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Shortcut Trick

Identify Pythagorean Triplets and Quadrants:

For θ (Q1): sin θ = 3/5 → (3, 4, 5) triplet ⇒ cos θ = 4/5 (positive in Q1).

For φ (Q3): cos φ = −5/13 → (5, 12, 13) triplet ⇒ sin φ = −12/13 (negative in Q3).

Apply expansion: sin(θ + φ) = sin θ cos φ + cos θ sin φ = (3/5 × −5/13) + (4/5 × −12/13) = −15/65 − 48/65 = −63/65.

∴ The correct answer is −63/65.

Alternate Method

Given: sin θ = 3/5 (1^st Quadrant), cos φ = −5/13 (3^rd Quadrant).

Formula: sin(A + B) = sin A cos B + cos A sin B

Step 1: Calculate cos θ and sin φ

⇒ In 1^st Quadrant (Aircraft A): All ratios are positive.

⇒ cos θ = √(1 − sin²θ) = √(1 − (3/5)²) = √(1 − 9/25) = √(16/25) = 4/5.

⇒ In 3^rd Quadrant (Aircraft B): sin and cos are negative.

⇒ sin φ = −√(1 − cos²φ) = −√(1 − (−5/13)²) = −√(1 − 25/169) = −√(144/169) = −12/13.

Step 2: Substitute in Addition Formula

⇒ sin(θ + φ) = (sin θ × cos φ) + (cos θ × sin φ)

⇒ sin(θ + φ) = (3/5 × −5/13) + (4/5 × −12/13)

⇒ sin(θ + φ) = (−15/65) + (−48/65)

⇒ sin(θ + φ) = −63/65.

∴ The correct answer is −63/65.

Additional Information

Trigonometric Signs (ASTC Rule)

Quadrant 1: All positive | Quadrant 2: sin positive | Quadrant 3: tan positive | Quadrant 4: cos positive.

Compound Angle Formulas

sin(A ± B) = sin A cos B ± cos A sin B | cos(A ± B) = cos A cos B ∓ sin A sin B.

Standard Pythagorean Triplets

Common integer sides for right triangles: (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), and (9, 40, 41).

Shortcut Trick

When two equal distances are covered at different speeds v₁ and v₂, the Average Speed is the harmonic mean of the speeds.

Average Speed = (2 × v₁ × v₂) ÷ (v₁ + v₂)

⇒ (2 × 5 × 6) ÷ (5 + 6) = 60 ÷ 11

⇒ 5.4545... km/h

∴ The correct answer is 5.45 km/h.

Alternate Method

Given:

Distance going = 30 km, Speed going (v₁) = 5 km/h

Distance returning = 30 km, Speed returning (v₂) = 6 km/h

Formula Used:

Average Speed = Total Distance ÷ Total Time

Calculations:

⇒ Time taken going (t₁) = Distance ÷ Speed = 30 ÷ 5 = 6 hours

⇒ Time taken returning (t₂) = Distance ÷ Speed = 30 ÷ 6 = 5 hours

⇒ Total Distance = 30 + 30 = 60 km

⇒ Total Time = 6 + 5 = 11 hours

⇒ Average Speed = 60 ÷ 11

⇒ Average Speed ≈ 5.4545 km/h

∴ The correct answer is 5.45 km/h.

Additional Information

Average Speed (Different Distances)

When distances are unequal, always use the fundamental formula: Average Speed = (d₁ + d₂ + ...) ÷ (t₁ + t₂ + ...).

Relative Speed

If two objects move toward each other, Relative Speed = v₁ + v₂. If they move in the same direction, Relative Speed = |v₁ − v₂|.

Conversion of Units

To convert km/h to m/s, multiply by 5/18. To convert m/s to km/h, multiply by 18/5.

Shortcut Trick

Total Price = 2 × 400 = ₹800

Net Payable = 95% of Total Price (since discount is 5%)

Payable Amount = 0.95 × 800 = ₹760

∴ The correct answer is ₹760.

Alternate Method

Given:

Price of one screen guard = ₹400

Number of items bought = 2

Discount rate = 5% on the total bill

Formula Used:

Total Bill = Price × Quantity

Final Amount = Total Bill − (Discount % × Total Bill)

Calculations:

⇒ Total Bill of two screen guards = 2 × 400 = ₹800

⇒ Discount Amount = 5% of 800

⇒ Discount Amount = (5 ÷ 100) × 800 = ₹40

⇒ Amount to pay = 800 − 40 = ₹760

∴ The correct answer is ₹760.

Additional Information

Selling Price and Discount

The price after a discount is the Selling Price (SP). It is calculated as SP = Marked Price (MP) − Discount.

Discount Percentage Formula

To find the Discount %, use the formula: Discount % = (Discount Amount ÷ Marked Price) × 100.

Successive Discounts

If two discounts d₁% and d₂% are applied one after another, the equivalent single discount is (d₁ + d₂ − (d₁ × d₂) ÷ 100)%.

Shortcut Trick

Directly divide the corresponding numbers: 9.5 ÷ 0.19 = 50 and 0.085 ÷ 0.0017 = 50.

The expression simplifies to √(50 × 50).

The square root of (50²) is 50.

∴ The correct answer is 50.

Alternate Method

Given: A pharmaceutical quality analyst is verifying a concentration index using the expression: √((9.5 × 0.085) / (0.0017 × 0.19))

Formula Used: Decimal removal and √(a × b) properties.

⇒ √((9.5 × 0.085) / (0.19 × 0.0017))

⇒ √((9.5 / 0.19) × (0.085 / 0.0017))

⇒ To remove decimals, shift the decimal points: (950 / 19) × (850 / 17)

⇒ √(50 × 50)

⇒ √2500

⇒ 50

∴ The correct answer is 50.

Additional Information

Decimal Adjustment Rule

When dividing decimals, multiply both the numerator and denominator by 10ⁿ, where n is the number of decimal places, to convert them into whole numbers.

Square Root of Products

The property √(a × b) = √a × √b allows for simplifying complex expressions by calculating roots of individual factors.

Uniformity in Dosage

In pharmaceutical production, maintaining a consistent concentration index is vital for ensuring that every tablet in a batch meets safety and efficacy standards.

Shortcut Trick

Use the direct formula for two people working together: (A × B) ÷ (A + B).

Here, Amit (A) = 30 and Babli (B) = 45.

Required time = (30 × 45) ÷ (30 + 45) = 1350 ÷ 75 = 18 days.

∴ The correct answer is 18 days.

Alternate Method

Given:

Time taken by Amit = 30 days

Time taken by Babli = 45 days

Formula Used:

Efficiency = Total Work ÷ Time

Total Work = LCM of individual days

Calculations:

⇒ Total Efficiency of Amit and Babli = 3 + 2 = 5 units/day

⇒ Time taken to complete the work together = Total Work ÷ Total Efficiency

⇒ Time = 90 ÷ 5 = 18 days

∴ The correct answer is 18 days.

Additional Information

Pipes and Cisterns

The same LCM logic is applied here. If an outlet pipe (leak) is present, its efficiency is subtracted from the total.

Work and Wages

Total wages are always distributed in the ratio of the efficiency (or actual work done) of the individuals.

Efficiency and Time Relation

Efficiency is inversely proportional to the Time taken. If the ratio of time taken by two people is x : y, then their efficiency ratio is y : x.

Shortcut Trick

Change in average weight = 65 − 64 = 1 kg (Increase).

Total students after new student joins = 6 + 1 = 7.

Total weight increase across all students = 7 × 1 = 7 kg.

Weight of new student = Old Average + Total Increase = 64 + 7 = 71 kg.

∴ The correct answer is 71 kg.

Alternate Method

Given:

Initial number of students = 6

Initial average weight = 64 kg

New average weight = 65 kg

Formula:

Total Sum = Average × Number of observations

New Observation = (New Average × New Count) − (Old Average × Old Count)

Calculations:

⇒ Total weight of 6 students = 6 × 64 = 384 kg

⇒ Total weight of 7 students (including new student) = 7 × 65 = 455 kg

⇒ Weight of new student = Total weight (7 students) − Total weight (6 students)

⇒ Weight of new student = 455 − 384

⇒ Weight of new student = 71 kg

∴ The correct answer is 71 kg.

Additional Information

New Member Formula

Value of new member = New Average + [Number of old members × Change in Average].

Member Leaving Formula

Value of member who left = Old Average − [Number of new members × Change in Average].

Weighted Average

Weighted Average = (n₁x₁ + n₂x₂) ÷ (n₁ + n₂), used when two groups with different averages are combined.

Shortcut Trick

Total Failed Percentage = (% Failed in Physics + % Failed in Chemistry) − % Failed in Both

⇒ Total Failed = (50% + 40%) − 10% = 80%

⇒ Percentage of candidates who Passed in Both = 100% − 80% = 20%

∴ The correct answer is 20%.

Alternate Method

Given:

Failed in Physics, n(P) = 50%

Failed in Chemistry, n(C) = 40%

Failed in Both, n(P ∩ C) = 10%

Formula Used:

Total Failed Percentage, n(P ∪ C) = n(P) + n(C) − n(P ∩ C)

Calculations:

⇒ Candidates failed in Physics only = 50% − 10% = 40%

⇒ Candidates failed in Chemistry only = 40% − 10% = 30%

⇒ Total percentage of candidates who failed in at least one subject = 40% (Physics only) + 30% (Chemistry only) + 10% (Both) = 80%

⇒ Percentage of candidates who passed in both sections = 100% − 80% = 20%

∴ The correct answer is 20%.

Additional Information

Inclusion-Exclusion Principle

For any two finite sets A and B, the cardinality of the union is given by n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

De Morgan's Laws

The percentage of those who passed in both subjects is the complement of the union of those who failed in either subject: n(Pass Both) = 100 − n(Fail Physics ∪ Fail Chemistry).

Venn Diagram Regions

In a two-circle Venn diagram, the region outside both circles represents elements that do not belong to either set (e.g., passing in both when circles represent failures).

Shortcut Trick

Surface Area = 4πr² = 616 ⇒ r² = 616 ÷ (4 × 22/7) = 49.

Radius (r) = 7 cm.

Volume = (4/3)πr³ = (4/3) × (22/7) × 7 × 7 × 7 = 4312/3 cm³.

∴ The correct answer is 4312/3 cm³.

Alternate Method

Given:

Surface Area of the Sphere = 616 cm²

Formula Used:

1. Surface Area of a Sphere = 4πr²
2. Volume of a Sphere = (4/3)πr³

Take π = 22/7

Calculations:

⇒ 4 × (22/7) × r² = 616

⇒ (88/7) × r² = 616

⇒ r² = (616 × 7) ÷ 88

⇒ r² = 7 × 7 = 49

⇒ r = √49 = 7 cm

Now, finding the Volume:

⇒ Volume = (4/3) × (22/7) × 7³

⇒ Volume = (4/3) × (22/7) × 7 × 7 × 7

⇒ Volume = (4/3) × 22 × 49

⇒ Volume = (4 × 1078) ÷ 3

⇒ Volume = 4312/3 cm³

∴ The correct answer is 4312/3 cm³.

Additional Information

Volume of Hemisphere

The volume of a hemisphere is exactly half that of a sphere: V = (2/3)πr³.

Total Surface Area of Hemisphere

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

Curved Surface Area of Hemisphere

The CSA of a hemisphere is half the surface area of a sphere: CSA = 2πr².

Sphere Diameter

The diameter is the longest distance through the center: d = 2r.

Shortcut Trick

Find Radius (r) using Base Area: r = √(616 × 7 ÷ 22) = √(28 × 7) = 14 cm.

Use Curved Surface Area (CSA) formula: h = CSA ÷ (2 × π × r).

h = 1320 ÷ (2 × 22/7 × 14) = 1320 ÷ 88 = 15 cm.

∴ The correct answer is 15 cm.

Alternate Method

Given:

Base Area of Cylinder = 616 cm²

Curved Surface Area (CSA) = 1,320 cm²

π = 22/7

Calculations:

⇒ Base Area of Cylinder = πr²

⇒ 616 = (22/7) × r²

⇒ r² = (616 × 7) ÷ 22

⇒ r² = 28 × 7 = 196

⇒ Radius (r) = √196 = 14 cm

⇒ Curved Surface Area (CSA) = 2πrh

⇒ 1320 = 2 × (22/7) × 14 × h

⇒ 1320 = 2 × 22 × 2 × h

⇒ 1320 = 88 × h

⇒ Height (h) = 1320 ÷ 88

⇒ h = 15 cm

∴ The correct answer is 15 cm.

Additional Information

Total Surface Area (TSA) of a Cylinder

The Total Surface Area includes the curved surface and the areas of the two circular bases: TSA = 2πr(h + r).

Volume of a Cylinder

The Volume represents the capacity or space occupied by the cylinder: Volume = πr²h.

Base Circumference

The perimeter of the base of the cylinder is given by the formula for the Circumference of a circle: C = 2πr.

Shortcut Trick

Find the Radius squared (r²): r² = (3 × Volume) ÷ (π × h)

r² = (3 × 3300) ÷ (22/7 × 14) = 9900 ÷ 44 = 225

Calculate Slant height (l) using l = √(r² + h²)

l = √(225 + 14²) = √(225 + 196) = √421 cm

∴ The correct answer is √421 cm.

Alternate Method

Given:

Height (h) of the cone = 14 cm

Volume (V) of the cone = 3300 cm³

Formula Used:

Volume of a right circular cone = (1/3) × π × r² × h

Slant height (l) = √(r² + h²)

Calculations:

⇒ 3300 = (1/3) × (22/7) × r² × 14

⇒ 3300 = (22 × 2 × r²) ÷ 3

⇒ 3300 = (44 × r²) ÷ 3

⇒ r² = (3300 × 3) ÷ 44

⇒ r² = 75 × 3 = 225

⇒ Radius (r) = √225 = 15 cm

⇒ Slant height (l) = √(15² + 14²)

⇒ l = √(225 + 196)

⇒ l = √421 cm

∴ The correct answer is √421 cm.

Additional Information

Curved Surface Area (CSA)

The area of the curved surface of a cone is given by π × r × l.

Total Surface Area (TSA)

The total area including the base is π × r × (l + r).

Relation between Dimensions

In a right circular cone, the slant height, radius, and height form a right-angled triangle: l² = r² + h².