Quantitative Aptitude PYQs

Shortcut Trick

Let the pipes be A, B, and C (waste pipe).

Total Capacity (LCM of 100, 25, 40) = 200 units.

Efficiency of A = 2, B = 8, and A+B−C = 5.

Efficiency of C = (2 + 8) − 5 = 5 units/min.

Time taken by pipe C = 200 ÷ 5 = 40 min.

∴ The correct answer is 40 min.

Alternate Method

Calculations:

⇒ Let the efficiency of the waste pipe (C) be x.

⇒ Total efficiency = Efficiency of A + Efficiency of B − Efficiency of C

⇒ 5 = 2 + 8 − x

⇒ 5 = 10 − x

⇒ x = 10 − 5 = 5 units/min

⇒ Time taken by waste pipe to empty the tank = Total capacity ÷ Efficiency of C

⇒ Time = 200 ÷ 5 = 40 min

∴ The correct answer is 40 min.

Additional Information

Pipes and Cisterns Logic

Inlet pipes have positive efficiency, while outlet (leak/waste) pipes have negative efficiency.

Total Work Formula

Total Work = Efficiency × Time. Using the LCM method is the fastest way to solve multiple pipe problems.

Fractional Method

If a pipe fills a tank in 'x' minutes, its work in 1 minute is 1/x of the total tank volume.

Shortcut Trick

Net compound interest rate for 2 years at 20% = 20 + 20 + (20 × 20) ÷ 100 = 44%.

Total Amount = 100% (Principal) + 44% (Interest) = 144% of Principal.

Amount = 144% × 2400 = 1.44 × 2400 = ₹3456.

∴ The correct answer is ₹3456.

Alternate Method

Given: Principal (P) = ₹2,400, Rate (R) = 20% per annum, Time (n) = 2 years.

Formula Used: Amount (A) = P(1 + R/100)ⁿ

⇒ Amount (A) = 2400 × (1 + 20 ÷ 100)²

⇒ Amount (A) = 2400 × (1 + 1 ÷ 5)²

⇒ Amount (A) = 2400 × (6 ÷ 5) × (6 ÷ 5)

⇒ Amount (A) = 2400 × 36 ÷ 25

⇒ Amount (A) = 96 × 36

⇒ Amount (A) = 3456

∴ The correct answer is ₹3456.

Additional Information

Compound Interest (CI)

It is the interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods.

Difference between CI and SI

For 2 years, the difference between Compound Interest and Simple Interest is given by the formula: D = P × (R ÷ 100)².

Effective Rate for Half-Yearly Compounding

When interest is compounded half-yearly, the annual rate (R) is divided by 2, and the time period (n) is multiplied by 2.

Shortcut Trick

Total Amount = ₹1,287

Total Deductions = 36 + 22 + 29 = ₹87

Remaining Amount = 1,287 − 87 = ₹1,200

Remaining share ratio = 17 : 7 : 16 (Total parts = 40)

Monika's remaining share = (7 ÷ 40) × 1,200 = ₹210

Actual share = 210 + 22 = ₹232

∴ The correct answer is ₹232.

Alternate Method

Given:

Total amount = ₹1,287

Individual deductions: Rahul = ₹36, Monika = ₹22, Yuvraj = ₹29

Ratio after deduction = 17 : 7 : 16

Calculations:

⇒ Let the shares after deduction be 17x, 7x, and 16x.

⇒ (17x + 36) + (7x + 22) + (16x + 29) = 1287

⇒ 40x + 87 = 1287

⇒ 40x = 1287 − 87

⇒ 40x = 1200

⇒ x = 1200 ÷ 40 = 30

⇒ Monika's share after deduction = 7 × 30 = ₹210

⇒ Monika's actual share = 210 + 22 = ₹232

∴ The correct answer is ₹232.

Additional Information

Ratio and Proportion

If a total sum S is divided into a ratio a : b : c, then the share of b is calculated as (b ÷ (a + b + c)) × S.

Deduction Concept

When fixed amounts are deducted from shares to obtain a new ratio, subtract the total deduction from the original sum before applying the ratio parts.

Shortcut Trick

Speed ratio (Train : Car) = 160 : 100 = 8 : 5.

Time ratio (Train : Car) = 5 : 8 (since distance is constant).

Difference in time units = 8 − 5 = 3 units.

Given 3 units = 20 minutes ⇒ 8 units (Car's time) = (20 ÷ 3) × 8 = 160/3 minutes = 8/9 hours.

Speed of car = Distance ÷ Time = 60 ÷ (8/9) = 540 ÷ 8 = 67.5 km/h.

∴ The correct answer is 67.5 km/h.

Alternate Method

Given: Distance = 60 km, Train speed is 60% more than Car, Time delay = 20 mins.

Formula: Time = Distance ÷ Speed

⇒ Let Speed of car = x km/h.

⇒ Speed of train = 1.6x km/h.

⇒ Time taken by car − Time taken by train = 20/60 hours.

⇒ 60/x − 60/1.6x = 1/3

⇒ 60/x × (1 − 1/1.6) = 1/3

⇒ 60/x × (0.6/1.6) = 1/3

⇒ 60/x × (3/8) = 1/3

⇒ 22.5/x = 1/3

⇒ x = 22.5 × 3 = 67.5 km/h.

∴ The correct answer is 67.5 km/h.

Additional Information

Inverse Proportion of Speed and Time

When distance is constant, speed is inversely proportional to time (S₁/S₂ = T₂/T₁).

Relative Speed

In same direction, relative speed = S₁ − S₂. In opposite direction, relative speed = S₁ + S₂.

Average Speed

For constant distance segments, Average Speed = 2S₁S₂ ÷ (S₁ + S₂).

Shortcut Trick

Ratio of Income (Ashish : Kiran) = 5 : 8

Ratio of Expenditure (Ashish : Kiran) = 2 : 4 = 1 : 2

Since Savings (Income − Expenditure) are same (₹75,000), make the ratio difference equal: (5−1) ≠ (8−2). Adjust expenditure ratio by ×3 → 3 : 6. Now (5−3) = (8−6) = 2 units.

2 units = ₹75,000 ⇒ 1 unit = ₹37,500.

Income of Ashish = 5 units = 5 × ₹37,500 = ₹1,87,500.

∴ The correct answer is ₹1,87,500.

Alternate Method

Given: Ratio of incomes = 5 : 8, Ratio of expenditures = 2 : 4, Constant savings = ₹75,000.

Formula Used: Income − Savings = Expenditure

⇒ (5x − 75000) / (8x − 75000) = 2 / 4

⇒ (5x − 75000) / (8x − 75000) = 1 / 2

⇒ 2 × (5x − 75000) = 1 × (8x − 75000)

⇒ 10x − 1,50,000 = 8x − 75,000

⇒ 2x = 1,50,000 − 75,000

⇒ 2x = 75,000

⇒ x = 37,500

⇒ Monthly Income of Ashish = 5x = 5 × 37,500 = ₹1,87,500

∴ The correct answer is ₹1,87,500.

Additional Information

Income, Expenditure, and Savings Relationship

The fundamental equation for these problems is Income = Expenditure + Savings.

Ratio Balancing Method

When the difference (Savings) is constant, you can solve by making the vertical difference between the ratio parts equal for both individuals.

Cross-Multiplication Method

For different savings amounts, use the formula (I₁ × E₂ − I₂ × E₁) units = (S₁ × E₂ − S₂ × E₁).

Shortcut Trick

Identify the base value (A's marks) and the comparing value (B's marks).

Required percentage = (B ÷ A) × 100

Required percentage = (50 ÷ 75) × 100 = 2/3 × 100 = 66.666...

Rounding to two decimal places, we get 66.67%.

∴ The correct answer is 66.67%.

Alternate Method

Given:

Marks obtained by A = 75

Marks obtained by B = 50

Maximum Marks = 90 (Not required for this specific ratio calculation)

Formula Used:

Percentage = (Value / Base Value) × 100

Calculations:

⇒ Required Percentage = (Marks of B / Marks of A) × 100

⇒ Percentage = (50 / 75) × 100

⇒ Percentage = (2 / 3) × 100

⇒ Percentage = 200 / 3

⇒ Percentage = 66.6666...

⇒ Rounded Percentage ≈ 66.67%

∴ The correct answer is 66.67%.

Additional Information

Percentage Change

To find the percentage by which A is more than B: [(A − B) / B] × 100. Always divide by the 'than' value.

Fraction to Percentage Conversion

Common fractions: 1/2 = 50%, 1/3 = 33.33%, 2/3 = 66.67%, 1/4 = 25%, 1/5 = 20%.

Base Value Identification

The value following the word 'of' or 'than' is typically treated as the denominator (base value) in percentage problems.

Shortcut Trick

Area of a regular polygon with n sides inscribed in a circle of radius R is given by: Area = (n/2) × R² × sin(360°/n).

For Hexagon (n = 6): Area₆ = (6/2) × R² × sin(60°) = 3 × R² × (√3/2) = (3√3/2)R².

For Pentagon (n = 5): Area₅ = (5/2) × R² × sin(72°) = (5/2) × R² × (√(10 + 2√5)/4) = (5√(10 + 2√5)/8)R².

Ratio = (3√3/2) : (5√(10 + 2√5)/8) ⇒ Multiply by 8 ⇒ 12√3 : 5√(10 + 2√5).

∴ The correct answer is 12√3 : 5√(10 + 2√5).

Alternate Method

Given:

Radius of the circumcircle = R

Number of sides of first polygon (Hexagon) = 6

Number of sides of second polygon (Pentagon) = 5

Formula Used:

Area of regular n-gon = (n/2) × R² × sin(360°/n)

Calculations:

⇒ Area of Hexagon (A₁) = (6/2) × R² × sin(360°/6)

Shortcut Trick

For two successive discounts a% and b%, use the successive discount formula.

Equivalent Discount = a + b − (a × b) ÷ 100

Equivalent Discount = 12 + 13 − (12 × 13) ÷ 100

Equivalent Discount = 25 − 156 ÷ 100 = 25 − 1.56

Equivalent Discount = 23.44%

∴ The correct answer is 23.44%.

Alternate Method

Given: First discount (d₁) = 12%, Second discount (d₂) = 13%.

Formulas: Final SP = MP × (100 − d₁)% × (100 − d₂)% | Equivalent Discount% = (MP − SP) ÷ MP × 100

⇒ Let the Marked Price (MP) be 100.

⇒ Price after 1^st discount = 100 × (100 − 12) ÷ 100 = 88.

⇒ Price after 2^nd discount (Final SP) = 88 × (100 − 13) ÷ 100 = 88 × 0.87 = 76.56.

⇒ Single Equivalent Discount = MP − Final SP = 100 − 76.56 = 23.44%.

∴ The correct answer is 23.44%.

Additional Information

Successive Discount Concept

When a series of discounts are applied on the Marked Price, each subsequent discount is calculated on the reduced value obtained after the previous discount.

Net Discount for 3 values

For three successive discounts a, b, and c, calculate the equivalent of 'a' and 'b' first, then use that result with 'c'.

Shortcut Trick

Total Expenditure = 1120 + 910 = ₹2030

Total Books purchased = 52 + 47 = 99

Average Price = Total Cost ÷ Total Books = 2030 ÷ 99

Average Price ≈ ₹20.51

∴ The correct answer is ₹20.51.

Alternate Method

Given:

Number of books from Shop 1 = 52 | Cost = ₹1,120

Number of books from Shop 2 = 47 | Cost = ₹910

Formula Used:

Average = (Total Sum of Observations) ÷ (Total Number of Observations)

Calculations:

⇒ Total Cost Price = 1120 + 910 = ₹2030

⇒ Total Quantity (Books) = 52 + 47 = 99

⇒ Average price per book = 2030 ÷ 99

⇒ Average price per book = 20.50505...

⇒ Rounding off to two decimal places, we get ₹20.51

∴ The correct answer is ₹20.51.

Additional Information

Weighted Average

When items have different frequencies or counts, the average is calculated as (n₁x₁ + n₂x₂) ÷ (n₁ + n₂).

Rounding Off Rules

If the third decimal digit is 5 or more, the second decimal digit is increased by 1 (e.g., 20.505 → 20.51).

Arithmetic Mean

It represents the central value of a data set, calculated by dividing the sum of all values by the total count of values.

Shortcut Trick

Directly calculate the Total Surface Area (TSA) ratio.

TSA of one small cube = 6 × 4² = 6 × 16 = 96 cm².

TSA of original cuboid = 2 × (18×12 + 12×8 + 18×8) = 2 × (216 + 96 + 144) = 912 cm².

Required Ratio = 96 : 912 = 1 : 9.5 = 2 : 19.

∴ The correct answer is 2 : 19.

Alternate Method

Given:

Dimensions of cuboid: Length (L) = 18 cm, Width (B) = 12 cm, Height (H) = 8 cm.

Edge of small cube (a) = 4 cm.

Formula Used:

TSA of cuboid = 2(LB + BH + LH)

TSA of cube = 6a²

Calculations:

⇒ Total Surface Area of the cuboid = 2 × (18 × 12 + 12 × 8 + 8 × 18)

⇒ TSA of cuboid = 2 × (216 + 96 + 144) = 2 × (456) = 912 cm²

⇒ Total Surface Area of one small cube = 6 × (4)²

⇒ TSA of cube = 6 × 16 = 96 cm²

⇒ Required Ratio = TSA of cube : TSA of cuboid

⇒ Ratio = 96 : 912

⇒ Dividing both by 48: 96 ÷ 48 = 2 and 912 ÷ 48 = 19

⇒ Ratio = 2 : 19

∴ The correct answer is 2 : 19.

Additional Information

Volume of Solids

The volume of a cuboid is L × B × H and a cube is a³. When melting and recasting, the total volume remains constant: n × a³ = L × B × H.

Lateral Surface Area (LSA)

LSA of a cuboid = 2H(L + B), and LSA of a cube = 4a². LSA only considers the area of the four side walls.

Diagonal of Solids

Diagonal of a cuboid = √(L² + B² + H²), and diagonal of a cube = a√3.

Shortcut Trick

Side (a) = 20 cm, Half-diagonal (d₁/2) = 12 cm.

Using Pythagoras triplet (12, 16, 20), the other half-diagonal (d₂/2) = 16 cm.

Full diagonal d₂ = 16 × 2 = 32 cm.

Area = (1/2) × d₁ × d₂ = (1/2) × 24 × 32 = 384 cm².

∴ The correct answer is 384 cm².

Alternate Method

Given:

Side of rhombus (a) = 20 cm

Diagonal 1 (d₁) = 24 cm

Formula Used:

Area of rhombus = (1/2) × d₁ × d₂

Relation between side and diagonals: 4a² = d₁² + d₂²

Side = 20 d1 = 24 d2 = ?  12 cm 16 cm

Calculations:

⇒ In a rhombus, diagonals bisect each other at 90°.

⇒ Let half of d₂ be x.

⇒ Using Pythagoras theorem in the small right triangle: (d₁/2)² + x² = a²

⇒ (24/2)² + x² = 20²

⇒ 12² + x² = 400

⇒ 144 + x² = 400

⇒ x² = 400 − 144 = 256

⇒ x = √256 = 16 cm

⇒ Second diagonal d₂ = 2 × x = 2 × 16 = 32 cm

⇒ Area of rhombus = (1/2) × 24 × 32

⇒ Area = 12 × 32 = 384 cm²

∴ The correct answer is 384 cm².

Additional Information

Properties of Rhombus

All sides are equal, opposite angles are equal, and diagonals bisect each other at right angles (90°).

Perimeter of Rhombus

Perimeter = 4 × side = 4a. Also, 4a² = d₁² + d₂².

Area using Trigonometry

Area = a² × sin(θ), where θ is any internal angle of the rhombus.

Shortcut Trick

For a Mean Proportional in the form x : A :: A : y, the value is calculated as A = √(x × y).

Here, x = 54 and y = 150.

A = √(54 × 150) = √(8100) = 90.

∴ The correct answer is 90.

Alternate Method

Given: The continued proportion 54 : A :: A : 150.

Formula Used: Product of Means = Product of Extremes.

⇒ 54/A = A/150

⇒ A² = 54 × 150

⇒ A² = 8100

⇒ A = √8100

⇒ A = 90

∴ The correct answer is 90.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is √(a × b). It is the middle term in a continued proportion.

Third Proportional

If a : b :: b : c, then c is called the third proportional to a and b, where c = b² / a.

Fourth Proportional

In the ratio a : b :: c : d, d is the fourth proportional and is calculated as d = (b × c) / a.

Shortcut Trick

Initial Price = 100

Final Price = 100 × 0.65 × 0.60 × 0.50 × 1.45

Final Price = 28.275

Change % = 100 − 28.275 = 71.725% ≈ 71.73%

∴ The correct answer is Decreases by 71.73%.

Alternate Method

Given: Successive changes are −35%, −40%, −50%, and +45%.

Formula Used: Final Value = Initial Value × (100 ± r₁)/100 × (100 ± r₂)/100 × ...

⇒ Let original price be 100.

⇒ Price after 1^st month = 100 − 35% of 100 = 65.

⇒ Price after 2^nd month = 65 − 40% of 65 = 65 − 26 = 39.

⇒ Price after 3^rd month = 39 − 50% of 39 = 39 − 19.5 = 19.5.

⇒ Price after 4^th month = 19.5 + 45% of 19.5 = 19.5 + 8.775 = 28.275.

⇒ Total decrease = 100 − 28.275 = 71.725%.

⇒ Rounding off to two decimal places, we get 71.73%.

∴ The correct answer is Decreases by 71.73%.

Additional Information

Successive Percentage Formula

For two changes a% and b%, effective change is (a + b + ab/100)%. For decrease, use negative values.

Multiplier Method

A decrease of x% is equivalent to multiplying by (1 − x/100), and an increase is multiplying by (1 + x/100).

Rounding Off Rule

If the third decimal digit is 5 or more, increase the second decimal digit by 1 (e.g., 71.725 becomes 71.73).

Shortcut Trick

Ratio of Income (Vijay : Mohan) = 5 : 9

Ratio of Expenditure (Vijay : Mohan) = 2 : 4 = 1 : 2

Since Savings are equal (₹42,000), make the ratio difference equal: (5−1) ≠ (9−2).

Adjust Expenditure ratio by multiplying with (5−9) difference → 1:2 × 4 = 4 : 8.

Income 5 : 9 and Exp 4 : 8 → Savings = 1 unit = ₹42,000.

Vijay's Income = 5 units = 5 × 42,000 = ₹2,10,000.

∴ The correct answer is ₹2,10,000.

Alternate Method

Given: Ratio of incomes = 5 : 9, Each saving = ₹42,000, Ratio of expenditures = 2 : 4.

Formula: Income − Saving = Expenditure

⇒ (5x − 42,000) ÷ (9x − 42,000) = 2 ÷ 4

⇒ (5x − 42,000) ÷ (9x − 42,000) = 1 ÷ 2

⇒ 2 × (5x − 42,000) = 1 × (9x − 42,000)

⇒ 10x − 84,000 = 9x − 42,000

⇒ 10x − 9x = 84,000 − 42,000

⇒ x = 42,000

⇒ Vijay's Income = 5x = 5 × 42,000 = ₹2,10,000

∴ The correct answer is ₹2,10,000.

Additional Information

Income, Expenditure, and Saving Relationship

The fundamental equation is Income = Expenditure + Savings. For two people with fixed savings, the change in income units must correspond to the change in expenditure units.

Ratio Balancing Method

If Savings are equal, the numerical difference between the income ratio parts and expenditure ratio parts must be the same. If not, multiply the ratios by the cross-difference of the other ratio.

Cross Multiplication Trick

For problems with different savings: (I₁ × E₂ − I₂ × E₁) units = (S₁ × E₂ − S₂ × E₁) value.

Shortcut Trick

Relative Speed in same direction = 60 − 24 = 36 km/h.

Convert to m/s: 36 × (5/18) = 10 m/s.

Combined Length = Relative Speed × Time = 10 × 30 = 300 m.

∴ The correct answer is 300 m.

Alternate Method

Given:

Speed of Train 1 (S₁) = 60 km/h

Speed of Train 2 (S₂) = 24 km/h

Time taken to cross (t) = 30 sec

Formula Used:

Distance = Speed × Time

Relative Speed (Same Direction) = S₁ − S₂

Calculations:

⇒ Relative Speed = 60 − 24 = 36 km/h

⇒ Convert km/h to m/s: 36 × (5/18) m/s

⇒ Relative Speed = 2 × 5 = 10 m/s

⇒ Combined Length = Relative Speed × Time

⇒ Combined Length = 10 × 30

⇒ Combined Length = 300 m

∴ The correct answer is 300 m.

Additional Information

Relative Speed Concept

When two objects move in the same direction, relative speed is S₁ − S₂. When they move in opposite directions, it is S₁ + S₂.

Unit Conversion

To convert km/h to m/s, multiply by 5/18. To convert m/s to km/h, multiply by 18/5.

Crossing Objects

When a train crosses another train, the total distance covered is always the sum of the lengths of both trains.

Shortcut Trick

In a right-angled triangle with an angle of elevation of 45°, the height and the base are always equal.

Since the distance from the base (base) is 20 m, the height of the pole must also be 20 m.

Ratio of sides in 45°-45°-90° triangle is 1 : 1 : √2.

∴ The correct answer is 20 m.

Alternate Method

Given:

Distance from the base (BC) = 20 m

Angle of elevation (θ) = 45°

Formula Used:

tan θ = Perpendicular (Height) ÷ Base

Calculations:

⇒ tan 45° = Height (h) ÷ 20

⇒ 1 = h ÷ 20

⇒ h = 20 × 1

⇒ h = 20 m

∴ The correct answer is 20 m.

Additional Information

Trigonometric Ratios for Standard Angles

For 30°, tan 30° = 1/√3; For 60°, tan 60° = √3; For 45°, tan 45° = 1.

Angle of Elevation and Depression

The angle of elevation is the angle between the horizontal line and the line of sight when looking up at an object.

Right-Angled Triangle Properties

Pythagoras Theorem: Hypotenuse² = Base² + Perpendicular².

Shortcut Trick

Effective Rate of Interest for 2 years at 10% compounded annually = 10 + 10 + (10 × 10) ÷ 100 = 21%.

The Amount is 100% (Principal) + 21% (Interest) = 121% of the Principal.

Amount = 121% × 7,100 = 1.21 × 7,100.

Amount = 121 × 71 = ₹8,591.

∴ The correct answer is ₹8,591.

Alternate Method

Given:

Principal (P) = ₹7,100

Rate (R) = 10% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P × (1 + R/100)ⁿ

Calculations:

⇒ A = 7100 × (1 + 10/100)²

⇒ A = 7100 × (1 + 1/10)²

⇒ A = 7100 × (11/10) × (11/10)

⇒ A = 71 × 11 × 11

⇒ A = 71 × 121

⇒ A = ₹8,591

∴ The correct answer is ₹8,591.

Additional Information

Compound Interest Formula

Interest (CI) = Amount − Principal = P[(1 + R/100)ⁿ − 1].

Difference between CI and SI

For 2 years, the difference D = P(R/100)².

Successive Percentage Method

If interest is compounded annually, the effective rate for 't' years is the successive percentage change of 'R' for 't' times.

Shortcut Trick

Net discount = a + b − (a × b) ÷ 100 = 16 + 20 − (16 × 20) ÷ 100 = 32.8%.

Multiplier for SP = CP × (100 + P%) ÷ 100 = 7700 × 1.32.

Relationship: CP × (100 + P%) = MP × (100 − D%).

MP = 7700 × 132 ÷ (100 − 32.8) = 7700 × 132 ÷ 67.2.

MP = 7700 × 1320 ÷ 672 = 15125.

∴ The correct answer is ₹15,125.

Alternate Method

Given → Cost Price (CP) = ₹7,700, Gain% = 32%, Successive Discounts = 16% and 20%.
Formulas: SP = CP × (100 + P%) ÷ 100 | MP = SP × 100 ÷ (100 − D%) | Net Discount = a + b − ab ÷ 100.

⇒ Calculate Selling Price (SP): SP = 7700 × (132 ÷ 100) = 77 × 132 = ₹10,164.
⇒ Let Marked Price be MP. After 16% discount, Price = 0.84MP.
⇒ After second discount of 20%, SP = 0.84MP × 0.80 = 0.672MP.
⇒ Equating SP: 0.672MP = 10164.
⇒ MP = 10164 ÷ 0.672 = 10164000 ÷ 672.
⇒ MP = ₹15,125.
∴ The correct answer is ₹15,125.

Additional Information

Marked Price and Discount

Discount is always calculated on the Marked Price (MP). SP = MP − Discount.

Successive Discounts

Two discounts d₁ and d₂ are equivalent to a single discount of (d₁ + d₂ − d₁d₂/100)%.

Shortcut Trick

Convert the given equation x² − 3x − 1 = 0 into the form (x − 1/x) by dividing by x.

x − 3 − 1/x = 0 ⇒ x − 1/x = 3.

Use the algebraic shortcut: If (x − 1/x) = k, then (x² + 1/x²) = k² + 2.

Here, k = 3, so x² + 1/x² = 3² + 2 = 11.

Required value = (x² + 1/x²) − 6 = 11 − 6 = 5.

∴ The correct answer is 5.

Alternate Method

Given: x² − 3x − 1 = 0

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

Calculations:

⇒ x² − 1 = 3x

⇒ (x² − 1) ÷ x = 3

⇒ x − 1/x = 3

Squaring both sides of the equation:

⇒ (x − 1/x)² = 3²

⇒ x² + 1/x² − 2(x)(1/x) = 9

⇒ x² + 1/x² − 2 = 9

⇒ x² + 1/x² = 11

Now, find the value of the expression x² + 1/x² − 6:

⇒ (x² + 1/x²) − 6 = 11 − 6

⇒ 5

∴ The correct answer is 5.

Additional Information

Square Algebraic Identities

x² + 1/x² can be expressed as (x + 1/x)² − 2 or (x − 1/x)² + 2.

Cube Algebraic Identities

x³ + 1/x³ = (x + 1/x)³ − 3(x + 1/x) and x³ − 1/x³ = (x − 1/x)³ + 3(x − 1/x).

Relationship between Sum and Difference

(x + 1/x)² = (x − 1/x)² + 4, which is useful for switching between addition and subtraction forms.

Shortcut Trick

Loss of 10% means the Selling Price (SP) is 90% of the Cost Price (CP).

Ratio method: 10% = 1/10 (Loss/CP).

If CP = 10 units and Loss = 1 unit, then SP = 9 units.

Given 10 units = ₹1,200, so 1 unit = ₹120.

∴ SP = 9 × 120 = ₹1,080.

∴ The correct answer is ₹1,080.

Alternate Method

Given: Cost Price (CP) = ₹1,200, Loss% = 10%.
Formulas: Loss = (Loss% ÷ 100) × CP | SP = CP − Loss

⇒ Calculate Loss amount = (10 ÷ 100) × 1,200
⇒ Loss = 0.10 × 1,200 = ₹120
⇒ Selling Price (SP) = CP − Loss
⇒ SP = 1,200 − 120 = ₹1,080

∴ The correct answer is ₹1,080.

Additional Information

Relationship between CP and SP

When there is a loss, the Cost Price is always greater than the Selling Price (CP > SP). Formula: SP = CP × (100 − Loss%) ÷ 100.

Profit Scenario

If the watch was sold at a 10% profit, the SP would be CP × (100 + Profit%) ÷ 100 = 1,200 × 1.1 = ₹1,320.