Quantitative Aptitude PYQs

Shortcut Trick

The distance (d) from the foot of a building for a given angle of elevation θ is given by d = h × cot(θ).

Distance of point A from the foot = h × cot(30°) = h√3

Distance of point B from the foot = h × cot(60°) = h/√3

Ratio of distances = AF : BF = h√3 : (h / √3)

Ratio = √3 × √3 : 1 = 3 : 1

∴ The correct answer is 3 : 1.

Alternate Method

Given:

Angle of elevation from point A, θ₁ = 30°

Angle of elevation from point B, θ₂ = 60°

Formula Used:

In a right-angled triangle, tan(θ) = Perpendicular ÷ Base

30°

60°

T F A B h

AF = h√3

BF = h/√3

Calculation:

Let the height of the building be 'h' and its foot be 'F'.

Let the top of the building be 'T'.

⇒ In right-angled ΔTFA:

⇒ tan(30°) = TF ÷ AF

⇒ 1 / √3 = h ÷ AF

⇒ AF = h√3

⇒ In right-angled ΔTFB:

⇒ tan(60°) = TF ÷ BF

⇒ √3 = h ÷ BF

⇒ BF = h / √3

⇒ Required ratio = AF : BF

⇒ AF : BF = h√3 : (h / √3)

⇒ AF : BF = √3 × √3 : 1

⇒ AF : BF = 3 : 1

∴ The correct answer is 3 : 1.

Additional Information

Complementary Angles of Elevation

If the angles of elevation of the top of a tower from two points at distances 'a' and 'b' from its foot are complementary (sum to 90°), the height of the tower is given by h = √(a × b).

Angle of Elevation and Depression

The angle of elevation of an object as seen by an observer is numerically equal to the angle of depression of the observer as seen from the object, owing to alternating interior angles between parallel horizontal planes.

30°-60°-90° Triangle Theorem

In a right-angled triangle with angles 30°, 60°, and 90°, the sides opposite to these angles are strictly proportional to the ratio 1 : √3 : 2 respectively. This property frequently aids calculation speed in Heights and Distance problems.

Shortcut Trick

Diameter of hemisphere (d) = √(2 × CSA ÷ π)

⇒ d = √(2 × 154 × 7 ÷ 22)

⇒ d = √98 ≈ 9.9 cm

∴ The correct answer is 9.9 cm.

Alternate Method

 r    d = ? Curved Surface Area = 154 cm²

Given:

Curved Surface Area (CSA) of hemisphere = 154 cm²

Formula Used:

CSA = 2πr²

Diameter (d) = 2r

Calculations:

⇒ 2 × (22/7) × r² = 154

⇒ (44/7) × r² = 154

⇒ r² = (154 × 7) ÷ 44

⇒ r² = 24.5

⇒ r = √24.5 ≈ 4.95 cm

⇒ d = 2 × 4.95 = 9.9 cm

∴ The correct answer is 9.9 cm.

Additional Information

Total Surface Area of Hemisphere

The total surface area includes the curved surface and the flat circular base: TSA = 3πr².

Volume of a Hemisphere

The volume is exactly half of a full sphere: V = (2/3)πr³.

Relation between Areas

For a given radius, TSA of a hemisphere is 1.5 times its Curved Surface Area.

Shortcut Trick

For any cone, Volume ∝ r² × h.

New radius is 1/3 of the original, and new height is 1/2 of the original.

Ratio of new volume to original volume = (1/3)² × (1/2) = 1/9 × 1/2 = 1/18.

This means 1 part of the original volume is used out of 18 total parts.

Unused fraction = 1 − 1/18 = 17/18.

∴ The correct answer is 17/18.

Alternate Method

h r Original Cone

Melted

h/2 r/3 New Cone

Given:

Original cone height = h, radius = r

New smaller cone height = h/2, radius = r/3

Formula Used:

Volume of a cone = 1/3 × π × radius² × height

Calculations:

Let the volume of the original cone be V₁.

⇒ V₁ = 1/3 × π × r² × h

Let the volume of the new smaller cone be V₂.

⇒ V₂ = 1/3 × π × (r/3)² × (h/2)

⇒ V₂ = 1/3 × π × (r² / 9) × (h/2)

⇒ V₂ = (1/18) × (1/3 × π × r² × h)

⇒ V₂ = V₁ / 18

This means the new recast cone uses only 1/18^th of the original volume.

Unused volume = Original volume − Used volume

⇒ Unused volume = V₁ − V₁ / 18

⇒ Unused volume = 17/18 × V₁

∴ The correct answer is 17/18.

Additional Information

Volume of Frustum of a Cone

When a cone is cut by a plane parallel to its base, the remaining solid is called a frustum. Volume = 1/3 × π × h × (R² + r² + R × r).

Melting and Recasting Principle

When a 3D solid is melted and recast into a different shape, the total volume remains constant unless some specific portion is explicitly stated as wasted or unused.

Scaling Factor in 3D Objects

If all linear dimensions of a standard 3D solid are uniformly scaled by a factor 'k', its total surface area is scaled by k² and its volume is scaled by k³.

Shortcut Trick

A regular hexagon is constructed by joining 6 identical equilateral triangles at its center point.

If the side length of the regular hexagon is 'a', then each of these 6 interior triangles will also have a side length of 'a'.

Therefore, the area of the regular hexagon is exactly 6 times the area of an equilateral triangle having the same side length.

∴ The correct answer is 6:1.

Alternate Method

Given:

Side length of the regular hexagon = a

Side length of the equilateral triangle = a

Formulas Used:

Area of an equilateral triangle = (√3 ÷ 4) × (side)²

Area of a regular hexagon = 6 × (√3 ÷ 4) × (side)²

Regular Hexagon Equilateral Triangle

1 2 3 4 5 6

Side = a

1 Side = a

Area = 6 × A Area = A

Calculations:

⇒ Area of the regular hexagon = 6 × (√3 ÷ 4) × a²

⇒ Area of the equilateral triangle = (√3 ÷ 4) × a²

⇒ Required Ratio = [6 × (√3 ÷ 4) × a²] ÷ [(√3 ÷ 4) × a²]

⇒ Required Ratio = 6 ÷ 1 = 6:1

∴ The correct answer is 6:1.

Additional Information

Properties of a Regular Hexagon

A regular hexagon has 6 equal sides and 6 equal interior angles (each measuring 120°). The sum of all its interior angles is exactly 720°.

Circumradius

The distance from the center of a regular hexagon to any of its outward vertices is exactly equal to its side length. Thus, its circumradius (R) = a.

Inradius

The perpendicular distance from the center to the midpoint of any side acts as the height of the internal equilateral triangles. The inradius (r) = (√3 ÷ 2) × a.

Perimeter to Area Efficiency

For a specifically given perimeter, a regular hexagon encloses a significantly larger internal area compared to a square or an equilateral triangle.

Shortcut Trick

Total height of the prism = Sum of 5 layers = 5 + 6 + 7 + 8 + 9 = 35 cm.

Area of equilateral triangular base = (√3 ÷ 4) × side² = (√3 ÷ 4) × 4² = 4√3 cm².

Total Volume = Base Area × Total Height = 4√3 × 35 = 140√3 cm³.

Substituting √3 ≈ 1.732, Volume = 140 × 1.732 = 242.48 cm³.

∴ The correct answer is 242.48 cm³.

Alternate Method

Given:

⇒ Base of the prism = Equilateral triangle with side (a) = 4 cm.

⇒ Heights of the 5 layers = 5 cm, 6 cm, 7 cm, 8 cm, and 9 cm.

Formula Used:

⇒ Area of an equilateral triangle = (√3 ÷ 4) × a²

⇒ Volume of a prism = Base Area × Total Height

4 cm 4 cm 4 cm Area = 4√3 cm² Equilateral Base

Height Layers

9 cm

8 cm

7 cm

6 cm

5 cm

Total = 35 cm

Calculation:

⇒ Total height of the prism (H) = 5 + 6 + 7 + 8 + 9 = 35 cm.

⇒ Base Area = (√3 ÷ 4) × 4²

⇒ Base Area = (√3 ÷ 4) × 16 = 4√3 cm².

⇒ Total Volume = 4√3 × 35 = 140√3 cm³.

⇒ Substituting the given value of √3 ≈ 1.732:

⇒ Total Volume = 140 × 1.732 = 242.48 cm³.

∴ The correct answer is 242.48 cm³.

Additional Information

Volume and Surface Area of a Prism

Volume = Base Area × Height. Total Surface Area = Lateral Surface Area + 2 × Base Area.

Equilateral Triangle Properties

For a side length 'a', Area = (√3 ÷ 4)a², Height = (√3 ÷ 2)a, and Perimeter = 3a.

Sum of an Arithmetic Progression

The sum of 'n' terms is given by S_n = (n ÷ 2)[2a + (n−1)d], or simply (n ÷ 2) × (First term + Last term).

Shortcut Trick

The equation of a line passing through a given point must satisfy the coordinates of that point.

Substitute x = −1 and y = −2 in the given options to verify.

Option 1: −3(−1) − 5 = 3 − 5 = −2 (Satisfies the point)

Other options will not yield −2.

∴ The correct answer is y = −3x − 5.

Alternate Method

Given:

Coordinates of the point = (−1, −2)

Slope of the line (m) = −3

Formula Used:

The equation of a line passing through point (x₁, y₁) with slope m is given by: y − y₁ = m(x − x₁)

X Y O (-1, -2) y = -3x - 5

Calculations:

⇒ y − y₁ = m(x − x₁)

⇒ y − (−2) = −3[x − (−1)]

⇒ y + 2 = −3(x + 1)

⇒ y + 2 = −3x − 3

⇒ y = −3x − 3 − 2

⇒ y = −3x − 5

∴ The correct answer is y = −3x − 5.

Additional Information

Slope-Intercept Form

The equation of a straight line with slope m and y-intercept c is given by the formula y = mx + c.

Two-Point Form

The equation of a line passing through two distinct points (x₁, y₁) and (x₂, y₂) is y − y₁ = [(y₂ − y₁) ÷ (x₂ − x₁)](x − x₁).

Intercept Form

A line that has x-intercept a and y-intercept b on the coordinate axes follows the equation x/a + y/b = 1.

Shortcut Trick

For any regular polygon, the sum of all exterior angles is always exactly 360°.

Number of sides = Sum of all exterior angles ÷ Measure of each exterior angle

⇒ Number of sides = 360° ÷ 60°

⇒ Number of sides = 6

∴ The correct answer is 6 sides.

Alternate Method

Given:

Measure of each exterior angle = 60°

Formula Used:

Interior Angle + Exterior Angle = 180°

Sum of interior angles of a polygon = (n − 2) × 180°, where n is the number of sides

Each interior angle = [(n − 2) × 180°] ÷ n

60° Exterior Angle

120° Interior Angle

Regular Hexagon (6 Sides)

Calculations:

⇒ Each interior angle = 180° − 60° = 120°

⇒ [(n − 2) × 180°] ÷ n = 120°

⇒ (n − 2) × 180° = 120°n

⇒ 180°n − 360° = 120°n

⇒ 180°n − 120°n = 360°

⇒ 60°n = 360°

⇒ n = 360° ÷ 60°

⇒ n = 6

∴ The correct answer is 6 sides.

Additional Information

Sum of Interior Angles

For a polygon with n sides, the sum of all interior angles is always given by (n − 2) × 180°.

Number of Diagonals

The total number of distinct diagonals that can be drawn within a polygon of n sides is given by the formula n(n − 3) ÷ 2.

Linear Pair Property

At any given vertex of a convex polygon, the interior angle and its adjacent exterior angle form a straight line, summing exactly to 180°.

Shortcut Trick

If G is the centroid of a triangle PQR, the coordinates of the third vertex R can be calculated directly using the relation: R = 3 × G − (P + Q)

x-coordinate = 3 × 1 − (3 − 2) = 3 − 1 = 2

y-coordinate = 3 × 2 − (−1 + 4) = 6 − 3 = 3

∴ The correct answer is (2, 3).

Alternate Method

Given:

Centroid of the triangle, G(x, y) = (1, 2)

Vertices, P(x₁, y₁) = (3, −1) and Q(x₂, y₂) = (−2, 4)

Let the third vertex be R(x₃, y₃).

Formula Used:

Centroid G(x, y) = [(x₁ + x₂ + x₃) ÷ 3, (y₁ + y₂ + y₃) ÷ 3]

G(1, 2)

Q(−2, 4)

P(3, −1)

R(x, y)

Calculation:

For the x-coordinate:

⇒ 1 = (3 − 2 + x₃) ÷ 3

⇒ 3 = 1 + x₃

⇒ x₃ = 2

For the y-coordinate:

⇒ 2 = (−1 + 4 + y₃) ÷ 3

⇒ 6 = 3 + y₃

⇒ y₃ = 3

∴ The correct answer is (2, 3).

Additional Information

1. Centroid of a Triangle

It is the intersection point of all three medians of the triangle. The centroid divides each median in the ratio 2:1. The formula is [(x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3].

2. Incenter of a Triangle

It is the point of intersection of internal angle bisectors. The coordinates are [(ax₁+bx₂+cx₃)/(a+b+c), (ay₁+by₂+cy₃)/(a+b+c)] where a, b, and c are side lengths.

3. Area of a Triangle

The area of a triangle given its three vertices is (1/2) × |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|.

Shortcut Trick

We know the direct algebraic identity: (a + b)² − (a − b)² = 4ab

Here, let a = √11 and b = √7

⇒ Result = 4 × √11 × √7

⇒ Result = 4√77

∴ The correct answer is 4√77.

Alternate Method

Given:

Expression: (√11 + √7)² − (√11 − √7)²

Formula Used:

(a + b)² = a² + b² + 2ab

(a − b)² = a² + b² − 2ab

Calculations:

⇒ Expand the first term: (√11 + √7)² = (√11)² + (√7)² + 2 × √11 × √7

⇒ (√11 + √7)² = 11 + 7 + 2√77 = 18 + 2√77

⇒ Expand the second term: (√11 − √7)² = (√11)² + (√7)² − 2 × √11 × √7

⇒ (√11 − √7)² = 11 + 7 − 2√77 = 18 − 2√77

⇒ Subtract the second expression from the first: (18 + 2√77) − (18 − 2√77)

⇒ 18 + 2√77 − 18 + 2√77

⇒ 4√77

∴ The correct answer is 4√77.

Additional Information

Sum of Squares Identity

Another useful variation is the sum of the squares: (a + b)² + (a − b)² = 2(a² + b²).

Difference of Squares

A very commonly used identity in simplification is a² − b² = (a + b)(a − b).

Square Root Properties

For any positive numbers x and y, the product of their square roots is √x × √y = √(xy) and squaring a root gives (√x)² = x.

Shortcut Trick The length of a Direct Common Tangent (DCT) is defined by the relation DCT2 = d2 − (r1 − r2)2. We are given that the length of the tangent is equal to the distance between the centers, which means DCT = d. For DCT2 to be equal to d2, the subtracted term (r1 − r2)2 must be exactly zero. This implies there is no difference between the lengths of their radii. ⇒ r1 − r2 = 0 ⇒ r1 = r2 ∴ The correct answer is r1 = r2.

Alternate Method Given: Radii of the two circles = r1 and r2 Distance between their centers = d Length of the Direct Common Tangent (L) = d Formula Used: The formula for the length of a Direct Common Tangent (L) is: L = √[d2 − (r1 − r2)2]

r₁ r₂ DCT = d d C₁ C₂

Calculations: ⇒ L = d (given in the question) ⇒ d = √[d2 − (r1 − r2)2] Squaring both sides: ⇒ d2 = d2 − (r1 − r2)2 ⇒ (r1 − r2)2 = d2 − d2 ⇒ (r1 − r2)2 = 0 Taking the square root on both sides: ⇒ r1 − r2 = 0 ⇒ r1 = r2 ∴ The correct answer is r1 = r2.

Additional Information Transverse Common Tangent (TCT) The length of a Transverse Common Tangent is given by TCT = √[d2 − (r1 + r2)2]. This tangent completely crosses the line segment joining the centers of the two circles. Number of Common Tangents When two circles do not touch or intersect (d > r1 + r2), they have exactly 4 common tangents (2 direct and 2 transverse). If they touch externally (d = r1 + r2), there are exactly 3 common tangents. Tangent Perpendicularity Theorem A line drawn from the center of a circle to the point of tangency makes a perpendicular angle (90 degrees) with the tangent line at that exact point.

Shortcut Trick

In a circle, the perpendicular from the center to a chord bisects it.

⇒ Half-chord = 2√21 ÷ 2 = √21 cm

⇒ Radius = √(Distance² + Half-chord²)

⇒ Radius = √(5² + (√21)²) = √(25 + 21)

⇒ Radius = √46 ≈ 6.78 cm

∴ The correct answer is 6.78 cm.

Alternate Method

O A B D

5 cm r √21 cm

2√21 cm

Given:

Length of the chord (AB) = 2√21 cm

Distance of the chord from the center (OD) = 5 cm

Formula Used:

The perpendicular drawn from the center of a circle to a chord bisects the chord.

By Pythagoras Theorem in right-angled ΔODB:

OB² = OD² + DB²

Calculations:

⇒ DB = AB ÷ 2 = 2√21 ÷ 2 = √21 cm

⇒ In ΔODB, OB is the radius (r)

⇒ r² = 5² + (√21)²

⇒ r² = 25 + 21

⇒ r² = 46

⇒ r = √46

⇒ r ≈ 6.78 cm

∴ The correct answer is 6.78 cm.

Additional Information

Perpendicular to a Chord

A perpendicular drawn from the center of a circle to a chord bisects it into two equal halves. The converse is also true.

Equal Chords Property

Chords of equal length in a circle are always equidistant from the center of the circle.

Pythagorean Triplet

In right-angled triangles inscribed in circles, the radius, half-chord, and perpendicular distance always form a Pythagorean relation: r² = d² + (L÷2)².

Shortcut Trick

The given expression is sin²A + cos²A + 2sinAcosA.

Notice that this is the direct algebraic expansion of (sinA + cosA)².

Since we are given sinA + cosA = x, directly substitute x into the expression.

Thus, (x)² = x².

∴ The correct answer is x².

Alternate Method

Given:

sinA + cosA = x

Formula Used:

(a + b)² = a² + b² + 2ab

Calculations:

⇒ sinA + cosA = x

Squaring both sides of the equation, we get:

⇒ (sinA + cosA)² = x²

Expanding the left-hand side using the algebraic identity:

⇒ sin²A + cos²A + 2sinAcosA = x²

This matches the required expression perfectly, giving the direct value.

∴ The correct answer is x².

Additional Information

Fundamental Trigonometric Identities

sin²θ + cos²θ = 1 | sec²θ − tan²θ = 1 | cosec²θ − cot²θ = 1

Important Algebraic Identities

(a + b)² = a² + b² + 2ab | (a − b)² = a² + b² − 2ab | a² − b² = (a − b)(a + b)

Double Angle Formulas

sin(2θ) = 2sinθcosθ | cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ

Shortcut Trick

Given, sin x = cos x ⇒ tan x = 1 ⇒ x = 45°

We need to find the value of sin⁴x + cos⁴x.

⇒ (sin 45°)⁴ + (cos 45°)⁴

⇒ (1/√2)⁴ + (1/√2)⁴

⇒ 1/4 + 1/4 = 1/2

∴ The correct answer is 1/2.

Alternate Method

Given:

sin x = cos x

Formula Used:

sin²x + cos²x = 1

Calculation:

⇒ sin x = cos x

Squaring both sides:

⇒ sin²x = cos²x

Substitute cos²x with sin²x in the standard trigonometric identity:

⇒ sin²x + sin²x = 1

⇒ 2sin²x = 1 ⇒ sin²x = 1/2

Similarly, cos²x = 1/2

Now evaluate the required expression:

⇒ sin⁴x + cos⁴x = (sin²x)² + (cos²x)²

⇒ (1/2)² + (1/2)²

⇒ 1/4 + 1/4 = 2/4 = 1/2

∴ The correct answer is 1/2.

Additional Information

Key Trigonometric Identities

sin²θ + cos²θ = 1 | 1 + tan²θ = sec²θ | 1 + cot²θ = cosec²θ

Double Angle Formulas

sin 2θ = 2 sin θ cos θ | cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1

Trigonometric Values at 45°

sin 45° = 1/√2 | cos 45° = 1/√2 | tan 45° = 1 | cot 45° = 1

Shortcut Trick

If a + b + c = 0, the standard algebraic identity yields a³ + b³ + c³ = 3abc.

Here, a + b + c = 0.1 + 0.2 − 0.3 = 0, so the condition is satisfied.

We can directly substitute 3abc in place of the numerator.

⇒ Expression = (3abc) ÷ (3abc) = 1

∴ The correct answer is 1.

Alternate Method

Given:

a = 0.1, b = 0.2, c = −0.3

a + b + c = 0

Formula Used:

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

Calculations:

⇒ We need to evaluate: (a³ + b³ + c³) ÷ (3abc)

⇒ From the given values, a + b + c = 0.1 + 0.2 + (−0.3) = 0

⇒ Substituting a + b + c = 0 into the identity formula:

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

⇒ a³ + b³ + c³ − 3abc = 0

⇒ a³ + b³ + c³ = 3abc

⇒ Substitute a³ + b³ + c³ with 3abc in the given expression.

⇒ Value = (3abc) ÷ (3abc) = 1

∴ The correct answer is 1.

Additional Information

Standard Algebraic Identity

The foundational identity for sums of cubes is: a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca).

Special Case 1

If the sum of the variables is zero (a + b + c = 0), the right side simplifies to 0, which gives the direct result a³ + b³ + c³ = 3abc.

Alternative Form of the Identity

Another extremely useful form to quickly find values without extensive multiplication is: a³ + b³ + c³ − 3abc = 1/2 × (a + b + c) × [(a − b)² + (b − c)² + (c − a)²].

Shortcut Trick

In the given expression, we can simply factor out the constant from the numerator.

Expression = [(4p)³ + (4q)³] ÷ (p³ + q³)

⇒ [64p³ + 64q³] ÷ (p³ + q³)

⇒ 64 × (p³ + q³) ÷ (p³ + q³) = 64

Since the variable terms cancel out, the exact values of p and q are not required.

∴ The correct answer is 64.

Alternate Method

Given:

p = 0.8

q = 0.02

Formula Used:

(a × b)³ = a³ × b³

Calculation:

⇒ Expression = [(4p)³ + (4q)³] ÷ (p³ + q³)

⇒ Expression = [4³p³ + 4³q³] ÷ (p³ + q³)

⇒ Expression = [64p³ + 64q³] ÷ (p³ + q³)

⇒ Expression = 64(p³ + q³) ÷ (p³ + q³)

⇒ Expression = 64

∴ The correct answer is 64.

Additional Information

BODMAS Rule

Always follow the sequence of Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction while solving simplifications.

Algebraic Identities

a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²).

Indices Rule

When two variables are multiplied and raised to a power, the power applies to both: (ab)ⁿ = aⁿ × bⁿ.

Shortcut Trick

By applying the Componendo and Dividendo rule on (x + y) / (x − y) = 5 / 2:

x / y = (5 + 2) / (5 − 2) = 7 / 3

Substitute x = 7 and y = 3 directly into the required expression:

(x³ + y³) : (x³ − y³) = (7³ + 3³) : (7³ − 3³)

= (343 + 27) : (343 − 27) = 370 : 316 = 185 : 158

∴ The correct answer is 185 : 158.

Alternate Method

Given:

(x + y) : (x − y) = 5 : 2

Ratio Visualization:x7 parts (x = 7)y3 parts (y = 3)x : y = 7 : 3

Calculations:

⇒ (x + y) / (x − y) = 5 / 2

⇒ By cross-multiplying:

⇒ 2 × (x + y) = 5 × (x − y)

⇒ 2x + 2y = 5x − 5y

⇒ 2y + 5y = 5x − 2x

⇒ 7y = 3x

⇒ x / y = 7 / 3

⇒ Using direct substitution, let x = 7 and y = 3 (since terms are homogeneous, common parts cancel out)

⇒ Substitute these into the expression: (x³ + y³) : (x³ − y³)

⇒ (7³ + 3³) : (7³ − 3³)

⇒ (343 + 27) : (343 − 27)

⇒ 370 : 316

⇒ Dividing both sides by 2:

⇒ 185 : 158

∴ The correct answer is 185 : 158.

Additional Information

Componendo and Dividendo Rule

If a/b = c/d, applying componendo and dividendo gives (a + b) / (a − b) = (c + d) / (c − d). This rule is widely used to solve complex ratio equations instantly.

Algebraic Identities for Cubes

Sum of cubes: a³ + b³ = (a + b)(a² − ab + b²)

Difference of cubes: a³ − b³ = (a − b)(a² + ab + b²)

Properties of Ratios

Invertendo: If a:b = c:d, then b:a = d:c.

Alternendo: If a:b = c:d, then a:c = b:d.

Shortcut Trick

Using the reverse calculation method from the final result:

⇒ Step 1: Divide the final result by 15 → 41340 ÷ 15 = 2756

⇒ Step 2: Subtract from the initial value → 4052 − 2756 = 1296

⇒ Step 3: Take the square root of the difference → √1296 = 36

∴ The correct answer is 36.

Alternate Method

Given:

When the square of a number is subtracted from 4052 and then multiplied by 15, the result is 41340.

Calculation:

Let the unknown number be x.

According to the given condition:

⇒ (4052 − x²) × 15 = 41340

⇒ 4052 − x² = 41340 ÷ 15

⇒ 4052 − x² = 2756

⇒ x² = 4052 − 2756

⇒ x² = 1296

⇒ x = √1296

⇒ x = 36

∴ The correct answer is 36.

Additional Information

BODMAS Rule

Always follow the strict order of Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction while simplifying algebraic equations.

Finding Square Roots Quickly

To find √1296 logically, note that 30² = 900 and 40² = 1600. Since 1296 ends in 6, its square root must end in 4 or 6. Hence, it can be 34 or 36. Verifying 36 × 36 = 1296 gives the answer instantly.

Shortcut Trick

To compare square roots and decimals quickly, simply square both sides of each relation.

(i) (√10)² = 10 and (3.2)² = 10.24 ⇒ 10 < 10.24 (True)

(ii) (√11)² = 11 and (√12)² = 12 ⇒ 11 < 12 (True)

(iii) (√13)² = 13 and (√14)² = 14 ⇒ 13 < 14 (True)

Since all three inequalities are mathematically correct, all statements are true.

∴ The correct answer is (i), (ii) and (iii).

Alternate Method

Given:

Three relations to evaluate:

(i) √10 < 3.2

(ii) √11 < √12

(iii) √13 < √14

Method Used:

The square root function is strictly increasing for positive numbers. Hence, for any positive real numbers x and y, if x < y, then √x < √y.

Calculations:

⇒ Checking relation (i):

⇒ We need to verify if √10 < 3.2

⇒ Squaring both sides: (√10)² < (3.2)²

⇒ 10 < 10.24

⇒ This relation holds true, so statement (i) is correct.

⇒ Checking relation (ii):

⇒ We know the base numbers inequality: 11 < 12

⇒ Taking square roots on both sides: √11 < √12

⇒ This relation holds true, so statement (ii) is correct.

⇒ Checking relation (iii):

⇒ We know the base numbers inequality: 13 < 14

⇒ Taking square roots on both sides: √13 < √14

⇒ This relation holds true, so statement (iii) is correct.

⇒ Since all statements are independently true, combining them makes the final option correctly encompass all.

∴ The correct answer is (i), (ii) and (iii).

Additional Information

Monotonicity of Square Roots

For any non-negative numbers a and b, if a < b, then √a < √b. This property allows you to compare surds directly by checking their internal radicands without finding their exact decimal values.

Square Root Estimation Formula

The square root of a non-perfect square N can be approximated quickly using the formula: √N ≈ x + (N − x²) ÷ (2 × x), where x² is the nearest perfect square to N.

Comparing Mixed Surds

To compare expressions like a√b and c√d, bring all outer numbers completely inside the root: √(a² × b) and √(c² × d). Then compare the resulting internal numbers directly.

Shortcut Trick

By using the ratio method for successive changes:

CP : MP = 100 : 140 = 5 : 7

MP : SP = 100 : 80 = 5 : 4

Overall ratio CP : MP : SP = (5 × 5) : (7 × 5) : (7 × 4) = 25 : 35 : 28

28 parts = ₹560 ⇒ 1 part = 20

Cost Price (CP) = 25 parts = 25 × 20 = 500

∴ The correct answer is ₹500.

Alternate Method

Given:

Markup = 40%

Discount = 20%

Selling Price (SP) = ₹560

CP = ?

MP

SP = ₹560

+40% Markup

-20% Discount

Net Profit = +12%

Formula Used:

Selling Price (SP) = Marked Price (MP) − Discount

Calculations:

⇒ Let the Cost Price (CP) be 100x.

⇒ Marked Price (MP) = 100x + (40% of 100x) = 140x

⇒ Discount = 20% of 140x = (20 ÷ 100) × 140x = 28x

⇒ Selling Price (SP) = 140x − 28x = 112x

⇒ Given SP = ₹560

⇒ 112x = 560

⇒ x = 560 ÷ 112 = 5

⇒ Cost Price (CP) = 100 × 5 = ₹500

∴ The correct answer is ₹500.

Additional Information

Marked Price (MP) and Discount

Discount is always calculated on the Marked Price. The selling price is determined by subtracting the discount from the marked price: SP = MP − Discount.

Direct Relation Formula

The ratio of Marked Price to Cost Price can be directly evaluated using the formula: MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%).

Successive Discounts

If two successive discounts of a% and b% are given, the single equivalent discount is calculated as: (a + b − (a × b) ÷ 100)%.

Shortcut Trick

Efficiency ratio of A to B = 1 : 2.

B works with A only for the remaining 75% of the job.

B's contribution = (2 ÷ 3) × 75% = 50% of the total work.

Wages are distributed in direct proportion to the actual work completed by each person.

B's share = 50% of ₹840 = ₹420.

∴ The correct answer is ₹420.

Alternate Method

Given:

Total wages = ₹840

B is twice as efficient as A.

A works the whole time; B joins when 25% of the work is complete.

Formula Used:

Total Work = Efficiency × Time

Wage Share = (Individual Work Done ÷ Total Work) × Total Wages

Calculations:

Let Total Work be 100 units. To construct the standard table, let us assume A completes the whole work in 100 days.

⇒ Work done by A alone for the first 25% = 25 units.

⇒ Remaining work = 100 − 25 = 75 units.

⇒ Combined efficiency of A and B = 1 + 2 = 3 units/day.

⇒ Time taken by A and B to complete the remaining work = 75 ÷ 3 = 25 days.

⇒ Total work done by B = Efficiency × Time = 2 × 25 = 50 units.

⇒ Total work done by A = 25 (working alone) + (1 × 25) (working together) = 50 units.

⇒ Ratio of work done by A and B = 50 : 50 = 1 : 1.

⇒ B's share = (1 ÷ 2) × 840 = ₹420.

∴ The correct answer is ₹420.

Additional Information

Work and Wages Relation

Wages are always distributed in proportion to the actual amount of work done by each individual, not strictly by their time or efficiency alone.

When everyone works for the same time

If all individuals work together from the start to finish, their wages are distributed directly in the ratio of their efficiencies.

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken to complete a job. If A is twice as efficient as B, A will take half the time B takes to finish the same work independently.