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Shortcut Trick

Let A = x − y, B = y − z, and C = z − x.

Sum = A + B + C = (x − y) + (y − z) + (z − x) = 0.

If A + B + C = 0, then directly A³ + B³ + C³ = 3ABC.

Substituting A, B, and C back directly yields 3(x − y)(y − z)(z − x).

∴ The correct answer is 3(x − y)(y − z)(z − x).

Alternate Method

Given:

⇒ x + y + z = 0

⇒ x³ + y³ + z³ = 3xyz

Formula Used:

⇒ If a + b + c = 0, then a³ + b³ + c³ = 3abc

Calculations:

⇒ We need to evaluate (x − y)³ + (y − z)³ + (z − x)³

⇒ Let a = x − y, b = y − z, and c = z − x

⇒ Checking the sum of these variables:

⇒ a + b + c = (x − y) + (y − z) + (z − x)

⇒ a + b + c = x − y + y − z + z − x = 0

⇒ Since the sum of a, b, and c is 0, the special algebraic identity applies directly:

⇒ a³ + b³ + c³ = 3abc

⇒ Substitute the values of a, b, and c back into the equation:

⇒ (x − y)³ + (y − z)³ + (z − x)³ = 3 × (x − y) × (y − z) × (z − x)

⇒ (x − y)³ + (y − z)³ + (z − x)³ = 3(x − y)(y − z)(z − x)

∴ The correct answer is 3(x − y)(y − z)(z − x).

Additional Information

Standard Cubic Identity

a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)

Modified Cubic Identity

a³ + b³ + c³ − 3abc = (1 ÷ 2) × (a + b + c) × [(a − b)² + (b − c)² + (c − a)²]

Condition for 3abc

If a + b + c = 0 OR if a = b = c, then a³ + b³ + c³ perfectly evaluates to 3abc.

Shortcut Trick

Overall percentage of interest = (390 ÷ 5000) × 100 = 7.8%

By alligation method for 6% and 9% interest rates:

Ratio of parts = (9 − 7.8) : (7.8 − 6) = 1.2 : 1.8 = 2 : 3

Sum lent at 6% = 5000 × [2 ÷ (2 + 3)] = 5000 × (2 ÷ 5) = ₹2000

∴ The correct answer is ₹2000.

Alternate Method

Given:

Total Sum (P) = ₹5000

Total simple interest (I) = ₹390

Rates of interest = 6% and 9%

Time (T) = 1 year

Formula Used:

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

₹ x at 6% ₹ (5000 − x) at 9%

Total Principal = ₹5000 Total Interest = ₹390

Calculations:

⇒ Let the sum lent at 6% be ₹x.

⇒ Then, the sum lent at 9% will be ₹(5000 − x).

⇒ Total Interest = Interest on first part + Interest on second part

⇒ [(x × 6 × 1) ÷ 100] + [((5000 − x) × 9 × 1) ÷ 100] = 390

⇒ 6x + 45000 − 9x = 39000

⇒ 3x = 45000 − 39000

⇒ 3x = 6000

⇒ x = 2000

∴ The correct answer is ₹2000.

Additional Information

Simple Interest (SI)

It is calculated only on the principal amount for the entire tenure. Formula: SI = (P × R × T) ÷ 100.

Compound Interest (CI)

Interest is calculated on the principal and also on the accumulated interest of previous periods. Formula: Amount = P × (1 + R ÷ 100)^T.

Difference between CI and SI

For a period of 2 years, the difference between Compound Interest and Simple Interest on a given principal is calculated as: D = P × (R ÷ 100)².

Alligation Method

A technique used to find the ratio in which two quantities at different rates or percentages are mixed to yield a mixture of a known average rate.

Shortcut Trick

Total interest for 1 year = 4950 ÷ 3 = ₹1650.

Assume the entire ₹15,000 is lent at X's rate of 10%. Expected interest = 10% of 15000 = ₹1500.

Extra interest generated = 1650 − 1500 = ₹150.

This ₹150 extra is due to the additional 2% rate (12% − 10%) on Y's share.

2% of Y = 150 ⇒ Y = 7500. Therefore, X's share = 15000 − 7500 = ₹7500.

∴ The correct answer is ₹7,500.

Alternate Method

Given:

Total principal = ₹15,000

Total Simple Interest (SI) for 3 years = ₹4,950

Rate for X = 10%, Rate for Y = 12%

Formula Used:

SI = (P × R × T) ÷ 100

Total = ₹15,000

Lent to X Rate = 10% p.a. Time = 3 years

Lent to Y Rate = 12% p.a. Time = 3 years

Total SI = ₹4,950

Calculation:

⇒ Let the amount lent to X be P.

⇒ Then, the amount lent to Y = 15000 − P.

⇒ Interest from X = (P × 10 × 3) ÷ 100 = 0.3P

⇒ Interest from Y = ((15000 − P) × 12 × 3) ÷ 100 = (15000 − P) × 0.36

⇒ Total Interest = 0.3P + 5400 − 0.36P = 4950

⇒ 5400 − 0.06P = 4950

⇒ 0.06P = 5400 − 4950

⇒ 0.06P = 450

⇒ P = 450 ÷ 0.06 = 7500

∴ The correct answer is ₹7,500.

Additional Information

Alligation Method for SI

Calculate the overall average rate of interest using the formula: Average Rate = (Total SI × 100) ÷ (Total Principal × Time). Use the alligation cross method to quickly find the ratio in which the total sum is divided.

Shift / Assume Method

Whenever a total amount is split into two parts at two different rates, assuming the entire amount is invested at the lower rate calculates the minimum expected interest. The excess interest is solely due to the higher rate part, directly revealing its value.

Simple Interest Key Fact

Unlike Compound Interest, Simple Interest remains constant every year if the principal, rate, and time do not change.

Shortcut Trick

In a 30°-60°-90° right-angled triangle, the ratio of sides opposite to the angles is 1 : √3 : 2.

The base adjacent to the 60° angle (which is opposite to 30°) corresponds to 1 part.

The length of the wire forms the hypotenuse, corresponding to 2 parts.

Given the base length is 10 m, we can multiply by 2 to get the hypotenuse directly.

Length of the wire = 10 × 2 = 20 m.

∴ The correct answer is 20 m.

Alternate Method

A B C 10 m 60° L = ?

Given:

Distance of the wire fixed on the ground from the base of the pole = 10 m

Angle of elevation of the wire from the ground = 60°

Formula Used:

cos(θ) = Base ÷ Hypotenuse

Calculation:

⇒ Let the length of the taut wire be L.

⇒ From the right-angled triangle formed, cos(60°) = 10 ÷ L

⇒ 1/2 = 10 ÷ L

⇒ L = 10 × 2

⇒ L = 20 m

∴ The correct answer is 20 m.

Additional Information

Trigonometric Ratios

For any right-angled triangle, sine = Perpendicular ÷ Hypotenuse, cosine = Base ÷ Hypotenuse, and tangent = Perpendicular ÷ Base.

Angle of Elevation

The angle formed by the line of sight with the horizontal plane when viewing an object located above the observer's eye level.

Standard 30°-60°-90° Triangle

The sides opposite to the angles 30°, 60°, and 90° respectively are always strictly in the ratio of 1 : √3 : 2.

Shortcut Trick

Volume of Cone = Volume of Sphere ⇒ (1/3)πr_c²h = (4/3)πr_s³

Substitute cone height h = 2r_c directly into the equation:

⇒ r_c² × 2r_c = 4r_s³

⇒ 2r_c³ = 4r_s³ ⇒ r_c³/r_s³ = 2/1

Taking cube roots on both sides gives the direct ratio of their radii.

∴ The correct answer is ∛2 : 1.

Alternate Method

Given:

Volume of the cone = Volume of the sphere

Height of the cone (h) = 2 × Radius of the cone (r_c)

Formula Used:

Volume of a Cone = (1/3)πr²h

Volume of a Sphere = (4/3)πr³

h = 2rc rc Cone

rs Sphere

Volume of Cone = Volume of Sphere

Calculations:

Let the radius of the cone be r_c and the radius of the sphere be r_s.

⇒ (1/3) × π × r_c² × h = (4/3) × π × r_s³

Substitute h = 2r_c in the above equation:

⇒ (1/3) × π × r_c² × (2r_c) = (4/3) × π × r_s³

Cancel out (π/3) from both sides of the equation:

⇒ 2r_c³ = 4r_s³

⇒ r_c³ / r_s³ = 4 / 2

⇒ (r_c / r_s)³ = 2 / 1

Taking the cube root on both sides:

⇒ r_c / r_s = ∛2 / 1

∴ The correct answer is ∛2 : 1.

Additional Information

Volume of a Hemisphere

A hemisphere is exactly half of a sphere. Its volume is calculated using the formula V = (2/3)πr³.

Surface Area of a Sphere

The total surface area of a sphere is equivalent to 4 times the area of its great circle: TSA = 4πr².

Curved Surface Area of a Cone

The lateral or curved surface area is CSA = πrl, where l = √(r² + h²) is the slant height.

Shortcut Trick

Total Surface Area (TSA) of Hemisphere = 3πr²

Total Surface Area (TSA) of Cylinder = 2πr(h + r)

Equating both: 3πr² = 2πr(h + r)

⇒ 3r = 2h + 2r ⇒ r = 2h

⇒ h/r = 1/2

∴ The correct answer is 1:2.

Alternate Method

Given:

Radius of hemisphere = Radius of cylinder = r

Total Surface Area (TSA) of hemisphere = TSA of cylinder

Formula Used:

TSA of a solid hemisphere = 3πr²

TSA of a solid cylinder = 2πrh + 2πr²

r Hemisphere

r

h

Cylinder

Calculation:

⇒ 3πr² = 2πr(r + h)

⇒ 3r = 2(r + h)

⇒ 3r = 2r + 2h

⇒ 3r − 2r = 2h

⇒ r = 2h

⇒ h/r = 1/2

⇒ Ratio of height to radius (h : r) = 1 : 2

∴ The correct answer is 1:2.

Additional Information

Cylinder Formulas

Curved Surface Area (CSA) = 2πrh | Total Surface Area (TSA) = 2πr(h + r) | Volume = πr²h

Hemisphere Formulas

Curved Surface Area (CSA) = 2πr² | Total Surface Area (TSA) = 3πr² | Volume = (2/3)πr³

Sphere Formulas

Total Surface Area (TSA) = 4πr² | Volume = (4/3)πr³

Cone Formulas

Curved Surface Area (CSA) = πrl | Total Surface Area (TSA) = πr(l + r) | Volume = (1/3)πr²h

Shortcut Trick

For a cube to be divided into n³ identical pieces, it must be cut into 'n' equal parts along all 3 dimensions.

Total identical pieces = 64 = 4³

⇒ Number of parts along each dimension (n) = 4

Number of cuts required for 'n' parts = n − 1 cuts per dimension.

⇒ Minimum total cuts = 3 × (4 − 1) = 3 × 3 = 9

∴ The correct answer is 9 cuts.

Alternate Method

Given:

Total number of identical smaller cubical pieces = 64

Formula Used:

Total number of pieces = (x + 1) × (y + 1) × (z + 1)

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

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

Large Cube cut into 64 pieces (4 × 4 × 4) 3 cuts along Height 3 cuts along Breadth 3 cuts along Length

Calculations:

⇒ 64 = 4 × 4 × 4

⇒ (x + 1) = 4 ⇒ x = 3

⇒ (y + 1) = 4 ⇒ y = 3

⇒ (z + 1) = 4 ⇒ z = 3

⇒ Minimum number of total cuts = x + y + z

⇒ Minimum cuts = 3 + 3 + 3 = 9

∴ The correct answer is 9 cuts.

Additional Information

Maximum Pieces from N Cuts

To get the maximum number of pieces from a given number of cuts (N), distribute the cuts equally among all three axes. Maximum pieces = (N/3 + 1)³.

Cubes with 3 Faces Painted

When a fully painted large cube is cut into n × n × n smaller identical pieces, the number of cubes with exactly 3 faces painted is always 8 (these are situated at the corners).

Cubes with 2 Faces Painted

The number of smaller cubes with exactly 2 faces painted lie precisely along the middle of the edges and is given by the formula 12 × (n − 2).

Cubes with 1 Face Painted

The number of smaller cubes with exactly 1 face painted lie in the center of the outer faces and is given by the formula 6 × (n − 2)².

Shortcut Trick

The volume of a cone is proportional to r² × h, so the percentage change depends on three successive changes: +20%, +20%, and −10%.

First, apply the successive percentage formula (x + y + xy/100) on the radius (+20% and +20%):

⇒ 20 + 20 + (20 × 20) ÷ 100 = 40 + 4 = +44%

Next, apply the formula on the combined radius change (+44%) and height change (−10%):

⇒ 44 − 10 + (44 × −10) ÷ 100 = 34 − 4.4 = +29.6%

Since the result is positive, the overall change is a 29.6% increase.

∴ The correct answer is 29.6% increase.

Alternate Method

Given:

Percentage increase in radius = 20%

Percentage decrease in height = 10%

Formula Used:

Volume of a cone = 1/3 × π × r² × h

r h

Original Cone Volume = 100%

+29.6%

1.2r 0.9h

New Cone Volume = 129.6%

Calculations:

⇒ Let the original radius (r) be 10 units and original height (h) be 10 units.

⇒ Original Volume (V₁) = 1/3 × π × (10)² × 10 = 1000π / 3 cubic units.

⇒ New radius = 10 + (20% of 10) = 10 + 2 = 12 units.

⇒ New height = 10 − (10% of 10) = 10 − 1 = 9 units.

⇒ New Volume (V₂) = 1/3 × π × (12)² × 9 = 1/3 × π × 144 × 9 = 1296π / 3 cubic units.

⇒ Change in volume = V₂ − V₁ = (1296π / 3) − (1000π / 3) = 296π / 3 cubic units.

⇒ Percentage change = (Change in volume ÷ Original Volume) × 100

⇒ Percentage change = [(296π / 3) ÷ (1000π / 3)] × 100

⇒ Percentage change = (296 ÷ 1000) × 100 = 29.6%

∴ The correct answer is 29.6% increase.

Additional Information

Volume of Cone vs Cylinder

The volume of a cone is exactly one-third the volume of a cylinder that shares the same base radius and height (V = 1/3 × πr²h).

Successive Percentage Change

When a value undergoes multiple percentage changes sequentially, the net percentage change for two variables is found using (a + b + ab/100)%. For formulas depending on three variables like r × r × h, apply this successive change formula twice.

Surface Area of a Cone

The curved surface area is given by πrl and the total surface area is πr(r + l), where l = √(r² + h²) is the slant height of the cone.

Shortcut Trick

Since A lies in the 2^nd quadrant, sine is positive and cosine is negative.

Using the 3-4-5 right angle triplet, if sinA = 3/5, then cosA = −4/5.

Expression: (sinA + cosA)²

⇒ (3/5 − 4/5)²

⇒ (−1/5)² = 1/25

∴ The correct answer is 1/25.

Alternate Method

Given:

sinA = 3/5

A lies in the second quadrant.

Formula Used:

Algebraic Identity: (a + b)² = a² + b² + 2ab

Trigonometric Identity: sin²A + cos²A = 1

A Base = −4 Perp = 3 Hyp = 5 y-axis x-axis 2^nd Quadrant

Calculation:

⇒ In the 2^nd quadrant, cosine values are negative.

⇒ cosA = −√(1 − sin²A)

⇒ cosA = −√(1 − 9/25) = −√(16/25) = −4/5

Now, expanding the given expression to solve it directly:

⇒ (sinA + cosA)² = sin²A + cos²A + 2 × sinA × cosA

⇒ (sinA + cosA)² = 1 + 2 × sinA × cosA

⇒ (sinA + cosA)² = 1 + 2 × (3/5) × (−4/5)

⇒ (sinA + cosA)² = 1 − 24/25

⇒ (sinA + cosA)² = (25 − 24) / 25 = 1/25

∴ The correct answer is 1/25.

Additional Information

ASTC Rule (Signs in Quadrants)

1^st Quadrant: All positive. 2^nd Quadrant: sin & csc positive. 3^rd Quadrant: tan & cot positive. 4^th Quadrant: cos & sec positive.

Pythagorean Identity

The foundational identity is sin²θ + cos²θ = 1 for any generic angle θ.

Common Pythagorean Triplets

Sets of integers that perfectly satisfy a² + b² = c². The most frequent sets in exams include (3, 4, 5), (5, 12, 13), and (8, 15, 17).

Shortcut Trick

According to the basic trigonometric identity, 1 + tan² A = sec² A.

We know that sec² A = 1 / cos² A = 1 / (1 − sin² A).

Substitute sin A = m / n into the equation:

⇒ sec² A = 1 / [1 − (m / n)²] = 1 / [(n² − m²) / n²]

⇒ sec² A = n² / (n² − m²)

∴ The correct answer is n² / (n² − m²).

Alternate Method

Given:

sin A = m / n = Perpendicular / Hypotenuse

A m (Perpendicular) n (Hypotenuse) Base = √(n² - m²)

Calculations:

⇒ Perpendicular (P) = m

⇒ Hypotenuse (H) = n

Using Pythagoras theorem:

⇒ Base (B) = √(H² − P²) = √(n² − m²)

⇒ tan A = P / B = m / √(n² − m²)

Now, we need to find the value of (1 + tan² A):

⇒ 1 + [m / √(n² − m²)]²

⇒ 1 + m² / (n² − m²)

⇒ (n² − m² + m²) / (n² − m²)

⇒ n² / (n² − m²)

∴ The correct answer is n² / (n² − m²).

Additional Information

Pythagorean Identities

Fundamental relations between trigonometric functions: sin² θ + cos² θ = 1, sec² θ − tan² θ = 1, and cosec² θ − cot² θ = 1.

Reciprocal Identities

Functions that are inverses of each other: sin θ = 1 / cosec θ, cos θ = 1 / sec θ, and tan θ = 1 / cot θ.

Quotient Identities

Relation of tangent and cotangent with sine and cosine: tan θ = sin θ / cos θ and cot θ = cos θ / sin θ.