Quantitative Aptitude PYQs

Shortcut Trick

Length of chord = 2 × √(Radius² − Distance²)

Length = 2 × √(10² − 6²)

Length = 2 × √(100 − 36) = 2 × √64

Length = 2 × 8 = 16 cm

∴ The correct answer is 16 cm.

Alternate Method

Given:

Radius of the circle (r) = 10 cm

Distance of the chord from the center (d) = 6 cm

Formula Used:

By Pythagoras theorem, Half-chord length = √(r² − d²)

Length of chord = 2 × Half-chord length (since the perpendicular from the center to a chord bisects it)

O A B M

10 cm 6 cm

Length of Chord AB = 16 cm

Calculations:

⇒ Let the chord be AB, center of the circle be O, and perpendicular from O to AB be OM.

⇒ Therefore, OM = 6 cm and OA = 10 cm.

⇒ In right-angled triangle OMA, by Pythagoras theorem:

⇒ OA² = OM² + AM²

⇒ 10² = 6² + AM²

⇒ 100 = 36 + AM²

⇒ AM² = 100 − 36 = 64

⇒ AM = √64 = 8 cm

⇒ Since OM is perpendicular to AB, it bisects the chord. So, AB = 2 × AM.

⇒ Length of chord AB = 2 × 8 = 16 cm

∴ The correct answer is 16 cm.

Additional Information

Perpendicular from the Centre

A perpendicular drawn from the center of a circle to a chord always bisects the chord into two equal parts.

Equal Chords Property

Chords of equal length in a circle are equidistant from the center. Conversely, chords that are equidistant from the center are equal in length.

Intersecting Chords Theorem

If two chords AB and CD intersect internally at a point P, then the product of their segments is equal: AP × PB = CP × PD.

Shortcut Trick

Given tanA + cotA = 4 ⇒ tanA + 1/tanA = 4 ⇒ tanA = 2 ± √3.

We need to find tan²A + sec²A = tan²A + (1 + tan²A) = 2tan²A + 1.

Taking the positive root, substitute tanA = 2 + √3 ⇒ tan²A = (2 + √3)² = 7 + 4√3.

Value = 2 × (7 + 4√3) + 1 = 14 + 8√3 + 1 = 15 + 8√3.

∴ The correct answer is 15 + 8√3.

Alternate Method

Given:

tanA + cotA = 4

Formula Used:

sec²A = 1 + tan²A

For ax² + bx + c = 0, roots are x = (−b ± √(b² − 4ac)) ÷ 2a

Calculations:

⇒ tanA + 1/tanA = 4

⇒ tan²A − 4tanA + 1 = 0

⇒ tanA = (−(−4) ± √((−4)² − 4 × 1 × 1)) ÷ (2 × 1)

⇒ tanA = (4 ± √(16 − 4)) ÷ 2

⇒ tanA = (4 ± √12) ÷ 2

⇒ tanA = (4 ± 2√3) ÷ 2 = 2 ± √3

Since the options contain an addition with radical terms, we evaluate for tanA = 2 + √3

⇒ tan²A = (2 + √3)²

⇒ tan²A = 4 + 3 + 2 × 2 × √3 = 7 + 4√3

Evaluating the required expression:

⇒ tan²A + sec²A

⇒ tan²A + (1 + tan²A)

⇒ 2tan²A + 1

⇒ 2 × (7 + 4√3) + 1

⇒ 14 + 8√3 + 1 = 15 + 8√3

∴ The correct answer is 15 + 8√3.

Additional Information

Trigonometric Identities

Fundamental Pythagorean identities include sin²A + cos²A = 1 and sec²A − tan²A = 1.

Standard Angle Equivalents

The equation tanA + cotA = 4 simplifies to sin2A = 1/2, meaning 2A = 30° or 150°. This yields A = 15° or 75°, where tan 75° = 2 + √3.

Algebraic Reciprocals

For any algebraic equation of the form x + 1/x = k, the squares can be directly found using x² + 1/x² = k² − 2.

Shortcut Trick

Substitute x = 1/2 in the given equation to find the value of sin A.

sin A = (2 × 1/2) ÷ (1 + 1/4) = 1 ÷ (5/4) = 4/5

For a right-angled triangle, if the perpendicular is 4 and the hypotenuse is 5, the base is 3 (Pythagorean triplet).

Thus, cos A = Base ÷ Hypotenuse = 3/5.

Checking the options by substituting x = 1/2, only Option 2 gives (1 − 1/4) ÷ (1 + 1/4) = 3/5.

∴ The correct answer is (1 − x²)/(1 + x²).

Alternate Method

Given:

sin A = 2x / (1 + x²)

Formula Used:

sin A = Perpendicular ÷ Hypotenuse

cos A = Base ÷ Hypotenuse

Base² + Perpendicular² = Hypotenuse²

A 2x 1 - x² 1 + x²

Calculations:

⇒ Let Perpendicular = 2x and Hypotenuse = 1 + x²

⇒ Base² = Hypotenuse² − Perpendicular²

⇒ Base² = (1 + x²)² − (2x)²

⇒ Base² = 1 + 2x² + x⁴ − 4x²

⇒ Base² = 1 − 2x² + x⁴

⇒ Base = √[(1 − x²)²]

⇒ Base = 1 − x²

⇒ cos A = Base ÷ Hypotenuse

⇒ cos A = (1 − x²) / (1 + x²)

∴ The correct answer is (1 − x²)/(1 + x²).

Additional Information

Trigonometric Ratios in Right Triangle

sin θ = Perpendicular ÷ Hypotenuse | cos θ = Base ÷ Hypotenuse | tan θ = Perpendicular ÷ Base

Pythagorean Identities

sin² θ + cos² θ = 1 | sec² θ − tan² θ = 1 | cosec² θ − cot² θ = 1

Half-Angle Substitution

If x = tan(θ/2), then sin A = 2x / (1 + x²) and cos A = (1 − x²) / (1 + x²). This standard identity directly leads to the required result.

Shortcut Trick

Notice the relationship between the bases in the numerator and denominator.

Each base in the denominator is exactly twice the corresponding base in the numerator (0.5 = 2 × 0.25 and 0.1 = 2 × 0.05).

Since the terms are cubed, the denominator is 2³ = 8 times the numerator.

Thus, the given expression simplifies directly to 1/8 = 0.125.

∴ The correct answer is 0.125.

Alternate Method

Given:

Expression = (0.25³ + 0.05³) ÷ (0.5³ + 0.1³)

Method Used:

Identify common factors and simplify algebraically by replacing the decimal values with variables.

Calculations:

⇒ Let a = 0.25 and b = 0.05

⇒ The denominator terms can be written as: 0.5 = 2 × 0.25 = 2a and 0.1 = 2 × 0.05 = 2b

⇒ Substituting these into the expression:

⇒ Expression = (a³ + b³) ÷ ((2a)³ + (2b)³)

⇒ Expression = (a³ + b³) ÷ (8a³ + 8b³)

⇒ Taking 8 common from the denominator:

⇒ Expression = (a³ + b³) ÷ [8 × (a³ + b³)]

⇒ The term (a³ + b³) cancels out from the numerator and denominator.

⇒ Expression = 1 ÷ 8

⇒ 1 ÷ 8 = 0.125

∴ The correct answer is 0.125.

Additional Information

Algebraic Identity for Cubes

Sum of cubes: a³ + b³ = (a + b)(a² − ab + b²). This is frequently used to simplify fractional algebraic expressions.

Factoring out from Powers

When a multiple is raised to a power, the constant can be factored out: (ka)ⁿ + (kb)ⁿ = kⁿ(aⁿ + bⁿ).

BODMAS Rule

Always simplify brackets first, followed by orders (powers/roots), division, multiplication, addition, and subtraction sequentially.

Shortcut Trick

Observe that each term in the denominator is exactly 5 times the corresponding term in the numerator.

We know that 0.25 = 5 × 0.05 and 0.025 = 5 × 0.005.

Factoring out 5³ from the denominator terms:

Expression = (0.05³ + 0.005³) ÷ [ 5³ × (0.05³ + 0.005³) ]

Expression = 1 ÷ 5³ = 1/125

∴ The correct answer is 1/125.

Alternate Method

Given:

Expression: (0.05³ + 0.005³) ÷ (0.25³ + 0.025³)

Calculations:

⇒ Numerator = 0.05³ + 0.005³

⇒ Denominator = 0.25³ + 0.025³

Notice that the terms in the denominator are multiples of the numerator:

⇒ 0.25 = 5 × 0.05

⇒ 0.025 = 5 × 0.005

Substituting these relation values back into the denominator:

⇒ Denominator = (5 × 0.05)³ + (5 × 0.005)³

⇒ Denominator = (5³ × 0.05³) + (5³ × 0.005³)

Taking 5³ common from both terms:

⇒ Denominator = 125 × (0.05³ + 0.005³)

Now, substituting the simplified denominator back into the original fraction:

⇒ Expression = (0.05³ + 0.005³) ÷ [ 125 × (0.05³ + 0.005³) ]

Canceling the common bracket (0.05³ + 0.005³) from the numerator and denominator:

⇒ Expression = 1 / 125

∴ The correct answer is 1/125.

Additional Information

Laws of Exponents

The exponent rule (a × b)ⁿ = aⁿ × bⁿ is heavily utilized in simplification to separate and factor out common multiples from base numbers.

Algebraic Identities

Often questions of this format use the sum of cubes identity (a³ + b³) = (a + b)(a² − ab + b²) to simplify large decimal fractions directly without full calculation.

BODMAS Rule

Always evaluate mathematical operations in the correct sequence: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction to arrive at the accurate result.

Shortcut Trick

Rewrite the expression in the form √(x + 2√y) to apply the direct splitting method.

√(9 + 4√5) = √(9 + 2√(4 × 5)) = √(9 + 2√20)

Find two numbers whose sum is 9 and product is 20 ⇒ The numbers are 5 and 4.

The square root is directly the sum of their individual square roots: √5 + √4.

⇒ √5 + 2

∴ The correct answer is √5 + 2.

Alternate Method

Given:

Expression: √(9 + 4√5)

Formula Used:

(a + b)² = a² + b² + 2ab

Calculations:

⇒ We need to express 9 + 4√5 as a perfect square to remove the outer root.

⇒ Compare the irrational part (4√5) with the 2ab term of the identity: 2ab = 4√5 ⇒ ab = 2√5.

⇒ By trial, possible values for a and b are 2 and √5.

⇒ Verify the sum of their squares: a² + b² = 2² + (√5)² = 4 + 5 = 9.

⇒ Since a² + b² = 9 perfectly matches the rational part, we can rewrite the expression:

⇒ 9 + 4√5 = 4 + 5 + 4√5 = 2² + (√5)² + 2 × 2 × √5 = (2 + √5)²

⇒ √(9 + 4√5) = √((2 + √5)²)

⇒ √(9 + 4√5) = 2 + √5 = √5 + 2

∴ The correct answer is √5 + 2.

Additional Information

Square Root of Surds

To find the square root of a mixed surd a ± 2√b, find two numbers whose sum is a and product is b. The answer will be √x ± √y.

Conjugate of a Surd

The conjugate of a + √b is a − √b. This property is highly useful when rationalising the denominator of an irrational fraction.

BODMAS Rule

Always simplify expressions following the correct order: Brackets, Orders (Powers/Roots), Division, Multiplication, Addition, and Subtraction.

Shortcut Trick

Given: 25% A = 20% B = 10% C

⇒ 25A = 20B = 10C ⇒ 5A = 4B = 2C

Find the LCM of the coefficients (5, 4, 2), which is 20.

⇒ A : B : C = (20 ÷ 5) : (20 ÷ 4) : (20 ÷ 2)

⇒ A : B : C = 4 : 5 : 10

∴ The correct answer is 4 : 5 : 10.

Alternate Method

Given:

25% of A = 20% of B = 10% of C

A B C 4 parts 5 parts 10 parts Ratio Comparison of A, B, and C

Calculation:

Convert the given percentages into fractions:

⇒ (25 ÷ 100) × A = (20 ÷ 100) × B = (10 ÷ 100) × C

⇒ A / 4 = B / 5 = C / 10

Let A / 4 = B / 5 = C / 10 = k (where k is a constant)

⇒ A = 4k

⇒ B = 5k

⇒ C = 10k

⇒ A : B : C = 4k : 5k : 10k

⇒ A : B : C = 4 : 5 : 10

∴ The correct answer is 4 : 5 : 10.

Additional Information

Compound Ratio

The ratio obtained by multiplying antecedent to antecedent and consequent to consequent of two or more ratios is called a compound ratio. For a : b and c : d, the compound ratio is (a × c) : (b × d).

Duplicate & Triplicate Ratio

The duplicate ratio of a : b is a² : b², and the triplicate ratio of a : b is a³ : b³.

Sub-duplicate Ratio

The sub-duplicate ratio of a : b is √a : √b, and the sub-triplicate ratio is a^1/3 : b^1/3.

Shortcut Trick When the sum of numbers under square roots is equal (7 + 2 = 9 and 6 + 3 = 9), the expression with the greater product is larger. Product for first expression: 7 × 2 = 14 Product for second expression: 6 × 3 = 18 Since 14 < 18, we have √7 + √2 < √6 + √3. ∴ The correct answer is (ii).

Alternate Method Given: Expression 1 = √7 + √2 Expression 2 = √6 + √3 Formula Used: (a + b)2 = a2 + b2 + 2ab Calculation: Let us compare the squares of both expressions. ⇒ (√7 + √2)2 = (√7)2 + (√2)2 + 2 × √7 × √2 ⇒ (√7 + √2)2 = 7 + 2 + 2√14 = 9 + 2√14 ⇒ (√6 + √3)2 = (√6)2 + (√3)2 + 2 × √6 × √3 ⇒ (√6 + √3)2 = 6 + 3 + 2√18 = 9 + 2√18 Comparing the two results, we have the common term 9 in both. We know that 14 < 18, so √14 < √18. ⇒ 9 + 2√14 < 9 + 2√18 ⇒ (√7 + √2)2 < (√6 + √3)2 Taking square roots on both sides (since both sums are positive): ⇒ √7 + √2 < √6 + √3 Thus, statement (ii) is correct. ∴ The correct answer is (ii).

Additional Information Comparing Surds with Constant Sums When the sum of the numbers inside the square roots is constant, simply check the product of the two numbers. The higher the product, the larger the value of the sum. Comparing Surds with Constant Differences If the difference between two surds is constant (e.g., √7 − √5 and √5 − √3), the expression with smaller numbers will always have a larger value. Rationalization of Surds Surds in the denominator are simplified by multiplying the numerator and denominator by the conjugate of the denominator, converting it into a rational number.

Shortcut Trick

Total amount spent = SP of Laptop + SP of Mouse

Total amount spent = (40000 − 4000) + 1500 × (100 − 20)%

Total amount spent = 36000 + (1500 × 0.8)

Total amount spent = 36000 + 1200 = 37200

∴ The correct answer is ₹37,200.

Alternate Method

Given:

Price of the laptop = ₹40,000

Discount on the laptop = ₹4,000

Marked price of the mouse = ₹1,500

Discount percentage on the mouse = 20%

Formula Used:

Selling Price (SP) = Marked Price (MP) − Discount Amount

Discount Amount = MP × (Discount % ÷ 100)

Laptop (MP: ₹40,000) SP: ₹36,000 Dis: ₹4,000 Mouse (MP: ₹1,500) SP: ₹1,200 Dis: 20% Total Spend = ₹36,000 + ₹1,200 = ₹37,200

Calculations:

⇒ Selling price of laptop = 40,000 − 4,000 = ₹36,000

⇒ Discount amount on mouse = 1,500 × (20 ÷ 100) = ₹300

⇒ Selling price of mouse = 1,500 − 300 = ₹1,200

⇒ Total amount spent = Selling price of laptop + Selling price of mouse

⇒ Total amount spent = 36,000 + 1,200 = 37,200

∴ The correct answer is ₹37,200.

Additional Information

Marked Price (MP) vs Selling Price (SP)

Discount is always calculated on the Marked Price of an item. Formula: SP = MP × (1 − Discount%/100).

Successive Discount

When two discounts d₁% and d₂% are given consecutively, the effective discount is (d₁ + d₂ − (d₁ × d₂)/100)%.

Cost Price (CP) vs Selling Price (SP)

Profit or loss is always calculated on the Cost Price. Profit = SP − CP.

Shortcut Trick

Total work = LCM(18, 24) = 72 units.

Efficiencies: A = 4 units/min, B = 3 units/min ⇒ Combined = 7 units/min.

Work done in 6 mins = 7 × 6 = 42 units ⇒ Remaining work = 72 − 42 = 30 units.

New combined efficiency = (4 ÷ 2) + (3 × 2) = 2 + 6 = 8 units/min.

Time for remaining work = 30 ÷ 8 = 3.75 minutes = 3 min 45 sec.

∴ The correct answer is 3 min 45 sec.

Alternate Method

Given:

Pipe A can fill the tank in 18 minutes.

Pipe B can fill the tank in 24 minutes.

Calculations:

⇒ Initial combined efficiency of A and B = 4 + 3 = 7 units/min

⇒ Work completed by both pipes in 6 minutes = 7 × 6 = 42 units

⇒ Remaining work to be completed = 72 − 42 = 30 units

⇒ New efficiency of Pipe A (halved) = 4 ÷ 2 = 2 units/min

⇒ New efficiency of Pipe B (doubled) = 3 × 2 = 6 units/min

⇒ New combined efficiency = 2 + 6 = 8 units/min

⇒ Time required to complete the remaining work = 30 ÷ 8 = 15 / 4 minutes

⇒ 15 / 4 minutes = 3 minutes + (3 / 4 × 60) seconds = 3 minutes 45 seconds

∴ The correct answer is 3 min 45 sec.

Additional Information

Pipes & Cisterns Concept

Filling pipes are considered to have a positive efficiency, while emptying pipes (leaks) have a negative efficiency. Total capacity is usually assumed as the LCM of individual timings.

Work and Efficiency Relation

Efficiency is inversely proportional to time. The foundational formula is Total Work = Efficiency × Time.

Unit Conversion

To convert a fractional part of a minute into exact seconds, simply multiply the fraction by 60 (e.g., 0.75 min = 0.75 × 60 = 45 sec).

Shortcut Trick

If x + (1/x) = −1, then a standard algebraic result is x³ = 1.

Reduce higher powers using this property: x⁴ = x³ × x = x.

⇒ x⁴ + (1/x⁴) = x + (1/x) = −1

⇒ 2(x² + 1/x²) = 2[(x + 1/x)² − 2] = 2[(−1)² − 2] = −2

⇒ Given expression value = −1 + (−2) = −3

∴ The correct answer is −3.

Alternate Method

Given:

x + (1/x) = −1

Formula Used:

a² + b² = (a + b)² − 2ab

Calculations:

Squaring the given equation on both sides:

⇒ x² + (1/x²) + 2(x)(1/x) = (−1)²

⇒ x² + (1/x²) + 2 = 1

⇒ x² + (1/x²) = 1 − 2 = −1

Now, squaring this new equation on both sides:

⇒ (x²)² + (1/x²)² + 2(x²)(1/x²) = (−1)²

⇒ x⁴ + (1/x⁴) + 2 = 1

⇒ x⁴ + (1/x⁴) = 1 − 2 = −1

We need to find the value of: x⁴ + (1/x⁴) + 2x² + (2/x²)

⇒ [x⁴ + (1/x⁴)] + 2[x² + (1/x²)]

Substitute the obtained values into the expression:

⇒ −1 + 2(−1)

⇒ −1 − 2 = −3

∴ The correct answer is −3.

Additional Information

Special Case of x + 1/x = 1

If x + (1/x) = 1, then multiplying both sides by x yields x² − x + 1 = 0, which gives x³ = −1.

Special Case of x + 1/x = −1

If x + (1/x) = −1, multiplying by x yields x² + x + 1 = 0, giving x³ = 1.

Sum of Squares Formula

If x + (1/x) = k, then x² + (1/x²) = k² − 2.

Sum of Cubes Formula

If x + (1/x) = k, then x³ + (1/x³) = k³ − 3k.

Shortcut Trick

Time from 10 January to 10 July is exactly 6 months (1/2 year).

Simple Interest for 6 months at 6% p.a. is effectively 3% of the Principal.

SI = 3% of 12000 = 3 × 120 = ₹360.

Total Amount = 12000 + 360 = ₹12360.

∴ The correct answer is ₹12360.

Alternate Method

Given:

Principal (P) = ₹12000

Rate (R) = 6% per annum

Time (T) = 10 January 2024 to 10 July 2024 = 6 months = 1/2 year

Formula Used:

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

Amount (A) = P + SI

10 Jan 2024 P = ₹12000

10 Jul 2024 A = ₹12360

Time = 6 months (1/2 year) Rate = 6% p.a.

SI = ₹360

Calculations:

⇒ SI = (12000 × 6 × 1) ÷ (100 × 2)

⇒ SI = (12000 × 3) ÷ 100

⇒ SI = ₹360

⇒ Amount (A) = P + SI

⇒ A = 12000 + 360 = ₹12360

∴ The correct answer is ₹12360.

Additional Information

Total Amount Formula

The total amount can be directly calculated using the formula: A = P × (1 + RT ÷ 100).

Simple vs Compound Interest

Simple Interest is calculated only on the initial principal value, whereas Compound Interest is calculated on the principal plus accumulated interest from previous periods.

Time Calculation Rule

When the exact same dates are given across different months (e.g., 10 Jan to 10 Jul), time is directly calculated in exact months (6 months), which translates directly to a fractional year (6 ÷ 12 = 1/2 year).

Shortcut Trick

Since the rate of interest and time period are the same, the ratio of Simple Interest to Principal remains constant.

Ratio for the first case = B ÷ A

Ratio for the second case = C ÷ B

Equating both ratios: B ÷ A = C ÷ B

By cross-multiplying, we get: B² = AC

∴ The correct answer is B² = AC.

Alternate Method

Given:

Simple interest on amount A = B

Simple interest on amount B = C

Rate of interest (R) and Time (T) are the same for both cases.

Formula Used:

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

A

SI = B

B

SI = C

C

Ratio of SI to P is constant → B/A = C/B

Calculations:

⇒ For the first case, the principal is A and simple interest is B.

⇒ B = (A × R × T) ÷ 100

⇒ (R × T) ÷ 100 = B ÷ A --- (Equation 1)

⇒ For the second case, the principal is B and simple interest is C.

⇒ C = (B × R × T) ÷ 100

⇒ (R × T) ÷ 100 = C ÷ B --- (Equation 2)

⇒ Equating Equation 1 and Equation 2, we get:

⇒ B ÷ A = C ÷ B

⇒ B × B = A × C

⇒ B² = AC

∴ The correct answer is B² = AC.

Additional Information

Simple Interest (SI)

SI = (P × R × T) ÷ 100, where P = Principal, R = Rate %, and T = Time.

Compound Interest (CI)

Amount A = P × (1 + R ÷ 100)^T and CI = A − P.

Difference between CI and SI

For 2 years, the difference is D = P × (R ÷ 100)².

Shortcut Trick

When the angle of elevation is 45°, the perpendicular and base of the right-angled triangle are equal.

Therefore, Height of the tower = Length of the shadow.

Length of shadow = 20 m.

∴ The correct answer is 20 m.

Alternate Method

Given:

Height of the tower (AB) = 20 m

Angle of elevation of the sun (θ) = 45°

45° A B C 20 m (Tower) Shadow Length = x

Formula Used:

In a right-angled triangle, tan(θ) = Perpendicular ÷ Base

Calculation:

⇒ Let the length of the shadow be BC.

⇒ tan(45°) = AB ÷ BC

⇒ 1 = 20 ÷ BC

⇒ BC = 20 × 1

⇒ BC = 20 m

∴ The correct answer is 20 m.

Additional Information

Angle of Elevation vs Angle of Depression

The angle of elevation is formed when looking up from a horizontal line, while the angle of depression is formed when looking down. Both are numerically equal for the same line of sight due to alternate interior angles.

Standard Trigonometric Ratios

In height and distance problems, the most commonly used tangent values are tan(30°) = 1 ÷ √3, tan(45°) = 1, and tan(60°) = √3.

Complementary Elevation Angles

If a structure casts shadows of lengths a and b at two different times such that the respective angles of elevation are complementary (sum to 90°), the height of the structure is given by h = √(a × b).

Shortcut Trick

The volume of a hemisphere is exactly half the volume of a complete sphere.

Volume = (2/3) × π × r³

Substitute r = 5 cm directly into the formula.

Volume = (2/3) × π × 125 = 250π/3 cm³

∴ The correct answer is 250π/3 cm³.

Alternate Method

Given:

Radius of the hemisphere (r) = 5 cm

Formula Used:

Volume of a hemisphere = (2/3) × π × r³

r = 5 cm Hemisphere

Calculations:

⇒ Volume = (2/3) × π × (5)³

⇒ Volume = (2/3) × π × 125

⇒ Volume = 250π/3 cm³

∴ The correct answer is 250π/3 cm³.

Additional Information

Volume of a Full Sphere

The total volume of a complete sphere is given by V = (4/3) × π × r³. A hemisphere's volume is exactly half of this value.

Curved Surface Area (CSA)

The curved surface area of a hemisphere only accounts for the outer bowl part and is calculated as CSA = 2 × π × r².

Total Surface Area (TSA)

The total surface area includes both the curved surface and the flat circular top surface. It is calculated as TSA = 3 × π × r².

Shortcut Trick

When a sphere is inscribed inside a cube, the ratio of the volume of the cube to the volume of the sphere is always 6 : π.

The ratio of the unoccupied volume to the total volume of the cube is (6 − π) : 6.

Percentage of unoccupied volume = [(6 − π) ÷ 6] × 100

Taking π ≈ 3.14: ⇒ [(6 − 3.14) ÷ 6] × 100 = (2.86 ÷ 6) × 100

⇒ 47.666...% ≈ 47.67%

∴ The correct answer is 47.67%.

Alternate Method

Given:

A solid sphere is placed inside a cube such that it touches all six faces.

r = a/2

a

2D Cross-section: Sphere inside a Cube

Formula Used:

Volume of cube = a³ (where a is the side of the cube)

Volume of sphere = (4 ÷ 3) × π × r³ (where r is the radius of the sphere)

Calculations:

⇒ Let the side of the cube be 'a'.

⇒ Since the sphere touches all six faces, its diameter is strictly equal to the side length of the cube.

⇒ Radius of the sphere (r) = a ÷ 2

⇒ Volume of the cube = a³

⇒ Volume of the sphere = (4 ÷ 3) × π × (a ÷ 2)³

⇒ Volume of the sphere = (4 ÷ 3) × π × (a³ ÷ 8) = (π ÷ 6) × a³

⇒ Volume not occupied by the sphere = a³ − (π ÷ 6) × a³ = a³ × (1 − π ÷ 6)

⇒ Percentage of unoccupied volume = [a³ × (1 − π ÷ 6) ÷ a³] × 100

⇒ Percentage = (1 − 3.14 ÷ 6) × 100

⇒ Percentage = [(6 − 3.14) ÷ 6] × 100 = (2.86 ÷ 6) × 100

⇒ Percentage = 286 ÷ 6 = 47.666... ≈ 47.67%

∴ The correct answer is 47.67%.

Additional Information

Cube Properties

For a cube of side 'a', the Total Surface Area = 6a², Lateral Surface Area = 4a², and the length of the longest body diagonal = a√3.

Sphere Properties

For a sphere of radius 'r', the Surface Area = 4πr². The sphere holds the maximum possible volume among all 3D solid shapes for any given surface area.

Cube and Inscribed Cylinder

If a right circular cylinder is inscribed inside a cube of side 'a', its volume will be (π ÷ 4) × a³. The percentage of occupied volume by the cylinder is roughly 78.5%.

Shortcut Trick

Let the sides of the rectangular prism be 4x, 5x, and 6x.

Volume = 4x × 5x × 6x = 120x³

Since the volume is 960 cm³, we have 120x³ = 960.

⇒ x³ = 8 ⇒ x = 2

The longest side is 6x = 6 × 2 = 12 cm.

∴ The correct answer is 12 cm.

Alternate Method

Given:

The ratio of the sides of the rectangular prism = 4 : 5 : 6

Volume of the prism = 960 cm³

Formula Used:

Volume of a rectangular prism = Length × Width × Height

Length = 6x Height = 4x Width = 5x Volume = 960 cm³

Calculations:

Let the exact dimensions of the sides be 4x, 5x, and 6x.

⇒ Volume = 4x × 5x × 6x

⇒ 120x³ = 960

⇒ x³ = 960 ÷ 120

⇒ x³ = 8

⇒ x = 2

Now, finding the exact lengths of all three sides:

⇒ First side = 4 × 2 = 8 cm

⇒ Second side = 5 × 2 = 10 cm

⇒ Third side = 6 × 2 = 12 cm

⇒ Out of the three, the longest side is clearly 12 cm.

∴ The correct answer is 12 cm.

Additional Information

Total Surface Area (TSA)

The total area of all 6 faces of a cuboid is calculated using the formula: TSA = 2(lw + wh + hl).

Lateral Surface Area (LSA)

The area of the 4 side vertical faces (excluding the top and bottom) is calculated as: LSA = 2h(l + w).

Length of Diagonal

The longest distance between two opposite corners in a rectangular prism is given by the formula: Diagonal = √(l² + w² + h²).

Shortcut Trick

When a cone is cut parallel to its base, the volumes of the corresponding parts are proportional to the cubes of their heights.

Ratio of heights of top cone to total cone = 1 : 3.

Ratio of volumes of top cone to total cone = 1³ : 3³ = 1 : 27.

Volume of bottom part = Total volume − Top volume = 27 − 1 = 26 units.

Ratio of volumes of top part to bottom part = 1 : 26.

∴ The correct answer is 1 : 26.

Alternate Method

Given:

A conical bucket is cut at one-third of its height from the top.

Let total height of the cone be H and base radius be R.

Height of the top cone (h) = H / 3.

Formula Used:

Volume of cone = (1/3) × π × radius² × height

H/3 2H/3

Base Radius = R

r = R/3

Top Volume Bottom Volume

Ratio = Top Volume : Bottom Volume = 1 : 26

Calculations:

⇒ By similar triangles, radius of the top cone (r) = R / 3.

⇒ Total volume of the cone (V) = (1/3) × π × R² × H

⇒ Volume of the top part (V₁) = (1/3) × π × r² × h

⇒ V₁ = (1/3) × π × (R/3)² × (H/3)

⇒ V₁ = (1/27) × [(1/3) × π × R² × H]

⇒ V₁ = V / 27

⇒ Volume of the bottom part (V₂) = Total Volume − Volume of top part

⇒ V₂ = V − V/27

⇒ V₂ = 26V / 27

⇒ Ratio of top part to bottom part = V₁ : V₂

⇒ Ratio = (V / 27) : (26V / 27)

⇒ Ratio = 1 : 26

∴ The correct answer is 1 : 26.

Additional Information

Volume of Frustum of a Cone

When a cone is sliced parallel to the base, the lower portion forms a frustum. Its volume is V = (1/3) × π × h × (R² + r² + R × r), where h is the height of the frustum.

Similar 3D Figures Property

For any two similar solid figures with a linear scale factor of k (ratio of heights or radii), the ratio of their surface areas is k² and the ratio of their volumes is k³.

Relationship Between Cone and Cylinder

A cone has exactly one-third the volume of a cylinder that shares the same base radius and height. V_cone = (1/3) × π × r² × h.

Shortcut Trick

If x + 1/x = 2, then x = 1.

Here, tanA + cotA = 2 ⇒ tanA + 1/tanA = 2, so tanA = 1.

Since A ∈ (0, π/2), we get A = 45°.

Now, sinA + cosA = sin(45°) + cos(45°)

⇒ 1/√2 + 1/√2 = 2/√2 = √2.

∴ The correct answer is √2.

Alternate Method

Given:

tanA + cotA = 2

A ∈ (0, π/2)

Formula Used:

tanA = sinA / cosA

cotA = cosA / sinA

(a + b)² = a² + b² + 2ab

sin²A + cos²A = 1

Calculations:

⇒ tanA + cotA = 2

⇒ (sinA / cosA) + (cosA / sinA) = 2

⇒ (sin²A + cos²A) / (sinA × cosA) = 2

⇒ 1 / (sinA × cosA) = 2

⇒ sinA × cosA = 1/2

We need to find the value of (sinA + cosA).

⇒ Let (sinA + cosA) = x

Squaring both sides:

⇒ (sinA + cosA)² = x²

⇒ sin²A + cos²A + 2 × sinA × cosA = x²

Substituting the known identities and values:

⇒ 1 + 2 × (1/2) = x²

⇒ 1 + 1 = x²

⇒ x² = 2

⇒ x = √2 (Since A ∈ (0, π/2), both sinA and cosA are positive)

∴ The correct answer is √2.

Additional Information

Basic Trigonometric Identities

Three foundational identities are: sin²θ + cos²θ = 1, sec²θ − tan²θ = 1, and cosec²θ − cot²θ = 1.

Algebraic Reciprocal Property

If the sum of a number and its reciprocal is 2, i.e., x + 1/x = 2, then x must be equal to 1. This is frequently used in trigonometric equations involving tanθ and cotθ.

Signs in the First Quadrant

For any angle A ∈ (0, π/2), all six trigonometric ratios are positive. Thus, sinA + cosA must strictly be a positive value.

Shortcut Trick

Apply the direct trigonometric compound angle identity:

sin(A + B) = sin A × cos B + cos A × sin B

For A = 20° and B = 10°, the expression reduces to sin(20° + 10°) = sin 30°.

The standard value of sin 30° is 1/2.

∴ The correct answer is 1/2.

Alternate Method

Given:

Trigonometric expression: sin 20° × cos 10° + cos 20° × sin 10°

Formula Used:

sin(A + B) = sin A × cos B + cos A × sin B

Calculation:

⇒ Let angle A = 20° and angle B = 10°

⇒ The given expression exactly matches the right hand side of the formula.

⇒ sin 20° × cos 10° + cos 20° × sin 10° = sin(20° + 10°)

⇒ sin(30°)

⇒ From the standard trigonometric table, sin 30° = 1/2

∴ The correct answer is 1/2.

Additional Information

Compound Angle Identity for Sine

sin(A − B) = sin A × cos B − cos A × sin B

Compound Angle Identity for Cosine

cos(A + B) = cos A × cos B − sin A × sin B

cos(A − B) = cos A × cos B + sin A × sin B

Double Angle Formula

sin 2A = 2 × sin A × cos A

cos 2A = cos² A − sin² A