length and the two radii forming the sector: Perimeter = l + 2r.

Area of a Segment

A segment is the region bounded by a chord and an arc. Its area is: Area of segment = Area of sector − Area of the corresponding triangle.

Shortcut Trick

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

⇒ Radian measure = 150° × (π / 180°)

⇒ Radian measure = 15π / 18

⇒ Radian measure = 5π / 6

∴ The correct answer is 5π/6.

Alternate Method

Given:

Angle measure = 150°

Formula Used:

Angle in radians = Angle in degrees × (π / 180°)

150° 0° 150° = 5π/6

Calculations:

⇒ Radian measure = 150° × (π / 180°)

⇒ Radian measure = (150 / 180) × π

⇒ Radian measure = (15 / 18) × π

⇒ Radian measure = 5π / 6

∴ The correct answer is 5π/6.

Additional Information

Degrees to Radians Conversion

The formula to convert degrees to radians is: Radians = Degrees × (π / 180°).

Radians to Degrees Conversion

The formula to convert an angle from radians back to degrees is: Degrees = Radians × (180° / π).

Standard Angles in Radians

Important values to remember: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, and 360° = 2π.

Shortcut Trick

Since the central angle is 120°, the sector represents exactly 120° ÷ 360° = 1/3 of the full circle.

Area of the full circle = π × r² = π × 8² = 64π cm².

Area of the sector = (1 / 3) × 64π = 64π/3 cm².

∴ The correct answer is 64π/3 cm².

Alternate Method

120° r = 8 cm r = 8 cm

Centre O Area of Sector

Given:

⇒ Radius of the sector (r) = 8 cm

⇒ Central angle (θ) = 120°

Formula Used:

⇒ Area of Sector = (θ ÷ 360°) × πr²

Calculations:

⇒ Area = (120° ÷ 360°) × π × (8)²

⇒ Area = (1 / 3) × π × 64

⇒ Area = 64π/3 cm²

∴ The correct answer is 64π/3 cm².

Additional Information

Area of a Circle

The total region enclosed by a circle is given by πr², which corresponds to a full central angle of 360°.

Length of an Arc

The length of the curved boundary of a sector is proportional to its central angle, given by the formula L = (θ ÷ 360°) × 2πr.

Perimeter of a Sector

The total boundary length of a sector is obtained by adding the arc length to twice the radius: Perimeter = L + 2r.

Area of a Segment

The region bounded by a chord and its corresponding arc. It is calculated by subtracting the area of the corresponding triangle from the total area of the sector.

Shortcut Trick

For any linear equation ax + by = c, all lines parallel to it are of the identical form ax + by = k (where k ≠ c).

The given line is 3x + 2y = 5.

By comparing the options, only the equation 3x + 2y = 7 matches this exact a and b coefficient format.

∴ The correct answer is 3x + 2y = 7.

Alternate Method

Given:

The equation of the original line is 3x + 2y = 5.

Concept Used:

Two lines are strictly parallel if their slopes are equal (m₁ = m₂).

For a line represented as ax + by = c, the slope m is determined by −a/b.

X Y O

3x + 2y = 5 3x + 2y = 7

Calculations:

⇒ Slope of the given line 3x + 2y = 5 is m = −3/2.

⇒ Let us calculate the slopes of all given options to find the identical one:

⇒ 1) 3x + 2y = 7 → Slope = −3/2 (Parallel to given line)

⇒ 2) 2x + 3y = 5 → Slope = −2/3 (Not parallel)

⇒ 3) 3x − 2y = 5 → Slope = 3/2 (Not parallel)

⇒ 4) x + y = 5 → Slope = −1 (Not parallel)

⇒ Only the first equation has the same slope as the original given line.

∴ The correct answer is 3x + 2y = 7.

Additional Information

Parallel Lines Condition

Two general lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are strictly parallel and do not touch when a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

Perpendicular Lines Condition

Two lines cross each other at exactly 90 degrees if the mathematical product of their slopes equals −1 (m₁ × m₂ = −1). The perpendicular inverse format is bx − ay + k = 0.

Intersecting Lines Condition

Linear graphs will always intersect at exactly one unique point if they possess structurally different slopes, proven when a₁/a₂ ≠ b₁/b₂.

Coincident Lines Condition

Two drawn lines represent the exact same path and hold infinite solutions universally if their constants are fully proportional: a₁/a₂ = b₁/b₂ = c₁/c₂.

Shortcut Trick

Using the identity sin(90° − θ) = cos(θ), we get cos(x) = cos(2x).

By comparing the angles directly for the principal value, we can write:

x = 2x ⇒ x = 0°

Checking the options, 0° satisfies the equation: sin(90°) = cos(0°) ⇒ 1 = 1.

∴ The correct answer is 0°.

Alternate Method

Given:

sin(90° − x) = cos(2x)

Formula Used:

sin(90° − θ) = cos(θ)

cos(2θ) = 2cos²(θ) − 1

Calculation:

⇒ sin(90° − x) = cos(2x)

⇒ cos(x) = cos(2x)

⇒ cos(x) = 2cos²(x) − 1

⇒ 2cos²(x) − cos(x) − 1 = 0

⇒ 2cos²(x) − 2cos(x) + cos(x) − 1 = 0

⇒ 2cos(x)(cos(x) − 1) + 1(cos(x) − 1) = 0

⇒ (2cos(x) + 1)(cos(x) − 1) = 0

⇒ cos(x) = 1 or cos(x) = −1/2

For x in the first quadrant (as per options), cos(x) = 1.

⇒ x = 0°

∴ The correct answer is 0°.

Additional Information

Complementary Angle Identities

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

General Solution of Cosine

If cos(x) = cos(y), then the general solution is x = 360° × n ± y, where n is any integer.

Double Angle Formula

cos(2θ) can be expressed as cos²(θ) − sin²(θ), or 2cos²(θ) − 1, or 1 − 2sin²(θ).

Shortcut Trick

Given x = √5 + 2

Square both sides directly: x² = 5 + 4 + 4√5 = 9 + 4√5

We need to find the value of x² − 4√5

Substitute x²: 9 + 4√5 − 4√5 = 9

∴ The correct answer is 9.

Alternate Method

Given:

x = √5 + 2

Formula Used:

(a + b)² = a² + b² + 2ab

Calculation:

⇒ x² = (√5 + 2)²

⇒ x² = (√5)² + (2)² + 2 × 2 × √5

⇒ x² = 5 + 4 + 4√5

⇒ x² = 9 + 4√5

Now, the required expression is x² − 4√5.

Substituting the value of x²:

⇒ (9 + 4√5) − 4√5

⇒ 9

∴ The correct answer is 9.

Additional Information

Algebraic Identities

Basic identities like (a + b)² = a² + b² + 2ab and (a − b)² = a² + b² − 2ab are fundamental in simplification questions involving surds and radicals.

Simplification of Surds

When squaring a binomial containing a surd, the rational parts and irrational parts are combined separately. For example, (√x + y)² = x + y² + 2y√x.

Conjugate Surds

The expressions (a + √b) and (a − √b) are conjugates. Their product (a + √b)(a − √b) = a² − b is always a rational number.

Shortcut Trick

The congruence of triangles is determined by matching their corresponding sides and angles.

Since the problem states that all three corresponding sides are exactly equal (AB = XY, BC = YZ, CA = ZX).

This perfectly matches the defining condition for the Side-Side-Side (SSS) congruence rule.

∴ The correct answer is SSS.

Alternate Method

Given:

⇒ In Triangle ABC and Triangle XYZ:

⇒ AB = XY

⇒ BC = YZ

⇒ CA = ZX

A B C

X Y Z

Concept Used:

⇒ SSS Congruence Rule: If three sides of one triangle are equal to the corresponding three sides of another triangle, then the two triangles are congruent.

Calculations:

⇒ We map the sides of both triangles to verify their relation.

⇒ AB = XY (First pair of sides is equal)

⇒ BC = YZ (Second pair of sides is equal)

⇒ CA = ZX (Third pair of sides is equal)

⇒ Since all three corresponding sides are exactly equal, Triangle ABC is identically congruent to Triangle XYZ under the Side-Side-Side property.

∴ The correct answer is SSS.

Additional Information

SAS (Side-Angle-Side) Congruence

Two triangles are congruent if two sides and the included angle of one triangle are exactly equal to two sides and the included angle of the other.

ASA (Angle-Side-Angle) Congruence

Two triangles are congruent if two angles and the included side of one triangle are exactly equal to two angles and the included side of the other.

AAS (Angle-Angle-Side) Congruence

Two triangles are congruent if any two angles and a non-included side of one triangle are equal to the corresponding angles and side of the other.

RHS (Right Angle-Hypotenuse-Side) Congruence

Two right-angled triangles are congruent if the hypotenuse and one side of one triangle are equal to the hypotenuse and the corresponding side of the other.

Shortcut Trick

A and B are complementary angles ⇒ A + B = 90° ⇒ B = 90° − A

tanB = tan(90° − A) = cotA

Given sinA = 3/5 = Perpendicular / Hypotenuse ⇒ Base = √(5² − 3²) = 4

cotA = Base / Perpendicular = 4/3

∴ The correct answer is 4/3.

Alternate Method

Given:

sinA = 3/5

A and B are complementary angles (A + B = 90°).

Formula Used:

Pythagoras Theorem: (Hypotenuse)² = (Perpendicular)² + (Base)²

tanθ = Perpendicular ÷ Base

ABBase = 4Perpendicular = 3Hypotenuse = 5

Calculations:

⇒ In a right-angled triangle, sinA = Perpendicular ÷ Hypotenuse = 3 ÷ 5.

⇒ By Pythagoras theorem: Base = √(Hypotenuse² − Perpendicular²)

⇒ Base = √(5² − 3²) = √(25 − 9) = √16 = 4.

⇒ Since A and B are complementary, they form the two acute angles of the same right-angled triangle.

⇒ For angle B, the perpendicular and base sides are swapped relative to angle A.

⇒ Perpendicular to B = Base to A = 4.

⇒ Base to B = Perpendicular to A = 3.

⇒ tanB = (Perpendicular to B) ÷ (Base to B)

⇒ tanB = 4/3

∴ The correct answer is 4/3.

Additional Information

Trigonometric Ratios of Complementary Angles

sin(90° − θ) = cosθ and tan(90° − θ) = cotθ. Complementary angles sum to exactly 90°.

Pythagorean Triplets

A set of positive integers satisfying a² + b² = c². Standard base triplets include (3, 4, 5) and (5, 12, 13).

Basic Trigonometric Ratios

sinθ = P/H, cosθ = B/H, tanθ = P/B, where P is Perpendicular, B is Base, and H is Hypotenuse.

Shortcut Trick

The sum of opposite angles in a cyclic quadrilateral is always 180°.

Angle A and Angle C are opposite angles in the cyclic quadrilateral ABCD.

∠C = 180° − ∠A

∠C = 180° − 100° = 80°

∴ The correct answer is 80°.

Alternate Method

A B C D

100° 80°

∠A + ∠C = 180°

Given:

∠A = 100°

ABCD is a cyclic quadrilateral.

Formula Used:

In a cyclic quadrilateral, the sum of a pair of opposite interior angles is equal to 180°.

Calculation:

⇒ ∠A + ∠C = 180°

⇒ 100° + ∠C = 180°

⇒ ∠C = 180° − 100°

⇒ ∠C = 80°

∴ The correct answer is 80°.

Additional Information

Cyclic Quadrilateral

A quadrilateral is called cyclic if all its four vertices lie on the circumference of a single circle. The primary property of a cyclic quadrilateral is that the sum of its opposite interior angles is always 180°.

Exterior Angle Property

If any one side of a cyclic quadrilateral is extended outwardly, the exterior angle formed is strictly equal to the interior opposite angle.

Ptolemy's Theorem

For a cyclic quadrilateral, the product of the lengths of the diagonals is equal to the sum of the products of the lengths of its opposite sides (AC × BD = AB × CD + AD × BC).

Shortcut Trick

Using the fundamental trigonometric identity: sin²A + cos²A = 1

Subtract the given equation (sin²A − cos²A = 1/4) from this identity:

(sin²A + cos²A) − (sin²A − cos²A) = 1 − 1/4

2cos²A = 3/4

cos²A = 3/8

∴ The correct answer is 3/8.

Alternate Method

Given:

sin²A − cos²A = 1/4

Formula Used:

sin²A + cos²A = 1 ⇒ sin²A = 1 − cos²A

Calculations:

⇒ Substitute the value of sin²A in the given equation.

⇒ (1 − cos²A) − cos²A = 1/4

⇒ 1 − 2cos²A = 1/4

⇒ 2cos²A = 1 − 1/4

⇒ 2cos²A = 3/4

⇒ cos²A = 3/4 × 1/2

⇒ cos²A = 3/8

∴ The correct answer is 3/8.

Additional Information

Trigonometric Square Identities

1. sin²θ + cos²θ = 1

2. sec²θ − tan²θ = 1

3. cosec²θ − cot²θ = 1

Double Angle Formula for Cosine

The double angle formula can be expressed in three different forms:

cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ