Quantitative Aptitude PYQs

Shortcut Trick

Combined efficiency of filling pipe (A) and emptying pipe (B) = (1/4 − 1/8) = 1/8 per minute.

Time to fill full tank = 8 minutes.

Time to fill one-half tank = 8 × (1/2) = 4 minutes.

∴ The correct answer is 4 min.

Alternate Method

Given: Filling time = 4 min, Emptying time = 8 min.

Formula Used: Total Work = LCM of time; Efficiency = Work ÷ Time.

⇒ Net efficiency of (A + B) = 2 − 1 = 1 unit/min.

⇒ Total capacity to be filled = 1/2 × 8 units = 4 units.

⇒ Time required = 4 units ÷ 1 unit/min = 4 minutes.

∴ The correct answer is 4 min.

Additional Information

Pipes and Cisterns Logic

Filling pipes have positive efficiency while emptying pipes (leakages) have negative efficiency.

Time-Work Relationship

If two pipes A and B fill/empty a tank in 'x' and 'y' hours respectively, then the net part filled/emptied in 1 hour = (1/x ± 1/y).

Shortcut Trick

Average of squares of first n natural numbers = (n + 1)(2n + 1) ÷ 6

Here n = 4 (for the first 4 natural numbers).

Average = (4 + 1) × (2 × 4 + 1) ÷ 6

Average = 5 × 9 ÷ 6 = 45 ÷ 6 = 7.5

∴ The correct answer is 7.5.

Alternate Method

Given: The first 4 natural numbers are 1, 2, 3, and 4.

Formula Used: Average = Sum of the squares ÷ Total count of numbers (n)

⇒ Square of 1 = 1² = 1

⇒ Square of 2 = 2² = 4

⇒ Square of 3 = 3² = 9

⇒ Square of 4 = 4² = 16

⇒ Sum = 1 + 4 + 9 + 16 = 30

⇒ Average = Sum ÷ n = 30 ÷ 4

⇒ Average = 7.5

∴ The correct answer is 7.5.

Additional Information

Sum of Squares of first n natural numbers

The formula for the sum is S_n = n(n + 1)(2n + 1) ÷ 6.

Sum of first n natural numbers

The formula for the sum of the first n natural numbers is S_n = n(n + 1) ÷ 2. The average is (n + 1) ÷ 2.

Sum of Cubes of first n natural numbers

The formula for the sum is S_n = [n(n + 1) ÷ 2]². The average is n(n + 1)² ÷ 4.

Shortcut Trick

When three numbers are in continued proportion, the middle term A is the mean proportional.

A² = Product of extremes ⇒ A = √(63 × 175)

A = √(9 × 7 × 25 × 7) = 3 × 7 × 5 = 105

∴ The correct answer is 105.

Alternate Method

Given: 63 : A :: A : 175

Formula Used: Product of Means = Product of Extremes

⇒ A × A = 63 × 175

⇒ A² = 11025

⇒ A = √11025

⇒ A = 105

∴ The correct answer is 105.

Additional Information

Mean Proportional

If a, b, c are in continued proportion, then b is the mean proportional such that b = √(ac).

Third Proportional

In the ratio a : b :: b : c, the term c is called the third proportional and is calculated as c = b² ÷ a.

Fourth Proportional

In the proportion a : b :: c : d, the term d is the fourth proportional where d = (b × c) ÷ a.

Shortcut Trick

The Selling Price (SP) is 80% of the Marked Price (MP) since the discount is 20%.

SP = 6,420 × (80 ÷ 100) = 6,420 × 0.8.

SP = 642 × 8 = 5,136.

∴ The correct answer is ₹5,136.

Alternate Method

Given → Marked Price (MP) = ₹6,420; Discount Percentage = 20%.
Formulas: Discount = (Discount% ÷ 100) × MP | SP = MP − Discount.

→ Discount Amount = (20 ÷ 100) × 6,420 = 0.2 × 6,420 = ₹1,284.
→ Selling Price (SP) = 6,420 − 1,284.
→ Selling Price (SP) = ₹5,136.

∴ The correct answer is ₹5,136.

Additional Information

Marked Price & Discount

The Marked Price is the price printed on the label. Discount is always calculated on the Marked Price unless stated otherwise. SP = MP × (100 − D%) ÷ 100.

Successive Discount

If two discounts d₁ and d₂ are given, the Equivalent Discount = d₁ + d₂ − (d₁ × d₂) ÷ 100.

Shortcut Trick

Let the Marked Price (MP) be 100.

First Discount (20%): 100 − 20 = 80.

Second Discount (5%): 80 − (5% of 80) = 80 − 4 = 76.

Third Discount (21%): 76 − (21% of 76) = 76 − 15.96 = 60.04.

Effective price percentage = (60.04 ÷ 100) × 100 = 60.04%.

∴ The correct answer is 60.04%.

Alternate Method

Given: Three consecutive discounts d₁ = 20%, d₂ = 5%, and d₃ = 21%.

Formula: Net Selling Price (SP) = MP × (1 − d₁/100) × (1 − d₂/100) × (1 − d₃/100)

⇒ Let MP = 100.

⇒ Effective price = 100 × (80/100) × (95/100) × (79/100)

⇒ Effective price = 100 × 0.8 × 0.95 × 0.79

⇒ Effective price = 76 × 0.79 = 60.04.

⇒ Effective price percentage = (60.04 ÷ 100) × 100 = 60.04%.

∴ The correct answer is 60.04%.

Additional Information

Effective Discount vs Effective Price

The effective price is the final ratio of SP to MP, while effective discount is (MP − SP)/MP × 100. Effective discount = 100 − 60.04 = 39.96%.

Successive Percentage Formula

For two discounts a and b, the single equivalent discount is [a + b − (ab ÷ 100)]. This can be applied iteratively for three or more discounts.

Shortcut Trick

Check divisibility of 8,33,525 by prime factors of the options.

For 15, factors are 3 and 5. Sum of digits = 8+3+3+5+2+5 = 26.

Since 26 is not divisible by 3, the number is not divisible by 15.

∴ The correct answer is 15.

Alternate Method

Given: Number = 8,33,525

Divisibility Rules:

⇒ Divisibility by 15: Number must be divisible by 3 and 5. Sum of digits of 8,33,525 = 8 + 3 + 3 + 5 + 2 + 5 = 26. Since 26 is not divisible by 3, 8,33,525 is NOT divisible by 15.

⇒ Divisibility by 25: Last two digits must be divisible by 25. Last two digits are 25, which is divisible by 25. So, it is divisible by 25.

⇒ Divisibility by 55: Number must be divisible by 5 and 11. Alternating sum for 11: (8+3+2) − (3+5+5) = 13 − 13 = 0. Since 0 is divisible by 11, it is divisible by 55.

⇒ Divisibility by 35: Number must be divisible by 5 and 7. 8,33,525 ÷ 7 = 1,19,075. Since it is divisible by 7 and 5, it is divisible by 35.

∴ The correct answer is 15.

Additional Information

Divisibility by 3

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

Divisibility by 11

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

Co-prime Factors

To check divisibility by a composite number, check if it is divisible by its co-prime factors (e.g., for 15, use 3 and 5).

Shortcut Trick
Use the effective multiplier method for successive transactions.
Final SP = Initial CP × (100 + P1)% × (100 − L2)%
Final SP = 12000 × (110 ÷ 100) × (85 ÷ 100)
Final SP = 12000 × 1.1 × 0.85 = 13200 × 0.85
Final SP = ₹11,220
∴ The correct answer is ₹11,220.

Alternate Method
Given → Initial Cost Price (CP) = ₹12,000, Profit% = 10%, Loss% = 15%.
Formulas: SP = CP × (100 + P%) ÷ 100 | SP = CP × (100 − L%) ÷ 100

⇒ Vendor's Selling Price = 12000 + (10% of 12000) = 12000 + 1200 = ₹13,200.
⇒ This 13,200 is the Retailer's Cost Price.
⇒ Retailer's Selling Price = 13200 − (15% of 13200).
⇒ 15% of 13200 = 1320 + 660 = ₹1,980.
⇒ Final Selling Price = 13200 − 1980 = ₹11,220.
∴ The correct answer is ₹11,220.

Additional Information
Successive Percentage Change
When a value changes by x% and then y%, the net change is (x + y + xy ÷ 100)%. For loss, use negative values.
Relationship between CP and SP
If there is a profit, SP > CP. If there is a loss, SP < CP. The selling price of the first person becomes the cost price for the second person.

Shortcut Trick

Calculate the total effective rate of interest: (6% × 4) + (5% × 2) = 24% + 10% = 34%.

Total Simple Interest = 34% of the Principal (₹5,500).

SI = (34 ÷ 100) × 5500 = 34 × 55 = 1870.

∴ The correct answer is ₹1,870.

Alternate Method

Given:

Principal (P) = ₹5,500

Rate 1 (R₁) = 6%, Time 1 (T₁) = 4 years

Rate 2 (R₂) = 5%, Time 2 (T₂) = 2 years

Formula Used:

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

Calculations:

⇒ Interest for first 4 years (SI₁) = (5500 × 6 × 4) ÷ 100

⇒ SI₁ = 55 × 24 = ₹1,320

⇒ Interest for next 2 years (SI₂) = (5500 × 5 × 2) ÷ 100

⇒ SI₂ = 55 × 10 = ₹550

⇒ Total Simple Interest = SI₁ + SI₂ = 1320 + 550 = ₹1,870

∴ The correct answer is ₹1,870.

Additional Information

Simple Interest Principle

Simple interest is always calculated only on the original principal amount, regardless of the time elapsed or changes in interest rates.

Total Amount Formula

The total amount (A) at the end of the tenure is given by: A = Principal + Total Simple Interest.

Variable Rates Rule

When rates change over time, the total interest is the sum of interest calculated for each period: SI_total = P × (R₁T₁ + R₂T₂ + ...) ÷ 100.

Shortcut Trick

Work done by P = Time ÷ Total Time = 15 ÷ 30 = 1/2 of cistern.

Remaining work to be filled by Q = 1 − 1/2 = 1/2 of cistern.

Since both filled equal parts (50% each), the ratio is 1 :

1. Percentage = (1/2 ÷ 1/2) × 100 = 100%.

∴ The correct answer is 100%.

Alternate Method

Given: Pipe P works for 15 hours.

⇒ Work done by Pipe P = Efficiency × Time = 4 × 15 = 60 units

⇒ Remaining work = Total Work − Work by P = 120 − 60 = 60 units

⇒ Work done by Pipe Q = 60 units

⇒ Required Percentage = (Work by P ÷ Work by Q) × 100

⇒ Percentage = (60 ÷ 60) × 100 = 100%

∴ The correct answer is 100%.

Additional Information

Pipes & Cisterns Work Logic

The portion of the tank filled by a pipe is calculated as (Time it worked) ÷ (Total time it takes to fill the tank alone).

Efficiency and Time

Efficiency is inversely proportional to the time taken. Total Work = Efficiency × Time.

Inlet and Outlet Pipes

Inlet pipes perform positive work (fill), while outlet pipes or leaks perform negative work (empty).

Shortcut Trick

Convert decimals: 0.49 = (7/10)², 0.343 = (7/10)³, 0.2401 = (7/10)⁴

Expression = [(7/10)²]⁴ × [(7/10)³]⁴ ÷ [(7/10)⁴]⁴

= (7/10)^{8 + 12 − 16} = (7/10)⁴

So, x + 2 = 4

x = 2

∴ Answer = 2

Alternate Method

Given:

(0.49)⁴ × (0.343)⁴ ÷ (0.2401)⁴ = (7/10)^x+2

Concept Used:

Convert decimals into fractional powers and apply index laws

Calculation:

⇒ 0.49 = (7/10)²

⇒ 0.343 = (7/10)³

⇒ 0.2401 = (7/10)⁴

⇒ Substitute:

⇒ [(7/10)²]⁴ × [(7/10)³]⁴ ÷ [(7/10)⁴]⁴

⇒ (7/10)⁸ × (7/10)¹² ÷ (7/10)¹⁶

⇒ (7/10)^{(8 + 12 − 16)}

⇒ (7/10)⁴

⇒ Compare powers: x + 2 = 4

⇒ x = 2

∴ The correct answer is 2.

Additional Information

Laws of Exponents

a^m × aⁿ = a^m+n, a^m ÷ aⁿ = a^m−n

Power of Power

(a^m)ⁿ = a^mn

Base Conversion

Convert decimals into fractional powers of same base.

Quick Insight

Always express terms in common base for easy simplification.

Shortcut Trick

Time taken by A = 4 hours | Time taken by B = 6 hours.

Meeting Time from start = (Product of times) ÷ (Sum of times)

⇒ (4 × 6) ÷ (4 + 6) = 24 ÷ 10 = 2.4 hours.

⇒ 2 hours + (0.4 × 60) minutes = 2 hours 24 minutes.

⇒ 9:00 AM + 2 hours 24 minutes = 11:24 AM.

∴ The correct answer is 11:24 AM.

Alternate Method

Given: A starts at 9:00 AM, arrives at 1:00 PM (4 hrs). B starts at 9:00 AM, arrives at 3:00 PM (6 hrs).

Formula: Distance = Speed × Time | Relative Speed (Opposite) = S₁ + S₂

⇒ Let Total Distance = LCM(4, 6) = 12 km.

⇒ Speed of A = 12 / 4 = 3 kmph.

⇒ Speed of B = 12 / 6 = 2 kmph.

⇒ Relative Speed = 3 + 2 = 5 kmph.

⇒ Meeting Time = Distance ÷ Relative Speed = 12 / 5 = 2.4 hours.

⇒ 2 hours + (0.4 × 60) min = 2 hours 24 minutes.

⇒ 9:00 AM + 2:24 = 11:24 AM.

∴ The correct answer is 11:24 AM.

Additional Information

Relative Speed in Opposite Direction

When two objects move towards each other, their Relative Speed is the sum of their individual speeds (S₁ + S₂).

Different Starting Times

If starting times are different, first calculate the distance covered by the first person before the second starts, then apply relative speed to the remaining distance.

Average Speed for Entire Journey

If a person travels from X to Y at speed 'a' and Y to X at speed 'b', the Average Speed is (2 × a × b) / (a + b).

Shortcut Trick

Deviation from old average = 176 − 90 = 86

Average increase = Total Deviation ÷ New Number of Innings

Increase = 86 ÷ 8 = 10.75

Rounding 10.75 to the nearest whole number = 11

∴ The correct answer is 11.

Alternate Method

Given: Average of 7 innings = 90, 8^th inning score = 176

Formula: Average = Total Sum ÷ Number of items

⇒ Total runs in 7 innings = 7 × 90 = 630

⇒ Total runs after 8 innings = 630 + 176 = 806

⇒ New Average = 806 ÷ 8 = 100.75

⇒ Increase in average = New Average − Old Average

⇒ Increase = 100.75 − 90 = 10.75

⇒ Nearest whole number ≈ 11

∴ The correct answer is 11.

Additional Information

Average of Deviation

The sum of deviations of all observations from their mean is always zero. This property is used to quickly find the new average when a new value is added.

Weighted Average Formula

When two groups with averages A₁ and A₂ and sizes n₁ and n₂ are combined, the new average is (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Rounding Rules

To round to the nearest whole number, look at the first decimal digit. If it is 5 or more, round up; if it is less than 5, round down.

Shortcut Trick

Use the formula for Successive Percentage: Net Profit% = Markup% − Discount% − (Markup × Discount) ÷ 100.

Let Markup percentage be M = 50 and Discount percentage be X.

35 = 50 − X − (50 × X) ÷ 100.

35 = 50 − X − 0.5X ⇒ 35 = 50 − 1.5X.

1.5X = 50 − 35 ⇒ 1.5X = 15 ⇒ X = 15 ÷ 1.5 = 10.

∴ The correct answer is 10.

Alternate Method

Given → Markup% = 50%, Profit% = 35%, Discount = X%.
Formulas: MP = CP × (100 + Markup%) ÷ 100 | SP = CP × (100 + Profit%) ÷ 100 | Discount% = (MP − SP) ÷ MP × 100

⇒ Let Cost Price (CP) = 100.
⇒ Marked Price (MP) = 100 + 50% of 100 = 150.
⇒ Selling Price (SP) = 100 + 35% of 100 = 135.
⇒ Discount Amount = MP − SP = 150 − 135 = 15.
⇒ Discount Percentage (X) = (15 ÷ 150) × 100 = 10%.

∴ The correct answer is 10.

Additional Information

Relation between CP and MP

The ratio of Marked Price to Cost Price can be directly found using: MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%).

Markup Percentage

It is the extra price added to the Cost Price to reach the Marked Price. Formula: Markup% = [(MP − CP) ÷ CP] × 100.

Shortcut Trick

Let number = x

20% of x + 66 = x ⇒ 0.2x + 66 = x

⇒ 0.8x = 66 ⇒ x = 82.5

60% of x = 0.6 × 82.5 = 49.5

∴ Answer = 49.5

Alternate Method

Given:

20% of a number + 66 = the number

Formula used:

Percentage = (part/whole) × 100

Calculation:

⇒ Let number = x

⇒ 20% of x = 0.2x

⇒ Equation: 0.2x + 66 = x

⇒ x − 0.2x = 66

⇒ 0.8x = 66

⇒ x = 66 ÷ 0.8 = 82.5

⇒ 60% of x = 0.6 × 82.5 = 49.5

∴ The correct answer is 49.5.

Additional Information

Percentage Basics

x% = x/100 of the number

Linear Equation

Translate word problems into algebraic equations

Shortcut Insight

Rearrange equation to isolate variable quickly

Quick Tip

Convert percentages into decimals for easy calculation

Shortcut Trick

If a, b, c are in continued proportion, then b² = a × c.

Given: a = 75, b = Z, and c = 12 (third proportional).

⇒ Z² = 75 × 12

⇒ Z² = 900 ⇒ Z = 30.

∴ The correct answer is 30.

Alternate Method

Given: First number = 75, Second number = Z, Third proportional = 12.

Formula Used: For three numbers a, b, c to be in proportion, a/b = b/c.

⇒ 75/Z = Z/12

⇒ Z × Z = 75 × 12

⇒ Z² = 900

⇒ Z = √900

⇒ Z = 30

∴ The correct answer is 30.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is given by √(a × b).

Fourth Proportional

If a, b, c, d are in proportion, then a/b = c/d, where d is the fourth proportional (d = (b × c) / a).

Continued Proportion

Numbers a, b, c, d... are in continued proportion if a/b = b/c = c/d...

Shortcut Trick

Identify External Radius (R) = 12 ÷ 2 = 6 cm and Internal Radius (r) = 10 ÷ 2 = 5 cm.

Apply Shell Volume formula: V = 4/3 × π × (R³ − r³).

Calculate difference of cubes: 6³ − 5³ = 216 − 125 = 91.

Final calculation: 4/3 × 22/7 × 91 = 4/3 × 22 × 13 = 1144/3 ≈ 381.3 cm³.

∴ The correct answer is 381.3 cm³.

Alternate Method

Given:

External Diameter (D) = 12 cm ⇒ External Radius (R) = 12/2 = 6 cm

Internal Diameter (d) = 10 cm ⇒ Internal Radius (r) = 10/2 = 5 cm

Formula Used:

The Volume of a Spherical Shell is given by: V = 4/3 × π × (R³ − r³)

Calculations:

⇒ Volume = 4/3 × 22/7 × (6³ − 5³)

⇒ Volume = 4/3 × 22/7 × (216 − 125)

⇒ Volume = 4/3 × 22/7 × 91

⇒ Volume = 4/3 × 22 × 13 (Since 91 ÷ 7 = 13)

⇒ Volume = 1144 / 3

⇒ Volume = 381.333... cm³

⇒ Rounding off to 1 decimal place, Volume ≈ 381.3 cm³

∴ The correct answer is 381.3 cm³.

Additional Information

Volume of a Solid Sphere

The total space occupied by a solid sphere is V = 4/3 × πr³, where r is the radius.

Total Surface Area of Spherical Shell

The total surface area includes both internal and external surfaces: A = 4πR² + 4πr² = 4π(R² + r²).

Volume of a Hemisphere

A hemisphere is exactly half of a sphere, so its volume is V = 2/3 × πr³.

Shortcut Trick

Radius of circle r = √(Area × 7/22) = √(39424 × 7/22) = 112 cm.

Perimeter of square = r/2 = 112 / 2 = 56 cm.

Side of square = 56 / 4 = 14 cm.

Area of square = 14² = 196 cm².

∴ The correct answer is 196 cm².

Alternate Method

Given: Area of the circle = 39424 cm² and π = 22/7.

Formula Used: Area of circle = πr²; Perimeter of square = 4 × side; Area of square = side².

⇒ Area of circle = π × r² = 39424

⇒ (22 / 7) × r² = 39424

⇒ r² = (39424 × 7) / 22

⇒ r² = 1792 × 7 = 12544

⇒ r = √12544 = 112 cm

⇒ Perimeter of square = (1 / 2) × r = 112 / 2 = 56 cm

⇒ 4 × side = 56

⇒ side = 56 / 4 = 14 cm

⇒ Area of square = side × side = 14 × 14 = 196 cm²

∴ The correct answer is 196 cm².

Additional Information

Properties of a Square

A square has four equal sides. If the side is 'a', the diagonal is a√2. Area can also be expressed as (diagonal²) / 2.

Circle Mensuration

The circumference of a circle is 2πr. The ratio of the area to the square of the radius is always constant (π).

Common Square Roots

It is helpful to remember that 110² = 12100 and 112² = 12544 for fast calculations in competitive exams.

Shortcut Trick

Using the successive multiplying factor method:

Final Price = Initial Price × 0.90 × 0.50 × 0.90 × 1.45

Final Price = Initial Price × 0.58725

Net change = (0.58725 − 1) × 100 = −41.275%

∴ The correct answer is Decreases by 41.28%.

Alternate Method

Given: Initial price decreases by 10%, 50%, and 10%, then increases by 45%.

⇒ Let the Original Price be 100.

⇒ Price after 1^st month = 100 × (90/100) = 90

⇒ Price after 2^nd month = 90 × (50/100) = 45

⇒ Price after 3^rd month = 45 × (90/100) = 40.5

⇒ Price after 4^th month = 40.5 × (145/100) = 58.725

⇒ Overall Change = 58.725 − 100 = −41.275%

⇒ Rounding to two decimal places = 41.28% decrease.

∴ The correct answer is Decreases by 41.28%.

Additional Information

Successive Percentage Formula

For two changes a% and b%, the effective change is [a + b + (a × b)/100]%. This can be extended to multiple values.

Multiplying Factor (MF)

A decrease of x% corresponds to a factor of (1 − x/100), and an increase of x% corresponds to (1 + x/100).

Net Percentage Change

Net % Change = [(Final Value − Initial Value) / Initial Value] × 100%.

Shortcut Trick

Current ratio of Renu and Kashish = 9 : 8 (Difference = 1 unit)

Ratio after 5 years = 10 : 9 (Difference = 1 unit)

Increase in ratio units = 10 − 9 = 1 unit (or 9 − 8 = 1 unit)

Since 1 unit represents 5 years, the difference (1 unit) is 5 years.

∴ The correct answer is 5 years.

Alternate Method

Given: Present ratio = 9 : 8, Ratio after 5 years = 10 : 9.

Let the present ages be 9x and 8x.

⇒ (9x + 5) ÷ (8x + 5) = 10 ÷ 9

⇒ 9 × (9x + 5) = 10 × (8x + 5)

⇒ 81x + 45 = 80x + 50

⇒ 81x − 80x = 50 − 45

⇒ x = 5

⇒ Difference = 9x − 8x = x = 5 years.

∴ The correct answer is 5 years.

Additional Information

Constant Difference Principle

The age difference between two people always remains the same regardless of the number of years passed. In ratio problems, ensure the difference between ratio parts is equalized.

Ratio Balancing

If the difference between ratios is not the same, multiply the ratios by the difference of the other ratio to balance them (e.g., multiply ratio 1 by the difference of ratio 2).

Shortcut Trick

Apply the fundamental trigonometric identity: sin² x + cos² x = 1.

Comparing this with the given equation sin² x + K = 1.

It is immediately evident that K = cos² x.

∴ The correct answer is cos² x.

Alternate Method

Given: sin² x + K = 1

Formula Used: sin² θ + cos² θ = 1 ⇒ cos² θ = 1 − sin² θ

⇒ sin² x + K = 1

⇒ K = 1 − sin² x

⇒ From the Pythagorean identity, we know that 1 − sin² x = cos² x

⇒ K = cos² x

∴ The correct answer is cos² x.

Additional Information

Fundamental Trigonometric Identities

The three primary identities are sin² θ + cos² θ = 1, 1 + tan² θ = sec² θ, and 1 + cot² θ = cosec² θ.

Reciprocal Relations

sin θ = 1/cosec θ, cos θ = 1/sec θ, and tan θ = 1/cot θ.

Complementary Angles

sin(90° − θ) = cos θ and cos(90° − θ) = sin θ.