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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).