Quantitative Aptitude PYQs

Shortcut Trick

Using the tangent ratio for both points A and B directly:

tan(x) = Height ÷ Distance A = 25√3 ÷ 25 = √3 ⇒ x = 60°

tan(y) = Height ÷ Distance B = 25√3 ÷ 75 = 1/√3 ⇒ y = 30°

Sum of angles = x + y = 60° + 30° = 90°.

∴ The correct answer is 90°.

Alternate Method

x°

y°

T F A B

25√3 m 25 m

75 m

Given:

Height of the vertical pole, TF = 25√3 m

Distance of point A from the base, FA = 25 m

Distance of point B from the base, FB = 75 m

Formula Used:

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

Calculations:

In right-angled triangle TFA:

⇒ tan(x°) = TF ÷ FA

⇒ tan(x°) = 25√3 ÷ 25

⇒ tan(x°) = √3

⇒ x° = 60°   (since tan 60° = √3)

In right-angled triangle TFB:

⇒ tan(y°) = TF ÷ FB

⇒ tan(y°) = 25√3 ÷ 75

⇒ tan(y°) = √3 ÷ 3 = 1 ÷ √3

⇒ y° = 30°   (since tan 30° = 1/√3)

We need to find the value of x° + y°:

⇒ x° + y° = 60° + 30° = 90°

∴ The correct answer is 90°.

Additional Information

Angle of Elevation

The angle formed by the line of sight with the horizontal plane when the object is placed above the horizontal level of the observer.

Complementary Angles in Heights & Distances

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

Standard Trigonometric Values

Important values to remember: tan(30°) = 1/√3, tan(45°) = 1, tan(60°) = √3. These are extremely useful for quick mental calculations in geometry.

Shortcut Trick

Two identical solid hemispheres perfectly combine to form exactly one full solid sphere of the same radius.

Therefore, the volume and dimensions match perfectly without any relative change in the radius.

Radius of the new sphere = Radius of the given hemisphere = 21 cm.

∴ The correct answer is 21 cm.

Alternate Method

Given:

Radius of each hemisphere (R) = 21 cm

Number of identical hemispheres melted = 2

R=21 Hemisphere 1 Vol: V

R=21 Hemisphere 2 Vol: V

→

r=? New Sphere Vol: 2V

Formula Used:

Volume of a hemisphere = (2/3) × π × R³

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

Calculations:

⇒ Total volume of the two melted hemispheres = 2 × [(2/3) × π × (21)³]

⇒ Total volume = (4/3) × π × (21)³

⇒ Let the radius of the newly cast sphere be r.

⇒ Volume of the new sphere = Total volume of the two hemispheres

⇒ (4/3) × π × r³ = (4/3) × π × (21)³

⇒ Canceling the common factor (4/3) × π from both sides:

⇒ r³ = 21³

⇒ r = 21 cm

∴ The correct answer is 21 cm.

Additional Information

Recasting of 3D Objects

When one or more solid shapes are melted and recast into a completely new shape, the total volume of the material strictly remains constant throughout the transformation.

Surface Area of a Hemisphere

Curved Surface Area (CSA) = 2 × π × R² | Total Surface Area (TSA) = 3 × π × R².

Surface Area of a Sphere

Total Surface Area (TSA) = 4 × π × R². A full sphere has only one continuous curved surface, so its CSA and TSA are exactly the same.

Shortcut Trick

Ratio of corresponding heights of the small cut cone to the large original cone = 10 / 30 = 1 / 3

For similar figures, the ratio of their volumes is the cube of the ratio of their heights.

Ratio of their volumes = (1 / 3)³ = 1 / 27

Volume of frustum = Volume of large cone − Volume of small cone = 27 − 1 = 26

Required ratio = Frustum Volume : Total Cone Volume = 26 : 27

∴ The correct answer is 26:27.

Alternate Method

Given:

Height of the original total cone (H) = 30 cm

A cut is made parallel to the base, 10 cm from the vertex.

Height of the smaller cone formed at the top (h) = 10 cm

Formula Used:

When a cone is cut parallel to its base, the smaller cone formed is similar to the original cone. The ratio of their volumes is equal to the cube of the ratio of their heights:

V₁ / V₂ = (h₁ / h₂)³

H = 30 cm

h = 10 cm

Small Cone Frustum Volume Ratio = Frustum : Total Cone

Calculations:

⇒ Volume of small cone (V_small) / Volume of original cone (V_large) = (10 / 30)³

⇒ V_small / V_large = (1 / 3)³

⇒ V_small / V_large = 1 / 27

⇒ Let the volume of the small cone be 1 unit and the large cone be 27 units.

⇒ Volume of the Frustum = V_large − V_small

⇒ Volume of the Frustum = 27 − 1 = 26 units

⇒ Ratio of the volume of the frustum to the entire cone = 26 : 27

∴ The correct answer is 26:27.

Additional Information

Similarity of Solid Figures

If two 3D shapes are similar with a scale factor of a:b, then the ratio of their corresponding surface areas is a² : b², and the ratio of their volumes is a³ : b³.

Volume of a Frustum

The volume of a frustum of a right circular cone is given by V = (1/3) × π × h × (R² + r² + R × r), where R and r are radii of the two parallel bases and h is the height of the frustum.

Slant Height of Frustum

The slant height of a frustum is computed as l = √(h² + (R − r)²), where h is the height and R, r are the radii of the larger and smaller circular bases respectively.

Shortcut Trick

Ratio of the number of sides of an octagon to a square = 8 : 4 = 2 : 1.

Ratio of their side lengths = 1 : 2 (since the octagon side is half of the square side).

Perimeter ratio = (Number of sides ratio) × (Side length ratio) = 2 × (1/2) = 1 : 1.

Since the perimeter ratio is exactly 1 : 1, the perimeters of both polygons are equal.

∴ The correct answer is 20 cm.

Alternate Method

Given:

Perimeter of the regular quadrilateral (square) = 20 cm

Side of the regular octagon = 1/2 × Side of the square

Formula Used:

Perimeter of a regular polygon = Number of sides × Length of one side

Perimeter = 20 cm Square (n=4) Side = 5 cm

Perimeter = 20 cm Octagon (n=8) Side = 2.5 cm

Side ÷ 2

Calculations:

⇒ Perimeter of the square = 4 × Side of square

⇒ 4 × Side of square = 20

⇒ Side of square = 5 cm

⇒ Side of regular octagon = 5 ÷ 2 = 2.5 cm

⇒ Perimeter of regular octagon = 8 × Side of octagon

⇒ Perimeter = 8 × 2.5 = 20 cm

∴ The correct answer is 20 cm.

Additional Information

Interior Angles of a Regular Polygon

Each interior angle = [(n − 2) × 180°] ÷ n, where n is the total number of sides.

Number of Diagonals

The total number of diagonals that can be drawn in an n-sided polygon = n(n − 3) ÷ 2.

Exterior Angles of a Polygon

The sum of all exterior angles of any convex polygon is always 360°. Each exterior angle of a regular polygon = 360° ÷ n.

Shortcut Trick

Total height of the column = Sum of the 4 sections in AP = 5 + 7 + 9 + 11 = 32 cm.

Base area of the regular hexagon = 6 × (√3 ÷ 4) × (Side)² = 6 × (√3 ÷ 4) × 100 = 150√3 cm².

Total Volume = Base Area × Total Height = 150√3 × 32 = 4800√3 cm³.

∴ The correct answer is 4800√3 cm³.

Alternate Method

Given:

Side length of the regular hexagon base, a = 10 cm

Heights of the 4 sections form an Arithmetic Progression (AP): First height = 5 cm, Common difference = 2 cm.

Formula Used:

Area of a regular hexagon = 3√3 ÷ 2 × a²

Volume of a prism = Area of base × Total height

a = 10 cm Hexagon Base

11 cm 9 cm 7 cm 5 cm

Total = 32 cm Column Heights

Calculations:

⇒ Total height (H) = Sum of heights of all 4 sections

⇒ H = 5 + 7 + 9 + 11 = 32 cm

⇒ Area of hexagonal base = 3√3 ÷ 2 × (10)²

⇒ Area of hexagonal base = 3√3 ÷ 2 × 100 = 150√3 cm²

⇒ Total Volume = Area of base × Total height

⇒ Volume = 150√3 × 32 = 4800√3 cm³

∴ The correct answer is 4800√3 cm³.

Additional Information

Volume of a Prism

The volume of any uniform prism is simply the cross-sectional (base) area multiplied by its perpendicular length or height. Formula: V = Area of base × Height.

Properties of a Regular Hexagon

A regular hexagon is composed of 6 congruent equilateral triangles. The area is calculated as Area = 3√3 ÷ 2 × a², and its perimeter is P = 6a.

Arithmetic Progression (AP)

A numerical sequence where the difference between consecutive terms is constant. The sum of the first n terms can be calculated using S_n = n/2 × [2a + (n−1)d].

Shortcut Trick

The coordinates of the midpoint are the simple average of the given x and y coordinates.

x-coordinate = (3 + 7) ÷ 2 = 5

y-coordinate = (4 + 8) ÷ 2 = 6

∴ The correct answer is (5, 6).

Alternate Method

Given:

First point (x₁, y₁) = (3, 4)

Second point (x₂, y₂) = (7, 8)

Formula Used:

Midpoint Formula: M(x, y) = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)

A (3, 4) B (7, 8) M (x, y)

Calculation:

⇒ x-coordinate = (3 + 7) ÷ 2 = 10 ÷ 2 = 5

⇒ y-coordinate = (4 + 8) ÷ 2 = 12 ÷ 2 = 6

⇒ The midpoint is (5, 6).

∴ The correct answer is (5, 6).

Additional Information

Distance Formula

The distance d between two points (x₁, y₁) and (x₂, y₂) is given by d = √((x₂ − x₁)² + (y₂ − y₁)²).

Section Formula

A point P(x, y) dividing a line segment joining two points internally in the ratio m:n has coordinates x = (mx₂ + nx₁) ÷ (m + n) and y = (my₂ + ny₁) ÷ (m + n).

Centroid of a Triangle

The centroid coordinates of a triangle with vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃) are given by ((x₁ + x₂ + x₃) ÷ 3, (y₁ + y₂ + y₃) ÷ 3).

Shortcut Trick

For any polygon with n sides, the direct formula for the number of diagonals is n(n − 3) ÷ 2.

For a hexagon, n = 6.

Number of diagonals = 6 × (6 − 3) ÷ 2 = 6 × 3 ÷ 2 = 9

∴ The correct answer is 9.

Alternate Method

Given:

A hexagon (number of sides, n = 6).

Formula Used:

Number of diagonals in a polygon of n sides = n(n − 3) ÷ 2

A B C D E F

Calculations:

⇒ Number of diagonals = 6 × (6 − 3) ÷ 2

⇒ Number of diagonals = 6 × 3 ÷ 2

⇒ Number of diagonals = 18 ÷ 2 = 9

∴ The correct answer is 9.

Additional Information

Sum of Interior Angles

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

Sum of Exterior Angles

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

Interior Angle of a Regular Polygon

Each interior angle of a regular polygon with n sides can be calculated as [(n − 2) × 180°] ÷ n.

Shortcut Trick

The centroid G divides the median AD internally in the ratio 2:1.

Using the inverse section formula: 3G = 2D + A ⇒ 2D = 3G − A

⇒ D_x = (3 × 4 − 2) ÷ 2 = 10 ÷ 2 = 5

⇒ D_y = (3 × 5 − 3) ÷ 2 = 12 ÷ 2 = 6

∴ The correct answer is (5, 6).

Alternate Method

Given:

Centroid G = (4, 5)

Vertex A = (2, 3)

D is the midpoint of BC, meaning AD is the median of triangle ABC.

A(2, 3) G(4, 5) D(x, y) B C 2 1

Formula Used:

The centroid divides the median in the ratio 2:1 internally from the vertex to the midpoint of the opposite side.

Section formula: P(x, y) = [(m × x₂ + n × x₁) ÷ (m + n), (m × y₂ + n × y₁) ÷ (m + n)]

Calculations:

⇒ Let the coordinates of D be (x, y).

⇒ Here, ratio m:n = 2:1, (x₁, y₁) = (2, 3) and (x₂, y₂) = (x, y).

⇒ x-coordinate of G = 4 = (2 × x + 1 × 2) ÷ (2 + 1)

⇒ 4 × 3 = 2x + 2

⇒ 12 − 2 = 2x

⇒ x = 10 ÷ 2 = 5

⇒ y-coordinate of G = 5 = (2 × y + 1 × 3) ÷ (2 + 1)

⇒ 5 × 3 = 2y + 3

⇒ 15 − 3 = 2y

⇒ y = 12 ÷ 2 = 6

∴ The correct answer is (5, 6).

Additional Information

Centroid of a Triangle

The centroid is the point of intersection of the three medians of the triangle. It divides each median in the ratio 2:1, with the larger part towards the vertex.

Coordinates of Centroid

If the vertices of a triangle are (x₁, y₁), (x₂, y₂), and (x₃, y₃), the centroid G is given by ((x₁ + x₂ + x₃) ÷ 3, (y₁ + y₂ + y₃) ÷ 3).

Midpoint Formula

If a line segment joins two points (x₁, y₁) and (x₂, y₂), its midpoint M is given by ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2).

Shortcut Trick

For equations involving algebraic square roots, substituting the given options is the fastest method.

Let us check Option 2: x = 6.

LHS = √(6 + 3) + √(6 − 2)

LHS = √9 + √4 = 3 + 2 = 5

Since LHS = RHS, x = 6 is the correct value.

∴ The correct answer is 6.

Alternate Method

Given:

√(x + 3) + √(x − 2) = 5

Method:

Let a = √(x + 3) and b = √(x − 2).

⇒ a + b = 5 . . . . (Equation 1)

Now, finding the difference of their squares:

⇒ a² − b² = (x + 3) − (x − 2)

⇒ a² − b² = x + 3 − x + 2 = 5

Using the algebraic identity a² − b² = (a − b)(a + b):

⇒ (a − b) × (a + b) = 5

⇒ (a − b) × 5 = 5

⇒ a − b = 1 . . . . (Equation 2)

Adding Equation 1 and Equation 2:

⇒ (a + b) + (a − b) = 5 + 1

⇒ 2a = 6

⇒ a = 3

Substituting the value of a back into our initial assumption:

⇒ √(x + 3) = 3

Squaring both sides:

⇒ x + 3 = 9

⇒ x = 9 − 3

⇒ x = 6

∴ The correct answer is 6.

Additional Information

Algebraic Identities

The difference of squares formula a² − b² = (a − b)(a + b) is extremely useful to eliminate surds when the difference of the terms under the root is a constant.

Extraneous Roots

When solving equations by squaring both sides, extraneous roots (false solutions) can be introduced. Always substitute the final answer back into the original equation to verify.

Surds and Indices

A surd is an unresolved radical, such as √2 or √3. Two surds of the form √a + √b and √a − √b are called conjugate surds, and their product is a rational number.

Shortcut Trick

For a Transverse Common Tangent, the distance (d), TCT length, and sum of radii (R + r) form a right-angled triangle.

By Pythagoras theorem: (R + r)² + TCT² = d²

Given TCT = 9 cm and d = 15 cm, this perfectly forms a 9-12-15 right triangle triplet.

Therefore, R + r = 12 cm.

Since R = 8 cm, r = 12 − 8 = 4 cm.

∴ The correct answer is 4 cm.

Alternate Method

Given:

Length of Transverse Common Tangent (TCT) = 9 cm

Distance between centers (d) = 15 cm

Radius of larger circle (R) = 8 cm

Let the radius of the smaller circle be r.

8 cm r TCT = 9 cm d = 15 cm

Formula Used:

Length of TCT = √(d² − (R + r)²)

Calculations:

⇒ 9 = √(15² − (8 + r)²)

⇒ 9² = 15² − (8 + r)²

⇒ 81 = 225 − (8 + r)²

⇒ (8 + r)² = 225 − 81

⇒ (8 + r)² = 144

⇒ 8 + r = 12

⇒ r = 12 − 8 = 4 cm

∴ The correct answer is 4 cm.

Additional Information

Direct Common Tangent (DCT)

A tangent that touches two circles and does not cross the line segment joining their centers. The length formula is DCT = √(d² − (R − r)²).

Transverse Common Tangent (TCT)

A tangent that crosses the line joining the centers of the two circles. Its length is always less than the length of the Direct Common Tangent for the same set of circles.

Number of Common Tangents

When two circles do not intersect and are strictly outside each other, there can be exactly 4 common tangents drawn between them (2 direct and 2 transverse).

Shortcut Trick

Angles subtended by the same arc (Arc AD) are always equal: ∠ABD = ∠ACD = 40°.

Angle formed in a semicircle by the diameter AB is a right angle, meaning ∠ADB = 90°.

In ΔABD, the sum of all interior angles must be 180°.

⇒ ∠BAD = 180° − (90° + 40°) = 50°.

∴ The correct answer is 50°.

Alternate Method

Cyclic Quadrilateral ABCD

A B C D O

40° 40° ?

Given:

⇒ AB is the diameter of the circle.

⇒ ABCD is a cyclic quadrilateral.

⇒ ∠ACD = 40°.

Formulas & Properties Used:

⇒ Angles subtended by the same arc at the circumference are equal.

⇒ The angle subtended by a diameter at any point on the circumference is 90°.

⇒ The sum of the interior angles of a triangle is always 180°.

Calculation:

⇒ Notice that both ∠ACD and ∠ABD are formed by joining the points A and D to the circumference.

⇒ Thus, they are in the same segment of the circle, making them equal.

⇒ ∠ABD = ∠ACD = 40°.

⇒ Since AB is the diameter passing through the center O, the angle it makes at the boundary is 90°.

⇒ Therefore, ∠ADB = 90°.

⇒ Now, in right-angled ΔABD, apply the angle sum property:

⇒ ∠BAD + ∠ABD + ∠ADB = 180°

⇒ ∠BAD + 40° + 90° = 180°

⇒ ∠BAD + 130° = 180°

⇒ ∠BAD = 180° − 130° = 50°.

∴ The correct answer is 50°.

Additional Information

Cyclic Quadrilateral Property

A quadrilateral is called a cyclic quadrilateral if all four of its vertices lie on the circumference of a single circle. The sum of any pair of opposite angles is exactly 180° (e.g., ∠A + ∠C = 180° and ∠B + ∠D = 180°).

Angle in a Semicircle

A foundational circle theorem dictates that the angle subtended by the diameter at any point on the boundary of the circle is always a right angle (90°).

Angles in the Same Segment

When two or more angles are subtended by the exact same arc at any point on the remaining part of the circle, those angles are completely identical in measure.

Shortcut Trick

Substitute sin² A = 1 − cos² A directly into the equation.

⇒ (1 − cos² A) − cos² A = 1/2

⇒ 1 − 2cos² A = 1/2

⇒ 2cos² A = 1/2

⇒ cos² A = 1/4

∴ The correct answer is 1/4.

Alternate Method

Given:

sin² A − cos² A = 1/2

Formula Used:

sin² A + cos² A = 1

Calculations:

⇒ sin² A + cos² A = 1

⇒ sin² A = 1 − cos² A

Substitute the value of sin² A into the given equation:

⇒ (1 − cos² A) − cos² A = 1/2

⇒ 1 − 2cos² A = 1/2

⇒ 2cos² A = 1 − 1/2

⇒ 2cos² A = 1/2

⇒ cos² A = (1/2) × (1/2)

⇒ cos² A = 1/4

∴ The correct answer is 1/4.

Additional Information

Basic Trigonometric Identities

1. sin² θ + cos² θ = 1

2. 1 + tan² θ = sec² θ

3. 1 + cot² θ = cosec² θ

Double Angle Formulas

cos 2θ = cos² θ − sin² θ = 2cos² θ − 1 = 1 − 2sin² θ

Reciprocal Identities

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

Shortcut Trick

Directly apply the identity: cosec²A = 1 + cot²A

Square the given value: cot²A = (x + 1/x)² = x² + 1/x² + 2

Substitute to find 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:

cosec²A − cot²A = 1 ⇒ cosec²A = 1 + cot²A

(a + b)² = a² + b² + 2ab

Calculations:

⇒ cosec²A = 1 + (x + 1/x)²

⇒ cosec²A = 1 + x² + 1/x² + (2 × x × 1/x)

⇒ cosec²A = 1 + x² + 1/x² + 2

⇒ cosec²A = x² + 1/x² + 3

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

Additional Information

Pythagorean Trigonometric Identities

sin²A + cos²A = 1, sec²A − tan²A = 1, and cosec²A − cot²A = 1.

Basic Algebraic Expansions

(a + b)² = a² + b² + 2ab and (a − b)² = a² + b² − 2ab.

Reciprocal Trigonometric Ratios

sin A = 1 ÷ cosec A | cos A = 1 ÷ sec A | tan A = 1 ÷ cot A.

Shortcut Trick

Observe that each term in the denominator is exactly twice the corresponding term in the numerator.

We know that (2x)³ = 8x³.

By factoring out 2³ = 8 from the denominator, the remaining expression perfectly cancels out the numerator.

Value = 1 ÷ 2³ = 1/8 = 0.125

∴ The correct answer is 0.125.

Alternate Method

Given:

Expression = (0.12³ + 0.06³) ÷ (0.24³ + 0.12³)

Calculations:

⇒ Let's express the denominator terms as multiples of the numerator terms.

⇒ Denominator = (2 × 0.12)³ + (2 × 0.06)³

⇒ Using the exponent rule (a × b)ⁿ = aⁿ × bⁿ:

⇒ Denominator = (2³ × 0.12³) + (2³ × 0.06³)

⇒ Taking 2³ common from both terms:

⇒ Denominator = 8 × (0.12³ + 0.06³)

⇒ Now substitute the denominator back into the given expression:

⇒ Expression = (0.12³ + 0.06³) ÷ [8 × (0.12³ + 0.06³)]

⇒ The term (0.12³ + 0.06³) cancels out entirely from both numerator and denominator.

⇒ Expression = 1 / 8

⇒ Expression = 0.125

∴ The correct answer is 0.125.

Additional Information

Laws of Exponents

When multiplying two terms with the same power, the rule (a × b)ⁿ = aⁿ × bⁿ is applied. This rule simplifies calculations by allowing you to factor out common bases efficiently.

Fraction to Decimal Conversion

It is helpful to remember common fraction conversions to save calculation time during exams: 1/2 = 0.5, 1/4 = 0.25, and 1/8 = 0.125.

Algebraic Identity for Cubes

Sum of cubes identity: a³ + b³ = (a + b)(a² − ab + b²). While not needed in this specific shortcut, it is frequently used to simplify complex polynomial expressions.

Shortcut Trick

Observe the relation between corresponding terms of the numerator and the denominator.

Each term in the numerator is exactly 5 times its corresponding term in the denominator (0.5 = 5 × 0.1, 0.05 = 5 × 0.01, etc.).

Since all terms are cubed, we can safely factor out 5³ from the entire numerator.

The remaining expression perfectly matches the denominator and completely cancels out.

Result = 5³ = 125.

∴ The correct answer is 125.

Alternate Method

Given:

[(0.5)³ + (0.05)³ + (0.005)³] ÷ [(0.1)³ + (0.01)³ + (0.001)³]

Calculations:

⇒ Let us express the numerator values as direct multiples of the denominator values:

⇒ 0.5 = 5 × 0.1

⇒ 0.05 = 5 × 0.01

⇒ 0.005 = 5 × 0.001

⇒ Substituting these rewritten values into the main expression:

⇒ [(5 × 0.1)³ + (5 × 0.01)³ + (5 × 0.001)³] ÷ [(0.1)³ + (0.01)³ + (0.001)³]

⇒ Expanding the cubes using the rule (a × b)³ = a³ × b³:

⇒ [5³ × (0.1)³ + 5³ × (0.01)³ + 5³ × (0.001)³] ÷ [(0.1)³ + (0.01)³ + (0.001)³]

⇒ Taking 5³ as a common factor from the numerator:

⇒ 5³ × [(0.1)³ + (0.01)³ + (0.001)³] ÷ [(0.1)³ + (0.01)³ + (0.001)³]

⇒ The common bracket term cancels out completely.

⇒ Remaining value = 5³ = 125

∴ The correct answer is 125.

Additional Information

Laws of Exponents

When a product of numbers is raised to a power, the power is smoothly distributed to each multiplier: (a × b)ⁿ = aⁿ × bⁿ.

BODMAS Rule

Simplification of arithmetic expressions always follows the strict sequence: Brackets, Orders (Powers/Roots), Division, Multiplication, Addition, and Subtraction.

Decimal Expansion Tricks

Avoid directly calculating large decimal powers, which consumes time and leads to errors. Always look for common ratios to extract as common factors first.

Shortcut Trick

Directly equate the percentages:

30 × m = 18 × n

m / n = 18 / 30

m / n = 3 / 5

∴ The correct answer is 3 : 5.

Alternate Method

Given:

30% of m = 18% of n

m (3 parts) n (5 parts)

Ratio m : n = 3 : 5

Calculations:

⇒ (30 / 100) × m = (18 / 100) × n

⇒ 30 × m = 18 × n

⇒ m / n = 18 / 30

Dividing the numerator and denominator by their greatest common divisor (6):

⇒ m / n = 3 / 5

⇒ m : n = 3 : 5

∴ The correct answer is 3 : 5.

Additional Information

Compound Ratio

For two given ratios a : b and c : d, the compound ratio is calculated as ac : bd.

Duplicate and Sub-duplicate Ratios

The duplicate ratio of a : b is a² : b². The sub-duplicate ratio is √a : √b.

Componendo and Dividendo

If two ratios are equal, such that a / b = c / d, then (a + b) / (a − b) = (c + d) / (c − d).

Shortcut Trick

Solve nested square roots step-by-step from the innermost root to the outside.

√289 = 17

√(127 + 17) = √144 = 12

√(388 + 12) = √400 = 20

∴ The correct answer is 20.

Alternate Method

Given:

Expression = √(388 + √(127 + √289))

Calculation:

⇒ First, evaluate the innermost square root: √289 = 17

⇒ Substitute this back into the expression:

⇒ Expression = √(388 + √(127 + 17))

⇒ Simplify the term inside the second square root: 127 + 17 = 144

⇒ Expression = √(388 + √144)

⇒ Evaluate the next square root: √144 = 12

⇒ Substitute this back into the expression:

⇒ Expression = √(388 + 12)

⇒ Expression = √400

⇒ Expression = 20

∴ The correct answer is 20.

Additional Information

Order of Operations for Nested Roots

Always start solving nested square roots from the innermost root and work your way outwards step by step.

Squares of Numbers ending in 3 or 7

Any perfect square that ends in the digit 9 will always have a square root ending in either 3 or 7. For example, 13² = 169 and 17² = 289.

Memorising Perfect Squares

Memorising perfect squares up to 30 helps to significantly speed up simplification problems in competitive exams.

Shortcut Trick For expressions like √a + √b and √c + √d where a + b = c + d, simply check their products ab and cd. Here, the sum under the roots is equal: 6 + 2 = 5 + 3 = 8. Now compare their products: 6 × 2 = 12 and 5 × 3 = 15. Since 12 < 15, the expression with the smaller product is smaller: √6 + √2 < √5 + √3. This means relationship (ii) is correct, while (i) and (iii) are incorrect. ∴ The correct answer is (i) and (iii). Alternate Method Given: Expression A = √6 + √2 Expression B = √5 + √3 Formula Used: (a + b)2 = a2 + b2 + 2ab Calculation: ⇒ Square both expressions to compare their magnitudes easily. ⇒ A2 = (√6 + √2)2 ⇒ A2 = 6 + 2 + 2√(6 × 2) ⇒ A2 = 8 + 2√12 ⇒ B2 = (√5 + √3)2 ⇒ B2 = 5 + 3 + 2√(5 × 3) ⇒ B2 = 8 + 2√15 ⇒ Both squared terms have the same constant sum 8. We only need to compare 2√12 and 2√15. ⇒ Since 12 < 15, we have √12 < √15. ⇒ Therefore, 8 + 2√12 < 8 + 2√15, which means A2 < B2. ⇒ Taking the square root of both sides gives A < B, so √6 + √2 < √5 + √3. ⇒ Thus, relationship (ii) is valid. The question asks to identify the incorrect relationships, which are (i) and (iii). ∴ The correct answer is (i) and (iii). Additional Information Comparison of Surds with Equal Sums When comparing expressions of the form √x + √y and √p + √q, if their sums are equal (x + y = p + q), the greater expression is the one with the larger product of the terms (xy vs pq). Comparison of Surds with Equal Differences When comparing √x − √y and √p − √q, if their differences are equal (x − y = p − q), the greater expression is the one with the smaller product of the terms. Squaring Inequalities For any positive real numbers a and b, if a2 > b2, then it is strictly true that a > b. This property allows converting surds into integer sums for direct and accurate comparison.

Shortcut Trick

Let the marked price (MP) of 1 novel be ₹100.

Cost price (CP) of 1 novel = (MP of 4 novels) ÷ 5 = 400 ÷ 5 = ₹80.

Selling price (SP) of 1 novel after 5% discount on MP = 100 − 5 = ₹95.

Profit on 1 novel = SP − CP = 95 − 80 = ₹15.

Profit % = (15 ÷ 80) × 100 = 1500 ÷ 80 = 18.75%.

∴ The correct answer is 18.75%.

Alternate Method

CP = ₹400 SP = ₹475 MP = ₹500

(For 5 Novels) (For 5 Novels) (For 5 Novels)

Profit = ₹75

Discount = 5%

Given:

⇒ Total purchased = 5 novels

⇒ CP of 5 novels = MP of 4 novels

⇒ Discount offered to customers = 5% on MP

Calculations:

⇒ Let the Marked Price (MP) of 1 novel be ₹100.

⇒ Total MP of 5 novels = 5 × 100 = ₹500.

⇒ Total CP of 5 novels = MP of 4 novels = 4 × 100 = ₹400.

⇒ Total discount on 5 novels = 5% of 500 = (5 ÷ 100) × 500 = ₹25.

⇒ Total SP of 5 novels = Total MP − Total discount = 500 − 25 = ₹475.

⇒ Profit earned = Total SP − Total CP = 475 − 400 = ₹75.

⇒ Profit % = (Profit ÷ Total CP) × 100

⇒ Profit % = (75 ÷ 400) × 100 = 75 ÷ 4 = 18.75%.

∴ The correct answer is 18.75%.

Additional Information

Marked Price and Discount

Discount is an allowance given to the buyer and is always calculated on the Marked Price (MP). SP = MP − Discount.

Successive Discounts

When two consecutive discounts of a% and b% are given, the single equivalent discount % = a + b − (a × b) ÷ 100.

Dishonest Dealer

If a trader sells goods at the cost price but uses false weights, his profit percentage is given by: (True Weight − False Weight) ÷ False Weight × 100.

Shortcut Trick

Let Total Work = LCM of 28 and 35 = 140 units.

Efficiency of A = 140 ÷ 28 = 5 units/day, Efficiency of B = 140 ÷ 35 = 4 units/day.

Work done by B in the last 17 days alone = 17 × 4 = 68 units.

Work done by A and B together before A left = 140 − 68 = 72 units.

Days A worked before leaving = 72 ÷ (5 + 4) = 72 ÷ 9 = 8 days.

∴ The correct answer is 8 days.

Alternate Method

Given:

Time taken by A alone = 28 days

Time taken by B alone = 35 days

B works alone for the final 17 days to finish the work.

Method Used: LCM Table Method

Calculations:

⇒ Total work to be completed = 140 units

⇒ Combined efficiency of A and B = 5 + 4 = 9 units/day

⇒ Work finished by B alone in the last 17 days = 17 × 4 = 68 units

⇒ Remaining work initially completed by both A and B together = 140 − 68 = 72 units

⇒ Number of days A and B worked together = Remaining Work ÷ Combined Efficiency

⇒ Days they worked together = 72 ÷ 9 = 8 days

⇒ Since A left after working together with B, A worked for exactly 8 days.

∴ The correct answer is 8 days.

Additional Information

Time and Efficiency Relationship

Efficiency is inversely proportional to time. If person A takes x days and person B takes y days, the ratio of their efficiencies is y : x.

Work and Wages

Wages are strictly distributed in proportion to the total work done by each individual. If workers work for the exact same duration, their wages can be directly distributed in the ratio of their respective efficiencies.

Alternate Days Work

When two people work on alternate days, group their combined effort into a 2-day cycle. Find the maximum integer multiple of this cycle that fits within the total work, then calculate the time for the remaining individual days.