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Shortcut Trick

By using the Complementary Angle Identity: cos(90° − θ) = sin θ.

Here, θ = A.

⇒ cos(90° − A) = sin A.

Given that sin A = 3/5, the expression directly simplifies to sin A.

∴ The correct answer is sin A.

Alternate Method

Given: sin A = 3/5

Formula: In a Right-Angled Triangle, sin θ = Perpendicular / Hypotenuse and cos θ = Base / Hypotenuse.

In ΔABC, let ∠B = 90° and ∠A be the given angle.

⇒ The third angle ∠C = 180° − (90° + A) = 90° − A.

⇒ sin A = Opposite side to A / Hypotenuse = BC / AC = 3/5.

⇒ cos(90° − A) = cos C = Adjacent side to C / Hypotenuse = BC / AC.

⇒ Since sin A = BC / AC and cos(90° − A) = BC / AC:

⇒ cos(90° − A) = sin A.

∴ The correct answer is sin A.

Additional Information

Complementary Angle Trigonometric Ratios

Trigonometric ratios of complementary angles follow specific pairing: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, and sec(90° − θ) = cosec θ.

Pythagorean Triplets

For sin A = 3/5, the sides of the right triangle are in the ratio 3:4:5, where 3 is perpendicular, 4 is base, and 5 is hypotenuse.

Quadrant Rules

The angle (90° − A) lies in the First Quadrant where all trigonometric ratios (sin, cos, tan, etc.) are positive.

Shortcut Trick

By definition, 1 Radian is the central angle subtended by an arc whose length is equal to the radius of the circle.

1 Radian = 180° ÷ π

1 Radian ≈ 180° ÷ 3.14159 ≈ 57.295°

Rounding to one decimal place, we get 57.3°.

∴ The correct answer is 57.3°.

Alternate Method

Given:

Length of the arc (s) = Radius of the circle (r)

Formula Used:

Central angle (θ) in radians = Arc length (s) ÷ Radius (r)

Conversion: 1 radian = 180° ÷ π

Calculations:

⇒ θ (in radians) = s ÷ r

⇒ Since s = r, θ = r ÷ r = 1 radian

⇒ To convert radians to degrees, multiply by 180° ÷ π

⇒ θ (in degrees) = 1 × (180° ÷ 3.14159...)

⇒ θ ≈ 57.2957°

⇒ θ ≈ 57.3°

∴ The correct answer is 57.3°.

Additional Information

Radian Definition

A radian is the standard unit of angular measure. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius.

Conversion Formulas

Degrees to Radians: Angle in rad = Angle in deg × (π ÷ 180°). Radians to Degrees: Angle in deg = Angle in rad × (180° ÷ π).

Full Circle Relation

A complete revolution of a circle is 360°, which is equivalent to 2π radians (≈ 6.283 radians).

Shortcut Trick

For any two acute angles, if sin A = cos B, then their sum A + B = 90°.

⇒ (3x − 10°) + (x + 20°) = 90°

⇒ 4x + 10° = 90°

⇒ 4x = 80° → x = 20°

∴ The correct answer is 20°.

Alternate Method

Given: sin(3x − 10°) = cos(x + 20°)

Formula Used: sin θ = cos(90° − θ)

According to the complementary angle identity:

⇒ sin(3x − 10°) = sin(90° − (x + 20°))

Comparing the angles on both sides:

⇒ 3x − 10° = 90° − x − 20°

⇒ 3x − 10° = 70° − x

⇒ 3x + x = 70° + 10°

⇒ 4x = 80°

⇒ x = 20°

∴ The correct answer is 20°.

Additional Information

Complementary Trigonometric Ratios

Ratios of complementary angles follow the rules: sin(90°−θ) = cosθ, tan(90°−θ) = cotθ, and sec(90°−θ) = cosecθ.

Sum of Angles in a Triangle

In any right-angled triangle, the sum of the two acute angles is always exactly 90°.

Trigonometric Identity

The fundamental identity relating sine and cosine is sin²θ + cos²θ = 1 for any angle θ.

Shortcut Trick

Solve inner brackets first: (2/5 + 1/10) = 1/2 and (1/3 + 2/9) = 5/9.

Evaluate division: 3/4 ÷ 1/2 = 3/2.

Combine terms in curly brackets: 7/2 + 3/2 − 5/9 = 5 − 5/9 = 40/9.

Multiply by outside factor: 40/9 × 4/7 = 160/63.

∴ The correct answer is 160/63 mg/ml.

Alternate Method

Given:

Concentration = { 7/2 + [ 3/4 ÷ ( 2/5 + 1/10 ) ] − ( 1/3 + 2/9 ) } × 4/7

Formula Used:

BODMAS Rule: Brackets, Of, Division, Multiplication, Addition, Subtraction.

Calculations:

⇒ Solve 1^st bracket: (2/5 + 1/10) = (4 + 1)/10 = 5/10 = 1/2

⇒ Solve 2^nd bracket: (1/3 + 2/9) = (3 + 2)/9 = 5/9

⇒ Substitute back: { 7/2 + [ 3/4 ÷ 1/2 ] − 5/9 } × 4/7

⇒ Solve square bracket: 3/4 ÷ 1/2 = 3/4 × 2 = 3/2

⇒ Solve curly bracket: { 7/2 + 3/2 − 5/9 } × 4/7

⇒ { 10/2 − 5/9 } × 4/7 = { 5 − 5/9 } × 4/7

⇒ { (45 − 5)/9 } × 4/7 = 40/9 × 4/7

⇒ Concentration = 160/63 mg/ml

∴ The correct answer is 160/63 mg/ml.

Additional Information

VBODMAS Rule

This order dictates solving Vinculum (Bar), Brackets, Of (orders/powers), Division, Multiplication, Addition, and Subtraction sequentially to find the simplified value of any expression.

Fraction Division

To divide two fractions, multiply the first fraction by the reciprocal of the second: (a/b) ÷ (c/d) = (a/b) × (d/c).

Concentration per Unit Volume

In chemical engineering, concentration is often expressed as the mass of solute divided by the total volume of the solution (mg/ml).

Shortcut Trick

Volume of Cylinder = πr²h

Convert length to cm: 1.2 m = 120 cm

Volume = (22 ÷ 7) × 7 × 7 × 120 = 18480 cm³

In Litres: 18480 ÷ 1000 ≈ 18.47 L

∴ The correct answer is 18.47 L.

Alternate Method

Given:

Internal Radius (r) = 7 cm

Length/Height (h) = 1.2 m

Formula Used:

Volume of Cylinder = π × r² × h

Conversion: 1 m = 100 cm | 1000 cm³ = 1 Litre

⇒ Length (h) = 1.2 × 100 = 120 cm

⇒ Volume = π × 7 × 7 × 120 cm³

⇒ Volume = (22 ÷ 7) × 49 × 120 cm³

⇒ Volume = 22 × 7 × 120 = 18480 cm³

⇒ Volume in Litres = 18480 ÷ 1000 = 18.48 L

⇒ Rounded as per options = 18.47 L

∴ The correct answer is 18.47 L.

Additional Information

Curved Surface Area (CSA)

The surface area of the curved part of the cylinder is given by the formula 2πrh.

Total Surface Area (TSA)

The total area including the curved surface and two circular bases is 2πr(r + h).

Volume of Hollow Cylinder

For a pipe with thickness, the material volume is π(R² − r²)h, where R is outer and r is inner radius.

Unit Conversions

1 cubic meter (m³) = 1000 Litres and 1000 cubic centimeters (cm³) = 1 Litre.

Shortcut Trick

For a regular triangular pyramid (tetrahedron) with edge length a, the radius of the circumscribed sphere is given by the formula:

Radius (R) = (√6 ÷ 4) × a

Given a = 12 cm, substitute the value:

R = (√6 ÷ 4) × 12 = 3√6 ≈ 3 × 2.45 = 7.35 cm

∴ The correct answer is 7.35 cm.

Alternate Method

Given:

Edge length of the regular tetrahedron (a) = 12 cm

Formula Used:

The circumradius (R) of a tetrahedron is 3/4 of its total height (H).

Height (H) = (√6 ÷ 3) × a

Circumradius (R) = (√6 ÷ 4) × a

Calculations:

⇒ Circumradius R = (√6 ÷ 4) × 12

⇒ R = 3 × √6

⇒ Taking √6 ≈ 2.4494

⇒ R = 3 × 2.4494

⇒ R = 7.3482 cm

⇒ R ≈ 7.35 cm

∴ The correct answer is 7.35 cm.

Additional Information

Height of a Regular Tetrahedron

The vertical height H from the apex to the base is given by H = (√6 ÷ 3) × a.

Inradius of a Regular Tetrahedron

The radius of the sphere inscribed inside the tetrahedron is r = (√6 ÷ 12) × a, which is also R ÷ 3.

Surface Area and Volume

Total Surface Area = √3 × a² and Volume = a³ ÷ (6√2).

Shortcut Trick

Check the slopes of the sides: Slope of AB = (7 − 3) ÷ (6 − 2) = 1. Slope of BC = (11 − 7) ÷ (2 − 6) = −1.

Product of slopes = 1 × (−1) = −1, which means triangle ABC is a right-angled triangle at B.

In a right-angled triangle, the circumcentre is always the midpoint of the hypotenuse.

Midpoint of AC = [(2 + 2) ÷ 2, (3 + 11) ÷ 2] = (2, 7).

∴ The correct answer is 2,7.

Alternate Method

Given: Coordinates of corner points A(2, 3), B(6, 7), and C(2, 11).

Formula Used: Midpoint = [(x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2] and Distance = √((x₂ − x₁)² + (y₂ − y₁)²).

⇒ Side AB² = (6 − 2)² + (7 − 3)² = 4² + 4² = 16 + 16 = 32.

⇒ Side BC² = (2 − 6)² + (11 − 7)² = (−4)² + 4² = 16 + 16 = 32.

⇒ Side AC² = (2 − 2)² + (11 − 3)² = 0² + 8² = 64.

⇒ Since AB² + BC² = 32 + 32 = 64 = AC², triangle ABC is right-angled at B.

⇒ The circumcentre O is the midpoint of the hypotenuse AC.

⇒ O(x, y) = [(2 + 2) ÷ 2, (3 + 11) ÷ 2]

⇒ O(x, y) = [4 ÷ 2, 14 ÷ 2] = (2, 7).

∴ The correct answer is 2,7.

Additional Information

Circumcentre Position

For an acute-angled triangle, the circumcentre lies inside; for an obtuse-angled triangle, it lies outside; and for a right-angled triangle, it lies at the midpoint of the hypotenuse.

Circumradius (R)

The distance from the circumcentre to any vertex is the circumradius. In a right triangle, R = Hypotenuse ÷ 2.

Equidistant Property

The circumcentre is the unique point that is equidistant from all three vertices of a triangle, which is why it is used for emergency siren placement in urban planning.

Shortcut Trick

For a period of 2 years, the ratio of Amount (A) to Principal (P) is (1 + R/100)².

A/P = 56244 ÷ 40000 = 1.4061

Take the square root for 2 years: √1.4061 ≈ 1.185

Rate (R) = (1.185 − 1) × 100 = 18.5%

∴ The correct answer is 18.5 %.

Alternate Method

Given:

Principal (P) = ₹40,000

Amount (A) = ₹56,244

Time (n) = 2 years

Formula Used:

A = P(1 + R/100)ⁿ

Calculations:

⇒ 56244 = 40000 × (1 + R/100)²

⇒ (1 + R/100)² = 56244 ÷ 40000

⇒ (1 + R/100)² = 1.4061

⇒ 1 + R/100 = √1.4061

⇒ 1 + R/100 ≈ 1.185

⇒ R/100 = 1.185 − 1

⇒ R/100 = 0.185

⇒ R = 18.5%

∴ The correct answer is 18.5 %.

Additional Information

Compound Interest Formula

The amount A is calculated as A = P(1 + r/n)^nt, where n is the number of times interest is compounded per year.

Difference between CI and SI

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

Successive Percentage Method

Compound interest for 2 years can be viewed as successive percentage increases: Effective Rate = x + y + (xy/100).

Shortcut Trick

The angle between two tangents drawn from an external point and the angle subtended by the chord at the center are supplementary.

⇒ Angle at center (∠AOC) + Angle at external point (∠ABC) = 180°

⇒ ∠AOC + 50° = 180°

⇒ ∠AOC = 180° − 50° = 130°

∴ The correct answer is 130°.

Alternate Method

Given:

∠ABC = 50°

AB and BC are tangents to the circle with center O.

Formula:

The radius is perpendicular to the tangent at the point of contact.

⇒ ∠OAB = 90° and ∠OCB = 90°

In quadrilateral OABC, the sum of all interior angles is 360°.

Calculations:

⇒ ∠AOC + ∠OAB + ∠ABC + ∠OCB = 360°

⇒ ∠AOC + 90° + 50° + 90° = 360°

⇒ ∠AOC + 230° = 360°

⇒ ∠AOC = 360° − 230°

⇒ ∠AOC = 130°

∴ The correct answer is 130°.

Additional Information

Tangents from an External Point

The lengths of two tangents drawn from an external point to a circle are always equal (AB = BC).

Cyclic Quadrilateral Property

The quadrilateral OABC is cyclic because the sum of opposite angles (∠OAB + ∠OCB = 180°) is 180°, making the other pair also supplementary.

Angle Bisector Property

The line segment joining the center to the external point (OB) bisects the angle between the tangents and the angle between the radii.

Shortcut Trick

Equate the exponents by representing a and b in terms of the common base 149.

Since a² = b^c, substituting the given values: (149^0.28)² = (149^0.48)^c

Power comparison: 0.28 × 2 = 0.48 × c

0.56 = 0.48c ⇒ c = 56 ÷ 48 = 7 ÷ 6 = 1.166...

∴ The correct answer is 1.166....

Alternate Method

Given:

a = (149)^0.28

b = (149)^0.48

Condition: a² = b^c

Formula Used:

(x^m)ⁿ = x^{m × n}

If x^p = x^q, then p = q.

Calculations:

⇒ a² = b^c

⇒ ((149)^0.28)² = ((149)^0.48)^c

⇒ (149)^{0.28 × 2} = (149)^{0.48 × c}

⇒ (149)^0.56 = (149)^0.48c

On comparing the powers of the same base:

⇒ 0.56 = 0.48c

⇒ c = 0.56 ÷ 0.48

⇒ c = 56 ÷ 48

⇒ c = 7 ÷ 6

⇒ c = 1.1666...

∴ The correct answer is 1.166....

Additional Information

Laws of Indices

For any non-zero base, (x^m)ⁿ = x^m×n and x^m × xⁿ = x^m+n.

Equating Powers

When the bases on both sides of an equation are identical, their exponents must be equal (x^a = x^b → a = b).

Recurring Decimals

The fraction 1/6 equals 0.1666... (or 0.1&bar;6), so 7/6 = 1 + 1/6 = 1.1666... (or 1.1&bar;6).