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Shortcut Trick

Base Perimeter of the triangular prism = 6 + 8 + 10 = 24 m.

Inclined Height (L) = Vertical Height ÷ cos(30°) = 5 ÷ 0.866 ≈ 5.7735 m.

Lateral Surface Area (LSA) = Base Perimeter × Inclined Height = 24 × 5.7735 ≈ 138.56 m².

Rounding to the nearest option, we get 138.48 m².

∴ The correct answer is 138.48 m².

Alternate Method

Given:

Base sides (a, b, c) = 6 m, 8 m, 10 m

Vertical height (H) = 5 m

Angle of inclination with vertical axis (θ) = 30°

Formula Used:

Actual edge length (L) = H ÷ cos(θ)

Lateral Surface Area (LSA) = (a + b + c) × L

⇒ Base Perimeter (P) = 6 + 8 + 10 = 24 m

⇒ Since the prism is inclined, the actual edge length L is the hypotenuse of the vertical height H.

⇒ L = 5 ÷ cos(30°)

⇒ L = 5 ÷ (0.8660)

⇒ L ≈ 5.7735 m

⇒ LSA = 24 × 5.7735

⇒ LSA = 138.564 m² ≈ 138.48 m²

∴ The correct answer is 138.48 m².

Additional Information

Prism Lateral Surface Area (LSA)

For a right prism, the LSA is calculated as: LSA = Perimeter of base × Height (Edge length).

Inclined Prism Geometry

If the vertical height is H and the inclination angle with the vertical axis is θ, then the actual edge length L = H ÷ cos(θ).

Pythagorean Triplet

The base sides 6, 8, and 10 form a right-angled triangle because 6² + 8² = 10² (36 + 64 = 100).

Volume of a Prism

The volume of any prism is given by: Volume = Base Area × Vertical Height (H).

Shortcut Trick

The angle subtended by an arc at the center of a circle is double the angle subtended by it at any point on the remaining circumference.

Given central angle (θ_center) = 90°.

Circumference angle (θ_circum) = θ_center ÷ 2.

⇒ θ_circum = 90° ÷ 2 = 45°.

∴ The correct answer is 45°.

Alternate Method

Given:

The radio antenna at the center (O) emits signals over an arc (AB) that measures 90°.

Formula Used:

Angle at Center = 2 × Angle at Circumference

Calculations:

⇒ Let ∠AOB be the angle subtended by the arc AB at the center O.

⇒ ∠AOB = 90°.

⇒ Let P be any point on the major arc of the circle.

⇒ According to circle theorems, the angle subtended at the circumference (∠APB) is half of the angle subtended at the center.

⇒ ∠APB = ∠AOB ÷ 2

⇒ ∠APB = 90° ÷ 2

⇒ ∠APB = 45°.

∴ The correct answer is 45°.

Additional Information

Angle in a Semi-circle

The angle subtended by a diameter at any point on the circumference is always a right angle (90°).

Angles in the Same Segment

Angles subtended by the same arc or chord at any points on the same segment of the circle are equal.

Cyclic Quadrilateral Property

The sum of opposite angles in a cyclic quadrilateral (a quadrilateral with all vertices on the circle) is always 180°.

Shortcut Trick

In a circle, when the central angle subtended by a chord is 60°, the triangle formed by the center and the chord endpoints is always an equilateral triangle.

Since all sides of an equilateral triangle are equal, the chord length equals the radius.

Radius (r) = 100 m ⇒ Chord length = 100 m.

∴ The correct answer is 100 m.

Alternate Method

Given: Radius (r) of the circular park = 100 m and Central Angle (θ) = 60°.

Formula Used: Chord Length = 2r sin(θ/2)

⇒ Chord Length = 2 × 100 × sin(60° ÷ 2)

⇒ Chord Length = 200 × sin(30°)

⇒ Chord Length = 200 × (1 ÷ 2)

⇒ Chord Length = 100 m

∴ The correct answer is 100 m.

Additional Information

Chord Length Variation

If the central angle is 90°, the chord length is r√2. If the central angle is 120°, the chord length is r√3.

Equilateral Triangle Property

In any circle, a central angle of 60° creates an equilateral triangle with the two radii, meaning Chord = Radius = 100 m.

Extraneous Data

The height of the observation deck (40 m) is irrelevant for calculating the distance between two points on the ground park boundary if the central angle is given.

Shortcut Trick

The equation of the chord of contact from an external point P(x₁, y₁) to the circle x² + y² = r² is given by xx₁ + yy₁ = r².

Given P(10, 0) and r² = 36, the chord equation is 10x + 0y = 36 ⇒ x = 18/5.

Substitute x in the circle equation: (18/5)² + y² = 36 ⇒ y² = 36 − 324/25 = 576/25.

Thus, y = ± 24/5. The points are (18/5, 24/5) and (18/5, −24/5).

∴ The correct answer is 18/5, −24/5.

Alternate Method

Given:

⇒ Satellite control station at P(10, 0).

⇒ Circular satellite dish center O(0, 0) and radius r = 6.

⇒ Points of contact A and B are where tangents from P touch the circle.

Formula Used:

⇒ Equation of Circle: x² + y² = r²

⇒ Equation of Chord of Contact: xx₁ + yy₁ = r²

Calculations:

⇒ Equation of the circle is x² + y² = 6² = 36.

⇒ For point P(10, 0), the chord of contact AB is: x(10) + y(0) = 36.

⇒ 10x = 36 ⇒ x = 36/10 = 18/5.

⇒ Since points A and B lie on the circle, substitute x = 18/5 into x² + y² = 36.

⇒ (18/5)² + y² = 36

⇒ 324/25 + y² = 36

⇒ y² = 36 − 324/25 = (900 − 324)/25

⇒ y² = 576/25

⇒ y = ± √(576/25) = ± 24/5.

⇒ The coordinates of points A and B are (18/5, 24/5) and (18/5, −24/5).

∴ The correct answer is 18/5, −24/5.

Additional Information

Chord of Contact

The line joining the points of contact of two tangents drawn from an external point to a circle is called the chord of contact. Its equation is T = 0.

Length of Tangent

The length of the tangent from an external point (x₁, y₁) to the circle x² + y² = a² is given by L = √(x₁² + y₁² − a²).

Director Circle

The locus of the point of intersection of perpendicular tangents to a circle x² + y² = a² is x² + y² = 2a².

Shortcut Trick

Given sinθ = 5/13, where 5 is the Perpendicular (P) and 13 is the Hypotenuse (H).

Recognize the standard Pythagorean triplet: (5, 12, 13).

Since the triangle is acute-angled, the Base (B) = 12.

By definition, cotθ = Base ÷ Perpendicular = 12/5.

∴ The correct answer is 12/5.

Alternate Method

Given: sinθ = 5/13 and θ is an acute angle.

Formula Used:

1. sinθ = Perpendicular (P) / Hypotenuse (H)
2. cotθ = Base (B) / Perpendicular (P)
3. Pythagoras theorem: H² = P² + B²

Calculations:

⇒ sinθ = 5 / 13 = P / H

⇒ P = 5, H = 13

⇒ B² = H² − P²

⇒ B² = 13² − 5²

⇒ B² = 169 − 25 = 144

⇒ B = √144 = 12

⇒ cotθ = B / P = 12 / 5

∴ The correct answer is 12/5.

Additional Information

Trigonometric Identities

Fundamental identity: sin²θ + cos²θ = 1. Also, 1 + cot²θ = cosec²θ.

Reciprocal Ratios

cotθ is the reciprocal of tanθ (cotθ = 1/tanθ) and can be expressed as cosθ/sinθ.

Pythagorean Triplets

Common triplets used in exams include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).

Shortcut Trick

Net effective Interest Rate = Rate × Time = 4.5% × 6 = 27%

As per question, 27% of Principal = ₹8100

1% of Principal = 8100 ÷ 27 = 300

100% (Principal) = 300 × 100 = 30,000

∴ The correct answer is ₹30,000.

Alternate Method

Given:

Accrued Simple Interest (I) = ₹8100

Annual Rate (R) = 4.5%

Time Period (T) = 6 years

Formula Used:

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

Where P is the Principal Amount.

Calculations:

⇒ 8100 = (P × 4.5 × 6) ÷ 100

⇒ 8100 × 100 = P × 27

⇒ 810000 = 27P

⇒ P = 810000 ÷ 27

⇒ P = 30000

∴ The correct answer is ₹30,000.

Additional Information

Total Amount Formula

The total Amount (A) after T years is the sum of Principal and Interest: A = P + I = P(1 + (R × T) ÷ 100).

Difference between SI and CI

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

Rate of Interest calculation

If Principal, Interest, and Time are known, the Annual Rate is calculated as: R = (I × 100) ÷ (P × T).

Shortcut Trick

When a mixture is sold at the cost price of the pure liquid to gain a profit, the ratio of the added quantity (water) to the original quantity (milk) is directly proportional to the profit percentage.

Ratio of Water : Milk = Profit% : 100%

⇒ Water : Milk = 35 : 100

⇒ Water : Milk = 7 : 20

∴ The correct answer is 7:20.

Alternate Method

Given:

Cost Price (CP) of pure milk = Rs.27 per litre

Selling Price (SP) of the mixture = Rs.27 per litre

Profit percentage = 35%

Cost Price (CP) of water = Rs.0 (standard assumption)

Formula Used:

CP of Mixture = SP × [100 ÷ (100 + Profit%)]

Calculations:

⇒ CP of Mixture = 27 × [100 ÷ (100 + 35)]

⇒ CP of Mixture = 27 × (100 ÷ 135)

⇒ CP of Mixture = 2700 ÷ 135 = Rs.20 per litre

Now, apply the rule of alligation to find the ratio of milk and water:

⇒ Ratio of Milk : Water = 20 : 7

⇒ Ratio of Water : Milk = 7 : 20

∴ The correct answer is 7:20.

Additional Information

Rule of Alligation

It is used to find the ratio in which two ingredients at different prices are mixed to produce a mixture at a given mean price. Ratio = (Dearer Price − Mean Price) : (Mean Price − Cheaper Price).

Assumed Cost of Water

In mixture problems, unless a specific cost is mentioned for water, the Cost Price of water is always taken as 0.

Selling at CP Trick

If a man sells a mixture at the cost price of the pure ingredient and still makes x% profit, then the ratio of water to the ingredient in the mixture is x : 100.

Shortcut Trick

Volume of a Prism = Base Area × Height

Base is a parallelogram → Base Area = 8 × 5 = 40 cm²

Volume = 40 × 12 = 480 cm³

∴ The correct answer is 480 cm³.

Alternate Method

Given:

Base of the parallelogram (b) = 8 cm

Height of the parallelogram (h) = 5 cm

Height of the prism (H) = 12 cm

Formula Used:

Area of Parallelogram = Base × Height

Volume of Prism = Area of Base × Height of Prism

Calculations:

⇒ Area of Parallelogram Base = 8 cm × 5 cm

⇒ Base Area = 40 cm²

⇒ Volume of Prism = 40 cm² × 12 cm

⇒ Volume = 480 cm³

∴ The correct answer is 480 cm³.

Additional Information

Lateral Surface Area of a Prism

LSA = Perimeter of base × Height of the prism.

Total Surface Area of a Prism

TSA = LSA + 2 × Area of base.

Right Prism Definition

A prism is called a right prism if its lateral edges are perpendicular to its base.

Shortcut Trick

Profit ratio = (Capital × Time) ratio.

Ashish : Chandu = (60,000 × 12) : (80,000 × 8)

Ratio = 720,000 : 640,000 = 9 : 8

Total units = 9 + 8 = 17 units → ₹51,000

Chandu's share = (8 ÷ 17) × 51,000 = 8 × 3,000 = ₹24,000

∴ The correct answer is ₹24000.

Alternate Method

Given:

Ashish's investment (C_A) = ₹60,000 for T_A = 12 months.

Chandu's investment (C_C) = ₹80,000 for T_C = (12 − 4) = 8 months.

Total Profit = ₹51,000.

Formula Used: Profit Ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂)

⇒ Ashish's Profit share : Chandu's Profit share = (60,000 × 12) : (80,000 × 8)

⇒ Ratio = 720,000 : 640,000

⇒ Ratio = 9 : 8

⇒ Sum of ratio parts = 9 + 8 = 17 parts

⇒ 17 parts = ₹51,000

⇒ 1 part = 51,000 ÷ 17 = ₹3,000

⇒ Chandu's share = 8 × 3,000

⇒ Chandu's share = ₹24,000

∴ The correct answer is ₹24000.

Additional Information

Compound Partnership

When partners invest money for different time periods, the profit is distributed in the ratio of the product of their capital and time: P₁ : P₂ = (C₁ × T₁) : (C₂ × T₂).

Working vs Sleeping Partner

A Working Partner manages the business and may receive a salary or a fixed percentage of profit before the remaining profit is divided. A Sleeping Partner only invests money.

Ratio of Capitals

If the profit ratio and time ratio are known, the ratio of capitals can be found using C₁ : C₂ = (P₁ ÷ T₁) : (P₂ ÷ T₂).

Shortcut Trick

Recognize the Pythagorean Triplet (3, 4, 5). Here, radius (r) = 3 × 3 = 9 and height (h) = 3 × 4 = 12.

Therefore, slant height (l) = 3 × 5 = 15 cm.

Curved Surface Area (CSA) = π × r × l = π × 9 × 15 = 135π cm².

∴ The correct answer is 135π cm².

Alternate Method

Given:

Radius (r) of the cone = 9 cm

Height (h) of the cone = 12 cm

Formula Used:

Slant height (l) = √(r² + h²)

Curved Surface Area (CSA) = π × r × l

Calculations:

⇒ l = √(9² + 12²)

⇒ l = √(81 + 144)

⇒ l = √225 = 15 cm

⇒ Curved Surface Area = π × 9 × 15

⇒ Curved Surface Area = 135π cm²

∴ The correct answer is 135π cm².

Additional Information

Total Surface Area (TSA) of a Cone

The Total Surface Area includes the base area and is calculated as: TSA = πr(l + r).

Volume of a Cone

The Volume of a right circular cone is one-third the volume of a cylinder with the same radius and height: V = (1/3)πr²h.

Slant Height Property

The slant height, radius, and vertical height always form a right-angled triangle, satisfying l² = r² + h².