Search Previous Year Questions

Shortcut Trick

Expand the given expression using the formula (a + b)² = a² + b² + 2ab

⇒ (√3 + 2)² = (√3)² + (2)² + 2 × √3 × 2

⇒ 3 + 4 + 4√3 = ? + 4√3

⇒ 7 + 4√3 = ? + 4√3

⇒ ? = 7

∴ The correct answer is 7.

Alternate Method

Given:

(√3 + 2)² = ? + 4√3

Formula Used:

(a + b)² = a² + b² + 2ab

Calculations:

Let a = √3 and b = 2

⇒ LHS = (√3 + 2)²

⇒ LHS = (√3)² + (2)² + 2 × √3 × 2

⇒ LHS = 3 + 4 + 4√3

⇒ LHS = 7 + 4√3

Now, substituting this value back into the original equation:

⇒ 7 + 4√3 = ? + 4√3

Subtracting 4√3 from both sides:

⇒ ? = 7

∴ The correct answer is 7.

Additional Information

Algebraic Identities

Basic expansion formulas: (a − b)² = a² + b² − 2ab and a² − b² = (a + b)(a − b).

Rules of Surds

Squaring a square root gives the original base value: (√a)² = a. Also, √a × √b = √(ab).

Shortcut Trick

Square roots of numbers ending in 25 follow a simple pattern: n × (n+1) followed by 25 gives a root of (n5).

√7225 → 72 = 8 × 9 ⇒ Root is 85.

√1225 → 12 = 3 × 4 ⇒ Root is 35.

⇒ (85 − 35) × 1/5

⇒ 50 ÷ 5 = 10

∴ The correct answer is 10.

Alternate Method

Given:

(√7225 − √1225) × (1/5)

Calculations:

To calculate √7225:

We know that 80² = 6400 and 90² = 8100.

Since 7225 is between these and ends in 25, the square root must end in 5.

⇒ √7225 = 85

To calculate √1225:

We know that 30² = 900 and 40² = 1600.

Since 1225 is between these and ends in 25, the square root must end in 5.

⇒ √1225 = 35

Substituting the square root values back into the expression:

⇒ (85 − 35) × (1/5)

⇒ 50 × (1/5)

⇒ 50 ÷ 5

⇒ 10

∴ The correct answer is 10.

Additional Information

BODMAS Rule

It defines the standard order of operations: Brackets, Orders (powers and roots), Division, Multiplication, Addition, and Subtraction.

Square Root Shortcut for Ends in 25

For a perfect square ending in 25, remove the last two digits. Factor the remaining number as n × (n + 1). The square root will simply be n followed by 5.

Shortcut Trick

Given 25% of (P + Q) = 75% of (P − Q).

Taking the inverse ratio: (P + Q) / (P − Q) = 75% / 25% = 3 / 1

Applying the Componendo and Dividendo rule:

P / Q = (3 + 1) / (3 − 1)

⇒ P / Q = 4 / 2 = 2 / 1

∴ The correct answer is 2 : 1.

Alternate Method

Given:

25% of (P + Q) = 75% of (P − Q)

P (2 parts) Q (1 part) P + Q (3 parts) P − Q (1 part) Ratio P : Q = 2 : 1

Calculations:

⇒ (25 / 100) × (P + Q) = (75 / 100) × (P − Q)

⇒ 1 × (P + Q) = 3 × (P − Q)

⇒ P + Q = 3P − 3Q

Bringing all P terms to one side and Q terms to the other:

⇒ Q + 3Q = 3P − P

⇒ 4Q = 2P

⇒ P / Q = 4 / 2

⇒ P : Q = 2 : 1

∴ The correct answer is 2 : 1.

Additional Information

Componendo and Dividendo

If a / b = c / d, then (a + b) / (a − b) = (c + d) / (c − d). This rule is extremely useful for rapidly solving equations that involve sums and differences of the same two variables.

Inverse Ratio Logic

If A% of X = B% of Y, the ratio of the variables is inversely proportional to their respective percentages. Therefore, X / Y = B / A.

Percentage Simplification

Converting standard percentages to simple fractions (e.g., 25% = 1 / 4, 75% = 3 / 4) makes cross-multiplication faster and prevents errors with larger numerical values.

Shortcut Trick

Value ratio = (Coin Ratio) × (Coin Denomination)

Value ratio of ₹5, ₹2, and ₹1 coins = (2 × 5) : (3 × 2) : (4 × 1) = 10 : 6 : 4

Total value in ratio units = 10 + 6 + 4 = 20 units

20 units = ₹200 ⇒ 1 unit = 10

Number of coins = 2 × 10, 3 × 10, and 4 × 10 = 20, 30, and 40 respectively.

∴ The correct answer is 20 coins of ₹5, 30 coins of ₹2, and 40 coins of ₹1.

Alternate Method

Given:

Ratio of number of ₹5, ₹2, and ₹1 coins = 2 : 3 : 4

Total amount in the bag = ₹200

Formula Used:

Total Value = (Denomination of coin) × (Number of coins)

₹5

₹2

₹1

Ratio: 2x Ratio: 3x Ratio: 4x

Value: 10x Value: 6x Value: 4x

Total Value = 10x + 6x + 4x = 20x = ₹200

Calculations:

⇒ Let the number of ₹5, ₹2, and ₹1 coins be 2x, 3x, and 4x respectively.

⇒ Value of ₹5 coins = 5 × 2x = 10x

⇒ Value of ₹2 coins = 2 × 3x = 6x

⇒ Value of ₹1 coins = 1 × 4x = 4x

⇒ Total value of all coins = 10x + 6x + 4x = 20x

⇒ According to the question, 20x = 200

⇒ x = 200 ÷ 20 = 10

⇒ Number of ₹5 coins = 2x = 2 × 10 = 20

⇒ Number of ₹2 coins = 3x = 3 × 10 = 30

⇒ Number of ₹1 coins = 4x = 4 × 10 = 40

∴ The correct answer is 20 coins of ₹5, 30 coins of ₹2, and 40 coins of ₹1.

Additional Information

Case 1: When Total Amount is Given

If the ratio of the number of coins is given alongside the total value, always multiply the coin ratio by their respective denominations to find the ratio of their values.

Case 2: When Total Number of Coins is Given

If the ratio of the values of the coins is given alongside the total number of coins, always divide the value ratio by their respective denominations to find the ratio of the number of coins.

Direct Relation

Number of coins = Total Value ÷ Value of one coin.

Shortcut Trick

We know the direct ratio relation between Cost Price (CP) and Marked Price (MP):

⇒ CP / MP = (100 − Discount%) / (100 + Profit%)

⇒ CP / 800 = (100 − 15) / (100 + 20) = 85 / 120

⇒ CP = 800 × (85 / 120) = 1700 / 3

⇒ CP ≈ 566.67

∴ The correct answer is ₹566.67.

Alternate Method

Given:

Marked Price (MP) = ₹800

Discount = 15%

Profit = 20%

Formula Used:

Selling Price (SP) = MP − (Discount% of MP)

Cost Price (CP) = (SP × 100) / (100 + Profit%)

CP = ?

SP = ₹680

MP = ₹800

+20% Profit

−15% Discount

CP = ₹680 × (100 / 120) = ₹566.67

Calculations:

⇒ SP = 800 − (15 / 100) × 800

⇒ SP = 800 − 120 = ₹680

⇒ CP = (SP × 100) / (100 + 20)

⇒ CP = (680 × 100) / 120

⇒ CP = 68000 / 120

⇒ CP = 1700 / 3 ≈ ₹566.67

∴ The correct answer is ₹566.67.

Additional Information

Direct CP and MP Ratio

A very useful formula linking Cost Price and Marked Price directly without calculating the Selling Price intermediate is: CP / MP = (100 − Discount%) / (100 + Profit%).

Successive Discounts

If two successive discounts of x% and y% are offered, the equivalent single discount is given by the successive percentage formula: (x + y − xy / 100)%.

MarkUp Percentage

MarkUp is the initial amount added to the Cost Price to set the Marked Price. It is always calculated on the base of Cost Price: MarkUp% = ((MP − CP) / CP) × 100.

Shortcut Trick

Profit = 10% of CP = ₹300 ⇒ CP = ₹3000

First Selling Price (SP₁) = CP + Profit = 3000 + 300 = ₹3300

Since SP₁ is after a 20% discount, SP₁ corresponds to 80% of Marked Price (MP).

New Selling Price after a 30% discount will correspond to 70% of MP.

New SP = SP₁ × (70 ÷ 80) = 3300 × (7 ÷ 8) = ₹2887.50

∴ The correct answer is ₹2887.50.

Alternate Method

Given:

Profit % = 10%

Discount % = 20%

Profit earned = ₹300

Formula Used:

Profit = (Profit% ÷ 100) × Cost Price (CP)

Selling Price (SP) = CP + Profit = Marked Price (MP) × (100 − Discount%) ÷ 100

+10% (Profit)

-20% (Discount)

-30% Discount

CP ₹3000

SP (1^st) ₹3300

MP ₹4125

New SP ₹2887.50

Calculations:

⇒ 10% of CP = 300

⇒ CP = 300 × (100 ÷ 10) = ₹3000

⇒ SP₁ = CP + Profit = 3000 + 300 = ₹3300

Let Marked Price be MP. Given the initial discount is 20%:

⇒ SP₁ = MP × (100 − 20)%

⇒ 3300 = MP × 0.8

⇒ MP = 3300 ÷ 0.8 = ₹4125

If the product is sold at a 30% discount:

⇒ New SP = MP × (100 − 30)%

⇒ New SP = 4125 × 0.7 = ₹2887.50

∴ The correct answer is ₹2887.50.

Additional Information

Marked Price and Discount

Discount is always calculated on the marked price of the article. Formula: SP = MP × (100 − D%) ÷ 100.

Profit and Loss

Profit and loss are always calculated on the cost price unless specified otherwise. Formula: SP = CP × (100 ± P/L%) ÷ 100.

Direct Relation between CP and MP

The ratio of Cost Price to Marked Price is directly derived as CP : MP = (100 − D%) : (100 + P%).

Successive Discounts

If two successive discounts of x% and y% are given, the equivalent single discount is given by (x + y − xy ÷ 100)%.

Shortcut Trick

For a cubic equation Aa³ + Ba² + Ca + D = 0, the sum of its roots is −B/A.

Comparing the given equation a³ − 4a² + 5a − 2 = 0, we get A = 1 and B = −4.

Sum of roots (x + y + z) = −(−4) ÷ 1 = 4.

∴ The correct answer is 4.

Alternate Method

Given:

Cubic equation: a³ − 4a² + 5a − 2 = 0

Roots of the equation: x, y, and z

Method Used:

Factorisation Method

⇒ Let P(a) = a³ − 4a² + 5a − 2 = 0

⇒ By trial and error, put a = 1 → P(1) = 1³ − 4(1)² + 5(1) − 2 = 1 − 4 + 5 − 2 = 0

⇒ Since P(1) = 0, (a − 1) is a factor of the equation.

⇒ Dividing P(a) by (a − 1) gives a quotient of (a² − 3a + 2).

⇒ Factorising the quotient: a² − 3a + 2 = a² − 2a − a + 2 = a(a − 2) − 1(a − 2) = (a − 1)(a − 2)

⇒ The complete factorisation is: (a − 1)(a − 1)(a − 2) = 0

⇒ The three roots are a = 1, a = 1, and a = 2.

⇒ Let x = 1, y = 1, and z = 2.

⇒ x + y + z = 1 + 1 + 2 = 4

∴ The correct answer is 4.

Additional Information

Vieta's Formulas for Cubic Equations

For a cubic equation Aa³ + Ba² + Ca + D = 0 with roots x, y, z: Sum of roots (x + y + z) = −B/A, sum of roots taken two at a time (xy + yz + zx) = C/A, and product of roots (xyz) = −D/A.

Factor Theorem

If P(x) is a polynomial and P(c) = 0 for some value c, then (x − c) is a factor of the polynomial P(x).

Roots of Polynomials

A polynomial of degree 'n' always has exactly 'n' complex roots (including real roots), and the sum of its roots is always calculated directly as −(Coefficient of xⁿ⁻¹) ÷ (Coefficient of xⁿ).

Shortcut Trick

Time duration from 15 March to 15 September is exactly 6 months (1/2 year).

Simple Interest = Principal × Rate × Time ÷ 100

Interest = 6000 × 10 × (1/2) ÷ 100 = ₹300

Total Amount = Principal + Interest = 6000 + 300 = ₹6300

∴ The correct answer is ₹6300.

Alternate Method

Given:

Principal (P) = ₹6000

Rate of Interest (R) = 10% per annum

Time (T) = 15 March to 15 September

15 March 15 September

P = ₹6000 Amount = ?

Time = 6 months (1/2 year)

SI = ₹300

Formula Used:

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

Amount = P + SI

Calculations:

⇒ The period from 15 March to 15 September is exactly 6 months.

⇒ Time (T) in years = 6 ÷ 12 = 1/2 year

⇒ SI = (6000 × 10 × 1/2) ÷ 100

⇒ SI = (6000 × 5) ÷ 100

⇒ SI = 30000 ÷ 100 = ₹300

⇒ Total Amount = 6000 + 300 = ₹6300

∴ The correct answer is ₹6300.

Additional Information

Simple Interest vs Compound Interest

Simple interest is calculated on the original principal amount only, whereas compound interest is calculated on both the principal and the accumulated interest of previous periods.

Compound Interest Formula

Amount = P × (1 + R÷100)^T, where P is the principal, R is the rate of interest per annum, and T is the time in years.

Time Calculation by Exact Dates

When interest is calculated day-by-day, the starting day is excluded and the ending day is included. For an exact same date across months, the number of months is directly considered as the exact time period (e.g., exactly 6 months here).

Shortcut Trick

When the simple interest from different schemes is equal, the ratio of investments is inversely proportional to the product of Rate and Time.

Ratio of investments = 1/R₁ : 1/R₂ : 1/R₃ (since time is 1 year for all)

Ratio = 1/4 : 1/5 : 1/10 = 5 : 4 : 2

Total parts = 5 + 4 + 2 = 11 parts = ₹2200 ⇒ 1 part = ₹200

Amount invested at 5% corresponds to 4 parts = 4 × 200 = ₹800

∴ The correct answer is ₹800.

Alternate Method

Total = ₹2200

Part 1 @ 4% for 1 yr

Part 2 @ 5% for 1 yr

Part 3 @ 10% for 1 yr

Interest is EQUAL across all three parts (I1 = I2 = I3)

Given:

Total sum invested = ₹2200

Rate of interest for scheme 1 (R₁) = 4%

Rate of interest for scheme 2 (R₂) = 5%

Rate of interest for scheme 3 (R₃) = 10%

Time period (T) = 1 year

Formula Used:

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

Calculations:

Let the amounts invested in the three schemes be P₁, P₂, and P₃.

⇒ P₁ + P₂ + P₃ = 2200

According to the question, the simple interest earned from all three schemes is equal:

⇒ (P₁ × 4 × 1) ÷ 100 = (P₂ × 5 × 1) ÷ 100 = (P₃ × 10 × 1) ÷ 100

⇒ 4P₁ = 5P₂ = 10P₃

Dividing the equation by the LCM of 4, 5, and 10 (which is 20):

⇒ 4P₁ ÷ 20 = 5P₂ ÷ 20 = 10P₃ ÷ 20

⇒ P₁ ÷ 5 = P₂ ÷ 4 = P₃ ÷ 2

⇒ Ratio of investments (P₁ : P₂ : P₃) = 5 : 4 : 2

Sum of the ratio parts = 5 + 4 + 2 = 11 parts

⇒ 11 parts = 2200

⇒ 1 part = 2200 ÷ 11 = 200

The amount invested at 5% is P₂, which corresponds to 4 parts.

⇒ Amount invested at 5% = 4 × 200 = ₹800

∴ The correct answer is ₹800.

Additional Information

Equal Interest Condition in SI

If a total sum is divided into multiple parts such that the Simple Interest earned on each part is equal, the ratio of the principal amounts is directly given by: P₁ : P₂ : P₃ = (1/R₁T₁) : (1/R₂T₂) : (1/R₃T₃).

Equal Amount Condition in SI

If a sum is divided such that the Final Amount (Principal + Interest) earned on each part is equal, the ratio of the principal investments is given by: P₁ : P₂ : P₃ = 1/(100 + R₁T₁) : 1/(100 + R₂T₂) : 1/(100 + R₃T₃).

Simple vs Compound Interest Concept

In Simple Interest, the interest is calculated only on the original principal for the entire time period. In Compound Interest, the interest is calculated on both the principal and the accumulated interest from previous periods.

Shortcut Trick

When objects are on opposite sides of a tower of height h with angles of depression α and β:

Distance between them = h × (cot α + cot β)

⇒ Distance = 120 × (cot 30° + cot 45°)

⇒ Distance = 120 × (√3 + 1)

∴ The correct answer is 120(√3 + 1) m.

Alternate Method

Given:

Height of the tower (AB) = 120 m

Angle of depression of first car (C) = 30° ⇒ Angle of elevation from C = 30°

Angle of depression of second car (D) = 45° ⇒ Angle of elevation from D = 45°

30°

45°

30°

45°

A B C D

120 m

x = 120√3 m y = 120 m

Calculations:

In right-angled triangle ABC:

⇒ tan 30° = AB ÷ BC

⇒ 1 ÷ √3 = 120 ÷ BC

⇒ BC = 120√3 m

In right-angled triangle ABD:

⇒ tan 45° = AB ÷ BD

⇒ 1 = 120 ÷ BD

⇒ BD = 120 m

Total distance between the two cars = BC + BD

⇒ Total distance = 120√3 + 120 = 120(√3 + 1) m

∴ The correct answer is 120(√3 + 1) m.

Additional Information

Heights and Distances on the Same Side

If both objects are on the same side of the tower, the distance between them is given by d = h(cot α − cot β), where α is the smaller angle of elevation.

Angle of Elevation vs Depression

The angle of depression from the top of an object is always equal to the angle of elevation from the bottom due to alternate interior angles formed by parallel horizontal lines.

Standard Trigonometric Values

tan 30° = 1/√3, tan 45° = 1, tan 60° = √3. Cotangent is always the reciprocal of tangent.