Search Previous Year Questions

Shortcut Trick

Rate 20% = 1/5 ⇒ Multiplier = 6/5

Amount after 2 years = 68,000 × (6/5) × (6/5) = 68,000 × 1.44 = ₹97,920

New Sum after adding ₹16,000 = 97,920 + 16,000 = ₹1,13,920

Final Amount after 1 more year = 1,13,920 × (6/5) = 1,13,920 × 1.2 = ₹1,36,704

∴ The correct answer is Rs. 1,36,704.

Alternate Method

Given: Initial Principal (P₁) = ₹68,000, Rate (R) = 20%, Time (t₁) = 2 years, Addition = ₹16,000, Time (t₂) = 1 year.

Formula: Amount (A) = P(1 + R/100)ⁿ

Calculation for the first 2 years:

⇒ Amount (A₁) = 68,000 × (1 + 20/100)²

⇒ A₁ = 68,000 × (1.2)²

⇒ A₁ = 68,000 × 1.44 = ₹97,920

Total amount after adding more money:

⇒ New Principal (P₂) = 97,920 + 16,000 = ₹1,13,920

Calculation for the 3^rd year:

⇒ Final Amount = 1,13,920 × (1 + 20/100)¹

⇒ Final Amount = 1,13,920 × 1.2 = ₹1,36,704

∴ The correct answer is Rs. 1,36,704.

Additional Information

Compound Interest Formula

Amount A = P(1 + R/100)ⁿ, where P is Principal, R is Rate, and n is time in years. Compound Interest CI = A − P.

Effective Rate for 2 Years

If the rate is R% for two years, the effective compound interest rate is (R + R + R²/100)%. For 20%, it is 20 + 20 + 4 = 44%.

Successive Growth

Compound interest is an example of successive percentage growth. When additional capital is added, the new sum becomes the principal for the subsequent period.

Shortcut Trick

Angle between tangents from an external point and the angle subtended by the chord at the center are supplementary.

∠AOC + ∠ABC = 180^o

⇒ ∠AOC = 180^o − 50^o

⇒ ∠AOC = 130^o

∴ The correct answer is 130^o.

Alternate Method

Given: ∠ABC = 50^o, where AB and BC are tangents to the circle with center O.

Formula: Sum of angles in a quadrilateral is 360^o and the radius is perpendicular to the tangent at the point of contact.

Step-by-step Calculation:

⇒ In quadrilateral OABC, OA ⊥ AB and OC ⊥ BC (Radius ⊥ Tangent property)

⇒ ∠OAB = 90^o and ∠OCB = 90^o

⇒ ∠OAB + ∠ABC + ∠OCB + ∠AOC = 360^o

⇒ 90^o + 50^o + 90^o + ∠AOC = 360^o

⇒ 230^o + ∠AOC = 360^o

⇒ ∠AOC = 360^o − 230^o

⇒ ∠AOC = 130^o

∴ The correct answer is 130^o.

Additional Information

Tangent Properties

Tangents drawn from an external point to a circle are equal in length (AB = BC).

Angle at Circumference

The angle subtended by a chord at the center is double the angle subtended by it at any point on the remaining part of the circle.

Alternate Segment Theorem

The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

Shortcut Trick

Apply the Tangent-Secant Theorem: PQ² = PR × PS

Given PQ = 16 cm and PR = 8 cm.

16² = 8 × PS ⇒ 256 = 8 × PS ⇒ PS = 32 cm.

Chord RS = PS − PR = 32 − 8 = 24 cm.

∴ The correct answer is 24 cm.

Alternate Method

Given: Length of tangent PQ = 16 cm, length of external secant segment PR = 8 cm.

Formula Used: When a tangent and a secant are drawn from an external point to a circle, the square of the tangent length is equal to the product of the secant length and its external segment length (PQ² = PR × PS).

Calculations:

⇒ Let the length of the chord RS be x cm.

⇒ Total length of secant PS = PR + RS = 8 + x.

⇒ According to the theorem: PQ² = PR × PS

⇒ 16² = 8 × (8 + x)

⇒ 256 = 8(8 + x)

⇒ 256 ÷ 8 = 8 + x

⇒ 32 = 8 + x

⇒ x = 32 − 8

⇒ x = 24 cm.

∴ The correct answer is 24 cm.

Additional Information

Intersecting Chords Theorem

If two chords AB and CD intersect at point P inside a circle, then PA × PB = PC × PD.

Two Secants Theorem

If two secants PAB and PCD are drawn from an external point P, then PA × PB = PC × PD.

Tangent-Radius Property

A tangent at any point of a circle is perpendicular to the radius through the point of contact (OQ ⊥ PQ).

Shortcut Trick

Investment ratio Arun : Bablu : Chandan = 24 : 16 : 40 = 3 : 2 : 5.

Bablu commission = 30% of ₹75,000 = ₹22,500.

Remaining profit = 70% of 75,000 = ₹52,500.

Bablu's share from remaining = (2 ÷ 10) × 52,500 = ₹10,500.

Total share = 22,500 + 10,500 = ₹33,000.

∴ The correct answer is ₹33,000.

Alternate Method

Given:

Investment of Arun = ₹2,40,000

Investment of Bablu = ₹1,60,000

Investment of Chandan = ₹4,00,000

Working Partner Commission = 30%

Total Profit = ₹75,000

Formula: Profit Share Ratio = Ratio of Capitals × Time Period

⇒ Ratio of Capitals = 2,40,000 : 1,60,000 : 4,00,000

⇒ Ratio = 24 : 16 : 40 = 3 : 2 : 5

⇒ Bablu's Commission = (30 ÷ 100) × 75,000 = ₹22,500

⇒ Remaining Profit = 75,000 − 22,500 = ₹52,500

⇒ Total Ratio Parts = 3 + 2 + 5 = 10

⇒ Bablu's Profit Share = (2 ÷ 10) × 52,500 = ₹10,500

⇒ Bablu's Total Share = Commission + Profit Share

⇒ Total Share = 22,500 + 10,500 = ₹33,000

∴ The correct answer is ₹33,000.

Additional Information

Working vs Sleeping Partner

A Working Partner manages the business and receives a salary or commission, while a Sleeping Partner only invests capital and shares profit/loss.

Partnership Profit Distribution

Profit is distributed in the ratio of (Capital₁ × Time₁) : (Capital₂ × Time₂) after deducting any fixed commissions or salaries.

Commission Calculation

Commission is usually calculated on the Gross Profit unless the question specifies it is on the net profit after such commission.

Shortcut Trick

Initial Loss = 15% → Selling Price (SP₁) = 100% − 15% = 85%

Required Gain = 15% → Required Selling Price (SP₂) = 100% + 15% = 115%

Using the unitary method: If 85% corresponds to ₹85,000, then 115% corresponds to:

Required SP = (85,000 ÷ 85) × 115 = 1,000 × 115 = ₹1,15,000

∴ The correct answer is ₹1,15,000.

Alternate Method

Given:

First Selling Price (SP₁) = ₹85,000

Loss percentage (L%) = 15%

Desired Gain percentage (G%) = 15%

Formula Used:

Cost Price (CP) = [SP × 100] ÷ [100 − Loss%]

New Selling Price (SP₂) = [CP × (100 + Gain%)] ÷ 100

Calculations:

⇒ Cost Price (CP) = (85,000 × 100) ÷ (100 − 15)

⇒ CP = (85,000 × 100) ÷ 85

⇒ CP = 1,000 × 100 = ₹1,00,000

Now, to find the new selling price for a 15% gain:

⇒ SP₂ = [1,00,000 × (100 + 15)] ÷ 100

⇒ SP₂ = [1,00,000 × 115] ÷ 100

⇒ SP₂ = 1,000 × 115 = ₹1,15,000

∴ The correct answer is ₹1,15,000.

Additional Information

Cost Price and Selling Price Relationship

When there is a profit, SP = CP + Profit. When there is a loss, SP = CP − Loss. Profit or Loss is always calculated on the Cost Price (CP).

Successive Percentage Formula

If an item is sold at a loss of L% and we need a gain of G%, the ratio of the two selling prices is (100 − L) : (100 + G).

Marked Price and Discount

Discount is always calculated on the Marked Price (MP). SP = MP − Discount. The final Profit% depends on both the markup and the discount given.

Shortcut Trick

Apply the Tangent-Secant Theorem: PB² = PC × PA.

24² = 16 × PA ⇒ 576 = 16 × PA ⇒ PA = 36 cm.

Since PA = PC + AC, then AC = 36 − 16 = 20 cm.

∴ The correct answer is 20 cm.

Alternate Method

Given:

Length of tangent PB = 24 cm

Length of external secant segment PC = 16 cm

Formula Used:

For a circle, when a tangent and a secant are drawn from an external point P, the property is: PB² = PC × PA

⇒ PA = PC + AC

⇒ PB² = PC × (PC + AC)

⇒ 24² = 16 × (16 + AC)

⇒ 576 = 16 × (16 + AC)

⇒ 16 + AC = 576 ÷ 16

⇒ 16 + AC = 36

⇒ AC = 36 − 16

⇒ AC = 20 cm

∴ The correct answer is 20 cm.

Additional Information

Tangent-Secant Theorem

If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment part.

Secant-Secant Theorem

When two secant segments are drawn to a circle from an external point P, the product of the lengths of one secant segment and its external part equals the product of the other: PA × PB = PC × PD.

Intersecting Chords Theorem

If two chords intersect inside a circle, the product of the segments of one chord is equal to the product of the segments of the other: PA × PB = PC × PD.

Shortcut Trick

Ratio of radius (r) to height (h) = 4 : 5.

Let r = 4x and h = 5x.

Volume ∝ r² × h ⇒ 4² × 5 = 80 units³.

80 × (22/7) × x³ = 86,240 ⇒ x³ = (86,240 × 7) ÷ (80 × 22) = 343.

x = √³343 = 7 ⇒ Height (h) = 5 × 7 = 35 cm.

∴ The correct answer is 35 cm.

Alternate Method

Given:

Volume of cylinder (V) = 86,240 cm³

Ratio of radius (r) and height (h) = 4 : 5

Formula Used:

Volume of cylinder (V) = πr²h

Calculations:

⇒ Let radius (r) = 4x and height (h) = 5x.

⇒ 86,240 = (22/7) × (4x)² × (5x)

⇒ 86,240 = (22/7) × 16x² × 5x

⇒ 86,240 = (22/7) × 80x³

⇒ x³ = (86,240 × 7) ÷ (22 × 80)

⇒ x³ = 1078 × 7 ÷ 22

⇒ x³ = 49 × 7

⇒ x³ = 343

⇒ x = 7

⇒ Height of cylinder (h) = 5x = 5 × 7 = 35 cm.

∴ The correct answer is 35 cm.

Additional Information

Curved Surface Area of Cylinder

The Curved Surface Area (CSA) is given by 2πrh, where r is the radius and h is the height.

Total Surface Area of Cylinder

The Total Surface Area (TSA) includes the CSA and the areas of the two circular bases: TSA = 2πr(h + r).

Base Area of Cylinder

The area of the circular base is calculated using the formula A = πr².

Shortcut Trick

Initial classrooms = 20 − 4 = 16

Initial students = 16 × 40 = 640

Current students = 20 × 35 = 700

New students = 700 − 640 = 60

∴ The correct answer is 60 students.

Alternate Method

Given:

Initial students per room = 40

Final students per room = 35

Final classrooms = 20

New classrooms added = 4

Calculations:

⇒ Total original classrooms = Final classrooms − Added classrooms

⇒ Original classrooms = 20 − 4 = 16

⇒ Total original students = 16 × 40 = 640

⇒ Total current students = 20 × 35 = 700

⇒ Number of new students = Total current students − Total original students

⇒ New students = 700 − 640 = 60

∴ The correct answer is 60 students.

Additional Information

Total Value Formula

The total number of items is always the product of the number of groups and the average per group: Total = Count × Average.

Weighted Average Logic

When two groups are combined, the new average is (n₁x₁ + n₂x₂) ÷ (n₁ + n₂), where n is the quantity and x is the value per unit.

Change in Total

To find the addition or removal of members, subtract the initial total from the final total.

Shortcut Trick

Expand (3 + 4√7)² using the identity (a + b)²: 9 + 112 + 24√7 = 121 + 24√7.

By comparing coefficients of √7, we find k = 24.

Substitute k in the expression: ((24 + 2)²) ÷ 36 = 26² ÷ 6².

Simplify the ratio: (26/6)² = (13/3)².

∴ The correct answer is (13/3)².

Alternate Method

Given: (3 + 4√7)² = 121 + k√7

Formula Used: (a + b)² = a² + b² + 2ab

⇒ (3)² + (4√7)² + 2 × 3 × 4√7 = 121 + k√7

⇒ 9 + 112 + 24√7 = 121 + k√7

⇒ 121 + 24√7 = 121 + k√7

⇒ k√7 = 24√7

⇒ k = 24

Now, substitute the value of k in ((k + 2)²) / 36:

⇒ ((24 + 2)²) ÷ 36

⇒ (26)² ÷ 36

⇒ 676 ÷ 36

⇒ (13 × 13 × 4) ÷ (9 × 4)

⇒ 169 ÷ 9

⇒ (13/3)²

∴ The correct answer is (13/3)².

Additional Information

Square of a Binomial

The expansion follows the standard algebraic identity (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b².

Comparison of Surds

For rational numbers a, b, c, d and irrational √x, if a + b√x = c + d√x, then the rational parts (a = c) and irrational coefficients (b = d) must be equal.

Power Rules

The property (x/y)ⁿ = xⁿ / yⁿ allows simplifying squared fractions by reducing the base before or after squaring.

Shortcut Trick

Simplify the fraction directly: 1080 ÷ 360 = 3.

Rewrite the base 6: 3⁴ × 2⁵ × 3 = 3⁵ × 2⁵ = (3 × 2)⁵ = 6⁵.

Comparing powers of 6: p = 5.

Evaluate the expression: 5² − 5(5) + 1 = 25 − 25 + 1 = 1.

Since 1 is neither prime nor composite, the result is determined.

∴ The correct answer is Neither a prime nor a composite number.

Alternate Method

Given: (3⁴ × 2⁵ × 1080) / 360 = 6^p

Formula Used: a^m × aⁿ = a^m+n; aⁿ × bⁿ = (ab)ⁿ; If a^x = a^y, then x = y.

⇒ (3⁴ × 2⁵ × (1080 ÷ 360)) = 6^p

⇒ 3⁴ × 2⁵ × 3¹ = 6^p

⇒ 3^{(4 + 1)} × 2⁵ = 6^p

⇒ 3⁵ × 2⁵ = 6^p

⇒ (3 × 2)⁵ = 6^p

⇒ 6⁵ = 6^p

⇒ p = 5

Now, finding the value of p² − 5p + 1:

⇒ 5² − 5(5) + 1

⇒ 25 − 25 + 1

⇒ 1

Analysis of result: The number 1 is neither a prime number (it has only one factor) nor a composite number (it does not have more than two factors).

∴ The correct answer is Neither a prime nor a composite number.

Additional Information

Definition of 1

The number 1 is the multiplicative identity. It is excluded from the definitions of both prime and composite numbers because prime numbers must have exactly two distinct factors.

Prime and Composite Numbers

Prime numbers are natural numbers greater than 1 with exactly two factors (1 and itself). Composite numbers are natural numbers greater than 1 that have more than two factors.

Exponent Rules

The Power of a Product Rule states that (ab)ⁿ = aⁿ × bⁿ. This is essential for solving equations where bases can be broken into prime factors.