Search Previous Year Questions

Shortcut Trick

Test the options for the larger number.

If the larger number is 20, then the smaller number is 20 − 8 = 12.

Condition: Twice the larger number = 2 × 20 = 40.

Condition: Three times the smaller number + 4 = (3 × 12) + 4 = 36 + 4 = 40.

Since both conditions match (40 = 40), the larger number is 20.

∴ The correct answer is 20.

Alternate Method

Given:

Difference between two numbers = 8

Let the larger number be x and the smaller number be y.

Formula Used:

Linear equations in two variables: x − y = d and ax + by = c.

Calculations:

⇒ x − y = 8

⇒ y = x − 8

⇒ 2x = 3y + 4

⇒ 2x = 3(x − 8) + 4

⇒ 2x = 3x − 24 + 4

⇒ 2x = 3x − 20

⇒ 3x − 2x = 20

⇒ x = 20

∴ The correct answer is 20.

Additional Information

Linear Equation Formulation

Word problems are solved by translating phrases like 'twice' as 2×, 'more than' as +, and 'difference' as subtraction.

Substitution Method

Expressing one variable in terms of another (e.g., y = x − 8) and substituting it into the second equation helps reduce variables.

Verification of Results

Always verify the calculated values in both original conditions to ensure accuracy in algebraic solutions.

Shortcut Trick

The standard Trigonometric Identity is cos²θ + sin²θ = 1.

Another fundamental identity is 1 + tan²θ = sec²θ, which implies 1 + tan²θ − sec²θ = 0.

By substituting this into the expression: (1 + tan²θ) + 1 − sec²θ ⇒ sec²θ + 1 − sec²θ = 1.

Since both values equal 1, the expression matches the given condition.

∴ The correct answer is 1 + tan²(θ) + (1− sec²(θ)).

Alternate Method

Given: Primary value = cos²θ + sin²θ

Formula Used:

1. Pythagorean Identity: sin²θ + cos²θ = 1
2. Secant Identity: sec²θ − tan²θ = 1 ⇒ tan²θ − sec²θ = −1

Calculations:

⇒ Target value = cos²θ + sin²θ = 1

⇒ Evaluating the expression: 1 + tan²θ + (1 − sec²θ)

⇒ Rearranging terms: (1 + tan²θ) + 1 − sec²θ

⇒ Substitute (1 + tan²θ) = sec²θ

⇒ Result = sec²θ + 1 − sec²θ

⇒ Result = 1

⇒ Since 1 = 1, the values are identical.

∴ The correct answer is 1 + tan²(θ) + (1− sec²(θ)).

Additional Information

Fundamental Pythagorean Identities

The three main identities are sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Reciprocal Relations

Key relations: sinθ = 1/cosecθ, cosθ = 1/secθ, and tanθ = 1/cotθ.

Ratio Identities

Tanθ is the ratio of sinθ to cosθ (tanθ = sinθ/cosθ), while cotθ is the inverse (cotθ = cosθ/sinθ).

Shortcut Trick

Ratio of Expenditure : Savings = 75 : 25 = 3 : 1 (Total = 4 units)

Apply the weighted average rule for percentage increases:

(Total Units × % Income Increase) = (Exp Units × % Exp Increase) + (Sav Units × % Sav Increase)

(4 × 20) = (3 × 10) + (1 × x)

80 = 30 + x ⇒ x = 50%

∴ The correct answer is 50% increase.

Alternate Method

Given: Initial spending = 75% of income. Income increases by 20%, Expenditure by 10%.

Let Initial Income = ₹100

⇒ Initial Expenditure = 75% of 100 = ₹75

⇒ Initial Savings = Income − Expenditure = 100 − 75 = ₹25

⇒ New Income = 100 + (20% of 100) = ₹120

⇒ New Expenditure = 75 + (10% of 75) = 75 + 7.5 = ₹82.5

⇒ New Savings = New Income − New Expenditure = 120 − 82.5 = ₹37.5

⇒ Change in Savings = 37.5 − 25 = ₹12.5

⇒ Percentage Change = (12.5 ÷ 25) × 100 = 50%

∴ The correct answer is 50% increase.

Additional Information

Income-Expenditure Formula

Income = Expenditure + Savings. Any increase in income must be balanced by the sum of increases in expenditure and savings.

Weighted Average Method

If components A and B form total T, then % change in T = (A/T × % change in A) + (B/T × % change in B).

Percentage Change

Percentage Increase/Decrease = [(New Value − Old Value) ÷ Old Value] × 100.

Shortcut Trick

Equate the common term Female students using LCM of 3 and 8, which is 24.

New Ratio Male : Female = 32 : 24 (multiplied by 8).

New Ratio Female : Staff = 24 : 3 (multiplied by 3).

Combined Ratio Male : Female : Staff = 32 : 24 : 3.

Total students : Staff = (32 + 24) : 3 = 56 : 3.

∴ The correct answer is 56 : 3.

Alternate Method

Given: Ratio of Male to Female students = 4 : 3

Ratio of Female students to Teaching staff = 8 : 1

⇒ Let Male students (M) = 4x and Female students (F) = 3x.

⇒ From the second ratio, F : T = 8 : 1, so Teaching staff (T) = F ÷ 8.

⇒ Total Students = M + F = 4x + 3x = 7x.

⇒ T = 3x ÷ 8 = 3x/8.

⇒ Ratio of Total Students to Teaching Staff = 7x : (3x/8).

⇒ Multiply both sides by 8 to clear the fraction: (7 × 8) : 3 = 56 : 3.

∴ The correct answer is 56 : 3.

Additional Information

Compound Ratio

When two or more ratios are multiplied together (a/b × c/d), the result (ac : bd) is called the compound ratio.

Balancing Ratios

To merge ratios A : B and B : C, find the LCM of the common term B and adjust both ratios accordingly to create a unified A : B : C sequence.

Invertendo Property

If a : b = c : d, then b : a = d : c. This is useful when the denominator or second term of the ratio needs to be compared or simplified.

Shortcut Trick

Ratio of students in three classes = 60 : 40 : 50 = 6 : 4 : 5.

Assume an average marks of 60.

Net deviation = (6 × (65 − 60)) + (4 × (72 − 60)) + (5 × (60 − 60)) = 30 + 48 + 0 = 78.

Average deviation = 78 ÷ (6 + 4 + 5) = 78 ÷ 15 = 5.2.

Combined average = Assumed Average + Average deviation = 60 + 5.2 = 65.2.

∴ The correct answer is 65.2.

Alternate Method

Given:

Class 1: 60 students, Average = 65

Class 2: 40 students, Average = 72

Class 3: 50 students, Average = 60

Formula Used:

Combined Average = (n₁a₁ + n₂a₂ + n₃a₃) ÷ (n₁ + n₂ + n₃)

Calculations:

⇒ Total marks for Class 1 = 60 × 65 = 3900

⇒ Total marks for Class 2 = 40 × 72 = 2880

⇒ Total marks for Class 3 = 50 × 60 = 3000

⇒ Sum of all marks = 3900 + 2880 + 3000 = 9780

⇒ Total number of students = 60 + 40 + 50 = 150

⇒ Combined Average = 9780 ÷ 150

⇒ Combined Average = 978 ÷ 15 = 65.2

∴ The correct answer is 65.2.

Additional Information

Weighted Average

It is an average where each quantity is given a specific weight according to its importance or size. Formula: Average = (Σ w_ix_i) ÷ Σ w_i.

Average Property

If each value in a data set is increased or decreased by a constant k, the new average also increases or decreases by k.

Net Deviation Rule

The sum of deviations of all observations from their arithmetic mean is always zero.

Shortcut Trick

A number divisible by 24 must be divisible by both 3 and 8.

For divisibility by 8, last three digits 44B must be divisible by 8 ⇒ B = 0 or 8.

For divisibility by 3, the sum of digits (26 + A + B) must be divisible by 3.

To maximize A + B, we take the largest possible value B = 8.

Now, 26 + A + 8 = 34 + A must be divisible by 3 ⇒ largest digit A = 8.

∴ The correct answer is 16.

Alternate Method

Given: The 7-digit number is 77A444B and it is divisible by 24.

Formula: If a number is divisible by 24, it must satisfy the divisibility rules for its co-prime factors 3 and 8.

⇒ Divisibility by 8: The last three digits 44B must be divisible by 8.

⇒ 440 + B ÷ 8 ⇒ Since 440 is divisible by 8 (55 × 8), B must be 0 or 8.

⇒ Divisibility by 3: The sum of digits 7 + 7 + A + 4 + 4 + 4 + B must be divisible by 3.

⇒ 26 + A + B must be a multiple of 3.

⇒ To find the maximum value of (A + B), let us test B = 8 (the maximum possible value for B).

⇒ Sum = 26 + A + 8 = 34 + A.

⇒ For 34 + A to be divisible by 3, possible values for A are 2, 5, 8.

⇒ To maximize the expression, we take A = 8.

⇒ Maximum (A + B) = 8 + 8 = 16.

∴ The correct answer is 16.

Additional Information

Divisibility Rule of 8

A number is divisible by 8 if its last three digits are divisible by 8 or if the last three digits are all zeros.

Divisibility Rule of 3

A number is divisible by 3 if the sum of all its individual digits is a multiple of 3.

Composite Divisibility

To check divisibility by a composite number like 24, check divisibility by its co-prime factors (3 and 8 in this case).

Shortcut Trick

Net Profit Ratio = (Weight actually received while buying / Standard weight) × (Standard weight / Weight actually given while selling)

Net Ratio = (1200 / 1000) × (1000 / 840) = 1200 : 840

Simplified Profit Ratio (SP : CP) = 10 : 7

Profit % = (Profit / CP) × 100 = (10 − 7) / 7 × 100 = 3/7 × 100

Profit % = 42.857...% ≈ 42.86%

∴ The correct answer is 42.86%.

Alternate Method

Given:

Weight obtained by dishonest shopkeeper while purchasing = 1200 g (instead of 1000 g)

Weight given by shopkeeper while selling = 840 g (instead of 1000 g)

Let the Cost Price (CP) of 1 g of rice be ₹1.

While purchasing, he pays for 1000 g but gets 1200 g.

⇒ Actual CP for 1200 g = ₹1000

⇒ CP of 1 g = 1000 / 1200 = ₹5/6

While selling, he charges for 1000 g (at CP) but gives only 840 g.

⇒ Selling Price (SP) of 840 g = ₹1000

⇒ SP of 1 g = 1000 / 840 = ₹25/21

Overall percentage profit:

⇒ Profit = SP − CP = (25/21) − (5/6)

⇒ Profit = (50 − 35) / 42 = 15/42 = 5/14

⇒ Profit % = (Profit / CP) × 100

⇒ Profit % = (5/14) / (5/6) × 100

⇒ Profit % = (5/14) × (6/5) × 100 = 6/14 × 100 = 3/7 × 100

⇒ Profit % = 42.857...%

∴ The correct answer is 42.86%.

Additional Information

Dishonest Dealer Formula

When a dealer sells at cost price but uses a false weight, Profit % = [Error / (True Value − Error)] × 100.

Combined Cheating (Buying and Selling)

If a dealer cheats by x% while buying and y% while selling, the net gain ratio is [(100+x)/100] × [100/(100−y)].

Successive Percentage Change

For successive profits of a% and b%, the effective profit is [a + b + (a × b)/100]%, though the ratio method is safer for weight errors.

Shortcut Trick

Volume of material in a pipe = π × h × (R + r) × (R − r)

Given: R = 8 cm, r = 7 cm, h = 20 cm, π = 3.14

Volume = 3.14 × 20 × (8 + 7) × (8 − 7)

Volume = 3.14 × 20 × 15 × 1 = 3.14 × 300 = 942

∴ The correct answer is 942 cubic cm.

Alternate Method

Given:

External radius (R) = 8 cm

Internal radius (r) = 7 cm

Length of the pipe (h) = 20 cm

Formula Used:

Volume of cylindrical pipe = πR²h − πr²h = πh(R² − r²)

Calculations:

⇒ Volume = 3.14 × 20 × (8² − 7²)

⇒ Volume = 3.14 × 20 × (64 − 49)

⇒ Volume = 3.14 × 20 × 15

⇒ Volume = 3.14 × 300

⇒ Volume = 942 cubic cm

∴ The correct answer is 942 cubic cm.

Additional Information

Total Surface Area (TSA) of Hollow Cylinder

TSA = External CSA + Internal CSA + 2 × Area of Base Rings = 2πRh + 2πrh + 2π(R² − r²).

Curved Surface Area (CSA)

For a solid cylinder, CSA = 2πrh. For a hollow pipe, the total curved area is the sum of the inner and outer curved surfaces.

Volume of a Solid Cylinder

The volume of a solid cylinder is calculated as V = πr²h, where r is the radius and h is the vertical height.

Shortcut Trick

The Effective Interest Rate for 4 years at 12% is: 12% × 4 = 48%.

Since 48% of the Principal equals the Simple Interest of ₹1,320.

Principal (100%) = (1,320 ÷ 48) × 100 = 27.5 × 100 = 2,750.

∴ The correct answer is ₹2,750.

Alternate Method

Given:

Simple Interest (SI) = ₹1,320

Rate of Interest (R) = 12% per annum

Time Period (T) = 4 years

Formula Used:

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

Calculations:

⇒ 1,320 = (P × 12 × 4) ÷ 100

⇒ 1,320 = (P × 48) ÷ 100

⇒ P × 48 = 1,320 × 100

⇒ P = 132,000 ÷ 48

⇒ P = 2,750

∴ The correct answer is ₹2,750.

Additional Information

Total Amount in Simple Interest

The total Amount (A) is the sum of the Principal and the Simple Interest: A = P + SI = P × [1 + (R × T) ÷ 100].

Constant Interest Property

In Simple Interest, the interest earned remains the same every year because it is always calculated on the original Principal only.

Rate and Time Calculation

To find the Rate (R): R = (SI × 100) ÷ (P × T). To find the Time (T): T = (SI × 100) ÷ (P × R).

Shortcut Trick

Total number of pens = 4 + 1 = 5

Total cost = (4 × 47) + (50% of 47)

Total cost = 188 + 23.5 = ₹211.5

Effective price = 211.5 ÷ 5 = ₹42.30

∴ The correct answer is ₹42.30.

Alternate Method

Given:

Price of one pen = ₹47

Scheme: Buy 4 pens and get a 50% discount on 1 extra pen.

Formula:

Effective price = Total payment / Total number of items

Calculation:

⇒ Cost of 4 pens = 4 × 47 = ₹188

⇒ Discount on the 5^th pen = 50% of 47 = 47 × (50/100) = ₹23.50

⇒ Discounted price of the 5^th pen = 47 − 23.50 = ₹23.50

⇒ Total amount paid for 5 pens = 188 + 23.50 = ₹211.50

⇒ Effective price per pen = 211.50 ÷ 5

⇒ Effective price per pen = ₹42.30

∴ The correct answer is ₹42.30.

Additional Information

Effective Discount %

To find the effective discount in such schemes: (Total Discount / Total Marked Price) × 100. In this question, Discount = ₹23.5 and Total Marked Price = 5 × 47 = ₹235. Effective Discount % = (23.5 / 235) × 100 = 10%.

Buy X Get Y Free

If the scheme were 'Buy 4 Get 1 Free', then the Effective Discount % would be [1 / (4 + 1)] × 100 = 20%.

Marked Price and Selling Price

Selling Price (SP) = Marked Price (MP) − Discount. Here, ₹42.30 is the effective selling price of each pen.