Search Previous Year Questions

Shortcut Trick

The monthly incomes ₹8000, ₹9000, and ₹10000 form an Arithmetic Progression (AP).

For a series in Arithmetic Progression with an odd number of terms, the Average is always the Middle Term.

Here, the middle term is ₹9000.

∴ The correct answer is ₹9000.

Alternate Method

Given:

Income in January = ₹8000

Income in February = ₹9000

Income in March = ₹10000

Formula Used:

Average Income = (Sum of incomes) ÷ (Number of months)

Jan: ₹8000

Feb: ₹9000

Mar: ₹10000

Balance Point (Average)

Calculations:

⇒ Total Income = 8000 + 9000 + 10000

⇒ Total Income = ₹27000

⇒ Number of months = 3

⇒ Average Monthly Income = 27000 ÷ 3

⇒ Average Monthly Income = ₹9000

∴ The correct answer is ₹9000.

Additional Information

Average in Arithmetic Progression

For any series where the difference between consecutive terms is constant, Average = (First term + Last term) ÷ 2.

Sum of Observations

Total Sum = Average × Number of observations. This is useful for finding missing values in a data set.

Effect of Change on Average

If every observation is increased or decreased by a constant value 'k', the new average also increases or decreases by 'k'.

Shortcut Trick

Path Area (inside) = π × w × (2R − w), where R is outer radius and w is width.

⇒ Area = (22/7) × 7 × (2 × 28 − 7) = 22 × (56 − 7) = 22 × 49 = 1078 m².

Inner Circumference = 2 × π × (R − w) = 2 × (22/7) × (28 − 7) = 132 m.

Total Cost = 132 × 56.78 = ₹7494.96.

∴ The correct answer is 1078 m² and ₹7494.96.

Alternate Method

Given:

Outer Radius (R) = 28 m

Path Width (w) = 7 m

Fencing Rate = ₹56.78 per m

Formula Used:

Inner Radius (r) = Outer Radius − Path Width

Area of Path = πR² − πr² = π(R² − r²)

Inner Circumference = 2πr

Calculations:

⇒ Inner Radius (r) = 28 − 7 = 21 m

⇒ Area of Path = (22/7) × (28² − 21²)

⇒ Area of Path = (22/7) × (784 − 441) = (22/7) × 343

⇒ Area of Path = 22 × 49 = 1078 m²

⇒ Length of Inner Edge = 2 × (22/7) × 21

⇒ Length of Inner Edge = 2 × 22 × 3 = 132 m

⇒ Total Fencing Cost = 132 × 56.78

⇒ Total Fencing Cost = ₹7494.96

∴ The correct answer is 1078 m² and ₹7494.96.

Additional Information

Area of Circular Ring (Path)

The area between two concentric circles is given by A = π(R + r)(R − r), where R is the outer radius and r is the inner radius.

Fencing Problems

Fencing is always done along the perimeter or circumference. Cost = Rate × Perimeter.

Circumference of Circle

The distance around the boundary of a circle is 2πr or πd, where r is the radius and d is the diameter.

Shortcut Trick

For a cylinder and a cone having the same radius and height, the volume of the cylinder is always exactly three times the volume of the cone.

Number of cones = Volume of Cylinder ÷ Volume of Cone = 3.

∴ The correct answer is 3.

Alternate Method

Given:

Diameter of cylinder = 10 cm ⇒ Radius (r) = 10 ÷ 2 = 5 cm

Height (h) of cylinder and cone = 21 cm

Radius of cone = Radius of cylinder = 5 cm

Formula Used:

Volume of Cylinder = πr²h

Volume of Cone = (1/3) × πr²h

Calculations:

⇒ Volume of water in cylinder = π × 5² × 21

⇒ Volume of water in cylinder = 525π cm³

⇒ Volume of one cone = (1/3) × π × 5² × 21

⇒ Volume of one cone = 175π cm³

⇒ Number of cones = Volume of Cylinder ÷ Volume of Cone

⇒ Number of cones = 525π ÷ 175π

⇒ Number of cones = 3

∴ The correct answer is 3.

Additional Information

Volume of Sphere

The volume of a sphere with radius r is given by the formula V = (4/3)πr³.

Cylinder and Cone Relationship

If a cylinder and a cone share the same base and height, the cylinder contains three times the volume of the cone.

Surface Area of Cylinder

The total surface area of a closed cylinder is 2πr(r + h), where r is the radius and h is the height.

Slant Height of Cone

The slant height (l) of a cone is calculated using the Pythagorean theorem: l = √(r² + h²).

Shortcut Trick

The volume of a pyramid is exactly 1/3 of the volume of the cube when they share the same base and height.

Base Area of Cube = 12 × 12 = 144 cm².

Height of Pyramid = Edge of Cube = 12 cm.

Volume = (1/3) × 144 × 12 = 144 × 4 = 576 cm³.

∴ The correct answer is 576 cm³.

Alternate Method

Given:

⇒ Edge of the cube (a) = 12 cm.

⇒ Base of the pyramid is a square of side 12 cm.

⇒ Height of the pyramid (h) = 12 cm (as the tip is at the center of the top face).

Formula Used:

⇒ Volume of a Pyramid = (1/3) × Base Area × Height

Calculations:

⇒ Base Area of the pyramid = 12 × 12 = 144 cm²

⇒ Vertical Height of the pyramid = 12 cm

⇒ Volume of the pyramid = (1/3) × 144 × 12

⇒ Volume = 144 × (12 ÷ 3)

⇒ Volume = 144 × 4

⇒ Volume = 576 cm³

∴ The correct answer is 576 cm³.

Additional Information

Volume of a Regular Pyramid

The volume of any pyramid is given by V = (1/3) × B × h, where B is the area of the base and h is the perpendicular height from the base to the apex.

Slant Height of a Square Pyramid

The slant height (l) is the distance from the apex to the midpoint of a base edge: l = √(h² + (s/2)²), where s is the side of the square base.

Total Surface Area of a Pyramid

Total Surface Area = Base Area + 1/2 × Perimeter of Base × Slant Height. For a square pyramid: TSA = s² + 2 × s × l.

Shortcut Trick

Convert Time into years: 3 years 4 months = 3 + 4/12 = 3 + 1/3 = 10/3 years.

Total Interest Percentage = Rate × Time = 6% × 10/3 = 20%.

Simple Interest = 20% of ₹4,800 = 4800 ÷ 5 = ₹960.

∴ The correct answer is ₹960.

Alternate Method

Given:

Principal (P) = ₹4,800

Rate (R) = 6% per annum

Time (T) = 3 years 4 months

Formula Used:

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

Calculations:

⇒ Convert 4 months to years: 4 ÷ 12 = 1/3 years.

⇒ Total Time (T) = 3 + 1/3 = 10/3 years.

⇒ SI = (4800 × 6 × 10/3) ÷ 100

⇒ SI = (4800 × 2 × 10) ÷ 100

⇒ SI = 48 × 20

⇒ SI = ₹960

∴ The correct answer is ₹960.

Additional Information

Total Amount Formula

The total Amount (A) at the end of the term is the sum of the Principal and Interest: A = P + SI = P(1 + RT/100).

Time Conversion Rule

Always convert Time into years to match the rate per annum (p.a.). Months are divided by 12, and days are divided by 365.

Difference between SI and CI

For the first year (if compounded annually), Simple Interest and Compound Interest are equal. From the second year, CI > SI because CI includes interest on interest.

Shortcut Trick

Cost Price (CP) = ₹5,000

Desired Profit = 25%

Selling Price (SP) = 125% of CP

SP = 1.25 × 5,000 = ₹6,250

∴ The correct answer is ₹6250.

Alternate Method

Given:

Cost Price (CP) of the watch = ₹5,000

Profit required = 25% of CP

Discount offered = 20% on Marked Price (MP)

Formula Used:

Selling Price (SP) = CP × (100 + Profit%) ÷ 100

⇒ Profit amount = 25% of ₹5,000

⇒ Profit = (25 ÷ 100) × 5,000

⇒ Profit = 1,250

⇒ Selling Price (SP) = Cost Price + Profit

⇒ SP = 5,000 + 1,250 = ₹6,250

Note: Although a 20% discount is mentioned, the question asks for the price Rajesh sells it for (SP) to achieve a specific profit on CP.

∴ The correct answer is ₹6250.

Additional Information

Marked Price Calculation

If the Selling Price and Discount% are known, the Marked Price (MP) is given by: MP = SP × 100 ÷ (100 − Discount%). Here, MP = 6,250 × 100 ÷ 80 = ₹7,812.5.

Relationship between CP and MP

The ratio of MP to CP can be expressed as: MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%).

Profit and Loss on SP

Standard profit is calculated on CP. If profit is asked as a percentage of SP, the formula is (Profit ÷ SP) × 100.

Shortcut Trick

In replacement problems, focus on the component that is not being added (Blue paint).

Initial quantity of Blue paint = 5 parts; Final quantity needed = 4 parts (50% of 8).

Fraction of mixture removed = (Change in Blue) ÷ (Initial Blue) = (5 − 4) ÷ 5 = 1/5.

Total quantity to remove and replace = 1/5 × 8 = 1.6 L.

∴ The correct answer is 1.6 L.

Alternate Method

Given: Initial ratio of White paint to Blue paint = 3 : 5. Total volume = 3 + 5 = 8 parts (assumed as 8 L).

Requirement: Final mixture must be 50% White and 50% Blue (Ratio 1 : 1 or 4 L : 4 L).

Let x be the quantity of mixture removed and replaced with White paint.

⇒ Quantity of Blue paint remaining after removal = 5 − (5/8) × x

⇒ In the final mixture, Blue paint must be 4 L (50% of 8 L).

⇒ 5 − 5x/8 = 4

⇒ 5 − 4 = 5x/8

⇒ 1 = 5x/8

⇒ 5x = 8

⇒ x = 8 ÷ 5 = 1.6 L

∴ The correct answer is 1.6 L.

Additional Information

Replacement Formula

When a quantity x is removed from a total volume V and replaced with one component, the remaining quantity of the other component is Q_final = Q_initial × (1 − x/V).

Rule of Alligation

It is used to find the ratio in which two ingredients at given prices/concentrations must be mixed to produce a mixture of a desired mean price/concentration.

Successive Replacement

If the replacement process is repeated n times, the final concentration of the original liquid is C_n = C₀ × (1 − x/V)ⁿ.

Shortcut Trick

Since profits are shared equally, the product of Investment × Time must be the same for both.

Daren: 12,000 × 12 = 1,44,000

Elly: 9,000 × t = 1,44,000

t = 1,44,000 ÷ 9,000 = 16 months.

∴ The correct answer is 16 months.

Alternate Method

Given:

Investment of Daren (C₁) = ₹12,000

Time period for Daren (T₁) = 12 months

Investment of Elly (C₂) = ₹9,000

Profit ratio = 1 : 1 (shared equally)

Formula Used:

Profit Ratio = (C₁ × T₁) : (C₂ × T₂)

Calculations:

⇒ 1 / 1 = (12,000 × 12) / (9,000 × T₂)

⇒ 1 × 9,000 × T₂ = 12,000 × 12

⇒ 9,000 × T₂ = 1,44,000

⇒ T₂ = 1,44,000 ÷ 9,000

⇒ T₂ = 144 ÷ 9

⇒ T₂ = 16 months

∴ The correct answer is 16 months.

Additional Information

Compound Partnership

When partners invest different amounts of capital for different periods of time, the profit is distributed in the ratio of the products of their capital and time.

Ratio of Capitals

If profits and time periods are known, the ratio of capitals is given by C₁ : C₂ = (P₁/T₁) : (P₂/T₂).

Ratio of Time Periods

If profits and capitals are known, the ratio of time periods is given by T₁ : T₂ = (P₁/C₁) : (P₂/C₂).

Shortcut Trick

Discount = 25% = 1/4 of the Original Price.

Therefore, Sale Price = (1 − 1/4) = 3/4 of the Marked Price.

Sale Price = 6000 × 3/4 = 4500.

∴ The correct answer is ₹4500.

Alternate Method

Given:

Marked Price (MP) of Smartwatch = ₹6,000

Discount Rate = 25%

Formula Used:

Sale Price (SP) = Marked Price × [ (100 − Discount%) ÷ 100 ]

Calculations:

⇒ Sale Price = 6000 × [ (100 − 25) ÷ 100 ]

⇒ Sale Price = 6000 × (75 ÷ 100)

⇒ Sale Price = 6000 × 0.75

⇒ Sale Price = 4500

∴ The correct answer is ₹4500.

Additional Information

Marked Price and Selling Price

Marked Price (MP) is the price listed on the tag, while Selling Price (SP) is the actual amount paid after deducting the Discount.

Discount Percentage Formula

Discount% = (Discount Amount ÷ Marked Price) × 100, where Discount Amount = MP − SP.

Successive Discounts

If two discounts d₁% and d₂% are given, the equivalent single discount is [ d₁ + d₂ − (d₁ × d₂ ÷ 100) ] %.

Shortcut Trick

In Trigonometry, the Tangent of an angle represents the ratio of the Opposite side to the Adjacent side.

When tan(θ) = 1, the Opposite side must be equal to the Adjacent side (Ratio 1:1).

In a Right-Angled Triangle, if two sides are equal, the triangle is an Isosceles Right Triangle, where the acute angles are always 45°.

∴ The correct answer is 45°.

Alternate Method

Given: tan(θ) = 1

Method: Use the Standard Trigonometric Ratio Table or Right Triangle Geometry.

⇒ tan(θ) = 1

⇒ From the trigonometric table, we know that tan(45°) = 1.

⇒ Comparing both sides: θ = 45°

⇒ Also, sin(45°) = 1/√2 and cos(45°) = 1/√2

⇒ tan(45°) = sin(45°) / cos(45°) = (1/√2) / (1/√2) = 1

∴ The correct answer is 45°.

Additional Information

Standard Values of Tangent

Common values are: tan(0°) = 0, tan(30°) = 1/√3, tan(45°) = 1, tan(60°) = √3, and tan(90°) is Undefined.

Trigonometric Identity

The relation between tan, sin, and cos is given by the formula: tan(θ) = sin(θ) ÷ cos(θ).

Reciprocal Ratio

The reciprocal of tan(θ) is cot(θ). Since tan(45°) = 1, it follows that cot(45°) = 1 ÷ 1 = 1.