CBSE Class 10 Mathematics Introduction to Trigonometry MCQs Set 26

Answer: (c) 10

Explanation:
1. Let the hypotenuse be h. The sides are h/2, (√3/2)h and h.
2. Perimeter = h(1 + 1/2 + √3/2) = h(3 + √3)/2 = 5(3 + √3), so h = 10.

Teacher's Note:
1. The sides of a 30°-60°-90° triangle are in the ratio 1 : √3 : 2.
2. Factor out 5 on the right side to compare.

Answer: (b) T₁ − T₂ = 0

Explanation:
1. T₁ = (√2/2 − 1/2 + 0)/(√2/2 + 1/2) = (√2 − 1)/(√2 + 1).
2. T₂ = (√2 − 1)/(√2 + 1 + 0) = (√2 − 1)/(√2 + 1).
3. So T₁ = T₂, that is T₁ − T₂ = 0.

Teacher's Note:
1. Reduce to sin and cos, or use sin² + cos² = 1 and 1 + tan² = sec².
2. Check the result by substituting a convenient angle such as 30° or 45°.

Answer: (d) none of these

Explanation:
1. tan 18° tan 72° = 1 and tan 36° tan 54° = 1, so the product is 1.
2. sin 71° = cos 19°, so sin² 19° + sin² 71° = 1.
3. sin 62° = cos 28° and sec 42° = cosec 48°, so the expression is 2 − 1 = 1.
4. All three equal one, so the answer is none of these.

Teacher's Note:
1. Use complementary angle identities such as tan(90° − θ) = cot θ.
2. When asked which is not one, check each option.

Answer: (c) 2

Explanation:
1. cot 80° = tan 10°, so sec² 10° − cot² 80° = sec² 10° − tan² 10° = 1.
2. Numerator = sin(15° + 75°) = sin 90° = 1. Denominator = cos² θ + sin² θ = 1.
3. The total is 1 + 1 = 2.

Teacher's Note:
1. Reduce to sin and cos, or use sin² + cos² = 1 and 1 + tan² = sec².
2. Check the result by substituting a convenient angle such as 30° or 45°.

Answer: (a) \( \frac{sin²θ}{1 − cosθ} \)

Explanation:
1. (1 + sec θ)/sec θ = cos θ + 1.
2. Multiply and divide by (1 − cos θ): (1 + cos θ)(1 − cos θ)/(1 − cos θ) = sin² θ/(1 − cos θ).

Teacher's Note:
1. Reduce to sin and cos, or use sin² + cos² = 1 and 1 + tan² = sec².
2. Check the result by substituting a convenient angle such as 30° or 45°.

Answer: (d) tan⁷ θ + cot⁷ θ = 2

Explanation:
1. tan θ + 1/tan θ = 2 gives (tan θ − 1)² = 0, so tan θ = 1.
2. Then tan⁷ θ + cot⁷ θ = 1 + 1 = 2, while tan⁵ θ + cot⁵ θ = 2, not 32.

Teacher's Note:
1. Reduce to sin and cos, or use sin² + cos² = 1 and 1 + tan² = sec².
2. Check the result by substituting a convenient angle such as 30° or 45°.

Answer: (a) 1

Explanation:
1. The expression is a perfect square: (cos² θ + sin² θ)² = 1² = 1 for every θ.

Teacher's Note:
1. Reduce to sin and cos, or use sin² + cos² = 1 and 1 + tan² = sec².
2. Check the result by substituting a convenient angle such as 30° or 45°.

Answer: (d) −1

Explanation:
1. (cos θ − 1)(1 + cos θ) = cos² θ − 1 = −sin² θ.
2. 1 + cot² θ = cosec² θ.
3. The product = −sin² θ × 1/sin² θ = −1.

Teacher's Note:
1. Reduce to sin and cos, or use sin² + cos² = 1 and 1 + tan² = sec².
2. Check the result by substituting a convenient angle such as 30° or 45°.

Answer: (c) √2

Explanation:
1. Combine: sin x [(sec x − 1) + (sec x + 1)]/(sec² x − 1) = 2 sin x sec x/tan² x = 2 tan x/tan² x = 2/tan x.
2. 2/tan x = 2 gives tan x = 1, so x = 45°.
3. cosec 45° = √2.

Teacher's Note:
1. Reduce to sin and cos, or use sin² + cos² = 1 and 1 + tan² = sec².
2. Check the result by substituting a convenient angle such as 30° or 45°.

Answer: (c) √2 sin θ

Explanation:
1. sin θ = (√2 − 1) cos θ, so cos θ = sin θ/(√2 − 1) = (√2 + 1) sin θ.
2. cos θ − sin θ = (√2 + 1) sin θ − sin θ = √2 sin θ.

Teacher's Note:
1. Reduce to sin and cos, or use sin² + cos² = 1 and 1 + tan² = sec².
2. Check the result by substituting a convenient angle such as 30° or 45°.

Answer: (c) x² + y² = 1

Explanation:
1. From x sin θ = y cos θ, write x = k cos θ and y = k sin θ.
2. Substitute: k cos θ sin³ θ + k sin θ cos³ θ = k sin θ cos θ (sin² θ + cos² θ) = k sin θ cos θ.
3. This equals sin θ cos θ, so k = 1 and x² + y² = cos² θ + sin² θ = 1.

Teacher's Note:
1. Reduce to sin and cos, or use sin² + cos² = 1 and 1 + tan² = sec².
2. Check the result by substituting a convenient angle such as 30° or 45°.

Answer: (a) independent of both α and β

Explanation:
1. x² + y² = r² cos² α (cos² β + sin² β) = r² cos² α.
2. x² + y² + z² = r² cos² α + r² sin² α = r².

Teacher's Note:
1. Reduce to sin and cos, or use sin² + cos² = 1 and 1 + tan² = sec².
2. Check the result by substituting a convenient angle such as 30° or 45°.

Answer: (d) 4

Explanation:
1. From the first equation x = 2y cos θ/sin θ.
2. Substituting in the second: 4y/sin θ − y/sin θ = 3, so 3y/sin θ = 3 and y = sin θ.
3. Then x = 2 cos θ, so x² + 4y² = 4 cos² θ + 4 sin² θ = 4.

Teacher's Note:
1. Reduce to sin and cos, or use sin² + cos² = 1 and 1 + tan² = sec².
2. Check the result by substituting a convenient angle such as 30° or 45°.

Answer: (d) (1 + cot x − cosec x)(1 + tan x + sec x) = 1

Explanation:
1. (a) sin⁴ − cos⁴ = (sin² − cos²)(sin² + cos²), so the ratio is 1: correct.
2. (b) With t = tan x both sides equal 1/(t(1 + t)): correct. (c) (sin² + cos²)/(sin² cos²) = cosec² sec²: correct.
3. (d) (1 + cot x − cosec x)(1 + tan x + sec x) = 2, not 1, so (d) is not correct.

Teacher's Note:
1. Multiply out in terms of sin and cos to evaluate (d).
2. The identity (d) gives 2.

Answer: (c) (1 + cos x)²

Explanation:
1. 2 + 2 cos x − sin² x = 2 + 2 cos x − (1 − cos² x) = 1 + 2 cos x + cos² x.
2. This equals (1 + cos x)².

Teacher's Note:
1. Reduce to sin and cos, or use sin² + cos² = 1 and 1 + tan² = sec².
2. Check the result by substituting a convenient angle such as 30° or 45°.

Answer: (d) \( \frac{23}{32} \)

Explanation:
1. Squaring: 1 + 2 sin x cos x = 1/4, so sin x cos x = −3/8.
2. sin⁴ x + cos⁴ x = 1 − 2 sin² x cos² x = 1 − 2 × 9/64 = 1 − 9/32 = 23/32.

Teacher's Note:
1. Reduce to sin and cos, or use sin² + cos² = 1 and 1 + tan² = sec².
2. Check the result by substituting a convenient angle such as 30° or 45°.

Answer: (b) sec θ

Explanation:
1. tan² θ = sec² θ − 1 = (sec θ − 1)(sec θ + 1).
2. So tan² θ/(1 + sec θ) = sec θ − 1.
3. The expression is 1 + sec θ − 1 = sec θ.

Teacher's Note:
1. Reduce to sin and cos, or use sin² + cos² = 1 and 1 + tan² = sec².
2. Check the result by substituting a convenient angle such as 30° or 45°.

Answer: (a) 1

Explanation:
1. sin x = 1 − sin² x = cos² x.
2. cos⁸ x + 2 cos⁶ x + cos⁴ x = cos⁴ x (cos² x + 1)² = sin² x (sin x + 1)² = (sin x + sin² x)².
3. This equals 1² = 1.

Teacher's Note:
1. Reduce to sin and cos, or use sin² + cos² = 1 and 1 + tan² = sec².
2. Check the result by substituting a convenient angle such as 30° or 45°.

Answer: (d) sec² x + cosec² x = 16/3

Explanation:
1. Using cos² x = 1 − sin² x: 2s² − 5s + 2 = 0, so (2s − 1)(s − 2) = 0 and sin x = 1/2.
2. Then cos² x = 3/4, sec² x = 4/3, cosec² x = 4.
3. sec² x + cosec² x = 4/3 + 4 = 16/3.

Teacher's Note:
1. Reduce to sin and cos, or use sin² + cos² = 1 and 1 + tan² = sec².
2. Check the result by substituting a convenient angle such as 30° or 45°.

Answer: (b) 5

Explanation:
1. sec² θ + tan² θ = 1 + 2 tan² θ = 2, so tan² θ = 1/2 and cot² θ = 2.
2. cosec² θ = 1 + cot² θ = 3.
3. cosec² θ + cot² θ = 3 + 2 = 5.

Teacher's Note:
1. Reduce to sin and cos, or use sin² + cos² = 1 and 1 + tan² = sec².
2. Check the result by substituting a convenient angle such as 30° or 45°.

Answer: (c) 3

Explanation:
1. Square: cosec² θ + sin² θ − 2 = 5, so cosec² θ + sin² θ = 7.
2. (sin θ + cosec θ)² = 7 + 2 = 9, so sin θ + cosec θ = 3 (positive for an acute angle).

Teacher's Note:
1. Reduce to sin and cos, or use sin² + cos² = 1 and 1 + tan² = sec².
2. Check the result by substituting a convenient angle such as 30° or 45°.

Answer: (c) a² − b²

Explanation:
1. x² − y² = (a sec θ + b tan θ)² − (a tan θ + b sec θ)².
2. = a²(sec² θ − tan² θ) + b²(tan² θ − sec² θ) + 2ab sec θ tan θ − 2ab sec θ tan θ.
3. = a² − b².

Teacher's Note:
1. Reduce to sin and cos, or use sin² + cos² = 1 and 1 + tan² = sec².
2. Check the result by substituting a convenient angle such as 30° or 45°.

Answer: (c) 25

Explanation:
1. Square and add: (a cos θ + b sin θ)² + (a sin θ − b cos θ)² = a² + b².
2. = 9 + 16 = 25.

Teacher's Note:
1. Reduce to sin and cos, or use sin² + cos² = 1 and 1 + tan² = sec².
2. Check the result by substituting a convenient angle such as 30° or 45°.