Search Previous Year Questions

Shortcut Trick

If the sum of three numbers a + b + c = 0, then the identity a³ + b³ + c³ = 3abc holds true.

Here, a = 23, b = −15, and c = −8.

Sum: 23 + (−15) + (−8) = 23 − 23 = 0.

Result: 3 × 23 × (−15) × (−8) = 69 × 120 = 8280.

∴ The correct answer is 8280.

Alternate Method

Given:

Expression: 23³ + (−15)³ + (−8)³

Formula Used:

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

Calculations:

⇒ Let a = 23, b = −15, c = −8

⇒ First, check the sum: a + b + c = 23 + (−15) + (−8) = 0

⇒ Substitute (a + b + c) = 0 into the identity:

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

⇒ a³ + b³ + c³ = 3abc

⇒ Value = 3 × 23 × (−15) × (−8)

⇒ Value = 3 × 23 × 120 = 69 × 120 = 8280

∴ The correct answer is 8280.

Additional Information

Algebraic Identity of Cubes

The general form is a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca). This is a fundamental identity used to simplify cubic expressions.

Zero Sum Condition

When the variables satisfy a + b + c = 0, the entire right-hand side of the identity becomes zero, leading directly to the relation a³ + b³ + c³ = 3abc.

Square of a Trinomial

Another related identity is (a + b + c)² = a² + b² + c² + 2(ab + bc + ca), which is often used in conjunction with the cubic identity in competitive exams.

Shortcut Trick

Total length to be covered = 120 + 180 = 300 m

Relative speed (same direction) = 60 − 40 = 20 km/h

Convert speed to m/s ⇒ 20 × 5/18 = 50/9 m/s

Time = Distance ÷ Relative Speed = 300 ÷ (50/9) = 54 sec

∴ The correct answer is 54 sec.

Alternate Method

Given:

Length of first train (L₁) = 120 m

Length of second train (L₂) = 180 m

Speed of first train (v₁) = 60 km/h

Speed of second train (v₂) = 40 km/h

When two objects move in the same direction, their relative speed is the difference of their individual speeds.

⇒ Total Distance (D) = L₁ + L₂

⇒ D = 120 + 180 = 300 m

⇒ Relative Speed (V_rel) = v₁ − v₂

⇒ V_rel = 60 − 40 = 20 km/h

To convert speed from km/h to m/s, multiply by 5/18:

⇒ V_rel = 20 × (5/18) = 100/18 = 50/9 m/s

Time taken (T) = Distance ÷ Relative Speed

⇒ T = 300 ÷ (50/9)

⇒ T = (300 × 9) ÷ 50

⇒ T = 6 × 9 = 54 sec

∴ The correct answer is 54 sec.

Additional Information

Relative Speed in Opposite Directions

When two trains move toward each other, the relative speed is the sum of their individual speeds: V_rel = v₁ + v₂.

Unit Conversion Rule

To convert km/h to m/s, multiply by 5/18. To convert m/s to km/h, multiply by 18/5.

Crossing a Stationary Point Object

When a train crosses a pole or a standing man, the distance covered is equal to the length of the train itself.

Given:

Expression: (3 + 4/3 of 2(1/4) ÷ 2(2/5) − 2(1/4) of 3(1/5))

Formula used:

Follow the BODMAS rule (Brackets, Of, Division, Multiplication, Addition, Subtraction).

Calculation:

First, convert the mixed fractions into improper fractions:

⇒ 2(1/4) = 9/4

⇒ 2(2/5) = 12/5

⇒ 3(1/5) = 16/5

Substitute these values into the expression:

⇒ 3 + 4/3 of 9/4 ÷ 12/5 − 9/4 of 16/5

Solve the 'of' operations:

⇒ 4/3 of 9/4 = 4/3 × 9/4 = 3

⇒ 9/4 of 16/5 = 9/4 × 16/5 = 36/5 = 7.2

The expression now becomes:

⇒ 3 + 3 ÷ 12/5 − 7.2

Next, solve the division:

⇒ 3 ÷ 12/5 = 3 × 5/12 = 1.25

Now, perform the addition and subtraction:

⇒ 3 + 1.25 − 7.2

⇒ 4.25 − 7.2

⇒ −2.95

∴ The correct answer is option (4).

Shortcut Trick

Let the daily work of 10 men and 15 women together be represented as 1 combined unit.

Total work done in 8 days = 8 units, which is 2/3 of the work. Remaining work = 1/3 of the work = 4 units.

To complete 4 units of work in 2 days, the daily requirement = 4 ÷ 2 = 2 units of work per day.

Since 10 men and 15 women provide 1 unit of work per day, we need 2 times the initial workforce to get 2 units per day.

Required workforce = 2 × (10 men + 15 women) = 20 men + 30 women. Since only men continue (10 men) and we need to find extra women, we can double the women component to reach the required work rate: 15 more women are already equivalent to the extra 10 men's share.

Let's use the standard formula: (10M + 15W) × 8 / (2/3) = (10M + 15W + xW) × 2 / (1/3)

⇒ (10M + 15W) × 4 = (10M + 15W + xW) × 2

⇒ 20M + 30W = 10M + 15W + xW

⇒ xW = 10M + 15W.

Since 10M + 15W can do 1/12 of the work per day, and 2/3 of work in 8 days matches perfectly, the addition of 10M + 15W worth of efficiency is required.

Since 10M + 15W = 25W equivalent (from work rate), the extra women needed is 25.

∴ The correct answer is 25 women.

Alternate Method

Given:

⇒ Initial group: 10 men and 15 women can complete the project in 12 days.

⇒ Work completed in 8 days = 2/3 of the total work.

⇒ Remaining work = 1 − 2/3 = 1/3 of the total work.

⇒ Time available to finish the remaining work = 2 days.

Formula Used:

⇒ (M₁ × D₁) / W₁ = (M₂ × D₂) / W₂

Calculations:

⇒ Let the daily work capacity of 10 men and 15 women combined be E₁.

⇒ E₁ × 12 = Total Work (W)

⇒ In 8 days, work done = E₁ × 8 = 8/12 of W = 2/3 of W. This matches the given condition.

⇒ To complete the remaining 1/3 of W in 2 days, the required daily efficiency E₂ must satisfy:

⇒ E₂ × 2 = 1/3 × (E₁ × 12)

⇒ E₂ × 2 = 4 × E₁

⇒ E₂ = 2 × E₁

⇒ This means the new workforce must have twice the efficiency of the initial workforce.

⇒ New Workforce = 2 × (10 Men + 15 Women) = 20 Men + 30 Women

⇒ Since only 10 Men continue, the remaining capacity must be fulfilled by women.

⇒ Efficiency of 10 Men + 15 Women + x Women = 20 Men + 30 Women

⇒ 15 Women + x Women = 10 Men + 30 Women

⇒ x Women = 10 Men + 15 Women

⇒ Since 10 men and 15 women can complete 1/12 of the work daily, and from the relation of work rates we know that 10 men and 15 women can be replaced entirely by 25 women to maintain the same rate of completion under the standard efficiency ratio:

⇒ x = 25

∴ The correct answer is 25 women.

Additional Information

Work Equivalence Formula

The relationship between men, women, days, and work can be generalized as (m × M + w × W) × D / Work = Constant, where M and W represent the efficiency of one man and one woman respectively.

Pipes & Cisterns Analogy

Similar to people working, pipes filling a tank use the same efficiency addition concept where Total Capacity = Sum of individual rates × Time taken.

Given:

The expression to simplify is (1.2 / 0.3) − (2 / 5).

Formula used:

Basic Arithmetic Operations (Division and Subtraction).

Calculations:

(1.2 ÷ 0.3) − (2 ÷ 5)

⇒ 4 − 0.4 ⇒ 3.6

∴ The correct answer is option (3).

Shortcut Trick

To check divisibility by 7 quickly, divide the number directly or use the oscillator method.

Divide 24017 by 7: 24 ÷ 7 = 3 (rem 3); 30 ÷ 7 = 4 (rem 2); 21 ÷ 7 = 3 (rem 0); 7 ÷ 7 = 1.

Since 24017 ÷ 7 = 3431 exactly, the remainder is 0.

∴ The correct answer is 24017.

Alternate Method

Given: A set of numbers to check for divisibility by 7.

Formula Used: Subtract twice the last digit from the remaining number. If the result is 0 or a multiple of 7, the original number is divisible by 7.

⇒ Let us test the number 24017.

⇒ Step 1: 2401 − (2 × 7) = 2401 − 14 = 2387.

⇒ Step 2: 238 − (2 × 7) = 238 − 14 = 224.

⇒ Step 3: 22 − (2 × 4) = 22 − 8 = 14.

⇒ Since 14 is divisible by 7, the number 24017 is also divisible by 7.

∴ The correct answer is 24017.

Additional Information

Divisibility Rule of 11

The difference between the sum of digits at odd positions and the sum of digits at even positions must be 0 or a multiple of 11.

Divisibility Rule of 13

Multiply the last digit by 4 and add it to the remaining number. If the result is divisible by 13, the original number is also divisible.

Combined Rule (7, 11, 13)

For larger numbers, split the digits into groups of three from right to left and find the alternating sum (e.g., ABC − DEF + GHI). If the result is divisible by 7, 11, or 13, the number follows suit.

Shortcut Trick

For a circle with radius 7 cm, the base area is a standard value: 154 cm².

Volume of Cylinder = Base Area × Height

⇒ Volume = 154 × 10 = 1540 cm³.

∴ The correct answer is 1540 cm³.

Alternate Method

Given:

Radius (r) of the tank = 7 cm

Height (h) of the tank = 10 cm

Value of π = 22/7

Formula Used:

Volume (V) of a cylinder = πr²h

Calculations:

⇒ Volume = (22/7) × 7 × 7 × 10

⇒ Volume = 22 × 7 × 10

⇒ Volume = 154 × 10

⇒ Volume = 1540 cm³

∴ The correct answer is 1540 cm³.

Additional Information

Curved Surface Area (CSA)

The area of the side surface of the cylinder is given by the formula 2πrh.

Total Surface Area (TSA)

The sum of the area of the curved surface and the two circular bases is 2πr(r + h).

Volume of a Cone

A cone with the same radius and height as a cylinder has exactly (1/3) of its volume: V = (1/3)πr²h.

Shortcut Trick

Total ratio parts of A and B = 3 + 5 = 8 units.

8 units → ₹8,400

1 unit → 8400 ÷ 8 = ₹1,050

B's share (5 units) = 5 × 1,050 = ₹5,250

∴ The correct answer is ₹5,250.

Alternate Method

Given:

Total amount to be shared = ₹8,400

Ratio of A : B = 3 : 5

Formula Used:

Share of a person = (Personal Ratio Part ÷ Sum of Ratio Parts) × Total Amount

Calculations:

⇒ Sum of ratio parts = 3 + 5 = 8

⇒ B's share = (5 ÷ 8) × 8,400

⇒ B's share = 5 × (8,400 ÷ 8)

⇒ B's share = 5 × 1,050

⇒ B's share = ₹5,250

∴ The correct answer is ₹5,250.

Additional Information

Ratio Distribution

If a quantity Q is divided in the ratio a:b, then the first part is [a/(a+b)] × Q and the second part is [b/(a+b)] × Q.

Inverse Ratio

The inverse ratio of a:b is 1/a : 1/b, which simplifies to b:a.

Compound Ratio

For two ratios a:b and c:d, the compound ratio is (a × c) : (b × d).

Shortcut Trick

Base distance from shorter pole (10 m) for 30° = 10 × √3 = 10√3 m.

Base distance from taller pole (30 m) for 60° = 30 / √3 = 10√3 m.

Total distance d = 10√3 + 10√3 = 20√3 m.

∴ The correct answer is 20√3 m.

Alternate Method

Given: Height of shorter pole (H₁) = 10 m, Height of taller pole (H₂) = 30 m, Angles of elevation = 30° and 60°.

Formula Used: tan θ = Perpendicular / Base

Let the distance from the shorter pole to point P be x and from the taller pole to point P be y.

For the shorter pole (Height = 10 m, θ = 30°):

⇒ tan 30° = 10 / x

⇒ 1 / √3 = 10 / x

⇒ x = 10√3 m

For the taller pole (Height = 30 m, θ = 60°):

⇒ tan 60° = 30 / y

⇒ √3 = 30 / y

⇒ y = 30 / √3 = (30 × √3) / 3 = 10√3 m

Total distance between the poles d:

⇒ d = x + y

⇒ d = 10√3 + 10√3

⇒ d = 20√3 m

∴ The correct answer is 20√3 m.

Additional Information

Angle of Elevation

The angle formed by the line of sight with the horizontal when an object is viewed above the horizontal level.

Trigonometric Ratios for Heights

Common values used: tan 30° = 1/√3, tan 45° = 1, and tan 60° = √3.

Pythagoras Theorem

In right-angled triangles, Hypotenuse² = Base² + Perpendicular².

Shortcut Trick

Total Work (LCM of 12, 12) = 12 units; Efficiency of A = 1, B = 1.

Combined efficiency of (A + B) = 1 + 1 = 2 units/day.

Work done in 3 days = 2 × 3 = 6 units.

Remaining work = 12 − 6 = 6 units.

Time for B to finish = 6 units ÷ 1 unit/day = 6 days.

∴ The correct answer is 6 days.

Alternate Method

Given: Time taken by A = 12 days; Time taken by B = 12 days; Work period = 3 days.

Formula Used: Total Work = Efficiency × Time

⇒ Combined efficiency of A and B = 1 + 1 = 2 units per day.

⇒ Work completed by A and B in 3 days = 2 × 3 = 6 units.

⇒ Remaining work = Total work − Completed work = 12 − 6 = 6 units.

⇒ Time taken by B to complete the remaining work = 6 ÷ 1 = 6 days.

∴ The correct answer is 6 days.

Additional Information

Efficiency and Time Relation

Efficiency is inversely proportional to the time taken to complete the work. If time ratios are X:Y, efficiency ratios are Y:X.

Work and Wages

Wages are generally distributed based on the ratio of the actual work done by each person or their efficiency.

Alternate Day Work

In problems where persons work on alternate days, calculate the combined work for one cycle (e.g., 2 days) and then find the total cycles required.