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Shortcut Trick

Slope (m) is the ratio of the vertical change to the horizontal change between two points.

m = (Change in y) ÷ (Change in x)

m = (7 − 3) ÷ (4 − 2)

m = 4 ÷ 2 = 2

∴ The correct answer is 2.

Alternate Method

Given:

Points on the line: (x₁, y₁) = (2, 3) and (x₂, y₂) = (4, 7)

Formula Used:

Slope of a line, m = (y₂ − y₁) / (x₂ − x₁)

X Y 0

(2, 3) (4, 7) Run = 2 Rise = 4 Slope (m) = Rise ÷ Run = 4 ÷ 2 = 2

Calculations:

⇒ Substitute the given coordinates into the slope formula.

⇒ m = (7 − 3) / (4 − 2)

⇒ m = 4 / 2

⇒ m = 2

∴ The correct answer is 2.

Additional Information

Equation of a Straight Line

The point-slope form of the line equation is given by (y − y₁) = m(x − x₁), where m is the slope.

Parallel and Perpendicular Lines

Parallel lines always have equal slopes (m₁ = m₂). Perpendicular lines have negative reciprocal slopes (m₁ × m₂ = −1).

Distance Between Two Points

The straight-line distance d between (x₁, y₁) and (x₂, y₂) is calculated using d = √((x₂ − x₁)² + (y₂ − y₁)²).

Shortcut Trick

For sectors with the same radius, their areas are directly proportional to their central angles.

Angle of the first sector = 180° = π radians.

Angle of the second sector = π/2 radians.

Ratio of areas = π ÷ (π/2) = 2 : 1.

∴ The correct answer is 2:1.

Alternate Method

Given:

Radius of both sectors (r) = 12 cm

Central angle of first sector (θ₁) = 180°

Central angle of second sector (θ₂) = π/2 radians

180° r = 12 cm Sector 1

:

π/2 rad r = 12 cm Sector 2

Formula Used:

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

Area of a sector (in radians) = (1/2) × r² × θ

Calculations:

⇒ Area of first sector = (180° ÷ 360°) × π × (12)²

⇒ Area of first sector = (1/2) × π × 144 = 72π cm²

⇒ Area of second sector = (1/2) × (12)² × (π/2)

⇒ Area of second sector = (1/2) × 144 × (π/2) = 36π cm²

⇒ Ratio = Area of first sector : Area of second sector

⇒ Ratio = 72π : 36π = 2 : 1

∴ The correct answer is 2:1.

Additional Information

Area of a Sector

When the angle θ is in degrees, Area = (θ ÷ 360°) × π × r². When θ is in radians, Area = (1/2) × r² × θ.

Arc Length

The length of an arc is given by l = (θ ÷ 360°) × 2πr in degrees, and l = r × θ in radians.

Radians to Degrees Conversion

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

Shortcut Trick

The sum of all interior angles of any polygon depends directly on its number of sides (n).

Sum = (n − 2) × 180°

For a 10-sided polygon, substitute n = 10.

Sum = (10 − 2) × 180° = 8 × 180° = 1440°

∴ The correct answer is 1440°.

Alternate Method

Decagon (10 Sides) Number of triangles = n − 2 Triangles = 10 − 2 = 8 Each triangle sums to 180° Total = 8 × 180° = 1440°

Given:

The polygon is a decagon, which has 10 sides.

Formula Used:

Sum of interior angles of an n-sided polygon = (n − 2) × 180°

Calculations:

⇒ Number of sides (n) = 10

⇒ Sum of interior angles = (10 − 2) × 180°

⇒ Sum = 8 × 180°

⇒ Sum = 1440°

∴ The correct answer is 1440°.

Additional Information

Sum of Exterior Angles

The sum of all exterior angles for any convex polygon is always 360°, regardless of the number of sides.

Interior Angle of a Regular Polygon

Each interior angle of a regular n-sided polygon can be found using the formula: [(n − 2) × 180°] ÷ n.

Number of Diagonals

The total number of diagonals that can be drawn in an n-sided polygon is given by the formula: n(n − 3) ÷ 2.

Shortcut Trick

Midpoint of the line connecting (2, 8) and (6, 4) is ((2 + 6) ÷ 2, (8 + 4) ÷ 2) = (4, 6).

Slope of the given line segment = (4 − 8) ÷ (6 − 2) = −4 ÷ 4 = −1.

Slope of its perpendicular bisector = −1 ÷ (−1) = 1.

Equation of the bisector passing through (4, 6) with slope 1: y − 6 = 1 × (x − 4).

Simplifying the equation gives: y = x + 2.

∴ The correct answer is y = x + 2.

Alternate Method

Given:

Two points: A(2, 8) and B(6, 4)

Formula Used:

Midpoint formula: M = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)

Slope formula: m = (y₂ − y₁) ÷ (x₂ − x₁)

Perpendicular slope condition: m₁ × m₂ = −1

Equation of a straight line: y − y₁ = m × (x − x₁)

A (2, 8) B (6, 4) M (4, 6)

y = x + 2

Calculation:

⇒ Let the line segment be AB. The perpendicular bisector passes exactly through the midpoint M of AB.

⇒ Midpoint M = ((2 + 6) ÷ 2, (8 + 4) ÷ 2)

⇒ M = (8 ÷ 2, 12 ÷ 2) = (4, 6)

⇒ Slope of line AB (m₁) = (4 − 8) ÷ (6 − 2)

⇒ m₁ = −4 ÷ 4 = −1

⇒ Let the slope of the perpendicular bisector be m₂.

⇒ m₁ × m₂ = −1

⇒ −1 × m₂ = −1 ⇒ m₂ = 1

⇒ Now, substitute the point M(4, 6) and slope m₂ = 1 into the straight line equation.

⇒ y − 6 = 1 × (x − 4)

⇒ y − 6 = x − 4

⇒ y = x − 4 + 6

⇒ y = x + 2

∴ The correct answer is y = x + 2.

Additional Information

Distance Formula

The linear distance between two points (x₁, y₁) and (x₂, y₂) on a Cartesian plane is given by D = √((x₂ − x₁)² + (y₂ − y₁)²).

Section Formula

The coordinates of a point dividing a line segment internally in the ratio m:n are calculated as ((m×x₂ + n×x₁) ÷ (m+n), (m×y₂ + n×y₁) ÷ (m+n)).

Parallel and Perpendicular Lines

Two lines are parallel if their slopes are strictly equal (m₁ = m₂). Two lines are exactly perpendicular if the product of their respective slopes is negative one (m₁ × m₂ = −1).

Shortcut Trick

The ratio of the areas of similar triangles is the square of the ratio of their perimeters.

Ratio of areas = 3² : 5² = 9 : 25.

Total area in ratio units = 9 + 25 = 34 units.

Given total area = 34 cm², which means 34 units = 34 cm² ⇒ 1 unit = 1 cm².

Area of the larger triangle = 25 units × 1 = 25 cm².

∴ The correct answer is 25 cm².

Alternate Method

Given:

Ratio of perimeters of two similar triangles = 3 : 5

Sum of their areas = 34 cm²

Formula Used:

For two similar triangles, Ratio of Areas = (Ratio of Perimeters)²

Perimeter = 3x Area = 9a

Perimeter = 5x Area = 25a

Similar Triangles Area Relationship Total Area = 9a + 25a = 34 cm²

Calculations:

⇒ Let the areas of the smaller and larger triangles be A₁ and A₂ respectively.

⇒ A₁ : A₂ = 3² : 5²

⇒ A₁ : A₂ = 9 : 25

⇒ Let A₁ = 9a and A₂ = 25a

⇒ Total Area = 9a + 25a = 34a

⇒ 34a = 34

⇒ a = 1 cm²

⇒ Area of the larger triangle = 25a = 25 × 1 = 25 cm²

∴ The correct answer is 25 cm².

Additional Information

Areas of Similar Triangles Theorem

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides, altitudes, or perimeters.

Perimeters of Similar Triangles Theorem

The ratio of the perimeters of two similar triangles is exactly equal to the ratio of their corresponding sides or corresponding medians.

Volumes of Similar Figures

If two 3D geometrical figures are similar, the ratio of their volumes is equal to the cube of the ratio of their corresponding linear dimensions (sides or radii).

Shortcut Trick

If a + b + c = 0, then a³ + b³ + c³ = 3abc.

Here, a = 19, b = 20, and c = −39. Since 19 + 20 − 39 = 0, we can apply the trick.

19³ + 20³ − 39³ = 3 × 19 × 20 × (−39) = −44460.

Now, add 118 to this result: −44460 + 118 = −44342.

∴ The correct answer is −44342.

Alternate Method

Given:

19³ + 20³ − 39³ + 118

Formula Used:

If a + b + c = 0, then a³ + b³ + c³ = 3abc

Calculations:

⇒ Let a = 19, b = 20, and c = −39

⇒ Sum of terms: a + b + c = 19 + 20 − 39 = 0

⇒ Therefore, 19³ + 20³ + (−39)³ = 3 × 19 × 20 × (−39)

⇒ 3 × 19 = 57

⇒ 57 × 20 = 1140

⇒ 1140 × (−39) = −44460

⇒ Substituting this value back into the original expression:

⇒ −44460 + 118 = −44342

∴ The correct answer is −44342.

Additional Information

Algebraic Identity for Cubes

a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca). If a + b + c = 0, the right-hand side becomes 0, leading to the direct shortcut: a³ + b³ + c³ = 3abc.

Sum of Cubes Formula

a³ + b³ = (a + b)(a² − ab + b²)

Difference of Cubes Formula

a³ − b³ = (a − b)(a² + ab + b²)

BODMAS Rule

Always follow the correct order of operations in algebraic expressions: Brackets, Orders (powers/roots), Division × Multiplication, Addition − Subtraction.

Shortcut Trick

Using the trigonometric identity: cosec²A = 1 + cot²A

Substitute cot A = x + 1/x directly into the identity.

cosec²A = 1 + (x + 1/x)²

cosec²A = 1 + x² + 1/x² + 2 = x² + 1/x² + 3

∴ The correct answer is x² + 1/x² + 3.

Alternate Method

Given:

cot A = x + 1/x

Formula Used:

cosec²θ − cot²θ = 1

(a + b)² = a² + b² + 2ab

Calculations:

⇒ cosec²A = 1 + cot²A

⇒ Substitute the given value of cot A:

⇒ cosec²A = 1 + (x + 1/x)²

⇒ Expanding the term using the algebraic identity:

⇒ (x + 1/x)² = x² + (1/x)² + 2 × x × (1/x)

⇒ (x + 1/x)² = x² + 1/x² + 2

⇒ Now substitute this back into the expression for cosec²A:

⇒ cosec²A = 1 + x² + 1/x² + 2

⇒ cosec²A = x² + 1/x² + 3

∴ The correct answer is x² + 1/x² + 3.

Additional Information

Fundamental Trigonometric Identities

1. sin²θ + cos²θ = 1

2. sec²θ − tan²θ = 1

3. cosec²θ − cot²θ = 1

Important Algebraic Identities

1. (a + b)² = a² + b² + 2ab

2. (a − b)² = a² + b² − 2ab

Reciprocal Trigonometric Relations

sin θ = 1 / cosec θ | cos θ = 1 / sec θ | tan θ = 1 / cot θ

Shortcut Trick

A tangent line forms a 90° angle with the radius at the point of contact, creating a right-angled triangle.

The sides of this triangle are the radius (8 cm), the tangent length, and the distance to the center (17 cm).

These values form a standard Pythagorean triplet: (8, 15, 17).

Since the radius is 8 cm and the hypotenuse is 17 cm, the tangent must be the remaining leg, which is 15 cm.

∴ The correct answer is 15 cm.

Alternate Method

O P A

8 cm 17 cm PA = ?

ΔOAP forms a right-angled triangle

Given:

Radius of the circle, OA = 8 cm

Distance from the external point P to the center O, OP = 17 cm

Formula Used:

By Pythagoras Theorem: Hypotenuse² = Base² + Perpendicular²

Calculations:

⇒ In right-angled ΔOAP, OP is the hypotenuse because the angle at the point of tangency (angle OAP) is 90°.

⇒ OP² = OA² + PA²

⇒ 17² = 8² + PA²

⇒ 289 = 64 + PA²

⇒ PA² = 289 − 64

⇒ PA² = 225

⇒ PA = √225 = 15 cm

∴ The correct answer is 15 cm.

Additional Information

Tangent-Radius Theorem

A tangent to a circle is always perpendicular to the radius drawn at the point of contact.

Tangents from an External Point

The lengths of the two distinct tangents drawn from an external point to a circle are always equal.

Secant-Tangent Theorem

If a tangent segment PA and a secant segment PBC are drawn to a circle from an exterior point P, then PA² = PB × PC.

Shortcut Trick

When the angle between two tangents from an external point is 90°, the tangents and the radii to the points of contact form a square.

The side of this square is exactly equal to the radius of the circle (r = 5 cm).

The distance from the external point to the center is simply the diagonal of this square.

Distance OP = r × √2 = 5 × √2 = 5√2 cm.

∴ The correct answer is 5√2 cm.

Alternate Method

P O A B

5 cm 5 cm 90°

PO = 5√2 cm

Given:

⇒ Radius of the circle (OA = OB) = 5 cm.

⇒ Angle between tangents PA and PB = ∠APB = 90°.

Formula Used:

⇒ Diagonal of a square = Side × √2

Calculations:

⇒ A tangent to a circle is always perpendicular to the radius drawn through the point of contact.

⇒ ∴ ∠OAP = 90° and ∠OBP = 90°.

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

⇒ ∠AOB = 360° − (90° + 90° + 90°) = 90°.

⇒ Since all four angles of the quadrilateral are 90° and the adjacent sides are equal (OA = OB = 5 cm), OAPB is a square.

⇒ The distance from point P to the center O is represented by the diagonal OP of the square OAPB.

⇒ Length of diagonal OP = Side × √2 = 5 × √2 = 5√2 cm.

∴ The correct answer is 5√2 cm.

Additional Information

Tangents from an External Point

When two tangents are drawn to a circle from a single external point, their lengths are always strictly equal (PA = PB).

Tangent-Radius Theorem

The tangent line is always perpendicular to the radius of the circle drawn to the exact point of tangency.

Properties of a Square

A quadrilateral where all four angles are 90° and at least two adjacent sides are equal is a square. The diagonal bisects the internal angles and has a length of Side × √2.

Shortcut Trick

Step 1: Ignore the decimals and multiply the integers directly: 45 × 6 = 270.

Step 2: Count the total number of decimal places in the given numbers: 2 (in 0.45) + 1 (in 0.6) = 3 decimal places.

Step 3: Place the decimal point 3 positions from the right in the product: 0.270 = 0.27.

∴ The correct answer is 0.27.

Alternate Method

Given:

The expression to simplify is 0.45 × 0.6

Method Used:

Convert the decimal numbers into fractions, multiply the numerators and denominators, and then convert the result back to a decimal.

Calculations:

⇒ 0.45 = 45 / 100

⇒ 0.6 = 6 / 10

⇒ 0.45 × 0.6 = (45 / 100) × (6 / 10)

⇒ 0.45 × 0.6 = (45 × 6) / (100 × 10)

⇒ 0.45 × 0.6 = 270 / 1000

⇒ 0.45 × 0.6 = 27 / 100

⇒ 0.45 × 0.6 = 0.27

∴ The correct answer is 0.27.

Additional Information

Multiplication of Decimals

When multiplying decimal numbers, multiply them exactly as if they are whole numbers. The number of decimal places in the final product will always be equal to the total sum of the decimal places present in the original factors.

Converting Decimals to Fractions

To convert a decimal to a fraction, write the decimal number without the point as the numerator. For the denominator, write 1 followed by as many zeros as there are decimal places in the original number (e.g., 0.85 = 85/100).

BODMAS Rule

When simplifying mathematical expressions containing multiple operations, always follow the BODMAS sequence: Bracket, Order (Powers/Roots), Division, Multiplication, Addition, and Subtraction.