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Shortcut Trick

The ratio of diagonals is 3 : 4, so the ratio of half-diagonals is also 3 : 4.

In a rhombus, half-diagonals and the side form a right-angled triangle.

Using the Pythagorean triplet (3, 4, 5), the side corresponds to ratio 5.

Given side length 5 units → 12.5, so 1 unit → 2.5.

Shorter diagonal ratio is 3 × 2 = 6 units → 6 × 2.5 = 15 units.

∴ The correct answer is 15 units.

Alternate Method

Given:

Ratio of diagonals (d₁ : d₂) = 3 : 4

Length of each side (a) = 12.5 units

Formula Used:

In a rhombus, the relation between side and diagonals is: a² = (d₁/2)² + (d₂/2)²

Calculations:

⇒ Let the diagonals be d₁ = 3x and d₂ = 4x.

⇒ (12.5)² = (3x/2)² + (4x/2)²

⇒ 156.25 = (1.5x)² + (2x)²

⇒ 156.25 = 2.25x² + 4x²

⇒ 156.25 = 6.25x²

⇒ x² = 156.25 ÷ 6.25 = 25

⇒ x = √25 = 5

⇒ Length of shorter diagonal (d₁) = 3 × 5 = 15 units.

∴ The correct answer is 15 units.

Additional Information

Properties of Rhombus Diagonals

The diagonals of a rhombus bisect each other at right angles (90°).

Area of a Rhombus

The area can be calculated using diagonals: Area = 1/2 × d₁ × d₂.

Perimeter of a Rhombus

Since all sides are equal, the perimeter = 4 × side.

Given:

{18 + [24 ÷ (3 of 2)] × 4} − 10

Formula used:

BODMAS rule (Brackets, Of, Division, Multiplication, Addition, Subtraction)

Calculations:

{18 + [24 ÷ (3 of 2)] × 4} − 10

⇒ {18 + [24 ÷ (3 × 2)] × 4} − 10

⇒ {18 + [24 ÷ 6] × 4} − 10

⇒ {18 + 4 × 4} − 10

⇒ {18 + 16} − 10

⇒ 34 − 10

⇒ 24

∴ The correct answer is option (1).

Shortcut Trick

Given: 2 cot A = 4 → cot A = 2.

Using the Trigonometric Identity: cosec² A = 1 + cot² A.

Substituting the value: cosec² A = 1 + (2)² = 1 + 4 = 5.

Expression value: (5 + 1) ÷ (5 − 1) = 6 ÷ 4 = 1.5.

∴ The correct answer is 1.5.

Alternate Method

Given:

2 cot A = 4

⇒ cot A = 4 ÷ 2 = 2

Formula Used:

1. cot A = Base ÷ Perpendicular
2. Pythagoras Theorem: H² = P² + B²
3. cosec A = Hypotenuse ÷ Perpendicular

Calculations:

Let Perpendicular (P) = 1, then Base (B) = 2 (since cot A = B/P = 2/1).

⇒ H² = 1² + 2² = 1 + 4 = 5

⇒ H = √5

⇒ cosec A = H ÷ P = √5 ÷ 1 = √5

⇒ cosec² A = (√5)² = 5

Required Expression = (cosec² A + 1) ÷ (cosec² A − 1)

⇒ (5 + 1) ÷ (5 − 1)

⇒ 6 ÷ 4

⇒ 1.5

∴ The correct answer is 1.5.

Additional Information

Fundamental Trigonometric Identities

There are three main identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Reciprocal Ratios

Cosecant is the reciprocal of sine (cosec θ = 1/sin θ) and cotangent is the reciprocal of tangent (cot θ = 1/tan θ).

Pythagoras Theorem in Trigonometry

For any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (H² = P² + B²).

Shortcut Trick

Ratio of Passed students to Failed students = 11 : 2

Total units in the class = 11 + 2 = 13 units

Given, Failed students (2 units) = 44

⇒ 1 unit = 44 ÷ 2 = 22

Total students = 13 units × 22 = 286

∴ The correct answer is 286 students.

Alternate Method

Given:

Ratio (Passed : Failed) = 11 : 2

Number of Failed students = 44

Formula:

Total Students = Passed Students + Failed Students

⇒ Let the number of passed students be 11x and failed students be 2x.

⇒ According to the question, 2x = 44

⇒ x = 44 ÷ 2 = 22

⇒ Number of passed students = 11 × 22 = 242

⇒ Total number of students = Passed + Failed = 242 + 44 = 286

∴ The correct answer is 286 students.

Additional Information

Ratio and Total Value

If two quantities are in the ratio a:b, the total quantity can be represented as (a + b)k, where k is a constant multiplier.

Unitary Method in Ratios

To find the total, first divide the given numerical value by its corresponding ratio parts to find the value of 1 unit, then multiply by the total ratio parts.

Percentage Conversion

The fail percentage in this class is (Failed parts ÷ Total parts) × 100 = (2 ÷ 13) × 100 ≈ 15.38%.

Shortcut Trick

If the ratio of two numbers is a : b and their HCF is H, then their LCM is given by the formula:

LCM = H × a × b (provided a and b are coprime)

Given a = 3, b = 8, and H = 6.

⇒ LCM = 6 × 3 × 8 = 144.

∴ The correct answer is 144.

Alternate Method

Given:

Ratio of numbers = 3 : 8

Highest Common Factor (HCF) = 6

Formula Used:

Number = Ratio part × HCF

Product of two numbers = HCF × LCM

Calculations:

⇒ Let the two numbers be 3x and 8x.

⇒ The HCF of 3x and 8x is x (since 3 and 8 are coprime).

⇒ Given x = 6.

⇒ Number₁ = 3 × 6 = 18

⇒ Number₂ = 8 × 6 = 48

⇒ Using the product rule: 18 × 48 = 6 × LCM

⇒ LCM = (18 × 48) ÷ 6

⇒ LCM = 3 × 48 = 144

∴ The correct answer is 144.

Additional Information

HCF and LCM Relation

For any two numbers A and B, the product of the numbers is equal to the product of their HCF and LCM: A × B = HCF × LCM.

Co-prime Numbers

Two numbers are called co-prime if their HCF is 1. If the terms of a ratio are co-prime, the HCF of the actual numbers is the common multiplier.

LCM of Ratios

If the ratio of two numbers is x : y and their HCF is H, then their LCM = H × x × y and the numbers are Hx and Hy.

Shortcut Trick

Area of Trapezium = Average of bases × Height

Average of bases = (8 + 12) ÷ 2 = 10 cm

Area = 10 × 5 = 50 cm²

Therefore, the Area of the trapezium is 50 cm².

∴ The correct answer is 50 cm².

Alternate Method

Given:

Length of first base (a) = 8 cm

Length of second base (b) = 12 cm

Height (h) = 5 cm

Formula Used:

Area of Trapezium = 1/2 × (a + b) × h

Calculations:

⇒ Area = 1/2 × (8 + 12) × 5

⇒ Area = 1/2 × 20 × 5

⇒ Area = 10 × 5

⇒ Area = 50 cm²

Thus, the area of the trapezium is 50 cm², which corresponds to Option (2).

∴ The correct answer is 50 cm².

Additional Information

Properties of a Trapezium

A quadrilateral with at least one pair of parallel sides. If the non-parallel sides are equal, it is called an isosceles trapezium.

Median of a Trapezium

The line segment joining the midpoints of the non-parallel sides is the median. Length = (Base₁ + Base₂) ÷ 2.

Area of a Rhombus

Area = 1/2 × d₁ × d₂, where d₁ and d₂ are the lengths of its diagonals.

Given:

Expression: ((19/18) ÷ (19/6)) ÷ ((5/18) × (16/15) + (4/6)) + (6/8) ÷ (24/25) of (25/8)

Formula used:

The BODMAS rule is used to solve this expression, which stands for Brackets, Of, Division, Multiplication, Addition, and Subtraction.

Calculations:

((19/18) ÷ (19/6)) ÷ ((5/18) × (16/15) + (4/6)) + (6/8) ÷ (24/25) of (25/8)

⇒ (19/18 × 6/19) ÷ (8/27 + 4/6) + (6/8) ÷ (24/25 × 25/8)

⇒ (1/3) ÷ (8/27 + 2/3) + (6/8) ÷ 3

⇒ (1/3) ÷ (8/27 + 18/27) + (6/8 × 1/3)

⇒ (1/3) ÷ (26/27) + 1/4

⇒ (1/3 × 27/26) + 1/4

⇒ 9/26 + 1/4

⇒ (18 + 13)/52

⇒ 31/52

∴ The correct answer is option (3).

Shortcut Trick

Profit = 25% = 1/4 ⇒ Ratio of CP : SP = 4 : 5

Given SP = 5 units = ₹600 ⇒ 1 unit = ₹120

Total CP = 4 units = 4 × 120 = ₹480

Original Purchase Price = Total CP − Repair = 480 − 50 = ₹430

∴ The correct answer is ₹430.

Alternate Method

Given: Selling Price (SP) = ₹600, Profit Percentage = 25%, Repair Cost = ₹50.

Formula Used: Total Cost Price (CP) = Selling Price × [100 / (100 + Profit%)]

⇒ Total CP = 600 × [100 / (100 + 25)]

⇒ Total CP = 600 × (100 / 125)

⇒ Total CP = 600 × (4 / 5)

⇒ Total CP = 120 × 4 = ₹480

⇒ Original Purchase Price = Total CP − Repair Expenses

⇒ Original Purchase Price = 480 − 50 = ₹430

∴ The correct answer is ₹430.

Additional Information

Concept of Total Cost Price

The Total Cost Price includes the Original Purchase Price plus all overheads like repair, transportation, and taxes. Profit or Loss is always calculated on this final sum.

Profit Percentage Formula

Profit % = [(SP − Total CP) / Total CP] × 100. This measures the gain relative to the entire investment.

Effective Selling Price

If a profit of P% is made, then SP = Total CP × [(100 + P)/100]. Conversely, Total CP = SP × [100/(100 + P)].

Shortcut Trick

Average Speed = Total Distance ÷ Total Time

Total Distance = 120 + 180 + 100 = 400 km

Total Time = (120 ÷ 40) + (180 ÷ 60) + (100 ÷ 50) = 3 + 3 + 2 = 8 hours

Average Speed = 400 ÷ 8 = 50 km/h

∴ The correct answer is 50 km/h.

Alternate Method

Given:

Distance₁ = 120 km, Speed₁ = 40 km/h

Distance₂ = 180 km, Speed₂ = 60 km/h

Distance₃ = 100 km, Speed₃ = 50 km/h

Formula Used:

Time = Distance ÷ Speed

Average Speed = (Total Distance) ÷ (Total Time)

Calculations:

⇒ Time taken for first part (t₁) = 120 ÷ 40 = 3 hours

⇒ Time taken for second part (t₂) = 180 ÷ 60 = 3 hours

⇒ Time taken for third part (t₃) = 100 ÷ 50 = 2 hours

⇒ Total Time (T) = 3 + 3 + 2 = 8 hours

⇒ Total Distance (D) = 120 + 180 + 100 = 400 km

⇒ Average Speed = 400 ÷ 8 = 50 km/h

∴ The correct answer is 50 km/h.

Additional Information

Average Speed for Equal Distances

If a car travels two equal distances at speeds x and y, the Average Speed is given by the Harmonic Mean: 2xy ÷ (x + y).

Unit Conversion

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

Relative Speed

When two objects move in the same direction, relative speed is the difference of their speeds. In opposite directions, it is the sum of their speeds.

Shortcut Trick

Find the scaling factor between the first two terms: 5.6 ÷ 0.8 = 7.

Apply the same scaling factor to the second term to find the third proportional (n).

n = 5.6 × 7 = 39.2.

∴ The correct answer is 39.2.

Alternate Method

Given: First term (a) = 0.8, Second term (b) = 5.6, Third proportional = n.

Formula Used: If a, b, and n are in continued proportion, then a : b :: b : n, which implies n = b² ÷ a.

⇒ a : b :: b : n

⇒ 0.8 : 5.6 :: 5.6 : n

⇒ 0.8 ÷ 5.6 = 5.6 ÷ n

⇒ n = (5.6 × 5.6) ÷ 0.8

⇒ n = 31.36 ÷ 0.8

⇒ n = 39.2

∴ The correct answer is 39.2.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is given by x = √(a × b). It is the value that satisfies a : x = x : b.

Fourth Proportional

For three numbers a, b, and c, the fourth proportional d satisfies a : b = c : d. It is calculated as d = (b × c) ÷ a.

Continued Proportion

A series of numbers a, b, c, d... are in continued proportion if the ratio between successive terms is constant: a/b = b/c = c/d.