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Shortcut Trick

A central angle of 90° means the sector is exactly a quarter of the full circle.

Area of a full circle = π × r²

⇒ Full Area = (22/7) × 7 × 7 = 154 cm²

Area of the sector = 154 ÷ 4 = 38.5 cm²

∴ The correct answer is 38.5 cm².

Alternate Method

r = 7 cm 90° Area of Sector

Given:

Radius of the circle (r) = 7 cm

Central angle (θ) = 90°

Formula Used:

Area of a sector = (θ ÷ 360°) × π × r²

Calculations:

⇒ Area = (90° ÷ 360°) × (22/7) × 7²

⇒ Area = (1/4) × (22/7) × 49

⇒ Area = (1/4) × 154

⇒ Area = 38.5 cm²

∴ The correct answer is 38.5 cm².

Additional Information

Length of an Arc

The length of the arc bounding the sector is given by: l = (θ ÷ 360°) × 2πr.

Perimeter of a Sector

The perimeter of a sector includes the arc length and the two radii forming the sector: Perimeter = l + 2r.

Area of a Segment

A segment is the region bounded by a chord and an arc. Its area is: Area of segment = Area of sector − Area of the corresponding triangle.

Shortcut Trick

To convert any angle from degrees to radians, multiply the degree value by π / 180°.

⇒ Radian measure = 150° × (π / 180°)

⇒ Radian measure = 15π / 18

⇒ Radian measure = 5π / 6

∴ The correct answer is 5π/6.

Alternate Method

Given:

Angle measure = 150°

Formula Used:

Angle in radians = Angle in degrees × (π / 180°)

150° 0° 150° = 5π/6

Calculations:

⇒ Radian measure = 150° × (π / 180°)

⇒ Radian measure = (150 / 180) × π

⇒ Radian measure = (15 / 18) × π

⇒ Radian measure = 5π / 6

∴ The correct answer is 5π/6.

Additional Information

Degrees to Radians Conversion

The formula to convert degrees to radians is: Radians = Degrees × (π / 180°).

Radians to Degrees Conversion

The formula to convert an angle from radians back to degrees is: Degrees = Radians × (180° / π).

Standard Angles in Radians

Important values to remember: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, and 360° = 2π.

Shortcut Trick

Since the central angle is 120°, the sector represents exactly 120° ÷ 360° = 1/3 of the full circle.

Area of the full circle = π × r² = π × 8² = 64π cm².

Area of the sector = (1 / 3) × 64π = 64π/3 cm².

∴ The correct answer is 64π/3 cm².

Alternate Method

120° r = 8 cm r = 8 cm

Centre O Area of Sector

Given:

⇒ Radius of the sector (r) = 8 cm

⇒ Central angle (θ) = 120°

Formula Used:

⇒ Area of Sector = (θ ÷ 360°) × πr²

Calculations:

⇒ Area = (120° ÷ 360°) × π × (8)²

⇒ Area = (1 / 3) × π × 64

⇒ Area = 64π/3 cm²

∴ The correct answer is 64π/3 cm².

Additional Information

Area of a Circle

The total region enclosed by a circle is given by πr², which corresponds to a full central angle of 360°.

Length of an Arc

The length of the curved boundary of a sector is proportional to its central angle, given by the formula L = (θ ÷ 360°) × 2πr.

Perimeter of a Sector

The total boundary length of a sector is obtained by adding the arc length to twice the radius: Perimeter = L + 2r.

Area of a Segment

The region bounded by a chord and its corresponding arc. It is calculated by subtracting the area of the corresponding triangle from the total area of the sector.

Shortcut Trick

For any linear equation ax + by = c, all lines parallel to it are of the identical form ax + by = k (where k ≠ c).

The given line is 3x + 2y = 5.

By comparing the options, only the equation 3x + 2y = 7 matches this exact a and b coefficient format.

∴ The correct answer is 3x + 2y = 7.

Alternate Method

Given:

The equation of the original line is 3x + 2y = 5.

Concept Used:

Two lines are strictly parallel if their slopes are equal (m₁ = m₂).

For a line represented as ax + by = c, the slope m is determined by −a/b.

X Y O

3x + 2y = 5 3x + 2y = 7

Calculations:

⇒ Slope of the given line 3x + 2y = 5 is m = −3/2.

⇒ Let us calculate the slopes of all given options to find the identical one:

⇒ 1) 3x + 2y = 7 → Slope = −3/2 (Parallel to given line)

⇒ 2) 2x + 3y = 5 → Slope = −2/3 (Not parallel)

⇒ 3) 3x − 2y = 5 → Slope = 3/2 (Not parallel)

⇒ 4) x + y = 5 → Slope = −1 (Not parallel)

⇒ Only the first equation has the same slope as the original given line.

∴ The correct answer is 3x + 2y = 7.

Additional Information

Parallel Lines Condition

Two general lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are strictly parallel and do not touch when a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

Perpendicular Lines Condition

Two lines cross each other at exactly 90 degrees if the mathematical product of their slopes equals −1 (m₁ × m₂ = −1). The perpendicular inverse format is bx − ay + k = 0.

Intersecting Lines Condition

Linear graphs will always intersect at exactly one unique point if they possess structurally different slopes, proven when a₁/a₂ ≠ b₁/b₂.

Coincident Lines Condition

Two drawn lines represent the exact same path and hold infinite solutions universally if their constants are fully proportional: a₁/a₂ = b₁/b₂ = c₁/c₂.

Shortcut Trick

Using the identity sin(90° − θ) = cos(θ), we get cos(x) = cos(2x).

By comparing the angles directly for the principal value, we can write:

x = 2x ⇒ x = 0°

Checking the options, 0° satisfies the equation: sin(90°) = cos(0°) ⇒ 1 = 1.

∴ The correct answer is 0°.

Alternate Method

Given:

sin(90° − x) = cos(2x)

Formula Used:

sin(90° − θ) = cos(θ)

cos(2θ) = 2cos²(θ) − 1

Calculation:

⇒ sin(90° − x) = cos(2x)

⇒ cos(x) = cos(2x)

⇒ cos(x) = 2cos²(x) − 1

⇒ 2cos²(x) − cos(x) − 1 = 0

⇒ 2cos²(x) − 2cos(x) + cos(x) − 1 = 0

⇒ 2cos(x)(cos(x) − 1) + 1(cos(x) − 1) = 0

⇒ (2cos(x) + 1)(cos(x) − 1) = 0

⇒ cos(x) = 1 or cos(x) = −1/2

For x in the first quadrant (as per options), cos(x) = 1.

⇒ x = 0°

∴ The correct answer is 0°.

Additional Information

Complementary Angle Identities

For any angle θ, sin(90° − θ) = cos(θ), cos(90° − θ) = sin(θ), and tan(90° − θ) = cot(θ).

General Solution of Cosine

If cos(x) = cos(y), then the general solution is x = 360° × n ± y, where n is any integer.

Double Angle Formula

cos(2θ) can be expressed as cos²(θ) − sin²(θ), or 2cos²(θ) − 1, or 1 − 2sin²(θ).

Shortcut Trick

Given x = √5 + 2

Square both sides directly: x² = 5 + 4 + 4√5 = 9 + 4√5

We need to find the value of x² − 4√5

Substitute x²: 9 + 4√5 − 4√5 = 9

∴ The correct answer is 9.

Alternate Method

Given:

x = √5 + 2

Formula Used:

(a + b)² = a² + b² + 2ab

Calculation:

⇒ x² = (√5 + 2)²

⇒ x² = (√5)² + (2)² + 2 × 2 × √5

⇒ x² = 5 + 4 + 4√5

⇒ x² = 9 + 4√5

Now, the required expression is x² − 4√5.

Substituting the value of x²:

⇒ (9 + 4√5) − 4√5

⇒ 9

∴ The correct answer is 9.

Additional Information

Algebraic Identities

Basic identities like (a + b)² = a² + b² + 2ab and (a − b)² = a² + b² − 2ab are fundamental in simplification questions involving surds and radicals.

Simplification of Surds

When squaring a binomial containing a surd, the rational parts and irrational parts are combined separately. For example, (√x + y)² = x + y² + 2y√x.

Conjugate Surds

The expressions (a + √b) and (a − √b) are conjugates. Their product (a + √b)(a − √b) = a² − b is always a rational number.

Shortcut Trick

The congruence of triangles is determined by matching their corresponding sides and angles.

Since the problem states that all three corresponding sides are exactly equal (AB = XY, BC = YZ, CA = ZX).

This perfectly matches the defining condition for the Side-Side-Side (SSS) congruence rule.

∴ The correct answer is SSS.

Alternate Method

Given:

⇒ In Triangle ABC and Triangle XYZ:

⇒ AB = XY

⇒ BC = YZ

⇒ CA = ZX

A B C

X Y Z

Concept Used:

⇒ SSS Congruence Rule: If three sides of one triangle are equal to the corresponding three sides of another triangle, then the two triangles are congruent.

Calculations:

⇒ We map the sides of both triangles to verify their relation.

⇒ AB = XY (First pair of sides is equal)

⇒ BC = YZ (Second pair of sides is equal)

⇒ CA = ZX (Third pair of sides is equal)

⇒ Since all three corresponding sides are exactly equal, Triangle ABC is identically congruent to Triangle XYZ under the Side-Side-Side property.

∴ The correct answer is SSS.

Additional Information

SAS (Side-Angle-Side) Congruence

Two triangles are congruent if two sides and the included angle of one triangle are exactly equal to two sides and the included angle of the other.

ASA (Angle-Side-Angle) Congruence

Two triangles are congruent if two angles and the included side of one triangle are exactly equal to two angles and the included side of the other.

AAS (Angle-Angle-Side) Congruence

Two triangles are congruent if any two angles and a non-included side of one triangle are equal to the corresponding angles and side of the other.

RHS (Right Angle-Hypotenuse-Side) Congruence

Two right-angled triangles are congruent if the hypotenuse and one side of one triangle are equal to the hypotenuse and the corresponding side of the other.

Shortcut Trick

A and B are complementary angles ⇒ A + B = 90° ⇒ B = 90° − A

tanB = tan(90° − A) = cotA

Given sinA = 3/5 = Perpendicular / Hypotenuse ⇒ Base = √(5² − 3²) = 4

cotA = Base / Perpendicular = 4/3

∴ The correct answer is 4/3.

Alternate Method

Given:

sinA = 3/5

A and B are complementary angles (A + B = 90°).

Formula Used:

Pythagoras Theorem: (Hypotenuse)² = (Perpendicular)² + (Base)²

tanθ = Perpendicular ÷ Base

ABBase = 4Perpendicular = 3Hypotenuse = 5

Calculations:

⇒ In a right-angled triangle, sinA = Perpendicular ÷ Hypotenuse = 3 ÷ 5.

⇒ By Pythagoras theorem: Base = √(Hypotenuse² − Perpendicular²)

⇒ Base = √(5² − 3²) = √(25 − 9) = √16 = 4.

⇒ Since A and B are complementary, they form the two acute angles of the same right-angled triangle.

⇒ For angle B, the perpendicular and base sides are swapped relative to angle A.

⇒ Perpendicular to B = Base to A = 4.

⇒ Base to B = Perpendicular to A = 3.

⇒ tanB = (Perpendicular to B) ÷ (Base to B)

⇒ tanB = 4/3

∴ The correct answer is 4/3.

Additional Information

Trigonometric Ratios of Complementary Angles

sin(90° − θ) = cosθ and tan(90° − θ) = cotθ. Complementary angles sum to exactly 90°.

Pythagorean Triplets

A set of positive integers satisfying a² + b² = c². Standard base triplets include (3, 4, 5) and (5, 12, 13).

Basic Trigonometric Ratios

sinθ = P/H, cosθ = B/H, tanθ = P/B, where P is Perpendicular, B is Base, and H is Hypotenuse.

Shortcut Trick

The sum of opposite angles in a cyclic quadrilateral is always 180°.

Angle A and Angle C are opposite angles in the cyclic quadrilateral ABCD.

∠C = 180° − ∠A

∠C = 180° − 100° = 80°

∴ The correct answer is 80°.

Alternate Method

A B C D

100° 80°

∠A + ∠C = 180°

Given:

∠A = 100°

ABCD is a cyclic quadrilateral.

Formula Used:

In a cyclic quadrilateral, the sum of a pair of opposite interior angles is equal to 180°.

Calculation:

⇒ ∠A + ∠C = 180°

⇒ 100° + ∠C = 180°

⇒ ∠C = 180° − 100°

⇒ ∠C = 80°

∴ The correct answer is 80°.

Additional Information

Cyclic Quadrilateral

A quadrilateral is called cyclic if all its four vertices lie on the circumference of a single circle. The primary property of a cyclic quadrilateral is that the sum of its opposite interior angles is always 180°.

Exterior Angle Property

If any one side of a cyclic quadrilateral is extended outwardly, the exterior angle formed is strictly equal to the interior opposite angle.

Ptolemy's Theorem

For a cyclic quadrilateral, the product of the lengths of the diagonals is equal to the sum of the products of the lengths of its opposite sides (AC × BD = AB × CD + AD × BC).

Shortcut Trick

Using the fundamental trigonometric identity: sin²A + cos²A = 1

Subtract the given equation (sin²A − cos²A = 1/4) from this identity:

(sin²A + cos²A) − (sin²A − cos²A) = 1 − 1/4

2cos²A = 3/4

cos²A = 3/8

∴ The correct answer is 3/8.

Alternate Method

Given:

sin²A − cos²A = 1/4

Formula Used:

sin²A + cos²A = 1 ⇒ sin²A = 1 − cos²A

Calculations:

⇒ Substitute the value of sin²A in the given equation.

⇒ (1 − cos²A) − cos²A = 1/4

⇒ 1 − 2cos²A = 1/4

⇒ 2cos²A = 1 − 1/4

⇒ 2cos²A = 3/4

⇒ cos²A = 3/4 × 1/2

⇒ cos²A = 3/8

∴ The correct answer is 3/8.

Additional Information

Trigonometric Square Identities

1. sin²θ + cos²θ = 1

2. sec²θ − tan²θ = 1

3. cosec²θ − cot²θ = 1

Double Angle Formula for Cosine

The double angle formula can be expressed in three different forms:

cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ