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Shortcut Trick

Apply Product Law: a = 2 + 3 = 5

Apply Quotient Law: 2 − b = 1 ⇒ b = 1

Required sum a + b = 5 + 1 = 6

∴ The correct answer is 6.

Alternate Method

Given:

Equation 1: x² × x³ = x^a

Equation 2: x² ÷ x^b = x¹

Method Used:

Using Laws of Exponents for the same base x:

1. Product Rule: x^m × xⁿ = x^m+n
2. Quotient Rule: x^m ÷ xⁿ = x^m−n

Calculations:

From Equation 1:

⇒ x²⁺³ = x^a

⇒ x⁵ = x^a

⇒ a = 5

From Equation 2:

⇒ x^2−b = x¹

⇒ 2 − b = 1

⇒ b = 2 − 1 = 1

Final calculation for a + b:

⇒ a + b = 5 + 1 = 6

∴ The correct answer is 6.

Additional Information

Power of a Power Rule

When a base with an exponent is raised to another power, multiply the exponents: (x^m)ⁿ = x^{m × n}.

Zero Exponent Property

Any non-zero real number raised to the power of zero is always 1: x⁰ = 1.

Negative Exponent Property

A negative exponent indicates the reciprocal of the base: x⁻ⁿ = 1 ÷ xⁿ.

Shortcut Trick

Use the standard algebraic relation: (Difference)² = (Sum)² − 4 × (Product)

⇒ (Difference)² = 35² − 4 × 286

⇒ (Difference)² = 1225 − 1144 = 81

⇒ Difference = √81 = 9

∴ The correct answer is 9.

Alternate Method

Given: Sum of two numbers (x + y) = 35 and their Product (xy) = 286.

Formula Used: (x − y)² = (x + y)² − 4xy

⇒ Let the two positive numbers be x and y.

⇒ x + y = 35

⇒ xy = 286

⇒ (x − y)² = (35)² − 4(286)

⇒ (x − y)² = 1225 − 1144

⇒ (x − y)² = 81

⇒ x − y = √81 = 9

∴ The correct answer is 9.

Additional Information

Quadratic Form Relation

If sum (S) and product (P) are known, the numbers are roots of the equation: t² − St + P = 0.

Basic Algebraic Identities

(a + b)² = a² + b² + 2ab and (a − b)² = a² + b² − 2ab. Subtracting these gives (a + b)² − (a − b)² = 4ab.

Factorization Method

For xy = 286, find factors such that they add to 35. Factors of 286 are 1, 2, 11, 13, 22, 26, 143, 286. Here 22 + 13 = 35.

Shortcut Trick

Use the effective percentage change formula: Net % = x + y + (x × y)/100.

Combine Mark-up (+40%) and first Discount (−18%): 40 − 18 − (40 × 18)/100 = 14.8%.

Combine result (+14.8%) and second Discount (−5%): 14.8 − 5 − (14.8 × 5)/100 = 9.8 − 0.74 = 9.06%.

Since the net result is positive, it represents a Profit Percentage.

∴ The correct answer is 9.06%.

Alternate Method

Given:

Mark-up Percentage = 40%

Successive Discounts = 18% and 5%

Formula Used:

Marked Price (MP) = CP × (1 + Mark-up/100)

Selling Price (SP) = MP × (1 − d₁/100) × (1 − d₂/100)

Profit % = [(SP − CP) / CP] × 100

⇒ Let Cost Price (CP) = 100

⇒ Marked Price (MP) = 100 + 40% of 100 = 140

⇒ Price after 18% discount = 140 × (100 − 18)/100 = 140 × 0.82

⇒ Price after 18% discount = 114.8

⇒ Price after 5% discount (SP) = 114.8 × (100 − 5)/100 = 114.8 × 0.95

⇒ Selling Price (SP) = 109.06

⇒ Profit = SP − CP = 109.06 − 100 = 9.06

⇒ Profit % = (9.06 / 100) × 100 = 9.06%

∴ The correct answer is 9.06%.

Additional Information

Successive Discount Formula

Total discount for two rates d₁ and d₂ is given by: d_net = d₁ + d₂ − (d₁ × d₂)/100.

Profit and Discount Relation

The relationship between CP and MP is: MP/CP = (100 + Profit%)/(100 − Discount%).

Markup and Profit

Mark-up is the value added to CP to reach MP. Profit is the final gain after all discounts on MP.

Shortcut Trick

Calculate the marks corresponding to 16% of the total.

16% of Maximum Marks = 448 + 64 = 512 marks.

1% of Maximum Marks = 512 ÷ 16 = 32 marks.

Friend's Percentage = 672 ÷ 32 = 21%.

∴ The correct answer is 21%.

Alternate Method

Given:

Marks scored by Girish = 448

Shortfall to reach 16% = 64

Marks scored by Friend = 672

Formula Used:

Percentage = (Obtained Marks ÷ Maximum Marks) × 100

Calculations:

⇒ 16% of Maximum Marks = 448 + 64

⇒ 16% of Maximum Marks = 512

⇒ Maximum Marks = (512 ÷ 16) × 100

⇒ Maximum Marks = 32 × 100 = 3200

⇒ Friend's Percentage = (672 ÷ 3200) × 100

⇒ Friend's Percentage = 672 ÷ 32

⇒ Friend's Percentage = 21%

∴ The correct answer is 21%.

Additional Information

Maximum Marks and Base Value

The Maximum Marks acts as the base (100%) for all percentage calculations in an examination. Total Marks = (Score ÷ Percentage) × 100.

Percentage Shortfall

Shortfall is the difference between a target score and the actual score. Target Marks = Actual Marks + Shortfall.

Passing Marks Criteria

Many exam problems define Passing Marks as a specific percentage of the total. If a student fails by x marks, then Passing Marks = Student's Marks + x.

Shortcut Trick

Initial Speed of the bus = 90 ÷ 3 = 30 kmph

Increased Speed = 30 + 5 = 35 kmph

Required Time = 70 ÷ 35 = 2 hours

∴ The correct answer is 2 hr.

Alternate Method

Given:

Initial Distance (D₁) = 90 km

Initial Time (T₁) = 3 hr

Speed Increase = 5 kmph

New Distance (D₂) = 70 km

Formula Used:

Speed = Distance ÷ Time

Time = Distance ÷ Speed

Calculations:

⇒ Initial Speed = D₁ ÷ T₁

⇒ Initial Speed = 90 ÷ 3 = 30 kmph

⇒ New Speed = 30 + 5 = 35 kmph

⇒ New Time = D₂ ÷ New Speed

⇒ New Time = 70 ÷ 35

⇒ New Time = 2 hr

∴ The correct answer is 2 hr.

Additional Information

Unit Conversion

To convert speed from kmph to m/s, multiply by 5/18. To convert m/s to kmph, multiply by 18/5.

Average Speed

Average Speed = Total Distance ÷ Total Time. If distances are equal, Average Speed = (2 × S₁ × S₂) ÷ (S₁ + S₂).

Relative Speed

Objects in opposite direction: Relative Speed = S₁ + S₂. Objects in same direction: Relative Speed = |S₁ − S₂|.

Shortcut Trick

Rate 12.5% = 1/8 in fraction.

Ratio of Principal (P) to Amount (A) for 1 year = 8 : 9.

For 2 years, Ratio P : A = 8² : 9² = 64 : 81.

Compound Interest (CI) = 81 − 64 = 17 units.

64 units = ₹8,000 ⇒ 1 unit = 8,000 ÷ 64 = 125.

CI = 17 × 125 = ₹2,125.

∴ The correct answer is ₹2,125.

Alternate Method

Given:

Principal (P) = ₹8,000

Rate (R) = 12.5% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P × (1 + R/100)ⁿ

Compound Interest (CI) = Amount − Principal

Calculations:

⇒ A = 8,000 × (1 + 12.5/100)²

⇒ A = 8,000 × (1 + 1/8)²

⇒ A = 8,000 × (9/8) × (9/8)

⇒ A = 125 × 81

⇒ A = ₹10,125

⇒ CI = 10,125 − 8,000

⇒ CI = ₹2,125

∴ The correct answer is ₹2,125.

Additional Information

Effective Interest Rate

For two years at rate R%, the effective compound interest rate is R + R + (R × R)/100.

Difference between CI and SI

For 2 years, CI − SI = P × (R/100)².

Compounding Periods

If compounded half-yearly, the rate becomes R/2 and the time period becomes 2n.

Shortcut Trick

Use the direct formula for percentage reduction to restore the original value: [r / (100 + r)] × 100.

Here, r = 25.

Reduction = [25 / (100 + 25)] × 100

Reduction = (25 / 125) × 100 = (1 / 5) × 100 = 20%.

∴ The correct answer is 20%.

Alternate Method

Given:

⇒ Increase in price = 25%

Formula Used:

⇒ Percentage Reduction = (Difference / Increased Price) × 100

Calculations:

⇒ Let the original price of the article be ₹100.

⇒ Increased price = 100 + 25% of 100 = 100 + 25 = ₹125.

⇒ To return to the original price (₹100), the reduction required = 125 − 100 = ₹25.

⇒ Required percentage reduction = (25 / 125) × 100

⇒ Required percentage reduction = (1 / 5) × 100

⇒ Required percentage reduction = 20%

∴ The correct answer is 20%.

Additional Information

Base Change Rule

If quantity A is r% more than B, then B is [r / (100 + r)] × 100% less than A. If A is r% less than B, then B is [r / (100 − r)] × 100% more than A.

Fractional Equivalents

25% is equivalent to 1/4. When a value increases by 1/n, it must be decreased by 1/(n+1) to return to its original value. Here, n=4, so reduction = 1/(4+1) = 1/5 = 20%.

Net Percentage Change

For two successive percentage changes a% and b%, the effective change is [a + b + (ab / 100)]%. To result in zero net change, the formula must equate to 0.

Shortcut Trick

Combine the decimal parts first: 5.25 + 2.75 = 8.

Simplify the product of fractions: (2/5) × (10/20) = (2/5) × (1/2) = 1/5 = 0.2.

Convert the remaining fraction to decimal: 3/5 = 0.6.

Perform the final subtraction: 8 − 0.6 − 0.2 = 7.2.

∴ The correct answer is 7.2.

Alternate Method

Given: 5.25 + 2.75 − (3/5) − (2/5) × (10/20)

Method: Solve using the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction).

⇒ 5.25 + 2.75 − 0.6 − (2/5 × 1/2)

⇒ 8.00 − 0.6 − (1/5)

⇒ 8 − 0.6 − 0.2

⇒ 8 − 0.8

⇒ 7.2

∴ The correct answer is 7.2.

Additional Information

BODMAS Rule

It defines the order of operations: Brackets, Of (orders), Division, Multiplication, Addition, and Subtraction.

Fraction to Decimal Conversion

To convert a fraction to a decimal, divide the numerator by the denominator (e.g., 3 ÷ 5 = 0.6).

Simplification Shortcut

Always look for common factors in fractions before multiplying to reduce the complexity of the numbers.

Shortcut Trick

Convert the given trigonometric expression into tangent and secant forms: 15 + 12 tan² A − 12 sec² A.

Factor out the common term 12: 15 + 12(tan² A − sec² A).

Apply the Pythagorean identity: Since sec² A − tan² A = 1, then tan² A − sec² A = −1.

The expression simplifies to: 15 + 12(−1) = 15 − 12 = 3.

∴ The correct answer is 3.

Alternate Method

Given: 15 + 12/cot² A − 12/cos² A

Formula Used: 1/cot A = tan A and 1/cos A = sec A; sec² A − tan² A = 1

⇒ Expression = 15 + 12 × (1/cot² A) − 12 × (1/cos² A)

⇒ Expression = 15 + 12 tan² A − 12 sec² A

⇒ Expression = 15 + 12(tan² A − sec² A)

⇒ From the identity sec² A − tan² A = 1, we know tan² A − sec² A = −1

⇒ Expression = 15 + 12(−1)

⇒ Expression = 15 − 12 = 3

∴ The correct answer is 3.

Additional Information

Pythagorean Identities

There are three fundamental identities: sin² A + cos² A = 1, 1 + tan² A = sec² A, and 1 + cot² A = cosec² A.

Reciprocal Relations

Trigonometric ratios are related as follows: tan A = 1/cot A, sec A = 1/cos A, and cosec A = 1/sin A.

Simplification Rule

When simplifying trigonometric expressions, always look to convert terms into sine/cosine or use established identities to reduce variables.

Shortcut Trick

Find the LCM of divisors 6, 14, and 7, which is 42.

The largest 4-digit multiple of 42 is 9996 (calculated as 9999 − 3).

Adding the remainder: 9996 + 5 = 10001 (This is a 5-digit number).

Subtract the LCM to find the previous valid number: 10001 − 42 = 9959.

∴ The correct answer is 9959.

Alternate Method

Given:

Divisors: 6, 14, 7

Remainder: 5

Target: Largest 4-digit number

Formula Used:

Required Number = [LCM(Divisors) × k] + Remainder

Calculations:

⇒ LCM(6, 14, 7) = 42

⇒ Largest 4-digit number = 9999

⇒ 9999 ÷ 42 = 238 (Quotient), 3 (Remainder)

⇒ Largest 4-digit multiple of 42 = 9999 − 3 = 9996

⇒ Check 4-digit constraint: 9996 + 5 = 10001 (Not a 4-digit number)

⇒ Previous multiple of 42 = 9996 − 42 = 9954

⇒ Required number = 9954 + 5 = 9959

∴ The correct answer is 9959.

Additional Information

LCM and Remainder Concept

When a number N is divided by a, b, c and leaves the same remainder 'r' in each case, then N = LCM(a, b, c) × k + r, where k is an integer.

Divisibility by 14

A number is divisible by 14 if it is simultaneously divisible by its co-prime factors 2 and 7.

Successive Division vs Single Remainder

If remainders are different (r₁, r₂, r₃), and the difference (divisor − remainder) is constant 'd', then the number N = LCM(divisors) × k − d.