Quantitative Aptitude PYQs

Shortcut Trick

For given x + 1/x = a, standard derived formulas are:

x² + 1/x² = a² − 2 ⇒ 4² − 2 = 14

x⁴ + 1/x⁴ = (14)² − 2 = 196 − 2 = 194

x⁶ + 1/x⁶ = (14)³ − 3 × 14 = 2744 − 42 = 2702

Substitute these values into the given expression directly: 2702 − 3(194) + 4(14) = 2702 − 582 + 56 = 2176.

∴ The correct answer is 2176.

Alternate Method

Given:

x + 1/x = 4

Formula Used:

(a + b)² = a² + b² + 2ab

(a + b)³ = a³ + b³ + 3ab(a + b)

Calculations:

⇒ Squaring both sides of the given equation:

⇒ (x + 1/x)² = 4²

⇒ x² + 1/x² + 2 = 16

⇒ x² + 1/x² = 14

⇒ Now, squaring x² + 1/x² = 14 on both sides:

⇒ (x² + 1/x²)² = 14²

⇒ x⁴ + 1/x⁴ + 2 = 196

⇒ x⁴ + 1/x⁴ = 194

⇒ Cubing the expression x² + 1/x² = 14 on both sides:

⇒ (x² + 1/x²)³ = 14³

⇒ x⁶ + 1/x⁶ + 3(x²)(1/x²)(x² + 1/x²) = 2744

⇒ x⁶ + 1/x⁶ + 3(14) = 2744

⇒ x⁶ + 1/x⁶ = 2744 − 42 = 2702

⇒ Now, we need to find the value of: x⁶ + 1/x⁶ − 3(x⁴ + 1/x⁴) + 4(x² + 1/x²)

Substituting the corresponding values we obtained:

⇒ 2702 − 3 × 194 + 4 × 14

⇒ 2702 − 582 + 56

⇒ 2120 + 56 = 2176

∴ The correct answer is 2176.

Additional Information

Square Identities

If x + 1/x = a, then x² + 1/x² = a² − 2. Conversely, if x − 1/x = a, then x² + 1/x² = a² + 2.

Cube Identities

If x + 1/x = a, then x³ + 1/x³ = a³ − 3a.

Alternative Power Identites

Powers like x⁵ + 1/x⁵ can be derived directly by multiplying (x² + 1/x²)(x³ + 1/x³) and subtracting (x + 1/x).

Shortcut Trick

When the number of years (T) equals the rate of interest (R), the formula for Rate simplifies directly to R = √(SI × 100 ÷ P).

⇒ R = √(432 × 100 ÷ 1200)

⇒ R = √(36) = 6

∴ The correct answer is 6%.

Alternate Method

Given:

Principal (P) = ₹1200

Simple Interest (SI) = ₹432

Time (T) = Rate (R)

Principal ₹1200

Rate = R% Time = R years

Interest ₹432

Formula Used:

Simple Interest (SI) = (P × R × T) ÷ 100

Calculations:

⇒ Let the numerical value of the rate of interest be R% and time be R years.

⇒ 432 = (1200 × R × R) ÷ 100

⇒ 432 = 12 × R²

⇒ R² = 432 ÷ 12

⇒ R² = 36

⇒ R = √36 = 6

∴ The correct answer is 6%.

Additional Information

Compound Interest Formula

When interest is compounded annually, Amount (A) = P(1 + R÷100)^T. The compound interest is calculated as CI = A − P.

SI vs CI Difference

The difference between Compound Interest and Simple Interest for exactly 2 years can be calculated directly using the formula: Difference = P(R÷100)².

Rule of 72

It is a quick calculation method used to estimate the number of years required to double the invested money at a given annual fixed interest rate. Years ≈ 72 ÷ R.

Shortcut Trick

Effective interest rates for 4 years: Scheme 1 = 32%, Scheme 2 = 40%

Let Scheme 2 investment be x, then Scheme 1 investment is (20000 − x).

40% of x − 32% of (20000 − x) = 2240

72% of x − 6400 = 2240 ⇒ x = 8640 ÷ 0.72 = 12000

∴ The correct answer is ₹12,000.

Alternate Method

Given:

Total Investment = ₹20,000

Scheme 1: Rate (R₁) = 8%, Time (T) = 4 years

Scheme 2: Rate (R₂) = 10%, Time (T) = 4 years

Difference in Interest (I₂ − I₁) = ₹2,240

Formula Used:

Simple Interest (SI) = (P × R × T) ÷ 100

Total Investment = ₹20000

Scheme 1: ₹(20000 − x) R = 8%, T = 4 yrs

Scheme 2: ₹x R = 10%, T = 4 yrs

I₂ − I₁ = ₹2240

Calculations:

⇒ Let the amount invested in Scheme 2 be ₹x.

⇒ Amount invested in Scheme 1 = ₹(20000 − x).

⇒ Interest from Scheme 2 (I₂) = (x × 10 × 4) ÷ 100 = 40x ÷ 100

⇒ Interest from Scheme 1 (I₁) = ((20000 − x) × 8 × 4) ÷ 100 = (640000 − 32x) ÷ 100

⇒ Given, I₂ − I₁ = 2240

⇒ (40x ÷ 100) − ((640000 − 32x) ÷ 100) = 2240

⇒ (40x − 640000 + 32x) ÷ 100 = 2240

⇒ 72x − 640000 = 224000

⇒ 72x = 864000

⇒ x = 12000

∴ The correct answer is ₹12,000.

Additional Information

Effective Rate of Interest

When simple interest is calculated for T years at R% per annum, the effective interest rate can be computed directly as (R × T)%.

Total Interest from Multiple Investments

When a principal P is split into multiple parts P₁, P₂, ..., P_n invested at rates R₁, R₂, ..., R_n, total SI = (P₁R₁T₁ + P₂R₂T₂ + ... + P_nR_nT_n) ÷ 100.

Compound Interest vs Simple Interest

Simple Interest is calculated only on the principal amount, whereas Compound Interest is calculated on the principal and the accumulated interest. For the 1^st year, SI and CI are equal if compounded annually.

Shortcut Trick

For a right-angled triangle with angles 30°, 60°, and 90°, the sides are in the ratio 1 : √3 : 2.

The side opposite to 30° is the cliff's height, which corresponds to 1 ratio part = 35 m.

The rope acts as the hypotenuse, which corresponds to 2 ratio parts.

Length of rope = 35 × 2 = 70 m.

∴ The correct answer is 70 m.

Alternate Method

30° 35 m (Cliff)

L = ? (Rope)

C B A

Given:

Vertical height of the cliff (Perpendicular) = 35 m

Angle of elevation of the rope with the ground (θ) = 30°

Formula Used:

sin(θ) = Perpendicular ÷ Hypotenuse

Calculations:

⇒ sin(30°) = 35 ÷ Length of rope

⇒ 1/2 = 35 ÷ Length of rope

⇒ Length of rope = 35 × 2

⇒ Length of rope = 70 m

∴ The correct answer is 70 m.

Additional Information

Heights and Distances (Trigonometry)

In right-angled triangles, trigonometric ratios relate angles to side lengths. For example, sin(θ) is used when the perpendicular and hypotenuse are involved, while tan(θ) relates perpendicular and base.

30°-60°-90° Triangle Theorem

In any 30°-60°-90° triangle, the lengths of the sides opposite these angles are always in the ratio of 1 : √3 : 2. This allows for direct calculation of sides without explicit trigonometric functions.

45°-45°-90° Triangle Theorem

In an isosceles right-angled triangle (45°-45°-90°), the sides opposite to the angles are in the proportion 1 : 1 : √2.

Shortcut Trick

Unoccupied Volume = Volume of Cube − Volume of Marble

⇒ Unoccupied Volume = a³ − (4/3)πr³

⇒ 10³ − (4/3) × 3.14 × 4³

⇒ 1000 − 267.94 ≈ 732.06

∴ The correct answer is 732.06 cm³.

Alternate Method

Given:

Side of the cube (a) = 10 cm

Radius of the marble (r) = 4 cm

Formula Used:

Volume of a Cube = a³

Volume of a Sphere = (4/3)πr³

Cube Cross-section

Marble

4 cm

10 cm

Calculations:

⇒ Volume of the cube = 10³ = 1000 cm³

⇒ Volume of the spherical marble = (4/3) × 3.14 × (4)³

⇒ Volume of the marble = (4/3) × 3.14 × 64

⇒ Volume of the marble = 803.84 ÷ 3 = 267.946 cm³

⇒ Unoccupied Volume = Volume of the cube − Volume of the marble

⇒ Unoccupied Volume = 1000 − 267.946 = 732.054 cm³

⇒ Rounding off to two decimal places, we get 732.06 cm³

∴ The correct answer is 732.06 cm³.

Additional Information

Volume of a Sphere

The total space occupied by a sphere in three-dimensional space is given by V = (4/3)πr³, where r is the radius.

Surface Area of a Sphere

The total curved surface area of a sphere is given by 4πr². For a sphere, the total surface area and curved surface area are exactly the same.

Inscribed Sphere in a Cube

The largest sphere that can fit perfectly inside a cube of side 'a' has a radius r = a/2. In this case, the maximum possible radius is 5 cm, allowing the 4 cm marble to easily fit inside.

Shortcut Trick

Since the given hemispheres are melted and recast, the total volume remains constant.

R³ = r₁³ + r₂³ ⇒ R³ = 2³ + 4³ = 8 + 64 = 72

Total Surface Area of new hemisphere = 3 × π × R²

TSA = 3 × π × (72)^2/3 ≈ 3 × 3.1416 × 17.307 ≈ 163.11 cm²

∴ The correct answer is 163 cm².

Alternate Method

Given:

Radius of the first hemisphere (r₁) = 2 cm

Radius of the second hemisphere (r₂) = 4 cm

Formula Used:

Volume of a hemisphere = 2/3 × π × r³

Total Surface Area (TSA) of a hemisphere = 3 × π × r²

r₁ = 2 cm r₂ = 4 cm R = ? TSA = 3πR² Melted V = V₁ + V₂

Calculations:

⇒ Volume of newly formed hemisphere = Sum of volumes of the two smaller hemispheres

⇒ 2/3 × π × R³ = (2/3 × π × 2³) + (2/3 × π × 4³)

⇒ R³ = 2³ + 4³

⇒ R³ = 8 + 64 = 72

⇒ R = (72)^1/3 cm

⇒ Total Surface Area (TSA) = 3 × π × R²

⇒ TSA = 3 × π × (72)^2/3

⇒ TSA ≈ 3 × 3.1416 × 17.307

⇒ TSA ≈ 163.11 cm²

∴ The correct answer is 163 cm².

Additional Information

Volume of a Hemisphere

The volume of a hemisphere is exactly half the volume of a full sphere. Formula: V = 2/3 × π × r³

Curved Surface Area (CSA)

The curved surface area includes only the rounded outer surface of the hemisphere. Formula: CSA = 2 × π × r²

Total Surface Area (TSA)

The total surface area includes the curved surface and the flat circular base at the top. Formula: TSA = 3 × π × r²

Recasting Objects

When solid metallic objects are melted and recast into another shape, the net volume always remains strictly conserved.

Shortcut Trick

Volume depends on three dimensions: Length, Breadth, and Height.

A 10% increase gives a multiplying factor of 1.1 for each dimension.

Effective multiplying factor for Volume = 1.1 × 1.1 × 1.1 = 1.331

Net percentage increase = (1.331 − 1) × 100 = 33.1%

∴ The correct answer is 33.1%.

Alternate Method

Given:

All three dimensions (Length, Breadth, and Height) of a cuboid are increased by 10%.

Formula Used:

Volume of cuboid = Length × Breadth × Height

Successive percentage formula = x + y + (x × y) ÷ 100

L

H

B

Initial Volume V = 100%

+10% in L, B, H

1.1L

1.1H

1.1B

New Volume 1.331V = 133.1%

Calculations:

⇒ Let the initial dimensions be L, B, and H.

⇒ Initial Volume (V₁) = L × B × H

⇒ After a 10% increase, the new dimensions are 1.1L, 1.1B, and 1.1H.

⇒ New Volume (V₂) = 1.1L × 1.1B × 1.1H

⇒ V₂ = 1.331(L × B × H) = 1.331V₁

⇒ Increase in volume = 1.331V₁ − V₁ = 0.331V₁

⇒ Percentage increase = (0.331V₁ ÷ V₁) × 100 = 33.1%

Successive Percentage Approach:

⇒ Successive increase of L and B = 10 + 10 + (10 × 10) ÷ 100 = 21%

⇒ Combining this 21% with the 10% increase in Height:

⇒ Final increase = 21 + 10 + (21 × 10) ÷ 100

⇒ Final increase = 31 + 2.1 = 33.1%

∴ The correct answer is 33.1%.

Additional Information

Successive Percentage Change Formula

When a value is changed by x% and then y%, the net percentage change is calculated as x + y + (x × y) ÷ 100. For three successive changes, apply the formula step-by-step.

Fraction Equivalent Method

A 10% increase means multiplying the base by 11/10. For a volume dependent on three dimensions, the net factor is (11/10)³ = 1331/1000, which represents a 33.1% overall increase.

Impact on Area vs. Volume

If all dimensions of an object scale by x%, its two-dimensional area scales by the factor (1 + x/100)² and its three-dimensional volume scales by (1 + x/100)³.

Shortcut Trick

When a cone is sliced parallel to its base, the smaller cone formed is similar to the original cone.

For similar 3D figures, Height Ratio = ∛(Volume Ratio)

Given Volume Ratio = 1 : 8

⇒ Height Ratio = ∛(1/8) = 1/2

∴ The correct answer is 1 : 2.

Alternate Method

Given:

Volume of the upper section (smaller cone), V₁ = 1/8 × Volume of the original cone (V₂)

Formula Used:

Volume of a cone, V = 1/3 × π × r² × h

V₁ V₂

h₁

h₂

V₁ = 1/8 × V₂

Calculations:

Let the radius and height of the smaller cone be r₁ and h₁ respectively.

Let the radius and height of the original cone be r₂ and h₂ respectively.

Since the slicing plane is parallel to the base, the smaller cone is completely similar to the original cone.

⇒ r₁ / r₂ = h₁ / h₂

Volume of smaller cone, V₁ = 1/3 × π × (r₁)² × h₁

Volume of original cone, V₂ = 1/3 × π × (r₂)² × h₂

⇒ V₁ / V₂ = [(r₁ / r₂)²] × (h₁ / h₂)

Substituting r₁ / r₂ = h₁ / h₂ in the above equation:

⇒ V₁ / V₂ = (h₁ / h₂)³

Given that V₁ / V₂ = 1 / 8 :

⇒ (h₁ / h₂)³ = 1 / 8

⇒ h₁ / h₂ = ∛(1 / 8) = 1 / 2

∴ The correct answer is 1 : 2.

Additional Information

Similar 3D Figures

For any two similar 3D geometric figures, the ratio of their volumes is always the cube of the ratio of their corresponding linear dimensions (such as heights, radii, or slant heights).

Volume of a Frustum

When a cone is sliced parallel to its base and the upper part is removed, the remaining lower part is called a frustum. The volume of a frustum is given by V = 1/3 × π × h × (R² + r² + R × r).

Volume Ratio Partition

If the height of a cone is divided in a 1 : 1 ratio from the apex, the volume ratio of the smaller upper cone to the entire cone is 1 : 8. Consequently, the volume ratio of the upper cone to the lower frustum is strictly 1 : 7.

Shortcut Trick

Using Pythagorean triplets (3, 4, 5) and (5, 12, 13):

For angle A: sin A = 3/5 ⇒ Perpendicular = 3, Hypotenuse = 5 ⇒ Base = 4 ∴ cos A = 4/5

For angle B: cos B = 12/13 ⇒ Base = 12, Hypotenuse = 13 ⇒ Perpendicular = 5 ∴ sin B = 5/13

sin(A + B) = sin A × cos B + cos A × sin B

⇒ (3/5) × (12/13) + (4/5) × (5/13) = 36/65 + 20/65 = 56/65

∴ The correct answer is 56/65.

Alternate Method

Given:

sin A = 3/5

cos B = 12/13

Angles A and B lie in the first quadrant (0, π/2), so all trigonometric ratios are positive.

A543Triangle AB13125Triangle B

Formula Used:

sin² θ + cos² θ = 1

sin(A + B) = sin A × cos B + cos A × sin B

Calculations:

⇒ cos A = √(1 − sin² A)

⇒ cos A = √(1 − (3/5)²) = √(1 − 9/25) = √(16/25) = 4/5

⇒ sin B = √(1 − cos² B)

⇒ sin B = √(1 − (12/13)²) = √(1 − 144/169) = √(25/169) = 5/13

Substitute these values into the compound angle formula:

⇒ sin(A + B) = (3/5) × (12/13) + (4/5) × (5/13)

⇒ sin(A + B) = 36/65 + 20/65

⇒ sin(A + B) = 56/65

∴ The correct answer is 56/65.

Additional Information

Trigonometric Addition Formulas

• sin(A − B) = sin A × cos B − cos A × sin B

• cos(A + B) = cos A × cos B − sin A × sin B

• cos(A − B) = cos A × cos B + sin A × sin B

Tangent Addition Formulas

• tan(A + B) = (tan A + tan B) ÷ (1 − tan A × tan B)

• tan(A − B) = (tan A − tan B) ÷ (1 + tan A × tan B)

Common Pythagorean Triplets

Memorizing standard triplets such as (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25) helps you instantly find the missing side of a right-angled triangle in exams.

Shortcut Trick

When the angle of elevation is 45°, the height of the object and the length of its shadow are exactly equal.

Because tan(45°) = 1, the perpendicular (height) and base (shadow) form an isosceles right triangle.

Given the length of the shadow = 40 m.

⇒ Height of the tower = 40 m.

∴ The correct answer is 40 m.

Alternate Method

45° Tower (h) Shadow = 40 m

A

B

C

Given:

Length of the shadow (Base BC) = 40 m

Angle of elevation of the sun (∠ACB) = 45°

Formula Used:

tan θ = Perpendicular ÷ Base

Calculations:

⇒ Let the height of the tower be h meters (AB).

⇒ In right-angled ΔABC, tan 45° = AB ÷ BC

⇒ 1 = h ÷ 40

⇒ h = 40 × 1

⇒ h = 40 m

∴ The correct answer is 40 m.

Additional Information

Standard Trigonometric Values

tan 30° = 1 ÷ √3, tan 45° = 1, tan 60° = √3. These are frequently used to relate heights and shadows in elevation problems.

45-45-90 Triangle Ratio

In a right-angled triangle with two 45° angles, the sides opposite the angles 45°, 45°, and 90° are in the ratio 1 : 1 : √2. Thus, the two perpendicular legs are always equal.

30-60-90 Triangle Ratio

In a right-angled triangle with 30° and 60° angles, the sides opposite to the 30°, 60°, and 90° angles are in the ratio 1 : √3 : 2 respectively.

Angle of Elevation vs Depression

The angle of elevation is measured upwards from the horizontal, whereas the angle of depression is measured downwards. They are mathematically equal due to alternate interior angles.

Shortcut Trick

Substitute the given point coordinates directly into the line equation.

⇒ y = mx + 2

⇒ 8 = m × 3 + 2

⇒ 3m = 8 − 2 = 6

⇒ m = 2

∴ The correct answer is 2.

Alternate Method

0 X Y

y-intercept (0, 2) Point (3, 8) Run = 3 Rise = 6

y = 2x + 2

Given:

Equation of the line: y = mx + 2

Point on the line = (3, 8)

Method Used:

If a point (x₁, y₁) lies on a straight line, substituting its coordinates must perfectly satisfy the algebraic equation of the line.

Calculations:

⇒ Substitute the coordinates x = 3 and y = 8 into the given equation.

⇒ 8 = m × 3 + 2

⇒ 8 − 2 = 3m

⇒ 6 = 3m

⇒ m = 6 ÷ 3

⇒ m = 2

∴ The correct answer is 2.

Additional Information

Slope-Intercept Form

The equation of a straight line is written as y = mx + c, where m represents the slope (gradient) of the line and c is the y-intercept (the point where the line crosses the Y-axis).

Slope Formula (Two Points)

If a line passes through two specified points (x₁, y₁) and (x₂, y₂), the slope is calculated as m = (y₂ − y₁) ÷ (x₂ − x₁).

Point-Slope Form

When the slope m and a single point (x₁, y₁) on the line are known, the equation can be formed using y − y₁ = m(x − x₁).

Shortcut Trick

For identical circles, the area of a sector is directly proportional to its central angle.

Convert the second sector's angle to degrees: 2π/3 radians = (2 × 180°) ÷ 3 = 120°.

Since both sectors have identical central angles (120°) and belong to the same circle (same radius), their areas are equal.

Therefore, the ratio of their areas is exactly 1:1.

∴ The correct answer is 1:1.

Alternate Method

120° 5 cm

First Sector

2π/3 rad 5 cm

Second Sector

Given:

Radius of the circle (r) = 5 cm

Central angle of 1^st sector (θ₁) = 120°

Central angle of 2^nd sector (θ₂) = 2π/3 radians

Formula Used:

Area of a sector (in degrees) = (θ/360°) × πr²

Area of a sector (in radians) = (1/2)r²θ

Calculations:

⇒ Area of 1^st sector = (120°/360°) × π × (5)²

⇒ Area of 1^st sector = (1/3) × 25π = 25π/3 cm²

⇒ Area of 2^nd sector = (1/2) × (5)² × (2π/3)

⇒ Area of 2^nd sector = (25/2) × (2π/3) = 25π/3 cm²

⇒ Ratio of their areas = (25π/3) ÷ (25π/3) = 1/1

∴ The correct answer is 1:1.

Additional Information

Area of Sector

A sector is a portion of a circle enclosed by two radii and an arc. If the central angle θ is in degrees, Area = (θ/360°) × πr². If θ is in radians, Area = (1/2)r²θ.

Length of an Arc

The arc length is the distance along the curved line making up the sector. For an angle in degrees, Length = (θ/360°) × 2πr. For radians, Length = rθ.

Radian to Degree Conversion

To convert an angle from radians to degrees, multiply by 180°/π. To convert from degrees to radians, multiply by π/180°.

Shortcut Trick

For any regular polygon, the sum of an interior angle and an exterior angle is 180°.

Exterior angle of a regular polygon = 360° ÷ n

⇒ Exterior angle = 360° ÷ 8 = 45°

⇒ Interior angle = 180° − 45° = 135°

∴ The correct answer is 135°.

Alternate Method

Given:

Number of sides in the regular polygon (octagon), n = 8

Formula Used:

Measure of each interior angle = (n − 2) × 180° ÷ n

135°

45° Regular Octagon n = 8

Calculations:

⇒ Sum of all interior angles = (8 − 2) × 180°

⇒ Sum of all interior angles = 6 × 180° = 1080°

⇒ Measure of each interior angle = 1080° ÷ 8

⇒ Measure of each interior angle = 135°

∴ The correct answer is 135°.

Additional Information

Sum of Interior Angles

For any convex polygon with n sides, the sum of its interior angles is (n − 2) × 180°.

Number of Diagonals

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

Exterior Angle Property

The sum of the exterior angles (taking one at each vertex) of any convex polygon is always exactly 360°, regardless of the number of sides.

Shortcut Trick

Sum of parallel sides = (2 × Area) ÷ Height = (2 × 240) ÷ 10 = 48 cm.

Ratio of parallel sides = 3 : 5 (Total 8 parts).

Value of 1 part = 48 ÷ 8 = 6 cm.

Lengths are 3 × 6 = 18 cm and 5 × 6 = 30 cm.

∴ The correct answer is 18 cm and 30 cm.

Alternate Method

Given:

Ratio of parallel sides = 3 : 5

Height (h) = 10 cm

Total Area = 240 cm²

Formula Used:

Area of Trapezoid = 1/2 × (Sum of parallel sides) × Height

3x 5x h = 10 cm Area = 240 cm²

⇒ Let the parallel sides be 3x and 5x.

⇒ Area = 1/2 × (3x + 5x) × 10 = 240

⇒ 1/2 × 8x × 10 = 240

⇒ 40x = 240

⇒ x = 240 ÷ 40 = 6

⇒ Length of first parallel side = 3 × 6 = 18 cm

⇒ Length of second parallel side = 5 × 6 = 30 cm

∴ The correct answer is 18 cm and 30 cm.

Additional Information

Isosceles Trapezoid

A trapezoid in which the non-parallel sides are equal in length. The base angles are equal and the diagonals are of equal length.

Midsegment of a Trapezoid

The line connecting the midpoints of the non-parallel sides. Its length is the average of the two parallel sides: Midsegment = (a + b) ÷ 2.

Area of Parallelogram

Area = Base × Height. Opposite sides are equal and parallel, meaning a parallelogram is considered a special type of trapezoid.

Shortcut Trick

If two triangles are congruent, their corresponding sides and perimeters are identical.

Perimeter of ΔABC = AB + BC + AC = 5 + 6 + 7 = 18 cm.

Since ΔXYZ is exactly the same shape and size as ΔABC, its perimeter remains exactly the same.

∴ The correct answer is 18 cm.

Alternate Method

Given:

Sides of ΔABC: AB = 5 cm, BC = 6 cm, AC = 7 cm.

ΔXYZ is congruent to ΔABC (ΔXYZ ≅ ΔABC).

Formula Used:

Perimeter of a triangle = Sum of all three sides.

For congruent triangles, corresponding sides are equal (CPCT).

A B C 5 cm 6 cm 7 cm

X Y Z 5 cm 6 cm 7 cm

≅ Perimeter = 5 + 6 + 7 = 18 cm

Calculations:

⇒ Since ΔXYZ ≅ ΔABC, their corresponding side lengths strictly match.

⇒ XY = AB = 5 cm

⇒ YZ = BC = 6 cm

⇒ XZ = AC = 7 cm

⇒ Perimeter of ΔXYZ = XY + YZ + XZ

⇒ Perimeter of ΔXYZ = 5 + 6 + 7 = 18 cm

∴ The correct answer is 18 cm.

Additional Information

Congruence of Triangles

Two triangles are congruent if all three corresponding sides and all three corresponding angles are equal in measure, meaning they superimpose exactly when placed over each other.

Conditions for Congruence

Triangles can be formally proven congruent using established criteria such as SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), AAS, and RHS (Right Angle-Hypotenuse-Side).

Properties of Congruent Triangles (CPCT)

Because congruent triangles possess the exact same overall dimensions, their calculated perimeters, internal areas, corresponding altitudes, and circumradii are mathematically identical.

Shortcut Trick

If a + 1 ÷ a = k, then a² + 1 ÷ a² = k² − 2.

Let a = √x, then a² = x. We are given k = 4.

⇒ x + 1 ÷ x = 4² − 2

⇒ x + 1 ÷ x = 16 − 2 = 14

∴ The correct answer is 14.

Alternate Method

Given:

√x + 1 ÷ √x = 4

Formula Used:

(a + b)² = a² + b² + 2ab

Calculations:

⇒ Squaring both sides of the given equation:

⇒ (√x + 1 ÷ √x)² = 4²

⇒ (√x)² + (1 ÷ √x)² + 2 × √x × (1 ÷ √x) = 16

⇒ x + 1 ÷ x + 2 = 16

⇒ x + 1 ÷ x = 16 − 2

⇒ x + 1 ÷ x = 14

∴ The correct answer is 14.

Additional Information

Minus Identity Variant

If x − 1 ÷ x = k, then x² + 1 ÷ x² = k² + 2.

Cube Identity (Positive)

If x + 1 ÷ x = k, then x³ + 1 ÷ x³ = k³ − 3k.

Cube Identity (Negative)

If x − 1 ÷ x = k, then x³ − 1 ÷ x³ = k³ + 3k.

Power of 4 Variant

If x + 1 ÷ x = k, then x⁴ + 1 ÷ x⁴ = (k² − 2)² − 2.

Shortcut Trick

Given, cotA = cosA ÷ sinA = 2 ÷ 1

This gives a direct ratio of cosA to sinA as 2 : 1. Let Base = 2 and Perpendicular = 1.

Using Pythagorean triplet logic, Hypotenuse = √(2² + 1²) = √5

⇒ sinA = 1/√5 and cosA = 2/√5

⇒ sinA − cosA = 1/√5 − 2/√5 = −1/√5

∴ The correct answer is −1/√5.

Alternate Method

Given:

cotA = 2 = 2/1

Formula Used:

cotA = Base ÷ Perpendicular

sinA = Perpendicular ÷ Hypotenuse

cosA = Base ÷ Hypotenuse

ABase = 2Perp = 1Hyp = √5

Calculations:

⇒ Base (B) = 2, Perpendicular (P) = 1

⇒ Hypotenuse (H) = √(B² + P²) = √(2² + 1²) = √5

⇒ sinA = 1/√5

⇒ cosA = 2/√5

⇒ sinA − cosA = 1/√5 − 2/√5 = −1/√5

∴ The correct answer is −1/√5.

Additional Information

Trigonometric Identities

The primary Pythagorean identities are: sin²A + cos²A = 1, sec²A − tan²A = 1, and cosec²A − cot²A = 1.

Basic Trigonometric Ratios

For any right-angled triangle: sinA = Perpendicular/Hypotenuse, cosA = Base/Hypotenuse, and tanA = Perpendicular/Base.

Reciprocal Relations

The reciprocal ratios are: cosecA = 1/sinA, secA = 1/cosA, and cotA = 1/tanA = cosA/sinA.

Shortcut Trick

The radius, the tangent, and the line connecting the center to the external point always form a right-angled triangle.

The given lengths for the legs are 7 cm and 24 cm.

Using the standard Pythagorean triplet (7, 24, 25), the hypotenuse directly equals 25.

Thus, the distance from the point to the center is 25 cm.

∴ The correct answer is 25 cm.

Alternate Method

Given:

Radius of the circle (r) = 7 cm

Length of the tangent from external point P (t) = 24 cm

O

T

P

7 cm

24 cm

d = ?

Formula Used:

A tangent at any point of a circle is perpendicular to the radius through the point of contact.

By Pythagoras theorem: Hypotenuse² = Perpendicular² + Base²

⇒ OP² = OT² + PT²

Calculation:

⇒ OP² = 7² + 24²

⇒ OP² = 49 + 576

⇒ OP² = 625

⇒ OP = √625

⇒ OP = 25 cm

∴ The correct answer is 25 cm.

Additional Information

Tangent Perpendicularity Theorem

A tangent to a circle is always perpendicular to the radius drawn to the point of tangency. This forms a 90° angle, enabling the direct use of the Pythagorean theorem.

Number of Tangents from a Point

From any external point, exactly two tangents can be drawn to a circle. Both tangents will always be equal in length.

Common Pythagorean Triplets

Triplets are integer combinations that satisfy the a² + b² = c² rule. Common examples include (3, 4, 5), (5, 12, 13), (7, 24, 25), and (8, 15, 17).

Shortcut Trick

According to the fundamental properties of a circle, the lengths of tangents drawn from a single external point to a circle are always equal.

Since the length of one tangent from the external point is given as 12 cm, the other tangent must also be exactly equal in length.

∴ The correct answer is 12 cm.

Alternate Method

O P A B

12 cm ? cm

Given:

⇒ Let the circle have center O and the external point be P.

⇒ Two tangents PA and PB are drawn to the circle, where PA = 12 cm.

Calculations:

⇒ We know that the radius drawn to the point of tangency is perpendicular to the tangent line.

⇒ ∠OAP = ∠OBP = 90°

⇒ In right-angled triangles ΔOAP and ΔOBP:

⇒ OA = OB (Radii of the same circle)

⇒ OP = OP (Common hypotenuse)

⇒ By RHS (Right-angle Hypotenuse Side) congruence criterion, ΔOAP ≅ ΔOBP.

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

⇒ PB = 12 cm.

∴ The correct answer is 12 cm.

Additional Information

Properties of Tangents

A tangent to a circle is perpendicular to the radius at the point of tangency. Lengths of the two tangents drawn from a common external point to a circle are always equal.

Direct Common Tangent (DCT)

Length of DCT = √[d² − (R − r)²], where d is the distance between centers and R, r are the radii of the two circles.

Transverse Common Tangent (TCT)

Length of TCT = √[d² − (R + r)²].

Shortcut Trick

Observe the multiplicative relation between the numerator and denominator: 0.04 = 0.2 × 0.2 and 0.02 = 0.2 × 0.1

⇒ Numerator = (0.2 × 0.2)³ + (0.2 × 0.1)³ = 0.2³ × (0.2³ + 0.1³)

The term (0.2³ + 0.1³) is common in both the numerator and denominator and cancels out.

⇒ Result = 0.2³ = 0.008

∴ The correct answer is 0.008.

Alternate Method

Given:

(0.04³ + 0.02³) ÷ (0.2³ + 0.1³)

Calculations:

Evaluate the cubes of the given decimals directly:

⇒ 0.04³ = 0.04 × 0.04 × 0.04 = 0.000064

⇒ 0.02³ = 0.02 × 0.02 × 0.02 = 0.000008

⇒ Numerator sum = 0.000064 + 0.000008 = 0.000072

Now evaluate the denominator:

⇒ 0.2³ = 0.2 × 0.2 × 0.2 = 0.008

⇒ 0.1³ = 0.1 × 0.1 × 0.1 = 0.001

⇒ Denominator sum = 0.008 + 0.001 = 0.009

Divide the numerator sum by the denominator sum:

⇒ 0.000072 ÷ 0.009

Multiply both numerator and denominator by 1000000 to remove decimals:

⇒ 72 ÷ 9000

⇒ 8 ÷ 1000 = 0.008

∴ The correct answer is 0.008.

Additional Information

Laws of Exponents

When factors are multiplied and raised to a power, the power distributes to each factor: (a × b)ⁿ = aⁿ × bⁿ.

Sum of Cubes Formula

The sum of two cubes can be factorised using the algebraic identity: a³ + b³ = (a + b)(a² − ab + b²).

Decimal Multiplication Rules

When multiplying decimals, the result has a number of decimal places equal to the sum of the decimal places in the individual numbers being multiplied.