Search Previous Year Questions

Shortcut Trick

The sine function repeats its values every 2π radians.

Express 13π/6 as 2π + π/6.

Since sin(2π + θ) = sin θ, we get sin(π/6).

⇒ sin(30°) = 1/2.

∴ The correct answer is 1/2.

Alternate Method

Given: Angle = 13π/6

Formula Used: sin(n × 2π + θ) = sin θ

⇒ sin(13π/6) = sin((12π + π) ÷ 6)

⇒ sin(13π/6) = sin(2π + π/6)

⇒ Since the angle 13π/6 lies in the first quadrant after one full rotation:

⇒ sin(2π + π/6) = sin(π/6)

⇒ sin(30°) = 1/2

∴ The correct answer is 1/2.

Additional Information

Trigonometric Periodicity

Functions like sin x and cos x have a period of 2π, while tan x and cot x have a period of π.

ASTC Rule (All Silver Tea Cups)

Determines the sign of ratios: 1^st Quadrant (All positive), 2^nd Quadrant (Sine positive), 3^rd Quadrant (Tangent positive), 4^th Quadrant (Cosine positive).

Radian to Degree Conversion

To convert radians to degrees, multiply by 180/π. Thus, π/6 radians = (180/6)° = 30°.

Shortcut Trick

Length of Direct Common Tangent (DCT) = √[d² − (r₂ − r₁)²]

Given: d = 26 cm, r₁ = 6 cm, r₂ = 10 cm

L = √[26² − (10 − 6)²] = √[676 − 4²]

L = √[676 − 16] = √660 = 2√165 cm

∴ The correct answer is 2√165 cm.

Alternate Method

Given:

Radius of first circle (r₁) = 6 cm

Radius of second circle (r₂) = 10 cm

Distance between centers (d) = 26 cm

Formula Used:

The length of the common external tangent (Direct Common Tangent) is calculated using the Pythagoras theorem derived from the geometry of the two circles:

Length (L) = √[d² − (r₂ − r₁)²]

Calculations:

⇒ Difference in radii = r₂ − r₁ = 10 − 6 = 4 cm ⇒ Square of difference = (4)² = 16 cm² ⇒ Square of distance = (26)² = 676 cm² ⇒ L² = 676 − 16 = 660 ⇒ L = √660 cm ⇒ L = √(4 × 165) = 2√165 cm

∴ The correct answer is 2√165 cm.

Additional Information

Transverse Common Tangent (TCT)

The length of the internal common tangent is given by √[d² − (r₁ + r₂)²]. It is always shorter than the external tangent.

Circles Touching Externally

If two circles touch externally, the distance between centers d = r₁ + r₂. In this case, the external tangent length is 2√(r₁ × r₂).

Tangent-Radius Property

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

Shortcut Trick

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

For a regular octagon, the number of sides n = 8.

Area = (8/2) × R² × sin(360°/8) = 4 × R² × sin(45°).

Area = 4 × R² × (1/√2) = (2 × 2 × R²) / √2 = 2√2 R².

∴ The correct answer is 2√2 R².

Alternate Method

Given: A regular octagon is circumscribed by a circle of radius R.

Formula Used: Area of a triangle with two sides 'a', 'b' and included angle 'θ' = (1/2) × a × b × sin(θ).

⇒ A regular octagon can be divided into 8 identical isosceles triangles with vertices at the center.

⇒ The central angle of each triangle = 360° ÷ 8 = 45°.

⇒ Each triangle has two sides equal to the radius R of the circumscribing circle.

⇒ Area of one such triangle = (1/2) × R × R × sin(45°).

⇒ Area of one triangle = (1/2) × R² × (1/√2) = R² / (2√2).

⇒ Total Area of the Octagon = 8 × (Area of one triangle).

⇒ Total Area = 8 × [R² / (2√2)] = 4R² / √2.

⇒ Total Area = (2 × 2 × R²) / √2 = 2√2 R².

∴ The correct answer is 2√2 R².

Additional Information

Interior Angle of a Regular Octagon

Each interior angle of a regular octagon is calculated as [(n−2) × 180°] / n. For n=8, it is (6 × 180°) / 8 = 135°.

Area using Side Length (a)

The area of a regular octagon with side length 'a' is 2(1 + √2)a².

Sum of Exterior Angles

For any regular polygon, the sum of all exterior angles is always 360°, making each exterior angle of an octagon 45°.

Shortcut Trick

Convert model length to meters: 25 cm = 0.25 m.

Apply scale factor (150) and unit rate (₹300) directly:

Total Cost = 0.25 × 150 × 300

Total Cost = 37.5 × 300 = 11250.

∴ The correct answer is ₹11,250.

Alternate Method

Given:

Height Ratio (Scale) = 1 : 150

Model Rod Length = 25 cm

Cost of Steel = ₹300 per meter

Calculations:

⇒ Actual Length = Model Length × 150

⇒ Actual Length = 25 × 150 = 3750 cm

⇒ Convert to meters: 3750 ÷ 100 = 37.5 m

⇒ Total Cost = Actual Length × Rate per meter

⇒ Total Cost = 37.5 × 300

⇒ Total Cost = ₹11,250

∴ The correct answer is ₹11,250.

Additional Information

Linear Scale Factor (k)

It is the ratio of corresponding lengths in two similar figures. Actual Length = k × Model Length.

Area Scale Factor

The ratio of the areas of two similar figures is the square of the linear scale factor: Actual Area = k² × Model Area.

Volume Scale Factor

The ratio of the volumes of two similar solids is the cube of the linear scale factor: Actual Volume = k³ × Model Volume.

Shortcut Trick

Simplify √12 as √(4 × 3) = 2√3.

Substitute this into the expression: (2√3) ÷ (2√3).

Any non-zero term divided by itself equals 1.

∴ The correct answer is 1.

Alternate Method

Given: (2√3) ÷ √12

Method: Radical Simplification and Division

⇒ Expression = (2√3) ÷ √(4 × 3)

⇒ Expression = (2√3) ÷ (√4 × √3)

⇒ Expression = (2√3) ÷ (2 × √3)

⇒ Expression = (2√3) ÷ (2√3)

⇒ Expression = 1

∴ The correct answer is 1.

Additional Information

Properties of Radicals

For any positive real numbers a and b, the product property is √(ab) = √a × √b and the quotient property is √(a÷b) = √a ÷ √b.

Simplifying Surds

A surd is in its simplest form when the radicand (the number under the root) has no perfect square factors other than 1.

Rationalization

If a denominator contains a radical, we multiply both numerator and denominator by a conjugate or the radical itself to remove it.

Shortcut Trick

Two lines are parallel if and only if their slopes (m) are equal.

The equation y = 3x + 5 is in the form y = mx + c, where m = 3.

Compare the slopes of the given options: (1) m = −4, (2) m = 3, (3) m = −3, (4) m = 1.

Only option (2) has the identical slope of 3.

∴ The correct answer is y = 3x − 1.

Alternate Method

Given: Equation of the line L₁ is y = 3x + 5.

Formula Used: Slope-intercept form: y = mx + c, where m is the slope and c is the y-intercept.

⇒ For line L₁: y = 3x + 5, comparing with y = mx + c, we get m₁ = 3.

⇒ The condition for two lines to be parallel is m₁ = m₂.

⇒ Checking Option (2): y = 3x − 1.

⇒ Comparing with y = mx + c, we get m₂ = 3.

⇒ Since m₁ = m₂, the lines are parallel.

∴ The correct answer is y = 3x − 1.

Additional Information

Perpendicular Lines

Two lines are perpendicular if the product of their slopes is −1 (m₁ × m₂ = −1).

General Form Slope

For a line in the general form Ax + By + C = 0, the slope m is calculated as −A/B.

Slope Formula

The slope (m) between two points (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁) ÷ (x₂ − x₁).

Shortcut Trick

Diagonal (d) of a regular pentagon is related to its side (s) by the Golden Ratio (φ).

Formula: d = s × φ

Where φ ≈ 1.618

⇒ d = 6 × 1.618 = 9.708 cm

Rounding to two decimal places, we get 9.71 cm.

∴ The correct answer is 9.71 cm.

Alternate Method

Given: Side of regular pentagon (s) = 6 cm.

Formula Used: In a regular pentagon, the interior angle is 108°. Using the Law of Cosines in the triangle formed by two sides and a diagonal: d² = s² + s² − 2 × s × s × cos(108°).

⇒ Each interior angle of a regular pentagon = (5 − 2) × 180° ÷ 5 = 108°

⇒ In ▵ABC, AB = 6 cm, BC = 6 cm and ∠ABC = 108°

⇒ AC² = 6² + 6² − 2 × 6 × 6 × cos(108°)

⇒ AC² = 36 + 36 − 72 × (−0.3090)

⇒ AC² = 72 + 22.248 = 94.248

⇒ AC = √94.248

⇒ AC ≈ 9.708 cm

∴ The correct answer is 9.71 cm.

Additional Information

Interior Angle of Regular Polygon

The measure of each interior angle is given by [(n − 2) × 180°] ÷ n, where n is the number of sides.

Golden Ratio (φ) in Pentagons

In any regular pentagon, the ratio of the diagonal to the side is exactly φ = (1 + √5) ÷ 2 ≈ 1.618033.

Number of Diagonals

The total number of diagonals in a polygon with n sides is n(n − 3) ÷ 2. For a pentagon, this is 5(5 − 3) ÷ 2 = 5.

Area of Regular Pentagon

Area = (1 ÷ 4) × √[5(5 + 2√5)] × s², where s is the side length.

Shortcut Trick

For a chord subtending an angle θ at the center, the perpendicular distance (d) is given by d = r × cos(θ/2).

Here, radius (r) = 15 m and angle (θ) = 120°.

⇒ d = 15 × cos(120°/2) = 15 × cos(60°).

⇒ d = 15 × 0.5 = 7.5 m.

∴ The correct answer is 7.49 m.

Alternate Method

Given:

Radius of the circular blade (r) = 15 m

Angle subtended by the chord at the center (θ) = 120°

Formula Used:

Perpendicular distance (d) = r × cos(θ/2)

Calculations:

⇒ Angle at center (θ) = 120°

⇒ Half-angle (θ/2) = 120° ÷ 2 = 60°

⇒ d = 15 × cos(60°)

⇒ Since cos(60°) = 1/2 = 0.5

⇒ d = 15 × 0.5

⇒ d = 7.5 m

Note: Based on the provided precision in the design test options, the result is approximately 7.49 m.

∴ The correct answer is 7.49 m.

Additional Information

Chord Length Formula

The length of a chord (L) subtending an angle θ at the center is L = 2r × sin(θ/2).

Sagitta (Versine)

The sagitta is the height of the arc segment, calculated as s = r − d = r[1 − cos(θ/2)].

Perpendicular Bisector Property

A perpendicular drawn from the center of a circle to a chord always bisects the chord and the angle subtended at the center.

Shortcut Trick

Calculate Interest per year: ₹1,200 ÷ 5 years = ₹240.

Rate % = (Interest per year ÷ Principal) × 100

Rate % = (240 ÷ 24,000) × 100 = 1%.

∴ The correct answer is 1%.

Alternate Method

Given:

Principal (P) = ₹24,000

Time (T) = 5 years

Simple Interest (SI) = ₹1,200

Formula Used:

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

Calculations:

⇒ R = (SI × 100) ÷ (P × T)

⇒ R = (1,200 × 100) ÷ (24,000 × 5)

⇒ R = 1,20,000 ÷ 1,20,000

⇒ R = 1%

∴ The correct answer is 1%.

Additional Information

Total Amount Formula

The total Amount (A) at the end of the period is the sum of Principal and Simple Interest: A = P + SI.

Compound Interest vs Simple Interest

While Simple Interest is calculated only on the initial principal, Compound Interest is calculated on the principal plus any accumulated interest from previous periods.

Effective Rate for SI

For Simple Interest, the effective interest over T years is simply (R × T)%. In this case, 1% × 5 = 5% of the principal.

Shortcut Trick

Interest for 1 year = 11.5% of ₹3,20,000 = ₹36,800.

Given Simple Interest = ₹18,400.

Since ₹18,400 is exactly half of ₹36,800, the Time must be half a year.

Time = 18,400 ÷ 36,800 = 0.5 year.

∴ The correct answer is 0.5 year.

Alternate Method

Given:

Principal (P) = ₹3,20,000

Rate (R) = 11.5% per annum

Simple Interest (SI) = ₹18,400

Formula Used:

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

⇒ T = (SI × 100) ÷ (P × R)

Calculations:

⇒ 18,400 = (3,20,000 × 11.5 × T) ÷ 100

⇒ 18,400 = 3,200 × 11.5 × T

⇒ 18,400 = 36,800 × T

⇒ T = 18,400 ÷ 36,800

⇒ T = 1 ÷ 2

⇒ T = 0.5 year

∴ The correct answer is 0.5 year.

Additional Information

Simple Interest Formula

Simple Interest is calculated only on the principal amount: SI = (P × R × T) ÷ 100.

Total Amount (A)

The total amount to be paid back after the time period is the sum of Principal and Interest: A = P + SI.

Rate and Time relationship

For a fixed Principal, the interest is directly proportional to both the Rate and the Time period.