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Shortcut Trick

For any linear equation given in the standard form Ax + By = C, the slope of the line can be directly calculated as −A ÷ B.

Here, the given equation is 2x + 3y = 6.

Comparing it with Ax + By = C, we clearly get A = 2 and B = 3.

Slope = −A ÷ B = −2 ÷ 3 = −2/3.

∴ The correct answer is −2/3.

Alternate Method

Given:

The equation of the straight line is: 2x + 3y = 6

Formula Used:

The slope-intercept form of a linear equation is y = mx + c, where 'm' represents the slope of the line and 'c' represents the y-intercept.

X Y O

Run = +3 Rise = −2

2x + 3y = 6

(0, 2)

(3, 0)

Calculations:

⇒ 2x + 3y = 6

⇒ Transpose 2x to the right-hand side:

⇒ 3y = −2x + 6

⇒ Divide the entire equation by 3 to isolate y:

⇒ y = (−2/3)x + (6/3)

⇒ y = (−2/3)x + 2

⇒ Comparing this final equation with y = mx + c, we can identify that the slope m is −2/3.

∴ The correct answer is −2/3.

Additional Information

Slope from General Form

When a straight line is defined algebraically as Ax + By + C = 0, its constant slope can be rapidly determined using the ratio formula m = −A ÷ B.

Slope-Intercept Form y = mx + c

This specific arrangement cleanly maps line properties directly: 'm' dictates line steepness and direction (slope), while 'c' pinpoints exactly where the line crosses the y-axis.

Slope Formula Using Two Points

If a straight line passes through any two unique points (x₁, y₁) and (x₂, y₂), the standard formula for its slope is evaluated as m = (y₂ − y₁) ÷ (x₂ − x₁), representing standard "rise over run".

Parallel and Perpendicular Characteristics

Lines that are purely parallel will inherently have identically equal slopes (m₁ = m₂). Lines intersecting at exactly 90 degrees possess completely negative reciprocal slopes (m₁ × m₂ = −1).

Shortcut Trick

For a given circle, the area of a sector is directly proportional to its central angle.

Central angle of the first sector = 60°.

Central angle of the second sector = π/3 radians = (180° ÷ 3) = 60°.

Since both sectors have the exact same radius and central angle, their areas are identically equal.

Ratio of their areas = 60° : 60° = 1 : 1.

∴ The correct answer is 1:1.

Alternate Method

Given:

Radius of the circle (r) = 8 cm

Central angle of the first sector (θ₁) = 60°

Central angle of the second sector (θ₂) = π/3 radians

Formula Used:

Area of a sector = (θ ÷ 360°) × πr²

60° 8 cm Sector 1

π/3 rad 8 cm Sector 2

Calculations:

⇒ Area of first sector (A₁) = (θ₁ ÷ 360°) × πr²

⇒ A₁ = (60° ÷ 360°) × π × (8)² = (1 ÷ 6) × 64π = 32π ÷ 3 cm²

⇒ To find A₂, we can convert θ₂ to degrees = π/3 × (180° ÷ π) = 60°

⇒ A₂ = (60° ÷ 360°) × π × (8)² = 32π ÷ 3 cm²

⇒ Alternatively using radian formula for A₂ = (1 ÷ 2) × r² × θ₂

⇒ A₂ = (1 ÷ 2) × (8)² × (π ÷ 3) = 32π ÷ 3 cm²

⇒ Ratio of their areas = A₁ : A₂

⇒ Ratio = (32π ÷ 3) : (32π ÷ 3) = 1 : 1

∴ The correct answer is 1:1.

Additional Information

Area of a Sector in Radians

If the central angle θ is given in radians, the area can be calculated directly using the formula: Area = (1 ÷ 2) × r² × θ.

Radian to Degree Conversion

To convert an angle from radians to degrees, multiply by (180° ÷ π). Therefore, π radians is exactly equivalent to 180°.

Length of an Arc

The length of the arc of a sector with central angle θ in degrees is calculated as: L = (θ ÷ 360°) × 2πr.

Shortcut Trick The sum of all three interior angles of a triangle is always exactly 180°. A triangle cannot have two right angles (90° + 90° = 180°) as the third angle would be 0°. Similarly, having one right angle and one obtuse angle (>90°) would exceed the maximum sum of 180°. Therefore, a valid triangle can have at most one right angle or one obtuse angle, with the remaining two angles strictly acute. ∴ The correct answer is Only one right or one obtuse angle.

Alternate Method Given: A closed polygon with three sides and three interior angles. Formula Used: Angle Sum Property of a Triangle: Sum of all three interior angles = 180°

Angle Sum Property: ∠A + ∠B + ∠C = 180°

Right Angled

Obtuse Angled

90°

Acute Angled

Calculations: ⇒ Let the three interior angles of the triangle be ∠A, ∠B, and ∠C. ⇒ According to the angle sum property: ∠A + ∠B + ∠C = 180°. ⇒ Checking Option 1 (Two right angles): If ∠A = 90° and ∠B = 90°, then 90° + 90° + ∠C = 180° ⇒ ∠C = 0°. This cannot form a closed triangle. ⇒ Checking Option 2 (One obtuse, one right): If ∠A > 90° and ∠B = 90°, then ∠A + ∠B > 180°. The sum exceeds 180° before even adding ∠C, which is impossible. ⇒ Checking Option 4 (Three angles > 60°): If ∠A > 60°, ∠B > 60°, and ∠C > 60°, then ∠A + ∠B + ∠C > 180°. This violates the angle sum property. ⇒ Checking Option 3 (Only one right or one obtuse angle): If ∠A = 90° or ∠A > 90°, the remaining sum for (∠B + ∠C) is exactly 90° or strictly less than 90°. Thus, the other two angles must both be acute, which perfectly satisfies the 180° property. ∴ The correct answer is Only one right or one obtuse angle.

Additional Information Acute Angled Triangle A triangle in which all three interior angles are strictly less than 90°. For example, an equilateral triangle has all angles as 60°. Right Angled Triangle A triangle containing exactly one right angle (90°). The side opposite to the right angle is the hypotenuse, following Pythagoras theorem: a2 + b2 = c2. Obtuse Angled Triangle A triangle in which exactly one of the interior angles is strictly greater than 90°. The sum of the remaining two angles is always less than 90°. Exterior Angle Property The measure of any exterior angle of a triangle is always equal to the sum of the measures of its two remote (non-adjacent) interior angles.

Shortcut Trick

Midpoint of AB = ((2 + 8) ÷ 2, (5 − 1) ÷ 2) = (5, 2).

Slope of AB = (−1 − 5) ÷ (8 − 2) = −1, so the slope of its perpendicular bisector is 1.

Using the equation y = mx + c for the bisector passing through (5, 2):

2 = 1 × 5 + c ⇒ c = −3.

∴ The correct answer is −3.

Alternate Method

Given:

Points A(2, 5) and B(8, −1)

Line L is the perpendicular bisector of segment AB.

Formulas Used:

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

Slope m = (y₂ − y₁) ÷ (x₂ − x₁)

Point-slope equation: y − y₁ = m × (x − x₁)

X Y O

A(2, 5) B(8, -1) M(5, 2) (0, -3) Line L

Calculations:

⇒ Midpoint of AB, M = ((2 + 8) ÷ 2, (5 − 1) ÷ 2)

⇒ M = (10 ÷ 2, 4 ÷ 2) = (5, 2)

⇒ Slope of line segment AB, m_AB = (−1 − 5) ÷ (8 − 2)

⇒ m_AB = −6 ÷ 6 = −1

⇒ Since line L is perpendicular to AB, its slope m_L = −1 ÷ m_AB

⇒ m_L = −1 ÷ (−1) = 1

⇒ Equation of line L passing through M(5, 2) with slope 1:

⇒ y − y₁ = m_L × (x − x₁)

⇒ y − 2 = 1 × (x − 5)

⇒ y = x − 5 + 2

⇒ y = x − 3

⇒ To find the y-intercept, set x = 0:

⇒ y = 0 − 3 = −3

∴ The correct answer is −3.

Additional Information

Midpoint Formula

The coordinates of the midpoint of a line segment connecting (x₁, y₁) and (x₂, y₂) are given by ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2).

Slope of Perpendicular Lines

If two lines are perpendicular to each other, the product of their slopes is always −1 (m₁ × m₂ = −1).

Equation of a Line

The slope-intercept form is y = mx + c, where m is the slope and c is the y-intercept (the point where the line crosses the y-axis).

Shortcut Trick

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

Given the side ratio (DEF to ABC) is 2 : 1.

Area ratio = (2)² : (1)² = 4 : 1

Area of ΔDEF = 4 × Area of ΔABC

Area of ΔDEF = 4 × 16 = 64 cm²

∴ The correct answer is 64 cm².

Alternate Method

Given:

Area of ΔABC = 16 cm²

ΔABC is similar to ΔDEF (ΔABC ~ ΔDEF)

Side of ΔDEF = 2 × Corresponding side of ΔABC

A B C

D E F

16 cm² Area = ?

x 2x

Formula Used:

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

Area(ΔDEF) ÷ Area(ΔABC) = (DE ÷ AB)²

Calculations:

⇒ Area(ΔDEF) ÷ 16 = (2x ÷ x)²

⇒ Area(ΔDEF) ÷ 16 = (2)²

⇒ Area(ΔDEF) ÷ 16 = 4

⇒ Area(ΔDEF) = 16 × 4

⇒ Area(ΔDEF) = 64 cm²

∴ The correct answer is 64 cm².

Additional Information

Similarity Area Theorem

The ratio of the areas of two similar triangles is strictly equal to the square of the ratio of their corresponding sides, altitudes, or medians. Formula: Area₁ / Area₂ = (S₁ / S₂)².

Perimeter of Similar Triangles

Unlike area, the ratio of the perimeters of two similar triangles is directly proportional to the ratio of their corresponding sides without any squaring. Formula: P₁ / P₂ = S₁ / S₂.

Congruent vs Similar

Congruent triangles are identical in both shape and size (Area ratio is exactly 1:1), while similar triangles have the same shape but may differ in size depending on the scale factor.

Shortcut Trick

Using the Digital Root (DR) method:

DR of 44³ = 8³ = 512 ⇒ 8

DR of 35³ = 8³ = 512 ⇒ 8

DR of 53³ = 8³ = 512 ⇒ 8, and DR of 159 = 1 + 5 + 9 = 15 ⇒ 6

Expression DR = 8 + 8 − 8 + 6 = 14 ⇒ 5

Checking options, DR of −20659 is 9 − (2 + 0 + 6 + 5 + 9) = 9 − 22 ⇒ 9 − 4 = 5. (Matches perfectly)

∴ The correct answer is -20659.

Alternate Method

Given:

44³ + 35³ − 53³ + 159

Calculations:

First, calculate the cube of each number:

⇒ 44³ = 44 × 44 × 44 = 85184

⇒ 35³ = 35 × 35 × 35 = 42875

⇒ 53³ = 53 × 53 × 53 = 148877

Substitute the values back into the expression:

⇒ 85184 + 42875 − 148877 + 159

Summing the positive numbers:

⇒ (85184 + 42875 + 159) − 148877

⇒ 128218 − 148877

Subtracting the larger value from the smaller value:

⇒ −20659

∴ The correct answer is -20659.

Additional Information

BODMAS Rule

Simplification strictly follows the BODMAS sequence: Brackets, Orders (powers and roots), Division, Multiplication, Addition, and Subtraction.

Digital Root Method

The sum of the digits of a number until a single digit is obtained. It serves as an excellent shortcut to verify large calculations without doing the actual arithmetic.

Digital Root for Negative Numbers

If an algebraic calculation involves negative final roots, you find the positive single digit form by subtracting its absolute digital root from 9.

Shortcut Trick

Given, sinA + cosA = 5/4

Square both sides directly to use the relation (sinA + cosA)² = 1 + sin2A.

1 + sin2A = (5/4)² = 25/16

sin2A = 25/16 − 1 = 9/16

∴ The correct answer is 9/16.

Alternate Method

Given:

sinA + cosA = 5/4

Formula Used:

(a + b)² = a² + b² + 2ab

sin²A + cos²A = 1

sin2A = 2sinAcosA

Calculation:

⇒ sinA + cosA = 5/4

Squaring both sides,

⇒ (sinA + cosA)² = (5/4)²

⇒ sin²A + cos²A + 2sinAcosA = 25/16

Substitute sin²A + cos²A = 1 and 2sinAcosA = sin2A,

⇒ 1 + sin2A = 25/16

⇒ sin2A = 25/16 − 1

⇒ sin2A = (25 − 16) / 16

⇒ sin2A = 9/16

∴ The correct answer is 9/16.

Additional Information

Trigonometric Identities

The fundamental Pythagorean identity states that sin²A + cos²A = 1 for any angle A.

Double Angle Formula

The double angle formula for sine is sin2A = 2sinAcosA. Other common variations include cos2A = cos²A − sin²A.

Difference Identity

Similarly, squaring the difference gives the relation (sinA − cosA)² = 1 − 2sinAcosA = 1 − sin2A.

Shortcut Trick

According to the tangent property of a circle, the lengths of two tangents drawn from an external point to a circle are always equal.

⇒ Length of Tangent PA = Length of Tangent PB

⇒ PB = 12 cm

∴ The correct answer is 12 cm.

Alternate Method

Given:

Length of tangent PA = 12 cm.

PA and PB are tangents from an external point P to a circle with center O.

P A B O 12 cm ? cm

Concept Used:

A tangent to a circle is perpendicular to the radius drawn to the point of tangency. By proving the two triangles formed by the tangents, radii, and the line connecting the external point to the center are congruent, we can establish that the tangent lengths are equal.

Calculations:

⇒ Consider △OAP and △OBP:

⇒ OA = OB (Radii of the same circle)

⇒ OP = OP (Common hypotenuse)

⇒ ∠OAP = ∠OBP = 90° (Tangent is perpendicular to the radius at the point of contact)

⇒ By RHS (Right angle - Hypotenuse - Side) congruence criterion, △OAP ≅ △OBP.

⇒ Therefore, PA = PB (Corresponding parts of congruent triangles)

⇒ Since PA = 12 cm, it follows that PB = 12 cm.

∴ The correct answer is 12 cm.

Additional Information

Direct Common Tangent (DCT)

When two circles do not intersect, the length of their direct common tangent is given by √(d² − (R − r)²), where d is the distance between their centers and R, r are their radii.

Transverse Common Tangent (TCT)

The length of the transverse common tangent (which crosses the line segment joining the centers) is calculated using the formula √(d² − (R + r)²).

Tangent-Secant Theorem

If a tangent segment and a secant segment are drawn to a circle from an exterior point, the square of the length of the tangent is equal to the product of the external secant part and the entire secant length (PT² = PA × PB).

Number of Tangents Rule

From an external point, exactly two tangents can be drawn to a circle. From a point on the circle, exactly one tangent can be drawn, and from a point inside the circle, zero tangents can be drawn.

Shortcut Trick

For a right-angled triangle, the inradius (r) is calculated directly by:

r = (Perpendicular + Base − Hypotenuse) ÷ 2

Hypotenuse = √(6² + 8²) = √100 = 10 cm

r = (6 + 8 − 10) ÷ 2 = 4 ÷ 2 = 2 cm

∴ The correct answer is 2 cm.

Alternate Method

Given:

Base (b) = 8 cm

Height (p) = 6 cm

Formula Used:

Area = 1/2 × Base × Height

Semi-perimeter (s) = (a + b + c) ÷ 2

Inradius (r) = Area ÷ s

8 cm 6 cm 10 cm

r r r

Calculations:

⇒ c = √(6² + 8²) = √(36 + 64) = 10 cm

⇒ Area = 1/2 × 8 × 6 = 24 cm²

⇒ s = (6 + 8 + 10) ÷ 2 = 12 cm

⇒ r = 24 ÷ 12 = 2 cm

∴ The correct answer is 2 cm.

Additional Information

Inradius of an Equilateral Triangle

Formula: r = a ÷ (2√3), where a is the side length.

Circumradius of a Right Triangle

R = Hypotenuse ÷ 2. The circumcenter naturally lies at the exact midpoint of the hypotenuse.

General Inradius Formula

r = Δ ÷ s, applicable to all triangles without exception, where Δ is the area and s is the semi-perimeter.

Pythagorean Triplets

Common integer sides of right triangles to memorize include (3, 4, 5), (5, 12, 13), (7, 24, 25), and (8, 15, 17).

Shortcut Trick

For any real numbers, if a + b + c = 0, then a³ + b³ + c³ = 3abc.

Given expression: (a³ + b³ + c³) ÷ (3abc)

Substitute a³ + b³ + c³ = 3abc directly into the numerator.

⇒ 3abc ÷ 3abc = 1

∴ The correct answer is 1.

Alternate Method

Given:

a = 0.02

b = 0.03

c = −0.05

a + b + c = 0

Formula Used:

a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)

Calculations:

⇒ We are given that a + b + c = 0.

⇒ Substitute this value into the standard formula.

⇒ a³ + b³ + c³ − 3abc = 0 × (a² + b² + c² − ab − bc − ca)

⇒ a³ + b³ + c³ − 3abc = 0

⇒ a³ + b³ + c³ = 3abc

We need to find the value of (a³ + b³ + c³) ÷ (3abc)

⇒ Substitute the derived value of a³ + b³ + c³ into the given expression.

⇒ (3abc) ÷ (3abc) = 1

∴ The correct answer is 1.

Additional Information

Standard Algebraic Identity

a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)

Alternative Form

a³ + b³ + c³ − 3abc = 1/2 × (a + b + c) × [(a − b)² + (b − c)² + (c − a)²]

Conditional Identity

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