Search Previous Year Questions

Shortcut Trick

Total Work (LCM of 12 and 39) = 156 units.

Efficiencies: A = 13, B = 4.

Work done by (A + B) in 4 days = (13 + 4) × 4 = 68 units.

Remaining work = 156 − 68 = 88 units.

Time taken by B = 88 ÷ 4 = 22 days.

∴ The correct answer is 22 days.

Alternate Method

Given:

Time taken by A = 12 days

Time taken by B = 39 days

Formula Used:

Efficiency = Total Work ÷ Time

Remaining Work = Total Work − Work Completed

Calculations:

⇒ Total efficiency of A and B = 13 + 4 = 17 units/day

⇒ Work done by A and B together in 4 days = 17 × 4 = 68 units

⇒ Remaining work to be done = 156 − 68 = 88 units

⇒ A leaves, so B completes the 88 units of work.

⇒ Time taken by B = 88 ÷ 4 = 22 days

∴ The correct answer is 22 days.

Additional Information

Pipes & Cisterns

Similar logic applies to pipes where inlet pipes have positive efficiency and outlet pipes have negative efficiency.

Work & Wages

Wages are distributed among workers in the ratio of their efficiency or the total work done by each.

Alternate Day Work

When people work on alternate days, calculate the work done in one complete cycle (e.g., 2 days) to find the total time.

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken to complete the work (Efficiency ∝ 1 ÷ Time).

Shortcut Trick

Total interest percentage for 3 years = 7.5% × 3 = 22.5%

Total Amount percentage = 100% (Principal) + 22.5% (Interest) = 122.5%

Amount = 122.5% of ₹8,000

Amount = 1.225 × 8000 = ₹9,800

∴ The correct answer is ₹9800.

Alternate Method

Given:

Principal (P) = ₹8,000

Rate of Interest (R) = 7.5% per annum

Time (T) = 3 years

Formula Used:

Simple Interest (SI) = (P × R × T) ÷ 100

Amount (A) = Principal (P) + Simple Interest (SI)

Calculations:

⇒ SI = (8000 × 7.5 × 3) ÷ 100

⇒ SI = 80 × 22.5

⇒ SI = ₹1,800

⇒ Amount = 8000 + 1800

⇒ Amount = ₹9,800

∴ The correct answer is ₹9800.

Additional Information

Simple Interest Concept

Simple Interest is calculated only on the initial principal amount throughout the entire period. SI = P × R × T / 100.

Amount in SI

The total value after adding interest to the principal is called the Amount. A = P(1 + RT/100).

SI vs CI Difference

For the first year, SI and Compound Interest (CI) are equal if interest is compounded annually. From the second year, CI > SI because interest is earned on interest.

Shortcut Trick

Convert the given target ratio 3:2 into decimal form: 3 ÷ 2 = 1.5.

Check the decimal values of the options:

60/45 ≈ 1.33 | 45/20 = 2.25 | 35/25 = 1.4 | 120/80 = 1.5.

Since 120/80 matches the decimal value 1.5, it is equivalent to 3:2.

∴ The correct answer is 120:80.

Alternate Method

Given: Target Ratio = 3 : 2

Formula Used: Simplest form of a ratio is obtained by dividing both terms by their Highest Common Factor (HCF).

⇒ Option 1: 60 : 45 (HCF = 15) ⇒ (60 ÷ 15) : (45 ÷ 15) = 4 : 3

⇒ Option 2: 45 : 20 (HCF = 5) ⇒ (45 ÷ 5) : (20 ÷ 5) = 9 : 4

⇒ Option 3: 35 : 25 (HCF = 5) ⇒ (35 ÷ 5) : (25 ÷ 5) = 7 : 5

⇒ Option 4: 120 : 80 (HCF = 40) ⇒ (120 ÷ 40) : (80 ÷ 40) = 3 : 2

∴ The correct answer is 120:80.

Additional Information

Simplest Form of Ratio

A ratio a:b is in its simplest form if the HCF of a and b is 1. To simplify, divide both terms by their common factors.

Equivalent Ratios

Ratios that represent the same relationship between numbers are called equivalent ratios. They can be found by multiplying or dividing both terms by the same non-zero number.

Proportion

When two ratios are equal, they are said to be in proportion. For a:b = c:d, the product of extremes (a×d) equals the product of means (b×c).

Shortcut Trick

In a 30°−60°−90° right-angled triangle, the sides are in the ratio 1 : √3 : 2.

The base (adjacent to 60°) corresponds to 1 unit.

Given, 1 unit = 15 m.

The ladder (hypotenuse) corresponds to 2 units.

Length of ladder = 2 × 15 = 30 m.

∴ The correct answer is 30 m.

Alternate Method

Given:

Angle of elevation of the ladder with the ground (θ) = 60°

Distance of the foot of the ladder from the wall (Base) = 15 m

Formula Used:

In a right-angled triangle, cos θ = Base / Hypotenuse

Calculations:

Let the length of the ladder be L.

⇒ cos 60° = 15 / L

⇒ 1 / 2 = 15 / L

⇒ L = 15 × 2

⇒ L = 30 m

∴ The correct answer is 30 m.

Additional Information

Trigonometric Ratios

Fundamental ratios in a right triangle: sin θ = P/H, cos θ = B/H, and tan θ = P/B (where P = Perpendicular, B = Base, H = Hypotenuse).

Standard Values of Cosine

Common values include cos 0° = 1, cos 30° = √3/2, cos 45° = 1/√2, cos 60° = 1/2, and cos 90° = 0.

Pythagoras Theorem

In any right-angled triangle, the relationship between the sides is given by H² = P² + B².

30-60-90 Triangle Rule

For a triangle with angles 30°, 60°, and 90°, the side lengths are always in the ratio x : x√3 : 2x.

Shortcut Trick

When Distance is constant, Speed is inversely proportional to Time.

Ratio of Time = 5 : 6 → Ratio of Speed = 6 : 5.

Original Speed = 300 ÷ 5 = 60 km/h.

Here, 6 units = 60 km/h ⇒ 1 unit (change) = 10 km/h.

Since the ratio decreases from 6 to 5, the speed decreases.

∴ The correct answer is 10 km/h decrease.

Alternate Method

Given:

Distance (D) = 300 km

Initial Time (T₁) = 5 hr

New Time (T₂) = 6 hr

Formula Used:

Speed (S) = Distance (D) ÷ Time (T)

Calculations:

⇒ Initial Speed (S₁) = 300 ÷ 5 = 60 km/h

⇒ New Speed (S₂) = 300 ÷ 6 = 50 km/h

⇒ Change in Speed = Initial Speed − New Speed

⇒ Change = 60 − 50 = 10 km/h

⇒ Since the new speed is lower, there is a decrease of 10 km/h.

∴ The correct answer is 10 km/h decrease.

Additional Information

Speed-Time Relation

For a fixed distance, Speed and Time are inversely proportional. If time increases by a certain factor, speed must decrease by the same factor to cover the same distance.

Average Speed

Average Speed is calculated as the Total Distance divided by the Total Time taken for the entire journey. Formula: Average Speed = (Total Distance) ÷ (Total Time).

Unit Conversion

To convert Speed from km/h to m/s, multiply the value by 5/18. To convert from m/s to km/h, multiply the value by 18/5.

Shortcut Trick

A number is divisible by 11 if the difference between the sum of digits at odd places and the sum of digits at even places is 0 or a multiple of 11.

Check 3,762: (2 + 7) − (6 + 3) = 9 − 9 = 0.

Since the result is 0, the number 3,762 is divisible by 11.

∴ The correct answer is 3,762.

Alternate Method

Given: Numbers are 3,762, 7,393, 4,915, and 6,184.

Divisibility Rule of 11: Let the digits of a number be checked from right to left. The difference (Sum of digits at positions 1, 3, 5...) − (Sum of digits at positions 2, 4, 6...) must be 0, 11, 22, etc.

⇒ For 3,762: (2 + 7) − (6 + 3) = 9 − 9 = 0 (Divisible)

⇒ For 7,393: (3 + 3) − (9 + 7) = 6 − 16 = −10 (Not divisible)

⇒ For 4,915: (5 + 9) − (1 + 4) = 14 − 5 = 9 (Not divisible)

⇒ For 6,184: (4 + 1) − (8 + 6) = 5 − 14 = −9 (Not divisible)

∴ The correct answer is 3,762.

Additional Information

Divisibility by 3 and 9

A number is divisible by 3 or 9 if the sum of its digits is divisible by 3 or 9, respectively.

Divisibility by 4

A number is divisible by 4 if the last two digits of the number are divisible by 4.

Divisibility by 8

A number is divisible by 8 if the last three digits of the number are divisible by 8.

Shortcut Trick

Check if the sum of terms is zero: 23 + (−14) + (−9) = 23 − 23 = 0.

Apply the Algebraic Identity: If a + b + c = 0, then a³ + b³ + c³ = 3abc.

Calculation: 3 × 23 × (−14) × (−9) = 3 × 23 × 126.

Result: 69 × 126 = 8694.

∴ The correct answer is 8694.

Alternate Method

Given:

a = 23

b = −14

c = −9

Formula Used:

a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)

Calculations:

⇒ First, evaluate the sum (a + b + c):

⇒ 23 + (−14) + (−9) = 23 − 14 − 9 = 0

⇒ Since the sum is zero, the right side of the identity becomes zero.

⇒ a³ + b³ + c³ − 3abc = 0 × (a² + b² + c² − ab − bc − ca) = 0

⇒ a³ + b³ + c³ = 3abc

⇒ Substitute the values: 3 × 23 × (−14) × (−9)

⇒ 3 × 23 × 126

⇒ 69 × 126 = 8694

∴ The correct answer is 8694.

Additional Information

Conditional Algebraic Identity

The identity a³ + b³ + c³ = 3abc holds true if and only if a + b + c = 0 or a = b = c.

General Expansion

The expression a² + b² + c² − ab − bc − ca can be rewritten as 1/2[(a−b)² + (b−c)² + (c−a)²], which is always non-negative for real numbers.

Trinomial Square

Another useful identity is (a + b + c)² = a² + b² + c² + 2(ab + bc + ca).

Shortcut Trick

Surface area of a sphere = π × d²

Area = 3.14 × 10 × 10 = 314 m²

Total cost = Area × rate = 314 × 5

∴ The correct answer is ₹1570.

Alternate Method

Given: Internal Diameter (d) = 10 m, Rate of coating = ₹5 per m², π = 3.14

Formula Used: Surface Area of a Sphere = 4πr²; Total Cost = Area × Rate

Calculations:

⇒ Radius (r) = Diameter ÷ 2 = 10 ÷ 2 = 5 m

⇒ Internal Surface Area = 4 × 3.14 × (5)²

⇒ Area = 4 × 3.14 × 25

⇒ Area = 100 × 3.14 = 314 m²

⇒ Total Cost = 314 × 5

⇒ Total Cost = ₹1570

∴ The correct answer is ₹1570.

Additional Information

Volume of a Sphere

The volume of a sphere is given by the formula V = (4/3)πr³, where r is the radius.

Hemisphere Surface Area

For a hemisphere, the Curved Surface Area (CSA) is 2πr² and the Total Surface Area (TSA) is 3πr².

Surface Area in terms of Diameter

The total surface area of a sphere can also be written as πd², which often simplifies calculations when the diameter is known.

Shortcut Trick

Use the Ratio Method for successive changes:

Year 2015 (10% increase): 10 → 11

Year 2016 (15% decrease): 20 → 17 (since 15% = 3/20)

Overall Ratio: (10 × 20) : (11 × 17) = 200 : 187

Given final value: 187 units = 1,87,000 ⇒ 1 unit = 1,000

Initial Population = 200 × 1,000 = 2,00,000

∴ The correct answer is 2,00,000.

Alternate Method

Let the Initial Population in the beginning of 2015 be P.

⇒ Population after 10% increase in 2015 = P × (1 + 10/100) = 1.1P

⇒ Population after 15% decrease in 2016 = 1.1P × (1 − 15/100)

⇒ Population at the end of 2016 = 1.1P × 0.85 = 0.935P

⇒ According to the question: 0.935P = 1,87,000

⇒ P = 1,87,000 ÷ 0.935

⇒ P = (1,87,000 × 1,000) ÷ 935

⇒ P = 200 × 1,000 = 2,00,000

∴ The correct answer is 2,00,000.

Additional Information

Successive Percentage Change Formula

For two successive changes of a% and b%, the effective net change is [a + b + (a × b)/100]%. Use positive for increase and negative for decrease.

Multiplier Method

Final Value = Initial Value × (1 ± r₁/100) × (1 ± r₂/100). This is widely used in population growth and depreciation problems.

Population Growth

If the population increases at a constant rate r% per annum, the population after n years is P(1 + r/100)ⁿ.

Shortcut Trick

In an isosceles triangle, angles opposite to equal sides are equal.

Given XY = XZ, therefore the base angles ∠Y = ∠Z.

Sum of angles in a triangle = 180°.

∠X + 2 × ∠Z = 180° ⇒ 40° + 2 × ∠Z = 180°.

2 × ∠Z = 140° ⇒ ∠Z = 70°.

∴ The correct answer is 70°.

Alternate Method

Given:

In ΔXYZ, XY = XZ and vertex angle ∠X = 40°.

Formula Used:

1. Isosceles Triangle Property: Angles opposite to equal sides are equal.
2. Angle Sum Property: The sum of all interior angles of a triangle is 180°.

Calculations:

⇒ In ΔXYZ, since XY = XZ, it is an isosceles triangle.

⇒ ∠Y = ∠Z (Property of isosceles triangles)

⇒ ∠X + ∠Y + ∠Z = 180° (Angle sum property)

⇒ 40° + ∠Z + ∠Z = 180°

⇒ 40° + 2 × ∠Z = 180°

⇒ 2 × ∠Z = 180° − 40°

⇒ 2 × ∠Z = 140°

⇒ ∠Z = 140° ÷ 2

⇒ ∠Z = 70°

∴ The correct answer is 70°.

Additional Information

Isosceles Triangle Theorem

If two sides of a triangle are congruent, then the angles opposite those sides are congruent (Equal sides ↔ Equal angles).

Triangle Angle Sum Theorem

The sum of the measures of the interior angles of any triangle is exactly 180°.

Exterior Angle Property

The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.