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Shortcut Trick

When three equal distances are covered at speeds x, y, and z, the Average Speed is given by the harmonic mean formula: 3xyz ÷ (xy + yz + zx).

Given speeds are 30 km/h, 60 km/h, and 90 km/h.

⇒ Average Speed = (3 × 30 × 60 × 90) ÷ (30×60 + 60×90 + 90×30)

⇒ Average Speed = 486000 ÷ (1800 + 5400 + 2700) = 486000 ÷ 9900 ≈ 49.09 km/h.

∴ The correct answer is 49 km/h.

Alternate Method

Given:

Speed 1 (v₁) = 30 km/h

Speed 2 (v₂) = 60 km/h

Speed 3 (v₃) = 90 km/h

Distance parts are equal (each is one-third of the total).

Formula Used:

Average Speed = Total Distance Covered ÷ Total Time Taken

Assume the distance of each part is the LCM of speeds 30, 60, and 90, which is 180 km.

⇒ Total Distance = 180 + 180 + 180 = 540 km

⇒ Time for 1^st part (t₁) = 180 ÷ 30 = 6 hours

⇒ Time for 2^nd part (t₂) = 180 ÷ 60 = 3 hours

⇒ Time for 3^rd part (t₃) = 180 ÷ 90 = 2 hours

⇒ Total Time = 6 + 3 + 2 = 11 hours

⇒ Average Speed = 540 ÷ 11 ≈ 49.09 km/h

∴ The correct answer is 49 km/h.

Additional Information

Average Speed for Two Equal Distances

When a journey is divided into two equal halves with speeds x and y, Average Speed = 2xy ÷ (x + y).

Average Speed for Equal Time Intervals

If a body travels at speed v₁ for time t and speed v₂ for the same time t, Average Speed = (v₁ + v₂) ÷ 2.

Unit Conversion

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

Shortcut Trick

Find the third side: 34 − (9 + 13) = 12 cm.

Calculate semi-perimeter (s): 34 ÷ 2 = 17 cm.

Apply Heron's Formula: Area = √[17 × (17−9) × (17−13) × (17−12)]

Area = √[17 × 8 × 4 × 5] = √[17 × 160] = √[16 × 170] = 4√170 cm².

∴ The correct answer is 4√170 cm².

Alternate Method

Given:

Side 1 (a) = 9 cm

Side 2 (b) = 13 cm

Perimeter = 34 cm

Formula Used:

Heron's Formula: Area = √[s(s − a)(s − b)(s − c)]

where s is the semi-perimeter = (a + b + c) ÷ 2.

Calculations:

⇒ Third side (c) = Perimeter − (a + b)

⇒ c = 34 − (9 + 13) = 34 − 22 = 12 cm.

⇒ Semi-perimeter (s) = 34 ÷ 2 = 17 cm.

⇒ Area = √[17 × (17 − 9) × (17 − 13) × (17 − 12)]

⇒ Area = √[17 × 8 × 4 × 5]

⇒ Area = √[17 × 160]

⇒ Area = √[17 × 16 × 10]

⇒ Area = 4√170 cm².

∴ The correct answer is 4√170 cm².

Additional Information

Heron's Formula

Used to find the area of a triangle when all three sides are known: Area = √[s(s − a)(s − b)(s − c)].

Right-angled Triangle Area

Area = 1/2 × base × height. This is a special case when two sides are perpendicular.

Equilateral Triangle Area

When all sides are equal to 'a', Area = (√3 ÷ 4) × a².

Triangle Inequality Theorem

The sum of any two sides of a triangle must always be greater than the third side (a + c).

Shortcut Trick

Total Area = Area of Trapezium + Area of Semicircle

Area = [(Sum of parallel sides ÷ 2) × Height] + [1/2 × πr²]

Area = [(40 ÷ 2) × 10] + [1/2 × 22/7 × 7²] = 200 + 77 = 277 m²

∴ The correct answer is 277 m².

Alternate Method

Given:

⇒ Parallel sides of trapezium: a = 14 m, b = 26 m

⇒ Height of trapezium: h = 10 m

⇒ Diameter of semicircle: d = 14 m (attached to smaller base)

Formula Used:

⇒ Area of Trapezium = 1/2 × (a + b) × h

⇒ Area of Semicircle = 1/2 × πr²

Calculations:

⇒ Area of Trapezium = 1/2 × (14 + 26) × 10

⇒ Area of Trapezium = 1/2 × 40 × 10 = 200 m²

⇒ Radius of Semicircle (r) = 14 ÷ 2 = 7 m

⇒ Area of Semicircle = 1/2 × 22/7 × 7 × 7

⇒ Area of Semicircle = 11 × 7 = 77 m²

⇒ Total Area = 200 + 77 = 277 m²

∴ The correct answer is 277 m².

Additional Information

Area of Quadrilaterals

For a trapezium, the area is the product of the average of the parallel sides and the perpendicular distance (height) between them.

Semicircle Geometry

A semicircle is half of a full circle; its area is 1/2πr² and its perimeter is (πr + 2r) if the diameter is included.

Composite Figures

To find the total area of composite shapes, calculate the area of each individual basic shape and add them together.

Shortcut Trick

For a constant arc length, the radius (r) is inversely proportional to the central angle (θ).

Relation: r₁ × θ₁ = r₂ × θ₂

⇒ r₁ / r₂ = θ₂ / θ₁

⇒ r₁ / r₂ = 100° / 50° = 2 / 1

∴ The correct answer is 2:1.

Alternate Method

Given:

Central angle of first circle (θ₁) = 50°

Central angle of second circle (θ₂) = 100°

Arc length of first circle (l₁) = Arc length of second circle (l₂)

Formula Used:

Arc length (l) = (θ / 360°) × 2πr

Calculations:

⇒ l₁ = (50 / 360) × 2πr₁

⇒ l₂ = (100 / 360) × 2πr₂

Since l₁ = l₂:

⇒ (50 / 360) × 2πr₁ = (100 / 360) × 2πr₂

⇒ 50 × r₁ = 100 × r₂

⇒ r₁ / r₂ = 100 / 50

⇒ r₁ / r₂ = 2 / 1

∴ The correct answer is 2:1.

Additional Information

Radian Measure

The arc length formula in radians is s = rθ, where θ is the angle in radians. To convert degrees to radians, multiply by π/180.

Area of Sector

The area of a sector is given by A = (1/2)r²θ (in radians) or A = (θ/360) × πr² (in degrees).

Inverse Proportion in Circles

When arc length is constant, if the central angle doubles, the radius must be halved to maintain the same length.

Shortcut Trick

For any two similar triangles, the ratio of their areas is always the square of the ratio of their corresponding sides.

Given Side Ratio = 3 : 5

Area Ratio = (3)² : (5)² = 9 : 25

∴ The correct answer is 9:25.

Alternate Method

Given:

Ratio of corresponding sides of two similar triangles (s₁ : s₂) = 3 : 5

Formula Used:

Area(Δ1) / Area(Δ2) = (Side₁ / Side₂)²

Calculations:

⇒ Area₁ / Area₂ = (s₁ / s₂)²

⇒ Area₁ / Area₂ = (3 / 5)²

⇒ Area₁ / Area₂ = 9 / 25

⇒ Area Ratio = 9 : 25

∴ The correct answer is 9:25.

Additional Information

Ratio of Perimeters

For similar triangles, the ratio of their perimeters is exactly equal to the ratio of their corresponding sides (s₁ : s₂).

Ratio of Altitudes and Medians

The ratio of corresponding altitudes, medians, or angle bisectors of two similar triangles is also equal to the ratio of their corresponding sides.

Similarity Theorem

If ΔABC ∼ ΔPQR, then Area(ΔABC) / Area(ΔPQR) = (AB/PQ)² = (BC/QR)² = (AC/PR)².

Shortcut Trick

The radius, tangent, and the distance from the center form a right-angled triangle.

According to the Pythagorean triplet (5, 12, 13):

If Radius = 5 cm and Hypotenuse (OP) = 13 cm, then the Tangent length must be 12 cm.

∴ The correct answer is 12 cm.

Alternate Method

Given:

⇒ Radius of the circle (r) = 5 cm

⇒ Distance from point P to center O (d) = 13 cm

Formula Used:

⇒ Length of Tangent = √(d² − r²)

Calculations:

⇒ In right-angled triangle OAP, ∠OAP = 90°

⇒ By Pythagoras Theorem: OP² = OA² + AP²

⇒ 13² = 5² + AP²

⇒ 169 = 25 + AP²

⇒ AP² = 169 − 25

⇒ AP² = 144

⇒ AP = √144

⇒ AP = 12 cm

∴ The correct answer is 12 cm.

Additional Information

Tangent-Radius Property

A tangent at any point of a circle is perpendicular to the radius through the point of contact (∠OAP = 90°).

Length of Tangents

The lengths of two tangents drawn from an external point to a circle are equal.

Pythagorean Triplets

Common triplets used in exams include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).

Shortcut Trick Apply the Inverse Tangent Addition formula: tan−1(a) + tan−1(b) = tan−1((a + b) / (1 − ab)) when ab < 1. Here, a = 3/4 and b = 5/12. Calculating the product: 3/4 × 5/12 = 15/48 = 5/16, which is < 1. Direct calculation: tan(x + y) = (3/4 + 5/12) / (1 − 5/16) = (14/12) / (11/16) = (7/6) × (16/11) = 56/33. ∴ The correct answer is tan−1(56/33).

Alternate Method Given: x = tan−1(3/4) and y = tan−1(5/12) Formula: tan(x + y) = (tan x + tan y) / (1 − tan x × tan y)

⇒ tan x = 3/4 and tan y = 5/12 ⇒ tan(x + y) = (3/4 + 5/12) / (1 − (3/4 × 5/12)) ⇒ Numerator = (9 + 5) / 12 = 14/12 = 7/6 ⇒ Denominator = 1 − 15/48 = (48 − 15) / 48 = 33/48 = 11/16 ⇒ tan(x + y) = (7/6) ÷ (11/16) ⇒ tan(x + y) = (7/6) × (16/11) = 112 / 66 = 56/33 ⇒ x + y = tan−1(56/33) ∴ The correct answer is tan−1(56/33).

Additional Information Sum for xy > 1 If x > 0, y > 0 and xy > 1, then tan−1x + tan−1y = π + tan−1((x + y) / (1 − xy)). Difference Formula tan−1x − tan−1y = tan−1((x − y) / (1 + xy)) for xy > −1. Complementary Relation tan−1x + cot−1x = π/2 for all real values of x.

Shortcut Trick

Speed in m/s = 54 × (5/18) = 15 m/s.

Total distance = Speed × Time = 15 × 30 = 450 m.

Platform length = Total distance − Train length = 450 − 180 = 270 m.

∴ The correct answer is 270 m.

Alternate Method

Given:

Length of train (L_t) = 180 m

Time taken (t) = 30 sec

Speed of train (v) = 54 km/h

Formula Used:

Total Distance = Speed × Time

Total Distance = Length of train + Length of platform

Calculations:

⇒ First, convert speed from km/h to m/s:

⇒ Speed = 54 × (5/18) = 15 m/s

⇒ Let the length of the platform be L_p

⇒ Total distance = L_t + L_p

⇒ 180 + L_p = 15 × 30

⇒ 180 + L_p = 450

⇒ L_p = 450 − 180

⇒ L_p = 270 m

∴ The correct answer is 270 m.

Additional Information

Unit Conversion

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

Train crossing a point object

When a train crosses a pole or a stationary person, the distance covered is equal to the length of the train.

Train crossing an object with length

When a train crosses a bridge, tunnel, or platform, the total distance covered is the sum of the train's length and the object's length.

Shortcut Trick

Use the direct ratio relation: CP : MP = (100 − Discount%) : (100 + Profit%).

CP : MP = (100 − 10) : (100 + 20) = 90 : 120 = 3 : 4.

Given Marked Price (4 units) = ₹ 15,000 ⇒ 1 unit = ₹ 3,750.

Cost Price (3 units) = 3 × 3,750 = ₹ 11,250.

∴ The correct answer is ₹ 11250.

Alternate Method

Given:

Marked Price (MP) = ₹ 15,000

Discount = 10%

Profit = 20%

Formula Used:

1. Selling Price (SP) = MP × (100 − Discount%)/100
2. Cost Price (CP) = SP × 100 / (100 + Profit%)

Calculations:

⇒ SP = 15,000 × (100 − 10)/100

⇒ SP = 15,000 × 90/100 = ₹ 13,500

⇒ CP = 13,500 × 100 / (100 + 20)

⇒ CP = 13,500 × 100/120

⇒ CP = 13,500 × 5/6

⇒ CP = 2,250 × 5 = ₹ 11,250

∴ The correct answer is ₹ 11250.

Additional Information

Markup Percentage

It is the percentage by which the cost price is increased to reach the marked price: Markup% = [(MP − CP) / CP] × 100.

Successive Discounts

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

Profit and Loss Relation

Profit or loss is always calculated on the Cost Price unless stated otherwise, whereas discount is always calculated on the Marked Price.

Shortcut Trick

Apply the Successive Percentage formula: Net % change = x + y + (xy/100)

Here, x = −10 (decrease) and y = +15 (increase).

Net change = −10 + 15 + [(−10 × 15) / 100] = 5 − 1.5 = 3.5% increase.

Final Population = 10,000 + (3.5% of 10,000) = 10,000 + 350 = 10,350.

∴ The correct answer is 10350 birds.

Alternate Method

Initial Population = 10,000

First Year Change = 10% Decrease

Second Year Change = 15% Increase

⇒ Population after 1^st year = 10,000 − (10% of 10,000)

⇒ Population after 1^st year = 10,000 − 1,000 = 9,000

⇒ Population after 2^nd year = 9,000 + (15% of 9,000)

⇒ Population after 2^nd year = 9,000 + 1,350 = 10,350

∴ The correct answer is 10350 birds.

Additional Information

Successive Percentage Change

When a value changes by x% and then y%, the net change is given by [x + y + (xy/100)]%.

Multiplying Factor Method

Final Value = Initial Value × (1 ± r₁/100) × (1 ± r₂/100). For a 10% decrease, the factor is 0.90, and for a 15% increase, it is 1.15.

Population Growth Formula

If the rate is constant for n years, the final population is P(1 + r/100)ⁿ.