Search Previous Year Questions

Given:

The expression: 0.125 × 4⁴ × 2⁻⁵ × 0.25² × 8²

Formula used:

a^m × aⁿ = a^m+n

(a^m)ⁿ = a^m×n

Calculation:

Convert each term into base 2:

0.125 = 1/8 = 1/2³ = 2⁻³

4⁴ = (2²)⁴ = 2⁸

2⁻⁵ = 2⁻⁵

0.25² = (1/4)² = (1/2²)² = 2⁻⁴

8² = (2³)² = 2⁶

Substitute these values back into the expression:

Expression =

Shortcut Trick

Assume total capacity of cistern = 12 units (LCM of 4 and 12).

Efficiency of Tap 1 = 3, Tap 2 = 1, and Tap 3 (emptying) = −1.

Combined efficiency when all opened = 3 + 1 − 1 = 3 units/hr.

Target work = 1/4 of total capacity = 12 ÷ 4 = 3 units.

Time required = Target units ÷ Combined efficiency = 3 ÷ 3 = 1 hr.

∴ The correct answer is 1 hr.

Alternate Method

Given:

Time taken by Tap A to fill = 4 hr

Time taken by Tap B to fill = 12 hr

Time taken by Tap C to empty = 12 hr

Formula Used:

Efficiency = Total Work ÷ Time

Total Efficiency = Efficiency_A + Efficiency_B − Efficiency_C

⇒ Net efficiency of all taps = 3 + 1 − 1 = 3 units/hr

⇒ Target to fill = 1/4 of cistern = 1/4 × 12 = 3 units

⇒ Time required = 3 units ÷ 3 units/hr = 1 hr

∴ The correct answer is 1 hr.

Additional Information

Pipes and Cisterns Basics

Filling pipes have positive efficiency, while emptying pipes (leakage) have negative efficiency.

Capacity Relation

Total capacity = Time taken to fill/empty × Efficiency of the pipe.

Time and Work Analogy

The concept is identical to men and work, where an emptying pipe acts like a person destroying the work.

Shortcut Trick

The product of two numbers is always equal to the product of their HCF and LCM.

Other number = (HCF × LCM) ÷ Given number

Other number = (36 × 7,560) ÷ 540

Other number = 7,560 ÷ 15 = 504

∴ The correct answer is 504.

Alternate Method

Given:

HCF = 36

LCM = 7,560

First Number (a) = 540

Formula Used:

a × b = HCF × LCM

Calculations:

⇒ 540 × b = 36 × 7,560

⇒ b = (36 × 7,560) ÷ 540

⇒ b = 272,160 ÷ 540

⇒ b = 504

∴ The correct answer is 504.

Additional Information

HCF and LCM Relation

For any two positive integers x and y, the product of the numbers is always equal to the product of their HCF and LCM: x × y = HCF(x, y) × LCM(x, y).

HCF Property

The Highest Common Factor (HCF) of a group of numbers is always a factor of their Least Common Multiple (LCM). Thus, LCM ÷ HCF is always an integer.

Co-prime Numbers

If the HCF of two numbers is 1, they are called co-prime or relatively prime. In this case, their LCM is simply the product of the two numbers.

Shortcut Trick

Total interest percentage = Rate × Time = 5% × 3 = 15%.

Simple Interest = 15% of ₹8,000.

SI = (15 ÷ 100) × 8,000 = 15 × 80 = 1,200.

∴ The correct answer is ₹1,200.

Alternate Method

Given:

Principal (P) = ₹8,000

Rate of Interest (R) = 5% per annum

Time (T) = 3 years

Formula Used:

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

Calculations:

⇒ SI = (8000 × 5 × 3) ÷ 100

⇒ SI = 80 × 15

⇒ SI = 1,200

∴ The correct answer is ₹1,200.

Additional Information

Amount in Simple Interest

The total amount (A) is the sum of the principal and the interest: A = P + SI = P(1 + (R × T) ÷ 100).

SI vs CI for Year 1

For the first year, if the interest is compounded annually, the Simple Interest and Compound Interest are always equal.

Finding Principal from SI

If SI, Rate, and Time are known, the Principal can be calculated as: P = (SI × 100) ÷ (R × T).

Shortcut Trick

Speed in m/s = 40 ÷ 8 = 5 m/s

To convert Speed from m/s to km/hr, multiply by 18/5.

Speed = 5 × (18/5) = 18 km/hr

∴ The correct answer is 18 km/hr.

Alternate Method

Given:

Distance traveled by the cyclist = 40 m

Time taken = 8 sec

Formula Used:

Speed = Distance ÷ Time

To convert speed from m/s to km/hr: Speed (m/s) × (18/5)

Calculations:

⇒ Speed = 40 ÷ 8

⇒ Speed = 5 m/s

⇒ Speed in km/hr = 5 × (18/5)

⇒ Speed = 18 km/hr

∴ The correct answer is 18 km/hr.

Additional Information

Unit Conversion (km/hr to m/s)

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

Average Speed

Average Speed is defined as the Total Distance divided by the Total Time taken for the entire journey.

Relative Speed

When two objects move in the opposite direction, their Relative Speed is the sum of their individual speeds (u + v).

Shortcut Trick

In any right-angled triangle, if the angle of elevation is 45°, the Height (Perpendicular) is always equal to the Distance (Base).

Given Distance = 40 m.

∴ Height = 40 m.

∴ The correct answer is 40 m.

Alternate Method

Given:

Distance from the point to the building (Base) = 40 m

Angle of elevation (θ) = 45°

Formula Used:

tan θ = Perpendicular / Base

Let the height of the building be h.

⇒ tan 45° = h / 40

⇒ 1 = h / 40 (Since tan 45° = 1)

⇒ h = 40 × 1

⇒ h = 40 m

∴ The correct answer is 40 m.

Additional Information

Trigonometric Ratios Table

Key values for tan: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.

45-45-90 Triangle Property

In a triangle with angles 45°, 45°, and 90°, the sides are in the ratio 1 : 1 : √2.

Angle of Elevation vs Depression

The angle of elevation from an observer to an object is always equal to the angle of depression from the object to the observer.

Shortcut Trick

The mean proportional between any two numbers a and b is defined as √(a × b).

Given values: a = (x + y)² and b = (x − y)².

Mean Proportional = √[(x + y)² × (x − y)²] = (x + y) × (x − y).

Using the algebraic identity (a + b)(a − b) = a² − b², we get x² − y².

∴ The correct answer is x² − y².

Alternate Method

Given:

First term (a) = (x + y)²

Third term (c) = (x − y)²

Formula Used:

If b is the mean proportional between a and c, then a : b = b : c or b² = ac ⇒ b = √ac.

Calculations:

⇒ Let the mean proportional be M.

⇒ M = √[(x + y)² × (x − y)²]

⇒ M = √[(x + y)(x − y)]²

⇒ M = (x + y)(x − y)

⇒ Applying identity: (a + b)(a − b) = a² − b²

⇒ M = x² − y²

∴ The correct answer is x² − y².

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is x such that a/x = x/b, giving x = √ab.

Third Proportional

If a, b, c are in continued proportion, then c is the third proportional and c = b²/a.

Fourth Proportional

For a, b, c, d in proportion (a:b :: c:d), d is the fourth proportional and d = (b × c)/a.

Algebraic Identity

Difference of squares: a² − b² = (a + b)(a − b).

Shortcut Trick

Given √x = 5, by squaring both sides we directly find x = 25.

The required value is √(x + 25).

Substituting x = 25: √(25 + 25) = √50.

Since 50 = 25 × 2, the value is 5√2.

∴ The correct answer is 5√2.

Alternate Method

Given:

The equation √x = 5.

Formula Used:

If √a = b, then a = b². Also, √(m × n) = √m × √n.

Calculations:

⇒ √x = 5

⇒ Squaring both sides: x = 5² = 25

⇒ Required expression = √(x + 25)

⇒ Substitute x = 25 into the expression: √(25 + 25)

⇒ Expression = √50

⇒ Expression = √(25 × 2)

⇒ 5√2

∴ The correct answer is 5√2.

Additional Information

Square Root Properties

For any non-negative numbers, the property √a × √b = √(ab) allows for the simplification of surds by factoring out perfect squares.

Solving Radical Equations

To solve for a variable under a radical, isolate the radical term and apply the inverse operation (squaring for square roots).

Perfect Squares

Memorizing perfect squares like 25, 36, 49, and 64 helps in quickly identifying factors that can be moved outside the √ symbol.

Shortcut Trick

The final price is calculated using successive multipliers.

Net Multiplier = (100 − 40)% × (100 − 22)% = 0.60 × 0.78 = 0.468

Amount paid = Marked Price × Net Multiplier = 1000 × 0.468 = 468

∴ The correct answer is ₹468.

Alternate Method

Given:

Marked Price (MP) of the oven = ₹1000

First Discount (d₁) = 40%

Additional Discount for credit card (d₂) = 22%

Formula Used:

Final Price = MP × (100 − d₁)/100 × (100 − d₂)/100

Calculations:

⇒ Price after the first 40% discount = 1000 − (40/100 × 1000)

⇒ Price after first discount = 1000 − 400 = ₹600

⇒ Additional 22% discount is applied on ₹600

⇒ Additional discount amount = 22/100 × 600 = ₹132

⇒ Final amount paid by Ritu = 600 − 132

⇒ Final amount paid = ₹468

∴ The correct answer is ₹468.

Additional Information

Single Equivalent Discount

For two successive discounts of x% and y%, the single equivalent discount is given by (x + y − xy/100)%.

Marked Price and Selling Price

The Selling Price (SP) is always Marked Price (MP) minus the Discount. Discount is always calculated on the Marked Price unless otherwise stated.

Successive Percentage Change

This concept is used when a value is changed multiple times consecutively. The order of discounts does not change the final selling price.

Shortcut Trick

Find the LCM of the divisors: LCM(12, 13, 7) = 1092.

Identify the largest multiple of 1092 below 9999: 1092 × 9 = 9828.

Add the common remainder to the multiple: 9828 + 3 = 9831.

∴ The correct answer is 9831.

Alternate Method

Given:

Divisors = 12, 13, 7

Remainder in each case = 3

Target = Largest 4-digit number

Calculations:

⇒ First, find the LCM of 12, 13, and 7.

⇒ 12 = 2² × 3

⇒ 13 = 13 (prime)

⇒ 7 = 7 (prime)

⇒ LCM = 12 × 13 × 7 = 1092

⇒ The largest 4-digit number is 9999.

⇒ Divide 9999 by the LCM: 9999 ÷ 1092

⇒ 9999 = (1092 × 9) + 171

⇒ The largest 4-digit multiple of 1092 = 9999 − 171 = 9828

⇒ Required number = Multiple + Remainder

⇒ Required number = 9828 + 3 = 9831

∴ The correct answer is 9831.

Additional Information

General Form for Constant Remainder

Any number that leaves a remainder 'r' when divided by x, y, and z can be expressed as: Number = [LCM(x, y, z) × k] + r, where k is an integer.

LCM of Prime Numbers

The LCM of a set of numbers that are co-prime (like 7, 13, and 12 in this case) is simply the product of those numbers.

Smallest n-digit Number

To find the smallest n-digit number satisfying a condition, divide the smallest n-digit base (like 1000 for 4-digits) by the LCM and find the next higher multiple.