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Shortcut Trick

If x + (1/x) = −1, then a standard algebraic result is x³ = 1.

Reduce higher powers using this property: x⁴ = x³ × x = x.

⇒ x⁴ + (1/x⁴) = x + (1/x) = −1

⇒ 2(x² + 1/x²) = 2[(x + 1/x)² − 2] = 2[(−1)² − 2] = −2

⇒ Given expression value = −1 + (−2) = −3

∴ The correct answer is −3.

Alternate Method

Given:

x + (1/x) = −1

Formula Used:

a² + b² = (a + b)² − 2ab

Calculations:

Squaring the given equation on both sides:

⇒ x² + (1/x²) + 2(x)(1/x) = (−1)²

⇒ x² + (1/x²) + 2 = 1

⇒ x² + (1/x²) = 1 − 2 = −1

Now, squaring this new equation on both sides:

⇒ (x²)² + (1/x²)² + 2(x²)(1/x²) = (−1)²

⇒ x⁴ + (1/x⁴) + 2 = 1

⇒ x⁴ + (1/x⁴) = 1 − 2 = −1

We need to find the value of: x⁴ + (1/x⁴) + 2x² + (2/x²)

⇒ [x⁴ + (1/x⁴)] + 2[x² + (1/x²)]

Substitute the obtained values into the expression:

⇒ −1 + 2(−1)

⇒ −1 − 2 = −3

∴ The correct answer is −3.

Additional Information

Special Case of x + 1/x = 1

If x + (1/x) = 1, then multiplying both sides by x yields x² − x + 1 = 0, which gives x³ = −1.

Special Case of x + 1/x = −1

If x + (1/x) = −1, multiplying by x yields x² + x + 1 = 0, giving x³ = 1.

Sum of Squares Formula

If x + (1/x) = k, then x² + (1/x²) = k² − 2.

Sum of Cubes Formula

If x + (1/x) = k, then x³ + (1/x³) = k³ − 3k.

Shortcut Trick

Time from 10 January to 10 July is exactly 6 months (1/2 year).

Simple Interest for 6 months at 6% p.a. is effectively 3% of the Principal.

SI = 3% of 12000 = 3 × 120 = ₹360.

Total Amount = 12000 + 360 = ₹12360.

∴ The correct answer is ₹12360.

Alternate Method

Given:

Principal (P) = ₹12000

Rate (R) = 6% per annum

Time (T) = 10 January 2024 to 10 July 2024 = 6 months = 1/2 year

Formula Used:

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

Amount (A) = P + SI

10 Jan 2024 P = ₹12000

10 Jul 2024 A = ₹12360

Time = 6 months (1/2 year) Rate = 6% p.a.

SI = ₹360

Calculations:

⇒ SI = (12000 × 6 × 1) ÷ (100 × 2)

⇒ SI = (12000 × 3) ÷ 100

⇒ SI = ₹360

⇒ Amount (A) = P + SI

⇒ A = 12000 + 360 = ₹12360

∴ The correct answer is ₹12360.

Additional Information

Total Amount Formula

The total amount can be directly calculated using the formula: A = P × (1 + RT ÷ 100).

Simple vs Compound Interest

Simple Interest is calculated only on the initial principal value, whereas Compound Interest is calculated on the principal plus accumulated interest from previous periods.

Time Calculation Rule

When the exact same dates are given across different months (e.g., 10 Jan to 10 Jul), time is directly calculated in exact months (6 months), which translates directly to a fractional year (6 ÷ 12 = 1/2 year).

Shortcut Trick

Since the rate of interest and time period are the same, the ratio of Simple Interest to Principal remains constant.

Ratio for the first case = B ÷ A

Ratio for the second case = C ÷ B

Equating both ratios: B ÷ A = C ÷ B

By cross-multiplying, we get: B² = AC

∴ The correct answer is B² = AC.

Alternate Method

Given:

Simple interest on amount A = B

Simple interest on amount B = C

Rate of interest (R) and Time (T) are the same for both cases.

Formula Used:

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

A

SI = B

B

SI = C

C

Ratio of SI to P is constant → B/A = C/B

Calculations:

⇒ For the first case, the principal is A and simple interest is B.

⇒ B = (A × R × T) ÷ 100

⇒ (R × T) ÷ 100 = B ÷ A --- (Equation 1)

⇒ For the second case, the principal is B and simple interest is C.

⇒ C = (B × R × T) ÷ 100

⇒ (R × T) ÷ 100 = C ÷ B --- (Equation 2)

⇒ Equating Equation 1 and Equation 2, we get:

⇒ B ÷ A = C ÷ B

⇒ B × B = A × C

⇒ B² = AC

∴ The correct answer is B² = AC.

Additional Information

Simple Interest (SI)

SI = (P × R × T) ÷ 100, where P = Principal, R = Rate %, and T = Time.

Compound Interest (CI)

Amount A = P × (1 + R ÷ 100)^T and CI = A − P.

Difference between CI and SI

For 2 years, the difference is D = P × (R ÷ 100)².

Shortcut Trick

When the angle of elevation is 45°, the perpendicular and base of the right-angled triangle are equal.

Therefore, Height of the tower = Length of the shadow.

Length of shadow = 20 m.

∴ The correct answer is 20 m.

Alternate Method

Given:

Height of the tower (AB) = 20 m

Angle of elevation of the sun (θ) = 45°

45° A B C 20 m (Tower) Shadow Length = x

Formula Used:

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

Calculation:

⇒ Let the length of the shadow be BC.

⇒ tan(45°) = AB ÷ BC

⇒ 1 = 20 ÷ BC

⇒ BC = 20 × 1

⇒ BC = 20 m

∴ The correct answer is 20 m.

Additional Information

Angle of Elevation vs Angle of Depression

The angle of elevation is formed when looking up from a horizontal line, while the angle of depression is formed when looking down. Both are numerically equal for the same line of sight due to alternate interior angles.

Standard Trigonometric Ratios

In height and distance problems, the most commonly used tangent values are tan(30°) = 1 ÷ √3, tan(45°) = 1, and tan(60°) = √3.

Complementary Elevation Angles

If a structure casts shadows of lengths a and b at two different times such that the respective angles of elevation are complementary (sum to 90°), the height of the structure is given by h = √(a × b).

Shortcut Trick

The volume of a hemisphere is exactly half the volume of a complete sphere.

Volume = (2/3) × π × r³

Substitute r = 5 cm directly into the formula.

Volume = (2/3) × π × 125 = 250π/3 cm³

∴ The correct answer is 250π/3 cm³.

Alternate Method

Given:

Radius of the hemisphere (r) = 5 cm

Formula Used:

Volume of a hemisphere = (2/3) × π × r³

r = 5 cm Hemisphere

Calculations:

⇒ Volume = (2/3) × π × (5)³

⇒ Volume = (2/3) × π × 125

⇒ Volume = 250π/3 cm³

∴ The correct answer is 250π/3 cm³.

Additional Information

Volume of a Full Sphere

The total volume of a complete sphere is given by V = (4/3) × π × r³. A hemisphere's volume is exactly half of this value.

Curved Surface Area (CSA)

The curved surface area of a hemisphere only accounts for the outer bowl part and is calculated as CSA = 2 × π × r².

Total Surface Area (TSA)

The total surface area includes both the curved surface and the flat circular top surface. It is calculated as TSA = 3 × π × r².

Shortcut Trick

When a sphere is inscribed inside a cube, the ratio of the volume of the cube to the volume of the sphere is always 6 : π.

The ratio of the unoccupied volume to the total volume of the cube is (6 − π) : 6.

Percentage of unoccupied volume = [(6 − π) ÷ 6] × 100

Taking π ≈ 3.14: ⇒ [(6 − 3.14) ÷ 6] × 100 = (2.86 ÷ 6) × 100

⇒ 47.666...% ≈ 47.67%

∴ The correct answer is 47.67%.

Alternate Method

Given:

A solid sphere is placed inside a cube such that it touches all six faces.

r = a/2

a

2D Cross-section: Sphere inside a Cube

Formula Used:

Volume of cube = a³ (where a is the side of the cube)

Volume of sphere = (4 ÷ 3) × π × r³ (where r is the radius of the sphere)

Calculations:

⇒ Let the side of the cube be 'a'.

⇒ Since the sphere touches all six faces, its diameter is strictly equal to the side length of the cube.

⇒ Radius of the sphere (r) = a ÷ 2

⇒ Volume of the cube = a³

⇒ Volume of the sphere = (4 ÷ 3) × π × (a ÷ 2)³

⇒ Volume of the sphere = (4 ÷ 3) × π × (a³ ÷ 8) = (π ÷ 6) × a³

⇒ Volume not occupied by the sphere = a³ − (π ÷ 6) × a³ = a³ × (1 − π ÷ 6)

⇒ Percentage of unoccupied volume = [a³ × (1 − π ÷ 6) ÷ a³] × 100

⇒ Percentage = (1 − 3.14 ÷ 6) × 100

⇒ Percentage = [(6 − 3.14) ÷ 6] × 100 = (2.86 ÷ 6) × 100

⇒ Percentage = 286 ÷ 6 = 47.666... ≈ 47.67%

∴ The correct answer is 47.67%.

Additional Information

Cube Properties

For a cube of side 'a', the Total Surface Area = 6a², Lateral Surface Area = 4a², and the length of the longest body diagonal = a√3.

Sphere Properties

For a sphere of radius 'r', the Surface Area = 4πr². The sphere holds the maximum possible volume among all 3D solid shapes for any given surface area.

Cube and Inscribed Cylinder

If a right circular cylinder is inscribed inside a cube of side 'a', its volume will be (π ÷ 4) × a³. The percentage of occupied volume by the cylinder is roughly 78.5%.

Shortcut Trick

Let the sides of the rectangular prism be 4x, 5x, and 6x.

Volume = 4x × 5x × 6x = 120x³

Since the volume is 960 cm³, we have 120x³ = 960.

⇒ x³ = 8 ⇒ x = 2

The longest side is 6x = 6 × 2 = 12 cm.

∴ The correct answer is 12 cm.

Alternate Method

Given:

The ratio of the sides of the rectangular prism = 4 : 5 : 6

Volume of the prism = 960 cm³

Formula Used:

Volume of a rectangular prism = Length × Width × Height

Length = 6x Height = 4x Width = 5x Volume = 960 cm³

Calculations:

Let the exact dimensions of the sides be 4x, 5x, and 6x.

⇒ Volume = 4x × 5x × 6x

⇒ 120x³ = 960

⇒ x³ = 960 ÷ 120

⇒ x³ = 8

⇒ x = 2

Now, finding the exact lengths of all three sides:

⇒ First side = 4 × 2 = 8 cm

⇒ Second side = 5 × 2 = 10 cm

⇒ Third side = 6 × 2 = 12 cm

⇒ Out of the three, the longest side is clearly 12 cm.

∴ The correct answer is 12 cm.

Additional Information

Total Surface Area (TSA)

The total area of all 6 faces of a cuboid is calculated using the formula: TSA = 2(lw + wh + hl).

Lateral Surface Area (LSA)

The area of the 4 side vertical faces (excluding the top and bottom) is calculated as: LSA = 2h(l + w).

Length of Diagonal

The longest distance between two opposite corners in a rectangular prism is given by the formula: Diagonal = √(l² + w² + h²).

Shortcut Trick

When a cone is cut parallel to its base, the volumes of the corresponding parts are proportional to the cubes of their heights.

Ratio of heights of top cone to total cone = 1 : 3.

Ratio of volumes of top cone to total cone = 1³ : 3³ = 1 : 27.

Volume of bottom part = Total volume − Top volume = 27 − 1 = 26 units.

Ratio of volumes of top part to bottom part = 1 : 26.

∴ The correct answer is 1 : 26.

Alternate Method

Given:

A conical bucket is cut at one-third of its height from the top.

Let total height of the cone be H and base radius be R.

Height of the top cone (h) = H / 3.

Formula Used:

Volume of cone = (1/3) × π × radius² × height

H/3 2H/3

Base Radius = R

r = R/3

Top Volume Bottom Volume

Ratio = Top Volume : Bottom Volume = 1 : 26

Calculations:

⇒ By similar triangles, radius of the top cone (r) = R / 3.

⇒ Total volume of the cone (V) = (1/3) × π × R² × H

⇒ Volume of the top part (V₁) = (1/3) × π × r² × h

⇒ V₁ = (1/3) × π × (R/3)² × (H/3)

⇒ V₁ = (1/27) × [(1/3) × π × R² × H]

⇒ V₁ = V / 27

⇒ Volume of the bottom part (V₂) = Total Volume − Volume of top part

⇒ V₂ = V − V/27

⇒ V₂ = 26V / 27

⇒ Ratio of top part to bottom part = V₁ : V₂

⇒ Ratio = (V / 27) : (26V / 27)

⇒ Ratio = 1 : 26

∴ The correct answer is 1 : 26.

Additional Information

Volume of Frustum of a Cone

When a cone is sliced parallel to the base, the lower portion forms a frustum. Its volume is V = (1/3) × π × h × (R² + r² + R × r), where h is the height of the frustum.

Similar 3D Figures Property

For any two similar solid figures with a linear scale factor of k (ratio of heights or radii), the ratio of their surface areas is k² and the ratio of their volumes is k³.

Relationship Between Cone and Cylinder

A cone has exactly one-third the volume of a cylinder that shares the same base radius and height. V_cone = (1/3) × π × r² × h.

Shortcut Trick

If x + 1/x = 2, then x = 1.

Here, tanA + cotA = 2 ⇒ tanA + 1/tanA = 2, so tanA = 1.

Since A ∈ (0, π/2), we get A = 45°.

Now, sinA + cosA = sin(45°) + cos(45°)

⇒ 1/√2 + 1/√2 = 2/√2 = √2.

∴ The correct answer is √2.

Alternate Method

Given:

tanA + cotA = 2

A ∈ (0, π/2)

Formula Used:

tanA = sinA / cosA

cotA = cosA / sinA

(a + b)² = a² + b² + 2ab

sin²A + cos²A = 1

Calculations:

⇒ tanA + cotA = 2

⇒ (sinA / cosA) + (cosA / sinA) = 2

⇒ (sin²A + cos²A) / (sinA × cosA) = 2

⇒ 1 / (sinA × cosA) = 2

⇒ sinA × cosA = 1/2

We need to find the value of (sinA + cosA).

⇒ Let (sinA + cosA) = x

Squaring both sides:

⇒ (sinA + cosA)² = x²

⇒ sin²A + cos²A + 2 × sinA × cosA = x²

Substituting the known identities and values:

⇒ 1 + 2 × (1/2) = x²

⇒ 1 + 1 = x²

⇒ x² = 2

⇒ x = √2 (Since A ∈ (0, π/2), both sinA and cosA are positive)

∴ The correct answer is √2.

Additional Information

Basic Trigonometric Identities

Three foundational identities are: sin²θ + cos²θ = 1, sec²θ − tan²θ = 1, and cosec²θ − cot²θ = 1.

Algebraic Reciprocal Property

If the sum of a number and its reciprocal is 2, i.e., x + 1/x = 2, then x must be equal to 1. This is frequently used in trigonometric equations involving tanθ and cotθ.

Signs in the First Quadrant

For any angle A ∈ (0, π/2), all six trigonometric ratios are positive. Thus, sinA + cosA must strictly be a positive value.

Shortcut Trick

Apply the direct trigonometric compound angle identity:

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

For A = 20° and B = 10°, the expression reduces to sin(20° + 10°) = sin 30°.

The standard value of sin 30° is 1/2.

∴ The correct answer is 1/2.

Alternate Method

Given:

Trigonometric expression: sin 20° × cos 10° + cos 20° × sin 10°

Formula Used:

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

Calculation:

⇒ Let angle A = 20° and angle B = 10°

⇒ The given expression exactly matches the right hand side of the formula.

⇒ sin 20° × cos 10° + cos 20° × sin 10° = sin(20° + 10°)

⇒ sin(30°)

⇒ From the standard trigonometric table, sin 30° = 1/2

∴ The correct answer is 1/2.

Additional Information

Compound Angle Identity for Sine

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

Compound Angle Identity for Cosine

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

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

Double Angle Formula

sin 2A = 2 × sin A × cos A

cos 2A = cos² A − sin² A