Quantitative Aptitude PYQs

Shortcut Trick

Using the duplex method a² | 2ab | b² for 74 (where a = 7, b = 4):

Step 1: b² = 4² = 16. Write 6, carry 1.

Step 2: 2ab + carry = (2 × 7 × 4) + 1 = 57. Write 7, carry 5.

Step 3: a² + carry = 7² + 5 = 49 + 5 = 54.

Combining the digits from left to right gives 5476.

∴ The correct answer is 5476.

Alternate Method

Given:

The expression to evaluate is 74².

Formula Used:

(a + b)² = a² + 2ab + b²

Calculations:

⇒ 74² = (70 + 4)²

⇒ (70 + 4)² = 70² + (2 × 70 × 4) + 4²

⇒ (70 + 4)² = 4900 + 560 + 16

⇒ (70 + 4)² = 5460 + 16

⇒ 74² = 5476

∴ The correct answer is 5476.

Additional Information

Squaring Numbers Near 50

For a number (50 + x), the square is calculated as: first two digits = (25 + x), last two digits = x². For example, 53² = (25 + 3) | 3² = 2809.

Squaring Numbers Near 100

For a number (100 − x), the square is: first two digits = (Number − x), last two digits = x². For example, 96² = (96 − 4) | 4² = 9216.

Squaring Numbers Ending in 5

For a number (n5), the square ends in 25 and the leading digits are n × (n + 1). For example, 65² = (6 × 7) | 25 = 4225.

Shortcut Trick

Let the Principal sum be 100%.

Interest for the 1^st year = 20% of Principal = 20%.

Interest for the 2^nd year = 20% + (20% of 20%) = 24%.

Given, 24% of the sum = ₹5,520

⇒ 1% of the sum = 5520 ÷ 24 = ₹230

⇒ 100% of the sum = 230 × 100 = ₹23,000

∴ The correct answer is ₹23,000.

Alternate Method

Given:

Rate of Interest (R) = 20% p.a.

Interest for the 2^nd year = ₹5,520

Formula Used:

Amount after 1^st Year (A₁) = P × (1 + R ÷ 100)

Interest for 2^nd Year = A₁ × (R ÷ 100)

Sum P (100%)

Amount after 1^st Yr

Amount (120%)

1^st Yr Int = 20%

2^nd Yr Int = 24%

24% = ₹5,520

Calculations:

⇒ Let the Principal sum be P.

⇒ Amount after the 1^st year (A₁) = P × (1 + 20 ÷ 100) = 1.2P

⇒ Interest for the 2^nd year is charged only on the amount after the 1^st year.

⇒ Interest for 2^nd year = 1.2P × (20 ÷ 100) = 0.24P

⇒ According to the question, 0.24P = 5520

⇒ P = 5520 ÷ 0.24

⇒ P = 552000 ÷ 24 = 23000

∴ The correct answer is ₹23,000.

Additional Information

Difference between CI and SI for 2 years

The difference between Compound Interest and Simple Interest for exactly 2 years is given by the formula: D = P × (R ÷ 100)².

Difference between CI and SI for 3 years

For 3 years, the difference formula extends to: D = P × (R ÷ 100)² × (3 + R ÷ 100).

Effective Rate of Compound Interest

For 2 years, the total effective compound interest rate can be found directly using the successive percentage formula: X + Y + (X × Y) ÷ 100.

Shortcut Trick

Let the second number be y, then the first number is 5y.

Product = 5y × y = 5y² = 9245

y² = 1849 ⇒ y = 43

Sum of the numbers = 5y + y = 6y = 6 × 43 = 258

∴ The correct answer is 258.

Alternate Method

Given:

Product of two positive numbers = 9245

First number = 5 × Second number

Ratio of Numbers = 5 : 1

First Number (5x)

5 parts

Second Number (x)

1 part

Product = 5x × x = 9245
Sum = 5x + x = 6x

Calculations:

⇒ Let the second number be x.

⇒ First number = 5x

⇒ Product of two numbers = 5x × x = 9245

⇒ 5x² = 9245

⇒ x² = 9245 ÷ 5 = 1849

⇒ x = √1849

⇒ x = 43 (since the numbers are positive)

⇒ Second number = 43

⇒ First number = 5 × 43 = 215

⇒ Sum of the numbers = 215 + 43 = 258

∴ The correct answer is 258.

Additional Information

Ratio and Proportion

When quantities are defined by a specific proportion, they can be expressed as common multiples of a single variable (e.g., 5x and x) which simplifies algebraic substitution and solving.

Finding Square Roots

To quickly find roots like √1849, check base bounds. Since 40² = 1600 and 50² = 2500, the root lies in the 40s. A number ending in 9 must have a root ending in 3 or 7. Verifying 43² gives exactly 1849.

Properties of Sum and Product

For any algebraic relation involving product (ab) and sum (a+b), establishing a ratio constraint removes the need for complex algebraic identities, allowing for a direct single-variable solution.

Shortcut Trick

From the first condition, 6P = Q + 46 ⇒ P = (Q + 46) ÷ 6.

Since ages are generally whole numbers, (Q + 46) must be a multiple of 6.

Checking the options: If Q = 2, then P = (2 + 46) ÷ 6 = 48 ÷ 6 = 8 (Integer).

For other options (Q = 1, 6, 5), P is not an integer.

Verifying Q = 2, P = 8 in the 2^nd condition: 6 × (2 + 10) = 72 and 5 × (8 + 10) − 18 = 72 (Matches!).

∴ The correct answer is 2 years.

Alternate Method

Given:

Six times the present age of P exceeds the present age of Q by 46 years.

After 10 years, six times Q's age will be 18 years less than five times P's age.

Present Ages
P's Age = p
Q's Age = q

After 10 Years
P's Age = p + 10
Q's Age = q + 10

• 10 Years

Eq 1: 6p − q = 46
Eq 2: 6(q + 10) = 5(p + 10) − 18

Calculation:

Let the present age of P be p and the present age of Q be q.

From the first condition:

⇒ 6p − q = 46

⇒ q = 6p − 46       ---- (Equation 1)

From the second condition (after 10 years):

⇒ 6 × (q + 10) = 5 × (p + 10) − 18

⇒ 6q + 60 = 5p + 50 − 18

⇒ 6q + 60 = 5p + 32

⇒ 5p − 6q = 28       ---- (Equation 2)

Substitute the value of q from Equation 1 into Equation 2:

⇒ 5p − 6 × (6p − 46) = 28

⇒ 5p − 36p + 276 = 28

⇒ −31p = 28 − 276

⇒ −31p = −248

⇒ p = −248 ÷ −31 = 8 years

Now, substitute p = 8 in Equation 1:

⇒ q = 6 × 8 − 46

⇒ q = 48 − 46 = 2 years

∴ The correct answer is 2 years.

Additional Information

Problems on Ages

If the current age is x, then age n years ago was (x − n) and age n years later will be (x + n).

Ratio in Ages

When the ratio of ages of two persons a and b is given as p : q, we can assume their ages to be pk and qk respectively, where k is a constant multiplier.

Linear Equations in Two Variables

An equation of the form ax + by + c = 0. Solving two independent equations simultaneously provides the unique values of the variables.

Shortcut Trick

For a number to be perfectly divisible by 3, the sum of its digits must be a multiple of 3.

Sum of digits of 934725 = 9 + 3 + 4 + 7 + 2 + 5 = 30.

Since 30 is perfectly divisible by 3, the number 934725 is also divisible by 3.

∴ The correct answer is 3.

Alternate Method

Given:

Number = 934725

Method Used:

Check the standard divisibility rules for the given options (2, 3, 6, 10).

Calculations:

⇒ Divisibility by 2: The last digit is 5. Since it is not an even number, 934725 is not divisible by 2.

⇒ Divisibility by 6: A number must be divisible by both 2 and 3 to be divisible by 6. Since it is not divisible by 2, it cannot be divisible by 6.

⇒ Divisibility by 10: The last digit must be 0 for a number to be divisible by 10. Since it is 5, 934725 is not divisible by 10.

⇒ Divisibility by 3: The sum of digits must be perfectly divisible by 3.

⇒ Sum = 9 + 3 + 4 + 7 + 2 + 5 = 30

⇒ 30 is divisible by 3 (30 ÷ 3 = 10).

⇒ Hence, the number 934725 is divisible by 3.

∴ The correct answer is 3.

Additional Information

Divisibility Rule of 4

A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

Divisibility Rule of 9

A number is divisible by 9 if the sum of all its digits is perfectly divisible by 9.

Divisibility Rule of 11

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

Shortcut Trick

Using the algebraic identity: (a + b)² + (a − b)² = 2(a² + b²).

Identify the values: a = 3x and b = 2.

Substitute into the formula: 2 × [(3x)² + (2)²]

⇒ 2 × [9x² + 4] = 18x² + 8.

∴ The correct answer is 18x² + 8.

Alternate Method

Given: The expression to simplify is (3x + 2)² + (3x − 2)².

Formula Used:

1. (a + b)² = a² + 2ab + b²
2. (a − b)² = a² − 2ab + b²

Calculations:

⇒ Expand the first term: (3x + 2)² = (3x)² + 2(3x)(2) + 2² = 9x² + 12x + 4

⇒ Expand the second term: (3x − 2)² = (3x)² − 2(3x)(2) + 2² = 9x² − 12x + 4

⇒ Add both expanded expressions: (9x² + 12x + 4) + (9x² − 12x + 4)

⇒ Combine like terms: (9x² + 9x²) + (12x − 12x) + (4 + 4)

⇒ 18x² + 0 + 8

⇒ 18x² + 8

∴ The correct answer is 18x² + 8.

Additional Information

Sum of Squares Identity

The sum of the squares of two binomials differing only in sign is given by (a + b)² + (a − b)² = 2(a² + b²).

Difference of Squares of Binomials

Subtracting the square of a difference from the square of a sum yields (a + b)² − (a − b)² = 4ab.

BODMAS in Algebra

Always expand exponents first before performing addition or subtraction of algebraic terms.

Shortcut Trick

For two numbers a and b, the third proportional x is calculated as b² ÷ a.

Here, a = 9 and b = 48.

x = (48 × 48) ÷ 9 = (16 × 3 × 16 × 3) ÷ 9 = 16 × 16 = 256.

∴ The correct answer is 256.

Alternate Method

Given:

First number (a) = 9

Second number (b) = 48

Third proportional = x

Formula Used:

If a, b, and x are in continued proportion, then a : b :: b : x, which implies x = b²/a.

Calculations:

⇒ 9 / 48 = 48 / x

⇒ 9 × x = 48 × 48

⇒ x = (48 × 48) / 9

⇒ x = 2304 / 9

⇒ x = 256

∴ The correct answer is 256.

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, then d is the fourth proportional, calculated as d = (b × c) / a.

Continued Proportion

Three numbers a, b, c are in continued proportion if a/b = b/c, meaning b² = ac.

Shortcut Trick

Let the age of R be 1 unit.

Then Q = 2 units and P = 2 units + 2 years.

Total age = 1 + 2 + 2 units + 2 years = 5 units + 2 = 27 years.

5 units = 25 ⇒ 1 unit = 5 years.

Q = 2 units = 10 years.

∴ The correct answer is 10 years.

Alternate Method

Given: Total age of P, Q, and R = 27 years.

Formula Used: Sum of individual ages = Total Age.

Let R age = x years.

⇒ Q age = 2x years.

⇒ P age = (2x + 2) years.

⇒ x + 2x + 2x + 2 = 27

⇒ 5x + 2 = 27

⇒ 5x = 25

⇒ x = 5 years.

⇒ Q = 2 × 5 = 10 years.

∴ The correct answer is 10 years.

Additional Information

Ratio of Ages

If the ratio of ages of A and B is a:b, their ages are taken as ax and bx.

Age after 'n' years

If current age is x, then age after n years = x + n and age n years ago = x − n.

Average Age

Average Age = (Sum of ages of all persons) ÷ (Number of persons).

Shortcut Trick

For a cuboid, the volume is the product of its three dimensions.

Volume = 13 × 4 × 9

⇒ 52 × 9 = 468 m³

∴ The correct answer is 468 m³.

Alternate Method

Given:

Length (L) of the cuboidal tank = 13 m

Breadth (B) of the cuboidal tank = 4 m

Height (H) of the cuboidal tank = 9 m

Formula Used:

Volume of a Cuboid = Length × Breadth × Height

Calculations:

⇒ Volume = 13 × 4 × 9

⇒ Volume = (13 × 4) × 9

⇒ Volume = 52 × 9

⇒ Volume = 468 m³

∴ The correct answer is 468 m³.

Additional Information

Total Surface Area of a Cuboid

The total surface area is the sum of the areas of all six faces: 2 × (LB + BH + HL).

Diagonal of a Cuboid

The longest rod that can be placed inside a cuboid is its diagonal, given by √(L² + B² + H²).

Lateral Surface Area of a Cuboid

The area of the four side walls (excluding top and bottom) is calculated as 2 × H × (L + B).

Shortcut Trick

In simple interest, interest earned = (Multiple − 1) × Principal.

To double (2 times), Interest = 2 − 1 = 1 unit. Time taken = 16 years.

To become 6 times, Interest = 6 − 1 = 5 units.

Since Interest ∝ Time, Time = 16 × 5 = 80 years.

∴ The correct answer is 80 years.

Alternate Method

Given: Principal (P) becomes 2P in 16 years.

Formula Used: Simple Interest (S.I.) = (P × R × T) / 100 | Amount = P + S.I.

⇒ Case 1: S.I.₁ = 2P − P = P

⇒ P = (P × R × 16) / 100

⇒ R = 100 / 16 = 6.25%

⇒ Case 2: New Amount = 6P

⇒ S.I.₂ = 6P − P = 5P

⇒ 5P = (P × 6.25 × T₂) / 100

⇒ 5 = (6.25 × T₂) / 100

⇒ T₂ = (5 × 100) / 6.25 = 500 / 6.25

⇒ T₂ = 80 years

∴ The correct answer is 80 years.

Additional Information

Simple Interest vs Time Relation

For a constant Principal and Rate, the time (T) required for a sum to become n times itself is given by T = (n − 1) × 100 / R.

Ratio Formula for S.I.

When the same sum is invested, the relationship between time periods is T₁ / (n₁ − 1) = T₂ / (n₂ − 1).

Rate of Interest Shortcut

If a sum doubles in T years at Simple Interest, the Rate of interest (R) is always 100 / T%.

Shortcut Trick

First Discounted Price = 24,500 × (100 − 20)% = 24,500 × 0.8 = 19,600.

Total Discount Remaining = 19,600 − 13,132 = 6,468.

Value of C = (Remaining Discount ÷ First Discounted Price) × 100.

C = (6,468 ÷ 19,600) × 100 = 33.

∴ The correct answer is 33.

Alternate Method

Given → Marked Price (MP) = ₹24,500, Selling Price (SP) = ₹13,132, D₁ = 20%, D₂ = C%.
Formulas: SP = MP × (100 − D₁)% × (100 − D₂)%

⇒ Step 1: Calculate price after first discount: 24,500 × (80/100) = 19,600.
⇒ Step 2: Use SP formula for second discount: 13,132 = 19,600 × (100 − C)/100.
⇒ Step 3: (13,132 ÷ 19,600) = (100 − C)/100.
⇒ Step 4: 0.67 = (100 − C)/100.
⇒ Step 5: 67 = 100 − C.
⇒ Step 6: C = 100 − 67 = 33.

∴ The correct answer is 33.

Additional Information

Successive Discounts

When two discounts a% and b% are given, the Equivalent Single Discount is (a + b − ab/100)%.

Marked Price & Selling Price

Selling Price (SP) is the price paid after applying all discounts to the Marked Price (MP).

Shortcut TrickTo find x when added to 25, 11, 34, 13 to make them proportional:Using the cross-difference ratio shortcut: (25 − 11) = 14 and (34 − 13) = 21.Ratio of differences = 14 : 21 = 2 : 3.Equating the scaled terms: 3 × (25 + x) = 2 × (34 + x) ⇒ 75 + 3x = 68 + 2x ⇒ x = −7.Substitute x = −7 into \( y^{2} = 9x(8x - 7) \) ⇒ \( y^{2} = 9(-7)(-56 - 7) = -63 \times -63 = 63^{2} \) ⇒ y = 63.∴ The correct answer is 63.Alternate MethodGiven: Numbers are 25, 11, 34, 13 and \( 9x : y :: y : (8x - 7) \) with \( y > 0 \).Formula Used: If a, b, c, d are in proportion, then \( \frac{a}{b} = \frac{c}{d} \). Also, if \( p : q :: q : r \), then \( q^{2} = p \times r \).Calculations:⇒ \( \frac{25 + x}{11 + x} = \frac{34 + x}{13 + x} \)⇒ \( (25 + x)(13 + x) = (34 + x)(11 + x) \)⇒ \( 325 + 38x + x^{2} = 374 + 45x + x^{2} \)⇒ \( 38x - 45x = 374 - 325 \)⇒ \( -7x = 49 \)⇒ x = −7Now, substituting the value of x in the mean proportion relation:⇒ \( y^{2} = 9x(8x - 7) \)⇒ \( y^{2} = 9(-7)[8(-7) - 7] \)⇒ \( y^{2} = -63 \times (-56 - 7) \)⇒ \( y^{2} = -63 \times -63 \)⇒ \( y^{2} = 63^{2} \)⇒ y = 63 (since \( y > 0 \))∴ The correct answer is 63.Additional InformationMean ProportionalIf three quantities a, b, and c are in continued proportion, then b is called the mean proportional between a and c, where b2 = ac.Proportionality PropertiesIf four terms a, b, c, d are in proportion, then product of extremes equals product of means: ad = bc.

Shortcut Trick
When the Profit earned at one price equals the Loss incurred at another, the Cost Price (CP) is simply the average of the two selling prices.
CP = (SP1 + SP2) ÷ 2
CP = (986 + 726) ÷ 2 = 1712 ÷ 2
CP = 856
∴ The correct answer is ₹856.

Alternate Method
Given: SP1 = ₹986 (Profit), SP2 = ₹726 (Loss). Formulas: Profit = SP − CP | Loss = CP − SP.

⇒ Let the Profit and Loss be x.
⇒ Case 1: CP = SP1 − Profit = 986 − x
⇒ Case 2: CP = SP2 + Loss = 726 + x
⇒ Equating CP: 986 − x = 726 + x
⇒ 2x = 986 − 726 = 260 ⇒ x = 130
⇒ CP = 726 + 130 = 856
∴ The correct answer is ₹856.

Additional Information
Profit and Loss Basics
Profit occurs when SP > CP, and Loss occurs when CP > SP. If Profit = Loss, the CP is the arithmetic mean of the two Selling Prices.
Linear Relationship
If the profit is 'n' times the loss, then CP = (SP1 + n × SP2) ÷ (n + 1).

Shortcut Trick

Let cost of TV = 100, then Washing Machine (WM) = 50.

Initial total cost of 8 WM and 2 TV = (8 × 50) + (2 × 100) = 400 + 200 = 600.

New WM cost = 50 × 1.53 = 76.5 | New TV cost = 100 × 0.45 = 45.

New total cost = (8 × 76.5) + (2 × 45) = 612 + 90 = 702.

Percentage change = (702 − 600) ÷ 600 × 100 = 102 ÷ 600 × 100 = 17% increase.

∴ The correct answer is Increase by 17%.

Alternate Method

Given: Cost of WM is 50% less than TV. WM cost increase = 53%, TV cost decrease = 55%.

Formulas: Total Cost = (Quantity₁ × Price₁) + (Quantity₂ × Price₂) | % Change = (New − Old) ÷ Old × 100

⇒ Initial Cost = (8 × 50) + (2 × 100) = 600

⇒ Final Cost = 8 × (50 + 53% of 50) + 2 × (100 − 55% of 100)

⇒ Final Cost = 8 × (50 + 26.5) + 2 × (45) = 8 × 76.5 + 90 = 612 + 90 = 702

⇒ % Change = [(702 − 600) ÷ 600] × 100 = (102 ÷ 600) × 100 = 17%

∴ The correct answer is Increase by 17%.

Additional Information

Percentage Increase/Decrease

To increase a value by x%, multiply by (100 + x) ÷ 100. To decrease by x%, multiply by (100 − x) ÷ 100.

Weighted Average Method

The net percentage change can also be found using the formula: % Change = (n₁c₁ + n₂c₂) ÷ (n₁w₁ + n₂w₂) where n is quantity and c is cost change.

Shortcut Trick

Let the complete number be 100%.

According to the question: 20% of the number + 60 = 100% of the number.

⇒ 100% − 20% = 60

⇒ 80% of the number = 60

The question directly asks for 80% of the same number, which is exactly 60.

∴ The correct answer is 60.

Alternate Method

Given:

20% of a number + 60 = the number

Concept Used:

Any complete quantity is always considered as 100% of itself.

20% 80% = 60  100% (The Complete Number)  Added Value = 60

Calculations:

Let the unknown number be x.

⇒ 20% of x + 60 = x

⇒ (20 ÷ 100) × x + 60 = x

⇒ x − 0.20x = 60

⇒ x(1 − 0.20) = 60

⇒ 0.80x = 60

⇒ 80% of x = 60

Since the question asks us to find 80% of the same number, we already have the required value.

∴ The correct answer is 60.

Additional Information

Base Value in Percentages

Every quantity in its original or complete state is always considered as 100%. When a part is given as p%, the remaining part is simply (100 − p)%.

Fraction to Percentage Conversion

Fractions can be easily converted to percentages by multiplying by 100. For example, 1/5 × 100 = 20%, and 4/5 × 100 = 80%.

Shortcut Trick

In a 30º−60º−90º right-angled triangle, the ratio of sides is 1 : √3 : 2.

Side opposite to 60º (Base) = 15 m = √3 units.

Side opposite to 30º (Height) = 1 unit = 15 ÷ √3.

Rationalizing: (15 × √3) ÷ 3 = 5√3 m.

∴ The correct answer is 5√3 m.

Alternate Method

Given:

Distance from flagpole (Base) = 15 m

Angle of elevation (θ) = 30º

Formula Used:

tan θ = Perpendicular (Height) ÷ Base

Calculations:

⇒ tan 30º = Height ÷ 15

⇒ 1/√3 = Height ÷ 15

⇒ Height = 15 / √3

⇒ Height = (15 × √3) / (√3 × √3)

⇒ Height = 15√3 / 3

⇒ Height = 5√3 m

∴ The correct answer is 5√3 m.

Additional Information

Trigonometric Ratios

Standard values: tan 30º = 1/√3, tan 45º = 1, and tan 60º = √3.

Angle of Elevation and Depression

The angle of elevation from an observer's eye to an object is equal to the angle of depression from the object to the observer due to alternate interior angles.

Pythagoras Theorem

In any right-angled triangle, Hypotenuse² = Base² + Perpendicular².

Shortcut Trick

Net filling efficiency of both pipes = (1 ÷ 8) − (1 ÷ 16) = 1 ÷ 16 of the tank per minute.

Time required to fill the entire tank = 16 minutes.

Time required to fill 1/4^th of the tank = 16 × (1 ÷ 4) = 4 minutes.

∴ The correct answer is 4 minutes.

Alternate Method

Given:

Time taken by filling pipe (A) = 8 minutes

Time taken by emptying pipe (B) = 16 minutes

Formula/LCM Table:

Calculations:

⇒ Total capacity of the tank = LCM(8, 16) = 16 units

⇒ Combined efficiency of both pipes when opened together = 2 − 1 = 1 unit/minute

⇒ Target volume to be filled = 1/4^th of total capacity = 16 ÷ 4 = 4 units

⇒ Time required to fill the target volume = 4 units ÷ 1 unit/minute = 4 minutes

∴ The correct answer is 4 minutes.

Additional Information

Pipes and Cisterns Net Efficiency

Filling pipes are considered as positive work (+ efficiency) and emptying pipes (leaks) are considered as negative work (− efficiency).

LCM Method for Work

Total capacity is assumed as the Least Common Multiple (LCM) of individual times, which simplifies the calculations of individual flow rates as integers.

Shortcut Trick

Radius (r) = 10.5 ÷ 2 = 5.25 cm.

Surface Area = 4 × π × r² = 4 × (22/7) × 5.25 × 5.25 = 346.5 cm².

Cost = (Total Area ÷ 100) × Rate = (346.5 ÷ 100) × 4 = 3.465 × 4 = ₹13.86.

∴ The correct answer is ₹13.86.

Alternate Method

Given:

Diameter of the solid brass sphere (d) = 10.5 cm.

Rate of tin-plating = ₹4 per 100 cm².

Formula Used:

Surface Area of Sphere = 4πr²

Total Cost = (Surface Area × Rate) ÷ Unit Area

Calculations:

⇒ Radius (r) = d ÷ 2 = 10.5 ÷ 2 = 5.25 cm = 21/4 cm

⇒ Surface Area = 4 × (22/7) × (21/4) × (21/4)

⇒ Surface Area = 4 × (22/7) × (441/16)

⇒ Surface Area = (22 × 63) ÷ 4 = 1386 ÷ 4 = 346.5 cm²

⇒ Cost for 100 cm² = ₹4

⇒ Total Cost = (346.5 ÷ 100) × 4

⇒ Total Cost = 3.465 × 4 = ₹13.86

∴ The correct answer is ₹13.86.

Additional Information

Volume of a Sphere

The volume of a sphere is given by V = (4/3)πr³, where r is the radius.

Hemisphere Properties

For a hemisphere: Curved Surface Area (CSA) = 2πr² and Total Surface Area (TSA) = 3πr².

Unit Conversions

1 m² = 10,000 cm². Always ensure units of area and rate are consistent before multiplication.

Shortcut Trick

For a pure recurring decimal in the form 0.XYXYXY..., the fraction is simply the repeating digits divided by the same number of 9s.

Here, the decimal is 0.595959... where the repeating block is 59.

Since there are exactly 2 repeating digits, we place 59 over two 9s.

Fraction = 59 ÷ 99

∴ The correct answer is 59/99.

Alternate Method

Given:

The recurring decimal number is 0.595959...

Method Used:

Algebraic multiplication method to eliminate the infinite recurring decimal portion.

Calculations:

⇒ Let x = 0.595959... (Equation 1)

⇒ Since there are 2 digits repeating, we multiply both sides by 10² (which is 100).

⇒ 100x = 59.595959... (Equation 2)

⇒ Subtract Equation 1 from Equation 2:

⇒ 100x − x = 59.595959... − 0.595959...

⇒ 99x = 59

⇒ x = 59 ÷ 99

∴ The correct answer is 59/99.

Additional Information

Pure Recurring Decimals

A decimal number where all digits immediately after the decimal point repeat continuously. To convert into a fraction, write the repeating block as the numerator and put as many 9s in the denominator as there are repeating digits. Example: 0.777... = 7/9.

Mixed Recurring Decimals

A decimal where some digits do not repeat before the repeating sequence begins. The formula is (Total number formed by all digits − Non-repeating part) ÷ (As many 9s as repeating digits followed by as many 0s as non-repeating digits). Example: 0.12555... = (125 − 12) ÷ 900 = 113/900.

Shortcut Trick

In any quadrilateral PQRS, the sum of the angles ∠P and ∠Q is always equal to twice the angle formed by the bisectors of the other two angles ∠R and ∠S.

⇒ ∠P + ∠Q = 2 × ∠STR

⇒ ∠P + ∠Q = 2 × 37.5° = 75°

Since the ratio of ∠P : ∠Q is 7 : 8, the difference between ∠Q and ∠P is:

⇒ Difference = (8 − 7) ÷ (7 + 8) × 75° = 1/15 × 75° = 5°

∴ The correct answer is 5°.

Alternate Method

Given:

• In quadrilateral PQRS, the bisectors of ∠R and ∠S meet at T.
• ∠STR = 37.5°
• Ratio of ∠P to ∠Q = 7 : 8

Formula Used:

• Sum of interior angles of a quadrilateral = 360°
• Sum of angles in a triangle = 180°

Calculations:

In triangle STR:

⇒ ∠STR + ∠TSR + ∠TRS = 180°

Since ST and RT are the angle bisectors of ∠S and ∠R respectively:

⇒ ∠TSR = ∠S ÷ 2

⇒ ∠TRS = ∠R ÷ 2

Substituting these values in the triangle angle sum equation:

⇒ 37.5° + (∠S ÷ 2) + (∠R ÷ 2) = 180°

⇒ (∠S + ∠R) ÷ 2 = 180° − 37.5°

⇒ (∠S + ∠R) ÷ 2 = 142.5°

⇒ ∠S + ∠R = 285°

In quadrilateral PQRS, the sum of all four interior angles is 360°:

⇒ ∠P + ∠Q + ∠R + ∠S = 360°

⇒ ∠P + ∠Q + 285° = 360°

⇒ ∠P + ∠Q = 360° − 285° = 75°

Let ∠P = 7x and ∠Q = 8x (since ∠P : ∠Q = 7 : 8):

⇒ 7x + 8x = 75°

⇒ 15x = 75°

⇒ x = 5°

The difference between the measures of ∠Q and ∠P is:

⇒ ∠Q − ∠P = 8x − 7x = x = 5°

∴ The correct answer is 5°.

Additional Information

Angle Bisector Theorem in Quadrilaterals

The angle formed by the bisectors of two adjacent angles of a quadrilateral is equal to half the sum of the remaining two angles. That is, ∠STR = (∠P + ∠Q) ÷ 2.

Sum of Interior Angles of a Polygon

For any n-sided polygon, the sum of interior angles is given by (n − 2) × 180°. For a quadrilateral (n = 4), the sum is always 360°.

Opposite Angle Bisectors

If the bisectors of opposite angles of a quadrilateral (say ∠P and ∠R) meet, the angle between them is equal to half the difference of the other two angles, i.e., |∠Q − ∠S| ÷ 2.