Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12

Tamilnadu State Board New Syllabus Samacheer Kalvi 11th Maths Guide Pdf Chapter 3 Trigonometry Ex 3.12 Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.12

Choose the correct or the most suitable answer:

Question 1.
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 1
(1) √2
(2) √3
(3) 2
(4) 4
Answer:
(4) 4

Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 2
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 3

Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12

Question 2.
If cos 28° + sin 28°= k3, then cos 17° is equal to
(1) \(\frac{\mathbf{k}^{3}}{\sqrt{2}}\)
(2) \(-\frac{\mathbf{k}^{3}}{\sqrt{2}}\)
(3) \(\pm \frac{\mathbf{k}^{3}}{\sqrt{2}}\)
(4) \(-\frac{\mathbf{k}^{3}}{\sqrt{3}}\)
Answer:
(1) \(\frac{\mathbf{k}^{3}}{\sqrt{2}}\)

Explaination:
cos 28° + sin 28° = k3
cos 28° + sin (90° – 62°) = k3
cos 28° + cos 62° = k3
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 4
2 cos 45° . cos 17° = k3
2 × \(\frac{1}{\sqrt{2}}\) cos 17° = k3
√2 cos 17° = k3
cos 17° = \(\frac{\mathrm{k}^{3}}{\sqrt{2}}\)

Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12

Question 3.
The maximum value of
4 sin2x + 3 cos2x + sin \(\) + cos \(\) is
(1) 4 + √2
(2) 3 + √2
(3) 9
(4) 4
Answer:
(1) 4 + √2

Explaination:
4 sin2x + 3 cos2x + sin \(\frac{x}{2}\) + cos \(\frac{x}{2}\)
= sin2x + 3 sin2x + 3 cos2x + sin \(\frac{x}{2}\) + cos \(\frac{x}{2}\)
= sin2x + 3(sin2x + cos2x) + sin \(\frac{x}{2}\) + cos \(\frac{x}{2}\)
= 3 + sin2x + sin \(\frac{x}{2}\) + cos \(\frac{x}{2}\) —– (1)
Maximum value of sin x = 1
sin x = 1 when x = \(\frac{\pi}{2}\)
Maximum value of sin2x = 1
Maximum value is obtained when x = \(\frac{\pi}{2}\)
∴ (1) ⇒ 4 sin2 x + 3 cos2 x + sin \(\frac{x}{2}\) + cos \(\frac{x}{2}\)
= 3 + 1 + sin \(\left(\frac{90^{\circ}}{2}\right)\) + cos \(\left(\frac{90^{\circ}}{2}\right)\)
= 4 + sin 5° + cos 45°
= 4 + \(\frac{1}{\sqrt{2}}\) + \(\frac{1}{\sqrt{2}}\) = 4 + \(\frac{2}{\sqrt{2}}\)
= 4 + √2

Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12

Question 4.
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 5
(1) \(\frac{1}{8}\)
(2) \(\frac{1}{2}\)
(3) \(\frac{1}{\sqrt{3}}\)
(4) \(\frac{1}{\sqrt{2}}\)
Answer:
(1) \(\frac{1}{8}\)

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 6

Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12

Question 5.
If π < 2θ < \(\frac{3 \pi}{2}\), \(\sqrt{2+\sqrt{2+2 \cos 4 \theta}}\) equals to
(1) – 2 cos θ
(2) – 2 sin θ
(3) 2 cos θ
(4) 2 sin θ
Answer:
(1) – 2 cos θ

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 7
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 8
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 9
∴ θ lies in the second quadrant, cos θ is negative in the IInd quadrant.
∴ \(\sqrt{2+\sqrt{2+2 \cos 4 \theta}}\) = – 2 cos θ

Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12

Question 6.
If tan 40° = λ, then
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 10
(1) \(\frac{1-\lambda^{2}}{\lambda}\)
(2) \(\frac{1+\lambda^{2}}{\lambda}\)
(3) \(\frac{1+\lambda^{2}}{2 \lambda}\)
(4) \(\frac{1-\lambda^{2}}{2 \lambda}\)
Answer:
(4) \(\frac{1-\lambda^{2}}{2 \lambda}\)

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 11

Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12

Question 7.
cos 1° + cos 2° + cos 3° + ….. + cos 179° =
(1) 0
(2) 1
(3) – 1
(4) 89
Answer:
(1) 0

Explaination:
cos 1° + cos 2° + cos 3° + ……………… + cos 179°
= (cos 1° + cos 179°) + (cos 2° + cos 178°) + (cos 3° + cos 177°) + …………..
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 12
= 2 cos 90° cos 89° + 2 cos 90° . cos 88° + …………….
= 2 × 0 × cos 89°+ 2 × 0 × cos 88° + …………..
= 0

Question 8.
Let fk(x) = \(\frac{1}{k}\)[sinkx + coskx] where x ∈ R and k ≥ 1. Then f4(x) – f6(x) =
(1) \(\frac{1}{4}\)
(2) \(\frac{1}{12}\)
(3) \(\frac{1}{6}\)
(4) \(\frac{1}{3}\)
Answer:
(2) \(\frac{1}{12}\)

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 13

Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12

Question 9.
Which of the following is not true?
(1) sin θ = – \(\frac{3}{4}\)
(2) cos θ = – 1
(3) tan θ = 25
(4) sec θ = \(\frac{1}{4}\)
Answer:
(4) sec θ = \(\frac{1}{4}\)

Explaination:
We know |cos θ| < 1
sec θ = \(\frac{1}{4}\)
⇒ \(\frac{1}{\cos \theta}\) = \(\frac{1}{4}\)
⇒ cos θ = 4
which is not possible.

Question 10.
cos 2θ cos 2Φ + sin2(θ – Φ) – sin2(θ + Φ) is equal to
(1) sin 2 (θ + Φ)
(2) cos 2 (8 + Φ)
(3) sin 2 (θ – Φ)
(4) cos 2(θ – Φ)
Answer:
(2) cos 2 (8 + Φ)

Explaination:
cos 2θ cos 2Φ + sin2(θ – Φ) – sin2(θ + Φ)
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 14
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 15
= cos 2θ cos 2Φ – sin 2θ sin 2Φ
= cos(2θ + 2Φ)
= cos 2(θ + Φ)

Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12

Question 11.
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 16
(1) sin A + sin B + sin C
(2) 1
(3) 0
(4) cos A + cos B + cos C
Answer:
(3) 0

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 17

Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12

Question 12.
If cos pθ + cos qθ = o and if p ≠ q then θ is equal to(n is any integer)
(1) Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 18
(2) Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 19
(3) Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 20
(4) Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 21
Answer:
Given cos pθ + cos qθ = o
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 22

Question 13.
If tan α and tan β are the roots of x2 + ax + b = 0 then \(\frac{\sin (\alpha+\beta)}{\sin \alpha \sin \beta}\) is equal to
(1) \(\frac{\mathbf{b}}{\mathbf{a}}\)
(2) \(\frac{\mathbf{a}}{\mathbf{b}}\)
(3) –\(\frac{\mathbf{a}}{\mathbf{b}}\)
(4) –\(\frac{\mathbf{b}}{\mathbf{a}}\)
Answer:
(3) –\(\frac{\mathbf{a}}{\mathbf{b}}\)

Explaination:
x2 + ax + b = 0
Given tan α and tan β are the roots of the above equation. Then
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 23
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 24

Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12

Question 14.
In a triangle ABC, sin2 A + sin2 B + sin2 C = 2 then the triangle is .
(1) equilateral triangle
(2) isosceles triangle
(3) right triangle
(4) scalene triangle
Answer:
(3) right triangle

Explaination:
Given sin2 A + sin2 B + sin2 C = 2
Suppose the given triangle is a right angle triangle with ∠C = 90°, then
sin2 C = sin 2 90° = 1 …….. (1)
∴ sin2 A + sin2 B + 1 = 2
sin2 A + sin2B = 1
Also A + B = 90° ⇒ A = 90° – B
sin A = sin (90° – B) = cos B ——- (2)
sin2 A + sin2 B + sin2 C = 2
Using equations (1) and (2)
⇒ cos2 B + sin2 B + sin2 90° = 2
1 + 1 = 2
2 = 2
∴ sin2 A + sin2 B + sin2 C = 2 is true.
∴ ∆ ABC is a right angle triangle.

Question 15.
If f(θ) = |sin θ| + |cos θ|, θ ∈ R then f(θ) is in the interval
(1) [0, 2]
(2) [1, √2]
(3) [1, 2]
(4) [0, 1]
Answer:
(2) [1, √2]

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 25
f(θ) = |sin θ| + |cos θ|
To find the point of intersection of the sine curve and cosine curve solving
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 26

Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12

Question 16.
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 27
(1) cos 2x
(2) cos x
(3) cos 3x
(4) 2 cos x
Answer:
(4) 2 cos x

Explaination:
Consider the numerator cos 6x + 6 cos 4x + 15 cos 2x + 10
cos 6x + 6 cos 4x + 15 cos 2x + 10 = cos 6x + cos 4x + 5 cos 4x + 5 cos 2x + 10 cos 2x + 10
= (cos 6x + cos 4x) + 5 (cos 4x + cos 2x) + 10(cos 2x + 1)
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 28
= 2 cos 5x cos x + 10 cos 3x . cos x + 20 cos2x
= 2 cos x (cos 5x + 5 cos 3x + 10 cos x)
Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12 29
= 2 cos x

Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12

Question 17.
The triangle of maximum area with constant perimeter 12m
(1) is an equilateral triangle with side 4m
(2) is an isosceles triangle with sides 2m, 5m ,5m
(3) is a triangle with sides 3m, 4m, 5m
(4) does not exists.
Answer:
(1) is an equilateral triangle with side 4m

Explaination:
Given the perimeter of the triangle is 12m
2s = 12 ⇒ s = 6
Maximum area is obtained when it is an equilateral triangle with side 4m each.

Question 18.
A wheel is spinning at 2 radians / second. How many seconds will it take to make 10 complete rotations.
(1) 10 π seconds
(2) 20 π seconds
(3) 5 π seconds
(4) 15 π seconds
Answer:
(1) 10 π seconds

Explaination:
In 1 second, it rotates = 2 radians
For 2 radians rotation time taken = 1 second
∴ For 1 complete rotation (2 π radians) time taken
= \(\frac { 1 }{ 2 }\) × 2π = π seconds.
∴ For 10 revolution time taken = π × 10
= 10 π seconds.

Question 19.
If sin α + cos α = b, then sin 2α is equal to
(1) b2 – 1, if b ≤ √2
(2) b2 – 1, if b > √2
(3) b2 – 1, if b ≥ √2
(4) b2 – 1, if b < √2
Answer:
(1) b2 – 1, if b ≤ √2

Explaination:
sin α + cos α = b
(sin α + cos α)2 = b2
sinv α + cos2 α + 2 sin α cos α = b2
1 + sin 2α = b2
sin 2α = b2 – 1
But – 1 ≤ sin 2α ≤ I
– 1 ≤ b2 – 1 ≤ 1
b2 – 1 ≤ 1 ⇒ b2 ≤ 2
⇒ b ≤ √2
∴ sin 2α = b2 – 1 if b ≤ √2

Samacheer Kalvi 11th Maths Guide Chapter 3 Trigonometry Ex 3.12

Question 20.
In a ∆ ABC
(i) sin \(\frac{\mathbf{A}}{2}\) sin \(\frac{\mathbf{B}}{2}\) sin \(\frac{\mathbf{C}}{2}\) > 0
(ii) sin A sin B sin C > 0,then
(1) Both (i) and (ii) are true
(2) only (1) is true
(3) only (ii) Is true
(4) neither (i) nor (ii) is true
Answer:
(1) Both (i) and (ii) are true

Explaination:
We know in a ∆ ABC
sin \(\frac{\mathbf{A}}{2}\) sin \(\frac{\mathbf{B}}{2}\) sin \(\frac{\mathbf{C}}{2}\) > 0
Also sin A sin B sin C > 0
∴ Statements (i) and (ii) are true.

Leave a Comment

Your email address will not be published. Required fields are marked *

Scroll to Top