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Shortcut Trick

Area of a circular path outside a circle = π × w × (2r + w)

Area = (22 ÷ 7) × 1.5 × (2 × 14 + 1.5)

Area = (33 ÷ 7) × 29.5 = 973.5 ÷ 7 ≈ 139.07 m²

∴ The correct answer is 139 m².

Alternate Method

O

14 m

1.5 m

R = 15.5 m

Given:

⇒ Radius of the circular park, r = 14 m

⇒ Width of the path built around it, w = 1.5 m

Formula Used:

⇒ Area of the path = Area of the outer circle − Area of the inner circle

⇒ Area of a circle = πR²

Calculations:

⇒ Outer radius, R = r + w

⇒ R = 14 + 1.5 = 15.5 m

⇒ Area of the path = πR² − πr²

⇒ Area = π × (15.5² − 14²)

⇒ Area = (22 ÷ 7) × (15.5 + 14) × (15.5 − 14)

⇒ Area = (22 ÷ 7) × 29.5 × 1.5

⇒ Area = (33 ÷ 7) × 29.5

⇒ Area = 973.5 ÷ 7 ≈ 139.07 m²

⇒ The approximate area of the path is 139 m².

∴ The correct answer is 139 m².

Additional Information

Area of Circular Path Outside

When a path of width 'w' is constructed outside a circle of radius 'r', the Area is calculated as πw(2r + w).

Area of Circular Path Inside

When a path of width 'w' is constructed inside a circle of radius 'r', the Area is calculated as πw(2r − w).

Area of a Circle

The total space occupied by a circular region is denoted by πr².

Shortcut Trick

To convert an angle from radians to degrees quickly, directly substitute π = 180°.

Given angle = 5π ÷ 4

⇒ Angle = (5 × 180°) ÷ 4

⇒ Angle = 5 × 45°

∴ The correct answer is 225°.

Alternate Method

Given:

Angle in radians = 5π ÷ 4

Formula Used:

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

0°90°180°270°5π/4 = 225°225°

Calculations:

⇒ Angle in degrees = (5π ÷ 4) × (180° ÷ π)

⇒ Angle in degrees = (5 × 180°) ÷ 4

⇒ Angle in degrees = 900° ÷ 4

⇒ Angle in degrees = 225°

∴ The correct answer is 225°.

Additional Information

Degrees to Radians Conversion

To convert an angle from degrees to radians, multiply the given angle by (π ÷ 180°).

Radian Measure

A radian is the angle subtended at the center of a circle by an arc whose length is exactly equal to the radius of the circle.

Common Angle Conversions

π ÷ 2 = 90°, π ÷ 3 = 60°, π ÷ 4 = 45°, and π ÷ 6 = 30°.

Shortcut Trick

Rewrite the equation y = -x + 4 as x + y = 4.

For a point to lie on the line, the sum of its coordinates must be 4.

Check the options: (5, 4) → 9; (1, 2) → 3; (3, 6) → 9; (2, 2) → 4.

Only the point (2, 2) gives a sum of 4.

∴ The correct answer is (2, 2).

Alternate Method

Given:

Equation of the line: y = -x + 4

X Y O

y = -x + 4

(2, 2)

(5, 4)

(1, 2)

(3, 6)

Calculations:

⇒ Let us substitute the given coordinates (x, y) into the equation x + y = 4.

⇒ Option 1: For (2, 2), LHS = 2 + 2 = 4. This satisfies the equation.

⇒ Option 2: For (5, 4), LHS = 5 + 4 = 9 ≠ 4. This does not satisfy.

⇒ Option 3: For (1, 2), LHS = 1 + 2 = 3 ≠ 4. This does not satisfy.

⇒ Option 4: For (3, 6), LHS = 3 + 6 = 9 ≠ 4. This does not satisfy.

∴ The correct answer is (2, 2).

Additional Information

Slope-Intercept Form

The equation of a straight line is often written as y = mx + c, where m is the slope of the line and c is the y-intercept. For y = -x + 4, the slope is -1 and the y-intercept is 4.

Checking a Point on a Line

A coordinate point (x₁, y₁) lies on a specific line if and only if its coordinates satisfy the equation of that line.

Intercepts of a Line

The x-intercept is calculated by setting y = 0, giving (4, 0). The y-intercept is calculated by setting x = 0, giving (0, 4). These are the points where the line crosses the respective axes.

Shortcut Trick

The relationship between arc length (l), radius (r), and central angle (θ) in radians is directly calculated using the formula:

θ = l ÷ r

⇒ θ = 10 ÷ 5 = 2 radians

∴ The correct answer is 2 radians.

Alternate Method

θ = ? r = 5 cm r = 5 cm Arc length (l) = 10 cm

Center O

Given:

Arc length of the sector (l) = 10 cm

Radius of the circle (r) = 5 cm

Formula Used:

The central angle in radians subtended by an arc at the center of a circle is the ratio of the arc length to its radius.

Angle (θ) = Arc length (l) ÷ Radius (r)

Calculations:

⇒ θ = 10 cm ÷ 5 cm

⇒ θ = 2

Since the formula directly yields the angle in radians, the central angle is 2 radians.

∴ The correct answer is 2 radians.

Additional Information

Conversion between Radians and Degrees

To convert an angle from radians to degrees, multiply by (180° ÷ π). Similarly, to convert degrees to radians, multiply by (π ÷ 180°). Thus, π radians = 180°.

Area of a Sector

The area of a sector with radius r and central angle θ (in radians) is determined by the formula A = (1 ÷ 2) × r² × θ. It can also be found using the arc length as A = (1 ÷ 2) × l × r.

Full Circle Arc Length

The total arc length encompassing a complete circle is its circumference, calculated as C = 2 × π × r. This aligns with a full rotation having a central angle of 2π radians.

Shortcut Trick

Given: tan(90° − A) = √3

We know from standard trigonometric values that tan 60° = √3.

Comparing both sides, we get: 90° − A = 60°

⇒ A = 30°

Therefore, sin A = sin 30° = 1/2.

∴ The correct answer is 1/2.

Alternate Method

A Base = √3 Perp = 1 Hyp = 2

Given:

tan(90° − A) = √3

Formula Used:

tan(90° − θ) = cot θ

cot θ = Base ÷ Perpendicular

sin θ = Perpendicular ÷ Hypotenuse

Calculations:

⇒ cot A = √3

⇒ Base ÷ Perpendicular = √3 ÷ 1

⇒ Let Base = √3 and Perpendicular = 1

⇒ By Pythagoras Theorem, Hypotenuse = √(Base² + Perpendicular²)

⇒ Hypotenuse = √((√3)² + 1²) = √(3 + 1) = √4 = 2

⇒ sin A = Perpendicular ÷ Hypotenuse = 1/2

∴ The correct answer is 1/2.

Additional Information

Complementary Angles

Two angles are complementary if their sum is 90°. Trigonometric ratios for complementary angles satisfy relations like sin(90°−θ) = cos θ and tan(90°−θ) = cot θ.

Pythagoras Theorem

In any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of its base and perpendicular: H² = P² + B².

Standard Trigonometric Values

Common values frequently used in exams include sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3, and tan 60° = √3.

Shortcut Trick

Substitute x = √7 directly into the expression x + 1/x.

Calculate 1/x = 1/√7 and rationalize it by multiplying the numerator and denominator by √7 to get √7/7.

Add the terms: √7 + √7/7 = (7√7 + √7) / 7.

Simplify the numerator to get 8√7/7.

∴ The correct answer is 8√7/7.

Alternate Method

Given:

x = √7

Formula Used:

For rationalization: 1/√a = √a / a

Calculations:

⇒ Substitute the given value of x in the expression x + 1/x

⇒ x + 1/x = √7 + 1/√7

⇒ Taking the common denominator as √7 for addition:

⇒ x + 1/x = (√7 × √7 + 1) / √7

⇒ x + 1/x = (7 + 1) / √7

⇒ x + 1/x = 8 / √7

⇒ Multiply the numerator and denominator by √7 to rationalize the denominator:

⇒ x + 1/x = (8 × √7) / (√7 × √7)

⇒ x + 1/x = 8√7 / 7

∴ The correct answer is 8√7/7.

Additional Information

Rationalizing the Denominator

When a fraction has an irrational number like √a in the denominator, multiplying both the numerator and denominator by √a removes the radical from the denominator. Formula: 1/√a = √a / a.

Operations with Surds

When multiplying square roots of the same base number, the square root symbol is eliminated: √a × √a = a. This basic property simplifies addition and multiplication of irrational fractions.

Standard Algebraic Identities

If x + 1/x = k, then x² + 1/x² = k² − 2. This rule is highly useful for finding higher powers of algebraic expressions quickly in quantitative aptitude exams.

Shortcut Trick

For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides.

Ratio of areas = (3 ÷ 5)²

Ratio of areas = 9 / 25

∴ The correct answer is 9:25.

Alternate Method

Side = 3 units Area 1

Side = 5 units Area 2

Given:

Ratio of corresponding sides of two similar triangles = 3:5

Formula Used:

If two triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides.

Area₁ ÷ Area₂ = (Side₁ ÷ Side₂)²

Calculations:

⇒ Area₁ ÷ Area₂ = (3 ÷ 5)²

⇒ Area₁ ÷ Area₂ = (3² ÷ 5²)

⇒ Area₁ ÷ Area₂ = 9 ÷ 25

⇒ The ratio of their areas = 9:25

∴ The correct answer is 9:25.

Additional Information

Similar Triangles

Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional. Similarity preserves the shape but not necessarily the size of the figure.

Ratio of Perimeters

The ratio of the perimeters of two similar triangles is identical to the ratio of their corresponding sides.

Ratio of Volumes

For similar three-dimensional objects, the ratio of their volumes is equal to the cube of the ratio of their corresponding sides (V₁ ÷ V₂ = s₁³ ÷ s₂³).

Shortcut Trick

Rationalize the denominator inside the root to find x.

x = √[(2 + √3)² ÷ (4 − 3)] = 2 + √3

Substitute x into the expression: x² + x − 9

⇒ (2 + √3)² + (2 + √3) − 9

⇒ 7 + 4√3 + 2 + √3 − 9 = 5√3

∴ The correct answer is 5√3.

Alternate Method

Given:

x = √[(2 + √3) ÷ (2 − √3)]

Calculations:

Rationalizing the denominator inside the square root:

⇒ x = √[((2 + √3) × (2 + √3)) ÷ ((2 − √3) × (2 + √3))]

⇒ x = √[(2 + √3)² ÷ (2² − (√3)²)]

⇒ x = √[(2 + √3)² ÷ (4 − 3)]

⇒ x = √[(2 + √3)² ÷ 1]

⇒ x = 2 + √3

Now, substitute the value of x in the given expression:

⇒ x² + x − 9

⇒ (2 + √3)² + (2 + √3) − 9

Using the algebraic formula (a + b)² = a² + 2ab + b²:

⇒ (2² + 2 × 2 × √3 + (√3)²) + 2 + √3 − 9

⇒ (4 + 4√3 + 3) + 2 + √3 − 9

⇒ 7 + 4√3 + 2 + √3 − 9

⇒ 9 + 5√3 − 9

⇒ 5√3

∴ The correct answer is 5√3.

Additional Information

Rationalization of Surds

To eliminate a square root from the denominator, multiply both the numerator and the denominator by its conjugate. The conjugate of a − √b is a + √b.

Algebraic Identity for Squares

The square of a binomial sum is calculated using the identity (a + b)² = a² + 2ab + b².

Difference of Squares

The product of the sum and difference of the same two terms yields the difference of their squares: (a − b)(a + b) = a² − b².

Shortcut Trick

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

The given ratio of angles is 1 : 2 : 3 : 4. Let the angles be x, 2x, 3x, and 4x.

Since the sum of opposite angle pairs must be equal, we must pair them as (x, 4x) and (2x, 3x) because x + 4x = 5x and 2x + 3x = 5x.

Equating the sum to 180°: 5x = 180° ⇒ x = 36°.

The smallest angle in the ratio is x, which is 36°.

∴ The correct answer is 36°.

Alternate Method

Given:

The angles of a cyclic quadrilateral are in the ratio of 1 : 2 : 3 : 4.

Formula Used:

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

O x 2x 3x 4x

Calculations:

⇒ Let the four angles of the cyclic quadrilateral be x, 2x, 3x, and 4x.

⇒ In a valid cyclic quadrilateral, the pairs of opposite angles must sum to the same value (180°).

⇒ Grouping the angles to find equal sums: (x + 4x) = 5x and (2x + 3x) = 5x.

⇒ Therefore, the opposite angle pairs are (x, 4x) and (2x, 3x).

⇒ Summing either opposite pair gives: 5x = 180°

⇒ x = 180° ÷ 5 = 36°

⇒ The measures of the four angles are 36°, 72°, 108°, and 144°.

⇒ Smallest angle = x = 36°

∴ The correct answer is 36°.

Additional Information

Cyclic Quadrilateral

A quadrilateral is called cyclic if all four of its vertices lie on the circumference of a circle. The sum of either pair of opposite angles is exactly 180°.

Exterior Angle Property

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

Ptolemy's Theorem

For any cyclic quadrilateral with side lengths a, b, c, d and diagonals p, q, the product of the diagonals equals the sum of the products of opposite sides: p × q = (a × c) + (b × d).

Brahmagupta's Formula

The area of a cyclic quadrilateral is given by Area = √(s − a)(s − b)(s − c)(s − d), where s is the semi-perimeter calculated as s = (a + b + c + d) ÷ 2.

Shortcut Trick

We know that sec A = Hypotenuse ÷ Adjacent = 13/5.

This directly forms the standard Pythagorean triplet (5, 12, 13).

Thus, the Opposite side to angle A is 12.

Since sin A = Opposite ÷ Hypotenuse, we get 12/13.

∴ The correct answer is 12/13.

Alternate Method

Given:

sec A = 13/5

Formula Used:

Pythagoras Theorem: Hypotenuse² = Opposite² + Adjacent²

sin A = Opposite ÷ Hypotenuse

A B C Adjacent = 5 Opposite = 12 Hypotenuse = 13

Let the sides of the right-angled triangle be:

⇒ Hypotenuse = 13k

⇒ Adjacent = 5k

Applying the Pythagoras theorem:

⇒ (13k)² = Opposite² + (5k)²

⇒ 169k² = Opposite² + 25k²

⇒ Opposite² = 169k² − 25k²

⇒ Opposite² = 144k²

⇒ Opposite = 12k

Now, substitute the values to find sin A:

⇒ sin A = Opposite / Hypotenuse

⇒ sin A = 12k / 13k

⇒ sin A = 12 / 13

∴ The correct answer is 12/13.

Additional Information

Trigonometric Ratios

In a right-angled triangle, sin A = Opposite ÷ Hypotenuse, cos A = Adjacent ÷ Hypotenuse, and tan A = Opposite ÷ Adjacent.

Pythagorean Triplets

A set of three positive integers a, b, and c that satisfy a² + b² = c². Common triplets include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).

Trigonometric Identities

The fundamental identities are: sin²A + cos²A = 1, sec²A − tan²A = 1, and cosec²A − cot²A = 1.