Quantitative Aptitude PYQs

Shortcut Trick

Given a + b = 5 and ab = 6, we can deduce the values are a = 3 and b = 2.

Substitute these directly into (a³ + b³)² − 9a²b²(a + b)²:

⇒ (3³ + 2³)² − 9(3 × 2)²(3 + 2)² ⇒ (27 + 8)² − 9(36)(25)

⇒ (35)² − 8100 ⇒ 1225 − 8100 = -6875

∴ The correct answer is -6875.

Alternate Method

Given:

a + b = 5

ab = 6

Formula Used:

a³ + b³ = (a + b)³ − 3ab(a + b)

Calculations:

⇒ a³ + b³ = (5)³ − 3 × 6 × 5

⇒ a³ + b³ = 125 − 90 = 35

Now, substitute the required values into the given expression:

⇒ (a³ + b³)² − 9a²b²(a + b)²

⇒ (35)² − 9 × (ab)² × (a + b)²

⇒ 1225 − 9 × (6)² × (5)²

⇒ 1225 − 9 × 36 × 25

⇒ 1225 − 8100

⇒ -6875

∴ The correct answer is -6875.

Additional Information

Sum of Cubes Formula

The sum of cubes can be factored as a³ + b³ = (a + b)(a² − ab + b²) or expressed as (a + b)³ − 3ab(a + b).

Difference of Cubes Formula

Similarly, the difference of cubes is given by a³ − b³ = (a − b)(a² + ab + b²) or (a − b)³ + 3ab(a − b).

Square of a Sum Formula

The expansion for the square of a sum is (a + b)² = a² + 2ab + b², which is frequently used to find sums of squares: a² + b² = (a + b)² − 2ab.

Shortcut Trick

If the entire sum of ₹10000 was lent at 7%, the interest would be 7% of 10000 = ₹700.

Extra interest received = ₹900 − ₹700 = ₹200.

This extra interest is purely due to the additional 5% (12% − 7%) on the second part.

⇒ 5% of the sum at 12% = ₹200

⇒ Sum at 12% = (200 × 100) ÷ 5 = ₹4000

∴ The correct answer is ₹4000.

Alternate Method

Given:

Total sum = ₹10000

Total annual interest = ₹900

Formula Used:

Overall Rate (R) = (Total Interest × 100) ÷ (Principal × Time)

Part 1 (7%)

Part 2 (12%)

Mean (9%)

12 − 9 = 3

9 − 7 = 2

Ratio of Parts = 3 : 2

Calculations:

⇒ Overall rate of interest = (900 × 100) ÷ (10000 × 1) = 9%

⇒ By rule of alligation, ratio of sum lent at 7% to sum lent at 12% = 3 : 2

⇒ Total ratio parts = 3 + 2 = 5 parts

⇒ Value of 5 parts = ₹10000

⇒ Value of 1 part = 10000 ÷ 5 = ₹2000

⇒ Sum lent at 12% (2 parts) = 2 × 2000 = ₹4000

∴ The correct answer is ₹4000.

Additional Information

Simple Interest vs Compound Interest

Simple Interest is calculated only on the principal amount, while Compound Interest is calculated on the principal and the accumulated interest of previous periods.

Rule of Alligation

When two ingredients or parts are mixed, the ratio in which they are distributed is given by: (Quantity of cheaper) / (Quantity of dearer) = (Price of dearer − Mean price) / (Mean price − Price of cheaper).

Effective Rate of Interest

The effective interest rate is the total interest generated over a year expressed as a percentage of the total principal. Formula: R = (Total SI × 100) / (Total Principal × Time).

Shortcut Trick

Find the effective principal for 1 year at rate R% by normalizing the second amount.

Effective P = P₁ + P₂ × (Rate multiplier) × (Time in years)

Effective P = 900 + 600 × 1.5 × (3 ÷ 12) = 900 + 225 = 1125

Now, apply the simple interest formula: (1125 × R × 1) ÷ 100 = 72

⇒ R = 7200 ÷ 1125 = 6.4%

∴ The correct answer is 6.4%.

Alternate Method

Given:

⇒ First Principal (P₁) = ₹900, Rate (R₁) = R%

⇒ Total Time for P₁ = 1 year = 12 months

⇒ Second Principal (P₂) = ₹600, Rate (R₂) = 1.5R%

⇒ Time for second sum (T₂) = 12 − 9 = 3 months = 1/4 year

⇒ Total Simple Interest = ₹72

0 Months 9 Months 12 Months ₹900 @ R% for 12 months ₹600 @ 1.5R% for 3 months Total Interest = ₹72

Calculations:

⇒ Simple Interest Formula = (P × R × T) ÷ 100

⇒ SI from P₁ = (900 × R × 1) ÷ 100 = 9R

⇒ SI from P₂ = (600 × 1.5R × 0.25) ÷ 100 = (600 × 1.5R × 1) ÷ 400

⇒ SI from P₂ = 150 × 1.5R ÷ 100 = 2.25R

⇒ Total SI = SI from P₁ + SI from P₂

⇒ 9R + 2.25R = 72

⇒ 11.25R = 72

⇒ R = 72 ÷ 11.25

⇒ R = 7200 ÷ 1125 = 6.4

∴ The correct answer is 6.4%.

Additional Information

Simple Interest Formula

Simple Interest (SI) is always calculated on the initial principal using I = (P × R × T) ÷ 100, where P is principal, R is rate of interest, and T is time in years.

Effective Principal Concept

When multiple investments are made at different rates and time periods, they can be merged into a single equivalent principal at the base rate: Effective P = Σ(P_i × Time factor × Rate multiplier).

Amount Formula

The total accumulated amount (A) after the given duration is simply the principal combined with the total interest earned: A = P + I.

Shortcut Trick

The tangent of the angle of elevation (θ) is the ratio of the flagpole's height to the shadow's length.

⇒ tan(θ) = 10√3 ÷ 10

⇒ tan(θ) = √3

Since tan(60°) = √3, the angle of elevation of the sun is 60°.

∴ The correct answer is 60°.

Alternate Method

θ 10√3 m 10 m A B C

Given:

⇒ Height of the flagpole (AB) = 10√3 m

⇒ Length of the shadow (BC) = 10 m

Formula Used:

⇒ tan(θ) = Perpendicular ÷ Base

Calculations:

Let the angle of elevation of the sun be θ (angle ACB).

In the right-angled ΔABC:

⇒ tan(θ) = AB ÷ BC

⇒ tan(θ) = 10√3 ÷ 10

⇒ tan(θ) = √3

We know from the standard trigonometric table that tan(60°) = √3.

⇒ θ = 60°

∴ The correct answer is 60°.

Additional Information

Trigonometric Tangent Ratio

In a right-angled triangle, the tangent function relates the opposite side to the adjacent side: tan(θ) = Perpendicular ÷ Base. It is fundamentally used to solve unknown heights and distances without calculating the hypotenuse.

Angle of Elevation vs Angle of Depression

The angle of elevation is the angle formed by the line of sight when looking upwards from the horizontal. Conversely, the angle of depression is formed when looking downwards. In height problems involving a single object and observer, both angles are geometrically equal due to alternate interior angles.

Standard Values of Tangent

It is highly beneficial to memorize standard tangent values for quick problem-solving: tan(30°) = 1÷√3, tan(45°) = 1, and tan(60°) = √3.

Shortcut Trick

When melting and recasting spheres, the constant terms (like (4/3) × π) cancel out.

Equate the effective volumes directly: R³ − r³ = n × r_small³

10³ − r³ = 50 × 2³

1000 − r³ = 400 ⇒ r³ = 600

r = ³√600 ≈ 8.43 cm (Since 8³ = 512 and 9³ = 729)

∴ The correct answer is 8.43 cm.

Alternate Method

Given:

Outer radius of the hollow sphere (R) = 10 cm

Radius of smaller solid spheres (r_small) = 2 cm

Number of smaller spheres (n) = 50

Formula Used:

Volume of hollow sphere = (4/3) × π × (R³ − r³)

Volume of solid sphere = (4/3) × π × r³

R=10

r=?

Hollow Sphere

Melted

2

50 Solid Spheres

Calculations:

Let the inner radius of the hollow sphere be r.

⇒ Volume of hollow sphere = Volume of 50 solid spheres

⇒ (4/3) × π × (10³ − r³) = 50 × (4/3) × π × 2³

⇒ Cancelling (4/3) × π from both sides:

⇒ 1000 − r³ = 50 × 8

⇒ 1000 − r³ = 400

⇒ r³ = 1000 − 400

⇒ r³ = 600

⇒ r = ³√600 ≈ 8.434 cm

∴ The correct answer is 8.43 cm.

Additional Information

Principle of Melting and Recasting

When a 3D object is melted and recast into another shape, the total volume of the material remains constant. Volume_initial = Volume_final.

Volume and Surface Area of a Solid Sphere

Volume = (4/3) × π × r³ | Total Surface Area = 4 × π × r²

Hemisphere Formulas

Volume = (2/3) × π × r³ | Curved Surface Area = 2 × π × r² | Total Surface Area = 3 × π × r²

Shortcut Trick

The ratio of Volume to Curved Surface Area of a hemisphere is [(2/3)πr³] ÷ [2πr²] = r / 3.

Given that Volume = 4 × CSA, we can write the ratio as 4.

⇒ r / 3 = 4

⇒ r = 12

∴ The correct answer is 12 cm.

Alternate Method

r Volume (V) = 2/3 × π × r³ Curved Surface Area (CSA) = 2 × π × r²

Given:

Volume of the hemisphere = 4 × Curved Surface Area of the hemisphere

Formula Used:

Volume of a hemisphere = 2/3 × πr³

Curved Surface Area (CSA) of a hemisphere = 2πr²

Calculations:

Let the radius of the hemisphere be r cm.

According to the given condition:

⇒ 2/3 × πr³ = 4 × (2πr²)

⇒ 2/3 × πr³ = 8πr²

Dividing both sides by πr²:

⇒ 2/3 × r = 8

⇒ r = 8 × (3 / 2)

⇒ r = 12

∴ The correct answer is 12 cm.

Additional Information

Total Surface Area of Hemisphere

It includes the curved surface area and the flat circular base area. Formula: TSA = 3πr².

Volume of a Sphere

A complete sphere has exactly twice the volume of a hemisphere of the same radius. Formula: V = (4/3)πr³.

Surface Area of a Sphere

The total curved surface of a complete sphere is four times the area of its great circle. Formula: SA = 4πr².

Shortcut Trick

Total pieces formed by making cuts along three perpendicular axes is (x + 1)(y + 1)(z + 1) = 8.

To minimize total cuts (x + y + z), the factors must be as close to each other as possible.

The closest integer factors for 8 are 2 × 2 × 2.

Thus, x + 1 = 2 ⇒ x = 1, y = 1, and z = 1.

Total minimum cuts = 1 + 1 + 1 = 3 cuts.

∴ The correct answer is 3 cuts.

Alternate Method

Given:

A cuboid is to be divided into 8 equal cuboids.

Formula Used:

Number of pieces = (x + 1) × (y + 1) × (z + 1)

Where x, y, and z are the number of cuts along the length, breadth, and height respectively.

3 Cuts = 8 Equal Cuboids Cut 1 Cut 2 Cut 3

Calculations:

⇒ Total pieces required = 8

⇒ (x + 1) × (y + 1) × (z + 1) = 8

⇒ To find the minimum number of cuts (x + y + z), we must express 8 as the product of three integers as close to each other as possible.

⇒ Let us evaluate the possible combinations for factors of 8:

⇒ Case 1: 8 × 1 × 1 → x = 7, y = 0, z = 0 → Total cuts = 7 + 0 + 0 = 7

⇒ Case 2: 4 × 2 × 1 → x = 3, y = 1, z = 0 → Total cuts = 3 + 1 + 0 = 4

⇒ Case 3: 2 × 2 × 2 → x = 1, y = 1, z = 1 → Total cuts = 1 + 1 + 1 = 3

⇒ The absolute minimum number of total cuts is obtained in Case 3.

⇒ Minimum cuts = 3.

∴ The correct answer is 3 cuts.

Additional Information

Maximum Pieces with N cuts

To maximize pieces for a given number of total cuts N, distribute the cuts equally among the three dimensions so that x ≈ y ≈ z ≈ N ÷ 3.

Total Painted Faces in a Cut Cube

If a fully painted larger cube is cut into n³ identical smaller cubes, the number of smaller cubes with exactly 3 painted faces is always 8 (located at the corners).

Zero Face Painted Cubes

The number of completely unpainted internal identical cubes is determined using the formula (n − 2)³, where n is the number of parts cut along one edge.

Shortcut Trick

The volume of a cone is directly proportional to the square of its radius and its height (V ∝ r²h).

Given the radius is halved (1/2) and the height is tripled (3).

Multiplying factor for Volume = (1/2)² × 3

New Volume factor = 1/4 × 3 = 3/4

∴ The correct answer is three-fourths of the original.

Alternate Method

Given:

⇒ Original radius = r, Original height = h

⇒ New radius = r/2, New height = 3h

Formula Used:

⇒ Volume of a cone (V) = 1/3 × πr²h

r h Original Cone

r/2 3h New Cone

Calculations:

⇒ Original Volume (V₁) = 1/3 × π × r² × h

⇒ New Volume (V₂) = 1/3 × π × (r/2)² × (3h)

⇒ V₂ = 1/3 × π × (r² / 4) × 3h

⇒ V₂ = 3/4 × (1/3 × π × r² × h)

⇒ V₂ = 3/4 × V₁

∴ The correct answer is three-fourths of the original.

Additional Information

Curved Surface Area of a Cone

CSA = πrl, where l is the slant height and is calculated as l = √(r² + h²).

Total Surface Area of a Cone

TSA = πr(r + l), combining the area of the circular base and the curved surface area.

Volume of a Cylinder

A cylinder with the same base radius and height as a cone has exactly 3 times the volume. The formula is V = πr²h.

Shortcut Trick

The standard Pythagorean triplet for a right-angled triangle is (5, 12, 13).

Given sin θ = 12/13, we identify Perpendicular = 12 and Hypotenuse = 13.

Using the triplet, the Base is directly found to be 5.

tan θ = Perpendicular ÷ Base = 12/5.

Since θ is in the 1^st quadrant, the ratio is positive.

∴ The correct answer is 12/5.

Alternate Method

Given:

sin θ = 12/13

θ ∈ (0, π/2)

Formula Used:

sin θ = Perpendicular (P) ÷ Hypotenuse (H)

Base (B) = √(H² − P²)

tan θ = Perpendicular (P) ÷ Base (B)

θ H = 13 P = 12 B = 5

Calculations:

⇒ Let P = 12 and H = 13

⇒ B = √(13² − 12²)

⇒ B = √(169 − 144)

⇒ B = √25

⇒ B = 5

⇒ tan θ = P ÷ B

⇒ tan θ = 12/5

Since θ lies in the first quadrant, all trigonometric functions are positive.

∴ The correct answer is 12/5.

Additional Information

Pythagorean Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (H² = P² + B²).

Trigonometric Ratios

The primary trigonometric ratios are defined as sin θ = P/H, cos θ = B/H, and tan θ = P/B.

Quadrant Sign Rule (ASTC)

In the 1^st quadrant (0 to π/2), all trigonometric ratios are positive. The ASTC rule (All, Sin, Tan, Cos) applies sequentially to the four quadrants.

Shortcut Trick

Given tanA = 2.

We know that cotA is the reciprocal of tanA, so cotA = 1/2.

Now, tanA + cotA = 2 + 1/2

⇒ (4 + 1) / 2 = 5/2

∴ The correct answer is 5/2.

Alternate Method

Given:

tanA = 2

Formula Used:

cotA = 1 / tanA

Calculations:

⇒ cotA = 1 / 2

⇒ tanA + cotA = 2 + 1/2

⇒ tanA + cotA = (4 + 1) / 2

⇒ tanA + cotA = 5/2

∴ The correct answer is 5/2.

Additional Information

Reciprocal Trigonometric Identities

tanθ and cotθ are reciprocals of each other. Therefore, tanθ × cotθ = 1.

Quotient Identities

tanθ = sinθ ÷ cosθ and cotθ = cosθ ÷ sinθ.

Pythagorean Identities

1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.

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.