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Shortcut Trick

Given a + b = 5 and ab = 6, we can deduce the values are a = 3 and b = 2.

Substitute these directly into (a³ + b³)² − 9a²b²(a + b)²:

⇒ (3³ + 2³)² − 9(3 × 2)²(3 + 2)² ⇒ (27 + 8)² − 9(36)(25)

⇒ (35)² − 8100 ⇒ 1225 − 8100 = -6875

∴ The correct answer is -6875.

Alternate Method

Given:

a + b = 5

ab = 6

Formula Used:

a³ + b³ = (a + b)³ − 3ab(a + b)

Calculations:

⇒ a³ + b³ = (5)³ − 3 × 6 × 5

⇒ a³ + b³ = 125 − 90 = 35

Now, substitute the required values into the given expression:

⇒ (a³ + b³)² − 9a²b²(a + b)²

⇒ (35)² − 9 × (ab)² × (a + b)²

⇒ 1225 − 9 × (6)² × (5)²

⇒ 1225 − 9 × 36 × 25

⇒ 1225 − 8100

⇒ -6875

∴ The correct answer is -6875.

Additional Information

Sum of Cubes Formula

The sum of cubes can be factored as a³ + b³ = (a + b)(a² − ab + b²) or expressed as (a + b)³ − 3ab(a + b).

Difference of Cubes Formula

Similarly, the difference of cubes is given by a³ − b³ = (a − b)(a² + ab + b²) or (a − b)³ + 3ab(a − b).

Square of a Sum Formula

The expansion for the square of a sum is (a + b)² = a² + 2ab + b², which is frequently used to find sums of squares: a² + b² = (a + b)² − 2ab.

Shortcut Trick

If the entire sum of ₹10000 was lent at 7%, the interest would be 7% of 10000 = ₹700.

Extra interest received = ₹900 − ₹700 = ₹200.

This extra interest is purely due to the additional 5% (12% − 7%) on the second part.

⇒ 5% of the sum at 12% = ₹200

⇒ Sum at 12% = (200 × 100) ÷ 5 = ₹4000

∴ The correct answer is ₹4000.

Alternate Method

Given:

Total sum = ₹10000

Total annual interest = ₹900

Formula Used:

Overall Rate (R) = (Total Interest × 100) ÷ (Principal × Time)

Part 1 (7%)

Part 2 (12%)

Mean (9%)

12 − 9 = 3

9 − 7 = 2

Ratio of Parts = 3 : 2

Calculations:

⇒ Overall rate of interest = (900 × 100) ÷ (10000 × 1) = 9%

⇒ By rule of alligation, ratio of sum lent at 7% to sum lent at 12% = 3 : 2

⇒ Total ratio parts = 3 + 2 = 5 parts

⇒ Value of 5 parts = ₹10000

⇒ Value of 1 part = 10000 ÷ 5 = ₹2000

⇒ Sum lent at 12% (2 parts) = 2 × 2000 = ₹4000

∴ The correct answer is ₹4000.

Additional Information

Simple Interest vs Compound Interest

Simple Interest is calculated only on the principal amount, while Compound Interest is calculated on the principal and the accumulated interest of previous periods.

Rule of Alligation

When two ingredients or parts are mixed, the ratio in which they are distributed is given by: (Quantity of cheaper) / (Quantity of dearer) = (Price of dearer − Mean price) / (Mean price − Price of cheaper).

Effective Rate of Interest

The effective interest rate is the total interest generated over a year expressed as a percentage of the total principal. Formula: R = (Total SI × 100) / (Total Principal × Time).

Shortcut Trick

Find the effective principal for 1 year at rate R% by normalizing the second amount.

Effective P = P₁ + P₂ × (Rate multiplier) × (Time in years)

Effective P = 900 + 600 × 1.5 × (3 ÷ 12) = 900 + 225 = 1125

Now, apply the simple interest formula: (1125 × R × 1) ÷ 100 = 72

⇒ R = 7200 ÷ 1125 = 6.4%

∴ The correct answer is 6.4%.

Alternate Method

Given:

⇒ First Principal (P₁) = ₹900, Rate (R₁) = R%

⇒ Total Time for P₁ = 1 year = 12 months

⇒ Second Principal (P₂) = ₹600, Rate (R₂) = 1.5R%

⇒ Time for second sum (T₂) = 12 − 9 = 3 months = 1/4 year

⇒ Total Simple Interest = ₹72

0 Months 9 Months 12 Months ₹900 @ R% for 12 months ₹600 @ 1.5R% for 3 months Total Interest = ₹72

Calculations:

⇒ Simple Interest Formula = (P × R × T) ÷ 100

⇒ SI from P₁ = (900 × R × 1) ÷ 100 = 9R

⇒ SI from P₂ = (600 × 1.5R × 0.25) ÷ 100 = (600 × 1.5R × 1) ÷ 400

⇒ SI from P₂ = 150 × 1.5R ÷ 100 = 2.25R

⇒ Total SI = SI from P₁ + SI from P₂

⇒ 9R + 2.25R = 72

⇒ 11.25R = 72

⇒ R = 72 ÷ 11.25

⇒ R = 7200 ÷ 1125 = 6.4

∴ The correct answer is 6.4%.

Additional Information

Simple Interest Formula

Simple Interest (SI) is always calculated on the initial principal using I = (P × R × T) ÷ 100, where P is principal, R is rate of interest, and T is time in years.

Effective Principal Concept

When multiple investments are made at different rates and time periods, they can be merged into a single equivalent principal at the base rate: Effective P = Σ(P_i × Time factor × Rate multiplier).

Amount Formula

The total accumulated amount (A) after the given duration is simply the principal combined with the total interest earned: A = P + I.

Shortcut Trick

The tangent of the angle of elevation (θ) is the ratio of the flagpole's height to the shadow's length.

⇒ tan(θ) = 10√3 ÷ 10

⇒ tan(θ) = √3

Since tan(60°) = √3, the angle of elevation of the sun is 60°.

∴ The correct answer is 60°.

Alternate Method

θ 10√3 m 10 m A B C

Given:

⇒ Height of the flagpole (AB) = 10√3 m

⇒ Length of the shadow (BC) = 10 m

Formula Used:

⇒ tan(θ) = Perpendicular ÷ Base

Calculations:

Let the angle of elevation of the sun be θ (angle ACB).

In the right-angled ΔABC:

⇒ tan(θ) = AB ÷ BC

⇒ tan(θ) = 10√3 ÷ 10

⇒ tan(θ) = √3

We know from the standard trigonometric table that tan(60°) = √3.

⇒ θ = 60°

∴ The correct answer is 60°.

Additional Information

Trigonometric Tangent Ratio

In a right-angled triangle, the tangent function relates the opposite side to the adjacent side: tan(θ) = Perpendicular ÷ Base. It is fundamentally used to solve unknown heights and distances without calculating the hypotenuse.

Angle of Elevation vs Angle of Depression

The angle of elevation is the angle formed by the line of sight when looking upwards from the horizontal. Conversely, the angle of depression is formed when looking downwards. In height problems involving a single object and observer, both angles are geometrically equal due to alternate interior angles.

Standard Values of Tangent

It is highly beneficial to memorize standard tangent values for quick problem-solving: tan(30°) = 1÷√3, tan(45°) = 1, and tan(60°) = √3.

Shortcut Trick

When melting and recasting spheres, the constant terms (like (4/3) × π) cancel out.

Equate the effective volumes directly: R³ − r³ = n × r_small³

10³ − r³ = 50 × 2³

1000 − r³ = 400 ⇒ r³ = 600

r = ³√600 ≈ 8.43 cm (Since 8³ = 512 and 9³ = 729)

∴ The correct answer is 8.43 cm.

Alternate Method

Given:

Outer radius of the hollow sphere (R) = 10 cm

Radius of smaller solid spheres (r_small) = 2 cm

Number of smaller spheres (n) = 50

Formula Used:

Volume of hollow sphere = (4/3) × π × (R³ − r³)

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

R=10

r=?

Hollow Sphere

Melted

2

50 Solid Spheres

Calculations:

Let the inner radius of the hollow sphere be r.

⇒ Volume of hollow sphere = Volume of 50 solid spheres

⇒ (4/3) × π × (10³ − r³) = 50 × (4/3) × π × 2³

⇒ Cancelling (4/3) × π from both sides:

⇒ 1000 − r³ = 50 × 8

⇒ 1000 − r³ = 400

⇒ r³ = 1000 − 400

⇒ r³ = 600

⇒ r = ³√600 ≈ 8.434 cm

∴ The correct answer is 8.43 cm.

Additional Information

Principle of Melting and Recasting

When a 3D object is melted and recast into another shape, the total volume of the material remains constant. Volume_initial = Volume_final.

Volume and Surface Area of a Solid Sphere

Volume = (4/3) × π × r³ | Total Surface Area = 4 × π × r²

Hemisphere Formulas

Volume = (2/3) × π × r³ | Curved Surface Area = 2 × π × r² | Total Surface Area = 3 × π × r²

Shortcut Trick

The ratio of Volume to Curved Surface Area of a hemisphere is [(2/3)πr³] ÷ [2πr²] = r / 3.

Given that Volume = 4 × CSA, we can write the ratio as 4.

⇒ r / 3 = 4

⇒ r = 12

∴ The correct answer is 12 cm.

Alternate Method

r Volume (V) = 2/3 × π × r³ Curved Surface Area (CSA) = 2 × π × r²

Given:

Volume of the hemisphere = 4 × Curved Surface Area of the hemisphere

Formula Used:

Volume of a hemisphere = 2/3 × πr³

Curved Surface Area (CSA) of a hemisphere = 2πr²

Calculations:

Let the radius of the hemisphere be r cm.

According to the given condition:

⇒ 2/3 × πr³ = 4 × (2πr²)

⇒ 2/3 × πr³ = 8πr²

Dividing both sides by πr²:

⇒ 2/3 × r = 8

⇒ r = 8 × (3 / 2)

⇒ r = 12

∴ The correct answer is 12 cm.

Additional Information

Total Surface Area of Hemisphere

It includes the curved surface area and the flat circular base area. Formula: TSA = 3πr².

Volume of a Sphere

A complete sphere has exactly twice the volume of a hemisphere of the same radius. Formula: V = (4/3)πr³.

Surface Area of a Sphere

The total curved surface of a complete sphere is four times the area of its great circle. Formula: SA = 4πr².

Shortcut Trick

Total pieces formed by making cuts along three perpendicular axes is (x + 1)(y + 1)(z + 1) = 8.

To minimize total cuts (x + y + z), the factors must be as close to each other as possible.

The closest integer factors for 8 are 2 × 2 × 2.

Thus, x + 1 = 2 ⇒ x = 1, y = 1, and z = 1.

Total minimum cuts = 1 + 1 + 1 = 3 cuts.

∴ The correct answer is 3 cuts.

Alternate Method

Given:

A cuboid is to be divided into 8 equal cuboids.

Formula Used:

Number of pieces = (x + 1) × (y + 1) × (z + 1)

Where x, y, and z are the number of cuts along the length, breadth, and height respectively.

3 Cuts = 8 Equal Cuboids Cut 1 Cut 2 Cut 3

Calculations:

⇒ Total pieces required = 8

⇒ (x + 1) × (y + 1) × (z + 1) = 8

⇒ To find the minimum number of cuts (x + y + z), we must express 8 as the product of three integers as close to each other as possible.

⇒ Let us evaluate the possible combinations for factors of 8:

⇒ Case 1: 8 × 1 × 1 → x = 7, y = 0, z = 0 → Total cuts = 7 + 0 + 0 = 7

⇒ Case 2: 4 × 2 × 1 → x = 3, y = 1, z = 0 → Total cuts = 3 + 1 + 0 = 4

⇒ Case 3: 2 × 2 × 2 → x = 1, y = 1, z = 1 → Total cuts = 1 + 1 + 1 = 3

⇒ The absolute minimum number of total cuts is obtained in Case 3.

⇒ Minimum cuts = 3.

∴ The correct answer is 3 cuts.

Additional Information

Maximum Pieces with N cuts

To maximize pieces for a given number of total cuts N, distribute the cuts equally among the three dimensions so that x ≈ y ≈ z ≈ N ÷ 3.

Total Painted Faces in a Cut Cube

If a fully painted larger cube is cut into n³ identical smaller cubes, the number of smaller cubes with exactly 3 painted faces is always 8 (located at the corners).

Zero Face Painted Cubes

The number of completely unpainted internal identical cubes is determined using the formula (n − 2)³, where n is the number of parts cut along one edge.

Shortcut Trick

The volume of a cone is directly proportional to the square of its radius and its height (V ∝ r²h).

Given the radius is halved (1/2) and the height is tripled (3).

Multiplying factor for Volume = (1/2)² × 3

New Volume factor = 1/4 × 3 = 3/4

∴ The correct answer is three-fourths of the original.

Alternate Method

Given:

⇒ Original radius = r, Original height = h

⇒ New radius = r/2, New height = 3h

Formula Used:

⇒ Volume of a cone (V) = 1/3 × πr²h

r h Original Cone

r/2 3h New Cone

Calculations:

⇒ Original Volume (V₁) = 1/3 × π × r² × h

⇒ New Volume (V₂) = 1/3 × π × (r/2)² × (3h)

⇒ V₂ = 1/3 × π × (r² / 4) × 3h

⇒ V₂ = 3/4 × (1/3 × π × r² × h)

⇒ V₂ = 3/4 × V₁

∴ The correct answer is three-fourths of the original.

Additional Information

Curved Surface Area of a Cone

CSA = πrl, where l is the slant height and is calculated as l = √(r² + h²).

Total Surface Area of a Cone

TSA = πr(r + l), combining the area of the circular base and the curved surface area.

Volume of a Cylinder

A cylinder with the same base radius and height as a cone has exactly 3 times the volume. The formula is V = πr²h.

Shortcut Trick

The standard Pythagorean triplet for a right-angled triangle is (5, 12, 13).

Given sin θ = 12/13, we identify Perpendicular = 12 and Hypotenuse = 13.

Using the triplet, the Base is directly found to be 5.

tan θ = Perpendicular ÷ Base = 12/5.

Since θ is in the 1^st quadrant, the ratio is positive.

∴ The correct answer is 12/5.

Alternate Method

Given:

sin θ = 12/13

θ ∈ (0, π/2)

Formula Used:

sin θ = Perpendicular (P) ÷ Hypotenuse (H)

Base (B) = √(H² − P²)

tan θ = Perpendicular (P) ÷ Base (B)

θ H = 13 P = 12 B = 5

Calculations:

⇒ Let P = 12 and H = 13

⇒ B = √(13² − 12²)

⇒ B = √(169 − 144)

⇒ B = √25

⇒ B = 5

⇒ tan θ = P ÷ B

⇒ tan θ = 12/5

Since θ lies in the first quadrant, all trigonometric functions are positive.

∴ The correct answer is 12/5.

Additional Information

Pythagorean Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (H² = P² + B²).

Trigonometric Ratios

The primary trigonometric ratios are defined as sin θ = P/H, cos θ = B/H, and tan θ = P/B.

Quadrant Sign Rule (ASTC)

In the 1^st quadrant (0 to π/2), all trigonometric ratios are positive. The ASTC rule (All, Sin, Tan, Cos) applies sequentially to the four quadrants.

Shortcut Trick

Given tanA = 2.

We know that cotA is the reciprocal of tanA, so cotA = 1/2.

Now, tanA + cotA = 2 + 1/2

⇒ (4 + 1) / 2 = 5/2

∴ The correct answer is 5/2.

Alternate Method

Given:

tanA = 2

Formula Used:

cotA = 1 / tanA

Calculations:

⇒ cotA = 1 / 2

⇒ tanA + cotA = 2 + 1/2

⇒ tanA + cotA = (4 + 1) / 2

⇒ tanA + cotA = 5/2

∴ The correct answer is 5/2.

Additional Information

Reciprocal Trigonometric Identities

tanθ and cotθ are reciprocals of each other. Therefore, tanθ × cotθ = 1.

Quotient Identities

tanθ = sinθ ÷ cosθ and cotθ = cosθ ÷ sinθ.

Pythagorean Identities

1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.