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Shortcut Trick

Identify the values of standard angles in degrees: π/4 = 45° and π/6 = 30°.

Apply the compound angle identity: cos(A − B) = cos A cos B + sin A sin B.

Substitute standard values: (1/√2 × √3/2) + (1/√2 × 1/2).

Combine terms: (√3 + 1) / 2√2.

∴ The correct answer is (√3 + 1) / 2√2.

Alternate Method

Given: cos (π/4 − π/6)

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

Step-by-step Calculation:

⇒ Let A = π/4 and B = π/6

⇒ cos(π/4 − π/6) = cos(π/4) × cos(π/6) + sin(π/4) × sin(π/6)

⇒ cos(45° − 30°) = (1/√2 × √3/2) + (1/√2 × 1/2)

⇒ cos 15° = √3 / 2√2 + 1 / 2√2

⇒ cos 15° = (√3 + 1) / 2√2

∴ The correct answer is (√3 + 1) / 2√2.

Additional Information

Compound Angle Identities

cos(A + B) = cos A cos B − sin A sin B and sin(A ± B) = sin A cos B ± cos A sin B.

Standard Values Table

sin 30° = 1/2, cos 30° = √3/2, sin 45° = 1/√2, cos 45° = 1/√2.

Radian to Degree Conversion

To convert radians to degrees, multiply by 180/π. Hence, π radians = 180°.

Shortcut Trick

Directly apply the volume formula for a cone: V = (1/3) × π × r² × h.

Substitute radius (r) = 4 cm and height (h) = 9 cm.

V = (1/3) × 3.14 × 4² × 9 = 3.14 × 16 × 3 = 150.72 cm³.

∴ The correct answer is 150.72 cm³.

Alternate Method

Given:

⇒ Radius (r) of the right circular cone = 4 cm

⇒ Height (h) of the right circular cone = 9 cm

Formula Used:

⇒ Volume of a cone (V) = (1/3) × π × r² × h

Calculations:

⇒ V = (1/3) × 3.14 × (4)² × 9

⇒ V = (1/3) × 3.14 × 16 × 9

⇒ V = 3.14 × 16 × 3

⇒ V = 3.14 × 48

⇒ V = 150.72 cm³

∴ The correct answer is 150.72 cm³.

Additional Information

Slant Height of a Cone

The distance from the apex to any point on the base circumference is l = √(r² + h²).

Curved Surface Area (CSA)

The area of the curved part of the cone is given by CSA = π × r × l.

Total Surface Area (TSA)

The sum of the curved area and the circular base area is TSA = πr(r + l).

Volume of a Cylinder

A cylinder with the same radius and height as a cone has three times its volume: V = πr²h.

Shortcut Trick

The Lateral Surface Area (LSA) of a cylinder is calculated using the formula: 2 × π × radius × height.

LSA = 2 × π × 4 × 6

LSA = 48π

∴ The correct answer is 48π cm².

Alternate Method

Given:

Radius (r) of the cylinder = 4 cm

Height (h) of the cylinder = 6 cm

Formula Used:

Lateral Surface Area of a cylinder = 2 × π × r × h

Calculations:

⇒ LSA = 2 × π × 4 × 6

⇒ LSA = 8 × 6 × π

⇒ LSA = 48π cm²

∴ The correct answer is 48π cm².

Additional Information

Total Surface Area (TSA) of a Cylinder

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

Volume of a Cylinder

The amount of space occupied by a right circular cylinder is calculated as: Volume = πr²h.

Base Area

The area of the base (or top) of a cylinder is simply the area of a circle: Area = πr².

Shortcut Trick

AAA (Angle-Angle-Angle) is a condition for similarity, not congruence.

For two triangles to be congruent, they must have the exact same shape and size.

While the angles are the same (same shape), the different side lengths mean the sizes are different.

Since no side of Triangle A is equal to a corresponding side of Triangle B, they cannot be congruent.

∴ The correct answer is Not Congruent.

Alternate Method

Given:

Triangle A angles: 40°, 60°, 80°

Triangle B angles: 40°, 60°, 80°

Side lengths of Triangle A ≠ Side lengths of Triangle B

Calculations:

⇒ According to the AAA (Angle-Angle-Angle) property, the two triangles are Similar (shape is identical).

⇒ Congruency requires both equal angles and equal corresponding sides (SSS, SAS, ASA, or RHS).

⇒ In this case, the side lengths are different, which means the scale factor is not 1.

⇒ If the scale factor is not 1, the triangles are of different sizes.

⇒ Therefore, they satisfy the conditions for Similarity but fail the condition for Congruence.

∴ The correct answer is Not Congruent.

Additional Information

Congruence Criteria

Triangles are congruent if they satisfy SSS (all sides equal), SAS (two sides and included angle equal), ASA/AAS (two angles and one side equal), or RHS (hypotenuse and one side of right triangles equal).

AAA Similarity Rule

The AAA property only ensures that the triangles are similar. Similarity means the corresponding sides are in the same ratio, but not necessarily of the same length.

Congruent vs Similar

All congruent triangles are similar (ratio 1:1), but all similar triangles are not necessarily congruent unless their corresponding sides are equal.

Shortcut Trick

For a transverse common tangent to exist, the distance between centers (d) must be greater than or equal to the sum of radii (r₁ + r₂).

Given: r₁ = 10 m, r₂ = 6 m ⇒ Sum of radii = 10 + 6 = 16 m.

Given: Distance between centers (d) = 12 m.

Since 12 m < 16 m (d < r₁ + r₂), the circles intersect at two points.

∴ The correct answer is No transverse tangent.

Alternate Method

Given: Radius of larger platform (r₁) = 10 m, Radius of smaller platform (r₂) = 6 m, Distance between centers (d) = 12 m.

The length of a transverse common tangent is calculated using the formula:

Length of TCT = √[d² − (r₁ + r₂)²]

⇒ r₁ + r₂ = 10 + 6 = 16 m

⇒ Length = √[12² − 16²]

⇒ Length = √[144 − 256]

⇒ Length = √[−112]

Since the value under the square root is negative, the length is not a real number, meaning the tangent does not exist.

∴ The correct answer is No transverse tangent.

Additional Information

Common Tangent Existence Rules

1. Separate circles (d > r₁+r₂): 4 tangents (2 Direct, 2 Transverse).

2. Touching externally (d = r₁+r₂): 3 tangents (2 Direct, 1 Transverse).

3. Intersecting (r₁−r₂ < d < r₁+r₂): 2 tangents (2 Direct, 0 Transverse).

Direct Common Tangent (DCT) Formula

The length of the direct common tangent is given by √[d² − (r₁ − r₂)²].

Touching Internally

If d = |r₁ − r₂|, the circles touch internally and have only 1 common tangent (Direct).

Shortcut Trick

For an angle of elevation of 45° in a right-angled triangle, the perpendicular (height) is always equal to the base (distance).

Ratio of sides in a 45°−45°−90° triangle is 1 : 1 : √2.

Since the base is 20 m, the height must also be 20 m.

∴ The correct answer is 20 m.

Alternate Method

Given:

Angle of elevation (θ) = 45°

Distance from base (Base) = 20 m

Formula Used:

tan θ = Perpendicular (Height) ÷ Base

Calculations:

⇒ tan 45° = Height (h) ÷ 20

⇒ 1 = h ÷ 20

⇒ h = 1 × 20

⇒ h = 20 m

∴ The correct answer is 20 m.

Additional Information

Trigonometric Ratios

Standard values: tan 30° = 1/√3, tan 45° = 1, and tan 60° = √3. These are frequently used in height and distance problems.

Angle of Elevation

It is the angle formed by the line of sight with the horizontal when looking upwards at an object.

Angle of Depression

It is the angle formed when looking downwards. Numerically, Angle of Elevation = Angle of Depression due to alternate interior angles.

Shortcut Trick

Assume the base average monthly income as ₹17,000.

Extra income in next 3 months = 19,000 − 17,000 = ₹2,000 per month.

Total excess income = 3 × 2,000 = ₹6,000.

Average net deviation for 9 months = 6,000 ÷ 9 = ₹666.67.

Required Average = 17,000 + 666.67 = ₹17,666.67.

∴ The correct answer is Rs. 17,666.67.

Alternate Method

Given: Average income for 6 months = ₹17,000; Income for next 3 months = ₹19,000.

Formula Used: Combined Average = (Total Income) ÷ (Total Months)

⇒ Total income for first 6 months = 6 × 17,000 = ₹1,02,000

⇒ Total income for next 3 months = 3 × 19,000 = ₹57,000

⇒ Total income for all 9 months = 1,02,000 + 57,000 = ₹1,59,000

⇒ Average income = 1,59,000 ÷ 9

⇒ Average income = ₹17,666.666...

⇒ Average income ≈ ₹17,666.67

∴ The correct answer is Rs. 17,666.67.

Additional Information

Weighted Average Formula

Used when different groups have different averages: Combined Avg = (n₁x₁ + n₂x₂) ÷ (n₁ + n₂), where n is the number of items and x is the average.

Net Deviation Method

This shortcut assumes a base average and calculates how much more or less the total sum is compared to that base, then redistributes the excess or deficit over the total count.

Basic Average Property

The Sum of Observations is always equal to the Average × Number of Observations. This is the fundamental basis for all Average problems.

Shortcut Trick Area of circle = 1.54 cm2 ⇒ πr2 = 1.54 ⇒ r = 0.7 cm. For a 90° sector, arc length (L) = 1/4 × (2πr) = 1/4 × 4.4 = 1.1 cm. Perimeter = Arc Length + 2 × Radius = 1.1 + 2(0.7) = 1.1 + 1.4 = 2.5 cm. ∴ The correct answer is 2.5 cm.

Alternate Method Given: Area of the circle = 1.54 cm2 Central angle (θ) = 90° Formula Used:

1. Area of circle = πr2
2. Perimeter of sector = (θ/360°) × 2πr + 2r

Calculations: ⇒ πr2 = 1.54 ⇒ (22/7) × r2 = 1.54 ⇒ r2 = (1.54 × 7) ÷ 22 ⇒ r2 = 0.07 × 7 = 0.49 ⇒ Radius (r) = √0.49 = 0.7 cm ⇒ Length of arc (L) = (90/360) × 2 × (22/7) × 0.7 ⇒ L = (1/4) × 4.4 = 1.1 cm ⇒ Perimeter = Arc Length + 2 × r ⇒ Perimeter = 1.1 + 2 × 0.7 = 1.1 + 1.4 ⇒ Perimeter = 2.5 cm ∴ The correct answer is 2.5 cm.

Additional Information Area of a Sector The space enclosed by two radii and an arc is calculated as A = (θ/360°) × πr2. Arc Length and Radius Relation The area of a sector can also be expressed in terms of arc length (L) as Area = (1/2) × L × r. Major vs Minor Sector A minor sector has a central angle θ < 180°, while a major sector has θ > 180°.

Shortcut Trick

Total units in the fraction 4/5 = 5 units.

Given: 5 units → 450 eatables.

So, 1 unit → 450 ÷ 5 = 90.

Sold items (4 units) → 4 × 90 = 360.

Since 360 = 120 × 3, it matches the option.

∴ The correct answer is 120 times of 3.

Alternate Method

Given:

Total eatables received by the provision store = 450

Fraction of eatables sold daily = 4/5

Calculation:

⇒ Number of eatables sold = Total eatables × Fraction sold

⇒ Sold = 450 × 4/5

⇒ Sold = (450 ÷ 5) × 4

⇒ Sold = 90 × 4

⇒ Sold = 360

Now, let us evaluate the given options:

⇒ 120 times of 3 = 120 × 3 = 360

∴ The correct answer is 120 times of 3.

Additional Information

Fraction to Percentage

To convert a fraction to a percentage, multiply by 100. Here, (4/5) × 100 = 80%.

Remainder Calculation

If 4/5 is sold, the remaining part is 1 − 4/5 = 1/5. For 450 items, the unsold quantity is 450 ÷ 5 = 90.

Unitary Method

Finding the value of a single unit (1/5) first makes calculating any other multiple (4/5) simpler.

Shortcut Trick

If cosecA + cotA = x, then cosecA = (x² + 1) / 2x.

Here, x = 11/4.

⇒ cosecA = ((11/4)² + 1) ÷ (2 × 11/4) = (137/16) ÷ (11/2) = 137/88.

Since sinA = 1/cosecA, we get sinA = 88/137.

∴ The correct answer is 88/137.

Alternate Method

Given: cosecA + cotA = 11/4

Formula Used: cosec²A − cot²A = 1

⇒ We know the algebraic identity: (cosecA − cotA)(cosecA + cotA) = 1

⇒ (cosecA − cotA) × (11/4) = 1

⇒ cosecA − cotA = 4/11

⇒ Adding both equations: (cosecA + cotA) + (cosecA − cotA) = 11/4 + 4/11

⇒ 2 cosecA = (121 + 16) / 44 = 137 / 44

⇒ cosecA = 137 / 88

⇒ sinA = 1 / cosecA = 1 / (137/88) = 88/137

∴ The correct answer is 88/137.

Additional Information

Fundamental Identity

The Pythagorean identity 1 + cot²θ = cosec²θ allows us to derive that cosecθ − cotθ is the reciprocal of cosecθ + cotθ.

Reciprocal Relations

Standard trigonometric reciprocals: sinθ = 1/cosecθ, cosθ = 1/secθ, and tanθ = 1/cotθ.

Secant-Tangent Identity

Similarly, sec²θ − tan²θ = 1. If secθ + tanθ = k, then secθ − tanθ = 1/k.