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Shortcut Trick

Total faces on a die = 6 (1, 2, 3, 4, 5, 6).

Even faces = 3 (2, 4, 6).

Probability = Even faces ÷ Total faces = 3 ÷ 6 = 1/2.

∴ Answer = 1/2

Alternate Method

Given:

A fair die is rolled once.

Total possible outcomes S = {1, 2, 3, 4, 5, 6}

Formula used:

Probability P(E) = n(E) ÷ n(S)

Die Roll Outcomes 1 2 3 4 5 6 Total Outcomes n(S) = 6 Even Outcomes (Green) n(E) = 3 P = 3/6 = 1/2

Calculations:

⇒ Sample space n(S) = 6

⇒ Even numbers E = {2, 4, 6}

⇒ Number of favorable outcomes n(E) = 3

⇒ Probability P(E) = 3 ÷ 6

⇒ P(E) = 1/2

∴ The correct answer is (1).

Shortcut Trick

Directly substitute ∠A = 30° into the expression: sin² 30° − cos² 30° + tan² 30°.

Value substitution: (1/2)² − (√3/2)² + (1/√3)²

Calculation: 1/4 − 3/4 + 1/3 = − 2/4 + 1/3 = − 1/2 + 1/3 = − 1/6.

∴ The correct answer is − 1/6.

Alternate Method

Given: In triangle ABC, ∠B = 90° and ∠A = 30°.

Formula Used: Standard trigonometric values: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3.

The expression to evaluate is sin² A − cos² A + tan² A.

⇒ Expression = (sin 30°)² − (cos 30°)² + (tan 30°)²

⇒ Expression = (1/2)² − (√3/2)² + (1/√3)²

⇒ Expression = 1/4 − 3/4 + 1/3

⇒ Expression = (1 − 3)/4 + 1/3

⇒ Expression = − 2/4 + 1/3

⇒ Expression = − 1/2 + 1/3

⇒ Expression = (− 3 + 2)/6

⇒ Expression = − 1/6

∴ The correct answer is − 1/6.

Additional Information

Trigonometric Identities

Fundamental identities include sin²θ + cos²θ = 1 and 1 + tan²θ = sec²θ.

Double Angle Relation

The term (sin²A − cos²A) is equal to − (cos²A − sin²A), which simplified is − cos 2A.

Standard Values at 30°

For θ = 30°: sin θ = 0.5, cos θ = 0.866, tan θ = 0.577.

Shortcut Trick

Net percentage change = a + b + (ab ÷ 100)

Here, a = −10% and b = +12%

Net % change = −10 + 12 + [(−10 × 12) ÷ 100] = 2 − 1.2 = 0.8% (increase)

Final Value = 62,000 + 0.8% of 62,000 = 62,000 + 496 = 62,496

∴ The correct answer is ₹62,496.

Alternate Method

Given:

Initial Investment = ₹62,000

First year decrease = 10%

Second year increase = 12%

Calculations:

⇒ Value after 1^st Year = 62,000 − (10% of 62,000)

⇒ Value after 1^st Year = 62,000 − 6,200 = ₹55,800

⇒ Value after 2^nd Year = 55,800 + (12% of 55,800)

⇒ 12% of 55,800 = (12 ÷ 100) × 55,800 = 12 × 558 = 6,696

⇒ Final Value = 55,800 + 6,696 = ₹62,496

∴ The correct answer is ₹62,496.

Additional Information

Successive Percentage Change

When a value changes by x% and then y%, the effective change is (x + y + xy/100)%. This is also applicable to Compound Interest calculations.

Multiplying Factor Method

For an x% increase, multiply by (1 + x/100). For an x% decrease, multiply by (1 − x/100). In this case: 62000 × 0.90 × 1.12.

Commutative Property of Change

The final value remains the same regardless of the order of percentage changes (i.e., 12% increase followed by 10% decrease yields the same result).

Given:

Curved surface area (CSA) of the cone = 195π cm².

Slant height (l) = 12 cm.

Formula Used:

Curved Surface Area (CSA) of a cone = π × r × l

Calculation:

CSA = π × r × l

⇒ 195π = π × r × 12

⇒ 195 = r × 12

⇒ r = 195 / 12

⇒ r = 16.25

The radius of the base is 16.25 cm.

Shortcut Trick

The numbers 8, 15, and 17 form a standard Pythagorean triplet.

Given sin θ = Perpendicular / Hypotenuse = 8 / 17.

This directly implies that the Base is 15.

In the 2^nd quadrant, the cosine ratio (Base / Hypotenuse) is always negative.

Therefore, cos θ = −15 / 17.

∴ The correct answer is −15/17.

Alternate Method

Given:

sin θ = 8 / 17

θ lies in the 2^nd quadrant.

Formula Used:

sin²θ + cos²θ = 1

X Y

θ r = 17 y = 8 x = −15

Calculations:

⇒ cos²θ = 1 − (8 / 17)²

⇒ cos²θ = 1 − 64 / 289

⇒ cos²θ = (289 − 64) / 289

⇒ cos²θ = 225 / 289

⇒ cos θ = ±√(225 / 289)

⇒ cos θ = ±15 / 17

Since θ lies in the 2^nd quadrant, the value of cos θ must be negative.

⇒ cos θ = −15 / 17

∴ The correct answer is −15/17.

Additional Information

ASTC Rule (Quadrant Signs)

Quadrant I: All positive | Quadrant II: sin & cosec positive | Quadrant III: tan & cot positive | Quadrant IV: cos & sec positive.

Pythagorean Triplets

Common triplets like (3, 4, 5), (5, 12, 13), and (8, 15, 17) help find missing sides of a right-angled triangle directly without manual calculations.

Trigonometric Identities

The three core Pythagorean identities are sin²θ + cos²θ = 1, sec²θ − tan²θ = 1, and cosec²θ − cot²θ = 1.

Shortcut Trick

The radius (24) and height (10) form a Pythagorean triplet with the slant height (l).

Triplet base (5, 12, 13) × 2 → (10, 24, 26). Thus, slant height l = 26 cm.

Curved Surface Area = π × r × l

CSA = π × 24 × 26 = 624π cm².

∴ The correct answer is 624π cm².

Alternate Method

Given:

Radius of the base (r) = 24 cm

Height of the cone (h) = 10 cm

Formula Used:

Slant height (l) = √(r² + h²)

Curved Surface Area (CSA) = π × r × l

h = 10 r = 24 l = ? Curved Surface Area = πrl

Calculations:

⇒ l = √(24² + 10²)

⇒ l = √(576 + 100)

⇒ l = √676

⇒ l = 26 cm

⇒ CSA = π × 24 × 26

⇒ CSA = 624π cm²

∴ The correct answer is 624π cm².

Additional Information

Total Surface Area of a Cone

TSA includes both the curved surface and the circular base: TSA = πrl + πr² = πr(l + r).

Volume of a Cone

The space occupied by the cone is given by: V = (1/3) × πr²h.

Frustum of a Cone

When the top of a cone is sliced off parallel to the base, the curved surface area of the resulting frustum is πl(r₁ + r₂).

Shortcut Trick

Using the Pythagorean triplet (8, 15, 17):

Height (h) = 15, Slant height (l) = 17 ⇒ Half of base side (a/2) = 8 cm.

Base side (a) = 8 × 2 = 16 cm.

Volume of pyramid = (1/3) × Area of base × h

Volume = (1/3) × 16² × 15 = 256 × 5 = 1280 cu.cm.

∴ The correct answer is 1280 cu.cm.

Alternate Method

Given:

Slant height (l) = 17 cm

Vertical height (h) = 15 cm

The base is a square with side 'a'.

For a pyramid with a square base, the relation between height, slant height, and side is:

l² = h² + (a/2)²

⇒ 17² = 15² + (a/2)²

⇒ 289 = 225 + (a/2)²

⇒ (a/2)² = 289 − 225

⇒ (a/2)² = 64

⇒ a/2 = √64 = 8 cm

⇒ a = 16 cm

The formula for the Volume of a Pyramid is:

Volume = (1/3) × (Base Area) × (Vertical Height)

⇒ Volume = (1/3) × a² × h

⇒ Volume = (1/3) × (16 × 16) × 15

⇒ Volume = 256 × 5

⇒ Volume = 1280 cu.cm.

∴ The correct answer is 1280 cu.cm.

Additional Information

Slant Height vs Slant Edge

Slant height (l) is the altitude of a triangular face, while slant edge (e) is the distance from the vertex to a corner of the base: e² = h² + (d/2)², where d is the diagonal of the base.

Lateral Surface Area (LSA)

For a regular pyramid, LSA = (1/2) × Perimeter of base × Slant height. For a square base: LSA = 2 × a × l.

Total Surface Area (TSA)

TSA = Base Area + Lateral Surface Area. For a square base: TSA = a² + 2al.

Given:

Cost Price = Rs. 9500

Loss % = 22%

Formula used:

SP = C.P(100 - Loss %)/100

Where,

S.P = Selling Price, C.P = Cost Price

Calculation:

By the using formula, we get

S.P = C.P(100 - Loss %)/100

⇒ S.P = 9500(100 - 22)/100

⇒ S.P = 7410

∴ Selling Price of the article is Rs. 7410

Shortcut Trick

Surface Area = 4πr² = 4π ⇒ r² = 1 ⇒ r = 1 cm.

Volume = (4/3)πr³ = (4/3)π(1)³ = 4π/3 cm³.

∴ The correct answer is 4π/3 cm³.

Alternate Method

Given: Surface area of the sphere = 4π cm².

Formula used: Surface area = 4πr²; Volume = (4/3)πr³.

⇒ 4πr² = 4π

⇒ r² = 1

⇒ Radius (r) = 1 cm

⇒ Volume = (4/3) × π × (1)³

⇒ Volume = 4π/3 cm³

∴ The correct answer is 4π/3 cm³.

Additional Information

Hemisphere formulas

For a hemisphere of radius r, Curved Surface Area = 2πr² and Total Surface Area = 3πr².

Volume of Hemisphere

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

Ratio of SA to Volume

For any sphere, the ratio of its Surface Area to its Volume is 3/r.

Given:

f(x) = 2x² + 7x - 4

Calculations:

2x² + 7x - 4 = 0

=> 2x² + 8x - x - 4 = 0

=> 2x(x + 4) - 1(x + 4) = 0

=> (2x - 1)(x + 4) = 0

=> 2x - 1 = 0, x + 4 = 0

=> x =  \(\dfrac{1}{2}\), - 4

Value of x cannot be negative, so, x =  \(\dfrac{1}{2}\)

∴ The answer is  \(\dfrac{1}{2}\)