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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.