Quantitative Aptitude PYQs

Given:

The line passes through the point (5, 7)

Slope of the line = −1

Calculation:

The general form of a straight line is:

y = mx + c

Substitute the given slope:

y = −x + c

Since the line passes through (5, 7), substitute x = 5 and y = 7:

7 = −5 + c

c = 12

Equation of the line:

y = −x + 12

Correct option: y = −x + 12

Given:

The quadratic equation is x² + kx + 16 = 0.

For the roots to be real and distinct:

Discriminant (D) > 0

Formula used:

Discriminant (D) = b² - 4ac

Here, a = 1, b = k, and c = 16.

Calculation:

D = k² - 4 × 1 × 16

⇒ D = k² - 64

For real and distinct roots, D > 0:

⇒ k² - 64 > 0

⇒ (k - 8)(k + 8) > 0

The critical points are k = 8 and k = -8.

Using the sign analysis of the quadratic inequality:

• For (k - 8)(k + 8) > 0, k < -8 or k > 8.

∴ The correct answer is option (1): k > 8.

Step 1: Simplify the first fraction

Given expression:

(1 + tan²x) / (1 + cot²x)

Using identities:

1 + tan²x = sec²x

1 + cot²x = csc²x

So, the expression becomes:

sec²x / csc²x

Now,

sec x = 1 / cos x

csc x = 1 / sin x

⇒ sec²x / csc²x = (1 / cos²x) ÷ (1 / sin²x)

⇒ (1 / cos²x) × (sin²x / 1)

⇒ sin²x / cos²x

⇒ tan²x

Step 2: Simplify the second fraction

(sec x − tan x) / (sec x + tan x)

Multiply numerator and denominator by (sec x − tan x)

Denominator:

(sec x + tan x)(sec x − tan x)

⇒ sec²x − tan²x

Using identity:

sec²x − tan²x = 1

So denominator = 1

Numerator:

(sec x − tan x)²

Substitute values:

sec x = 1 / cos x

tan x = sin x / cos x

⇒ ( (1 − sin x) / cos x )²

⇒ (1 − sin x)² / cos²x

Now,

cos²x = 1 − sin²x

⇒ cos²x = (1 − sin x)(1 + sin x)

So,

(1 − sin x)² / [(1 − sin x)(1 + sin x)]

Cancel (1 − sin x)

⇒ (1 − sin x) / (1 + sin x)

Step 3: Final Result

Multiply both simplified parts:

tan²x × (1 − sin x) / (1 + sin x)

∴ Final Answer = tan²x (1 − sin x) / (1 + sin x)

Given:

sinθ = 7 / 25

Calculation:

Using the identity:

sin²θ + cos²θ = 1

So,

cos²θ = 1 − sin²θ

cos²θ = 1 − (7 / 25)²

cos²θ = 1 − 49 / 625

cos²θ = (625 − 49) / 625

cos²θ = 576 / 625

Correct Answer: 576 / 625

Correct Option: Option 1

Shortcut Trick

Given tan A = 1, we know that A = 45° since tan 45° = 1.

The expression 2 sin A cos A is the trigonometric identity for sin 2A.

Substituting A = 45°: 2 sin 45° cos 45° = sin(2 × 45°) = sin 90°.

Since sin 90° = 1, the required value is 1.

∴ The correct answer is 1.

Alternate Method

Given: In ΔABC, ∠B = 90° and tan A = 1.

Formula Used: tan A = Perpendicular / Base; sin A = P / H; cos A = B / H

⇒ tan A = BC / AB = 1. Let BC = 1 and AB = 1.

⇒ By Pythagoras Theorem: AC = √(AB² + BC²)

⇒ AC = √(1² + 1²) = √2.

⇒ sin A = BC / AC = 1 / √2.

⇒ cos A = AB / AC = 1 / √2.

⇒ 2 sin A cos A = 2 × (1 / √2) × (1 / √2).

⇒ 2 sin A cos A = 2 × (1 / 2) = 1.

∴ The correct answer is 1.

Additional Information

Double Angle Identities

The expression 2 sin θ cos θ is identical to sin 2θ, which simplifies calculations for specific angles.

Standard Trigonometric Values

For θ = 45°, sin 45° = 1/√2, cos 45° = 1/√2, and tan 45° = 1.

Pythagorean Identity

For any angle A, the fundamental identity sin² A + cos² A = 1 always holds true.

Right Triangle Properties

In an isosceles right triangle (where tan A = 1), the two acute angles are both 45° and the legs are equal.

Shortcut Trick

Investment Ratio = (Profit Ratio) ÷ (Time Ratio)

Investment Ratio = (6/14) : (8/8) : (9/7)

Investment Ratio = (3/7) : 1 : (9/7)

Multiply the entire ratio by 7 to simplify: 3 : 7 : 9

∴ The correct answer is 3:7:9.

Alternate Method

Given:

Profit ratio of three partners = 6 : 8 : 9

Time of investment for each partner = 14 months, 8 months, and 7 months

Formula Used:

Profit = Investment × Time

⇒ Investment = Profit ÷ Time

Let the investments of the three partners be I₁, I₂, and I₃ respectively.

⇒ I₁ : I₂ : I₃ = (6/14) : (8/8) : (9/7)

⇒ I₁ : I₂ : I₃ = (3/7) : 1 : (9/7)

To eliminate the fractions, multiply the entire ratio by the LCM of denominators (7, 1, 7), which is 7:

⇒ I₁ : I₂ : I₃ = (3/7 × 7) : (1 × 7) : (9/7 × 7)

⇒ I₁ : I₂ : I₃ = 3 : 7 : 9

∴ The correct answer is 3:7:9.

Additional Information

Partnership Basic Rule

The profit earned in a business is distributed among partners in the ratio of the product of their capital and time duration (Profit = C × T).

Ratio of Capital/Investment

When the profit ratio and time ratio are given, the investment ratio is calculated as (P₁/T₁) : (P₂/T₂) : (P₃/T₃).

Working vs Sleeping Partner

A working partner manages the business and may receive a salary or extra profit share, while a sleeping partner only invests money.

Shortcut Trick

Combine the Pythagorean identity terms: sin² x + cos² x = 1.

Substitute this into the expression to get 1 + tan² x.

Recognize that 1 + tan² x is the standard identity for sec² x.

∴ The correct answer is sec² x.

Alternate Method

Given: sin² x + cos² x + tan² x

Formula Used:

1. sin² θ + cos² θ = 1
2. 1 + tan² θ = sec² θ

⇒ Expression = (sin² x + cos² x) + tan² x

⇒ Expression = 1 + tan² x

⇒ Expression = sec² x

∴ The correct answer is sec² x.

Additional Information

Primary Pythagorean Identity

The most fundamental identity in trigonometry is sin² θ + cos² θ = 1, which relates the sine and cosine of any angle.

Secant-Tangent Relationship

The identity sec² θ − tan² θ = 1 is derived by dividing the primary identity by cos² θ.

Cosecant-Cotangent Relationship

Similarly, dividing the primary identity by sin² θ yields the identity 1 + cot² θ = cosec² θ.

Given:

Profit to cost price ratio = 3:8

Formula Used:

Selling Price (SP) = Cost Price (CP) + Profit

Profit = (Profit to CP ratio) × CP

SP to CP ratio = SP ÷ CP

Calculation:

⇒ Profit = (3/8) × CP

⇒ SP = CP + Profit

⇒ SP = CP + (3/8) × CP

⇒ SP = (1 + 3/8) × CP

⇒ SP = (8/8 + 3/8) × CP

⇒ SP = (11/8) × CP

⇒ SP to CP ratio = 11:8

∴ The correct answer is 11:8.

Shortcut Trick

Ratio of apples to oranges = 7 : 9

According to the question, 7 units = 63 apples.

1 unit = 63 ÷ 7 = 9

Number of oranges = 9 units = 9 × 9 = 81

∴ The correct answer is 81.

Alternate Method

Given: Ratio of apples to oranges in a basket = 7 : 9

Given: Total number of apples = 63

⇒ Let the number of apples be 7x and the number of oranges be 9x.

⇒ Since the number of apples is 63, we can write:

⇒ 7x = 63

⇒ x = 63 ÷ 7

⇒ x = 9

⇒ Now, number of oranges = 9x

⇒ Number of oranges = 9 × 9 = 81

∴ The correct answer is 81.

Additional Information

Direct Proportion

Two quantities are in direct proportion if the ratio of their values remains constant. If x ∝ y, then x₁/y₁ = x₂/y₂.

Simplification of Ratios

A ratio a : b can be simplified by dividing both terms by their Highest Common Factor (HCF). For example, 63 : 81 = (63/9) : (81/9) = 7 : 9.

Compound Ratio

The compound ratio of two ratios a : b and c : d is defined as (a × c) : (b × d).

Given:

Difference between Compound Interest (CI) and Simple Interest (SI) = ₹360

Time = 2 years

Rate of Interest = 15% p.a.

Formula Used:

CI − SI = P × (r/100)²

Where P = Principal, r = Rate of Interest

Calculation:

⇒ 360 = P × (15/100)²

⇒ 360 = P × (225/10000)

⇒ P = 360 ÷ (225/10000)

⇒ P = 360 × (10000/225)

⇒ P = ₹16,000

∴ The correct answer is ₹16,000.

Given:

Principal = 200000

Rate of interest = 10%

Amount withdrawn after 1 year = 20000

Total time = 2 years

Formula Used:

Compound amount = Principal × (1 + Rate ÷ 100)

Calculation:

⇒ Amount after 1 year = 200000 × (1 + 10 ÷ 100)

⇒ Amount after 1 year = 200000 × 1.10

⇒ Amount after 1 year = 220000

⇒ Balance after withdrawal = 220000 − 20000

⇒ Balance = 200000

⇒ Amount after 2^nd year = 200000 × (1 + 10 ÷ 100)

⇒ Amount after 2^nd year = 200000 × 1.10

⇒ Amount after 2^nd year = 220000

∴ The correct answer is 220000.

Given:

Average of 50 numbers = 38

Wrongly taken numbers = 33 and 27

Correct numbers = 23 and 17

Total numbers = 50

Formula Used:

Total sum of numbers = Average × Total numbers

Corrected total sum = Original total sum - Wrong values + Correct values

Corrected average = Corrected total sum ÷ Total numbers

Calculation:

⇒ Total sum of numbers = 38 × 50 = 1900

⇒ Corrected total sum = 1900 - (33 + 27) + (23 + 17)

⇒ Corrected total sum = 1900 - 60 + 40

⇒ Corrected total sum = 1880

⇒ Corrected average = 1880 ÷ 50

⇒ Corrected average = 37.6

∴ The correct answer is 37.6.

Given:

First number = 13511

Second number = 13903

Third number = 14589

Same divisor = n

Same remainder = m

Formula Used:

If three numbers leave the same remainder when divided by a number, then that divisor is the H.C.F. of the pairwise differences of the numbers.

Remainder = Number mod Divisor

Calculation:

⇒ 13903 − 13511 = 392

⇒ 14589 − 13903 = 686

⇒ 14589 − 13511 = 1078

⇒ H.C.F. of 392 and 686

⇒ 686 = 392 × 1 + 294

⇒ 392 = 294 × 1 + 98

⇒ 294 = 98 × 3 + 0

⇒ H.C.F. = 98

⇒ n = 98

⇒ m = 13511 mod 98

⇒ 13511 = 98 × 137 + 85

⇒ m = 85

⇒ n + m = 98 + 85

⇒ n + m = 183

∴ The correct answer is 183.

Given:

Price increases by 20% in first year

Price decreases by 10% in second year

Final price = 59400

Formula Used:

New price = Original price × (1 + Increase%) × (1 − Decrease%)

Calculation:

⇒ Let original price = x

⇒ After 20% increase = x × 1.20

⇒ After 10% decrease = x × 1.20 × 0.90

⇒ Final price = x × 1.08

⇒ x × 1.08 = 59400

⇒ x = 59400 ÷ 1.08

⇒ x = 55000

∴ The correct answer is 55000.

Given:

Speed while going = 30 km/h

Speed while returning = 20 km/h

Total time = 5 hours

Formula Used:

Time = Distance ÷ Speed

Total time = Time going + Time returning

Calculation:

⇒ Let one-way distance = x

⇒ Time going = x ÷ 30

⇒ Time returning = x ÷ 20

⇒ x ÷ 30 + x ÷ 20 = 5

⇒ (2x + 3x) ÷ 60 = 5

⇒ 5x ÷ 60 = 5

⇒ x ÷ 12 = 5

⇒ x = 60

∴ The correct answer is 60.

Shortcut Trick

If A and B together take x days and B alone takes y days, then A alone takes:

Time = (x × y) ÷ (y − x)

Time taken by A = (10 × 15) ÷ (15 − 10)

Time taken by A = 150 ÷ 5 = 30 days

∴ The correct answer is 30 days.

Alternate Method

Given:

Time taken by A and B together = 10 days

Time taken by B alone = 15 days

Formula Used:

Total Work = LCM of individual days

Efficiency = Total Work ÷ Time Taken

Calculations:

⇒ Total work (LCM of 10 and 15) = 30 units

⇒ Efficiency of A + B = 3 units/day

⇒ Efficiency of B = 2 units/day

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

⇒ Efficiency of A = 3 − 2 = 1 unit/day

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

⇒ Time taken by A = 30 ÷ 1 = 30 days

∴ The correct answer is 30 days.

Additional Information

Pipes and Cisterns

The concept is identical to time and work; filling pipes act like positive work, and drainage pipes (leaks) act like negative work.

Work and Wages

Wages are always distributed among workers in the direct ratio of their efficiencies or the share of work done by each.

Efficiency-Time Relation

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

Given:

AB ∥ CD

AB = 2CD

Area of \( \Delta COD \) = 8

Formula Used:

In a trapezium, diagonals divide each other in the ratio of the parallel sides

Area of triangles with same altitude is proportional to their bases

Calculation:

⇒ AB : CD = 2 : 1

⇒ AO : OC = 2 : 1

⇒ BO : OD = 2 : 1

⇒ Area(\( \Delta AOB \)) : Area(\( \Delta COD \)) = (AB / CD)²

⇒ Area ratio = 2² : 1

⇒ Area ratio = 4 : 1

⇒ Area(\( \Delta AOB \)) = 4 × 8

⇒ Area(\( \Delta AOB \)) = 32

∴ The correct answer is 32.

Given:

Discount offered = 10%

Selling price after discount for two books = 540

Profit after discount = 20%

Formula Used:

Selling Price = Cost Price × (1 + Profit%)

Discounted price = Marked Price × (1 − Discount%)

Calculation:

⇒ 540 = 120% of Cost Price

⇒ Cost Price = 540 × 100 ÷ 120

⇒ Cost Price = 450

⇒ 540 = 90% of Marked Price

⇒ Marked Price = 540 × 100 ÷ 90

⇒ Marked Price = 600

⇒ Profit without discount = 600 − 450

⇒ Profit = 150

∴ The correct answer is 150.

Given:

Simple interest = Principal ÷ 4

Number of years = Rate of interest

Formula Used:

Simple Interest = (Principal × Rate × Time) ÷ 100

Calculation:

⇒ Let Principal = P

⇒ Rate = r

⇒ Time = r

⇒ Simple Interest = (P × r × r) ÷ 100

⇒ P ÷ 4 = (P × r²) ÷ 100

⇒ 1 ÷ 4 = r² ÷ 100

⇒ r² = 25

⇒ r = 5

∴ The correct answer is 5%.

Given:

Length of sheet = 44 cm

Height of cylinder = 10 cm

Formula Used:

Circumference of base = 2 × π × r

Volume of cylinder = π × r² × h

Calculation:

⇒ 2 × (22 ÷ 7) × r = 44

⇒ (44 ÷ 7) × r = 44

⇒ r = 7

⇒ Volume = (22 ÷ 7) × 7² × 10

⇒ Volume = (22 ÷ 7) × 49 × 10

⇒ Volume = 22 × 7 × 10

⇒ Volume = 1540

∴ The correct answer is 1540.