Trigonometric Functions Test 18

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Trigonometric Functions Test 18

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Weekly Quiz Competition

Question 1
1 / -0

The no of solution of the equation: $$1+\sin { x } \cos { x } =2\sin { x } \cos { x } $$ in $$x$$, $$\left[ 0,40 \right] $$ are

A
$$4$$

B
$$5$$

C
$$7$$

D
$$None\ of\ these$$

Solution

$$ 1+sinx cosx = 2sinx cosx $$
$$ x\epsilon [ 0,40]$$
$$ 1 = \dfrac{2}{2}sinx cosx $$
$$ [\because 2 sinx cosx - sin2x]$$
$$ 2 = sin 2x.$$
$$ 2x = \dfrac{1}{2}sin^{-1}(2)$$
$$ x = \dfrac{1}{2}sin^{-1}(2)$$
Question 2
1 / -0

$$\sin^2 20+\sin^270$$ is equal to_____

A
$$1$$

B
$$-1$$

C
$$0$$

D
$$2$$

Solution

$$\sin^2 20^o+\sin^270^o=\sin^2 20^o+\sin^2(90^o-20^o)=\sin^220^o+\cos^220^o=1$$
Question 3
1 / -0

Find number of solutions to the equation:$$\left[ \sin { x+\cos { x } } \right] =3+\left[ -\sin { x } \right] +\left[ -\cos { x } \right]$$

A
$$0$$

B
$$1$$

C
$$2$$

D
infinite

Solution
Question 4
1 / -0

The number of solutions of the equation $$|\cot x|=\cot x+\dfrac{1}{\sin x}(0\le x\le 2\pi)$$ is :

A
$$0$$

B
$$1$$

C
$$2$$

D
$$3$$

Solution

$$(\cot x)=\cot x+\dfrac{1}{\sin x}$$ $$0\le x\le 2\pi$$
$$0\pi\le x < \dfrac{3\pi}{2}$$ and $$0\le x\le \dfrac{\pi}{2}$$
$$\cot x=\cot x+\dfrac{1}{\sin x}$$
$$\sin x=0$$
$$x=0$$
$$2\cot x=\dfrac{1}{\sin x}$$
$$\dfrac{2}{\sin x}\cos x=\dfrac{1}{\sin x}$$
$$\sin x=1\Rightarrow x=\dfrac{\pi}{2}$$
$$0\cos x=\dfrac{1}{2}$$
$$x=\dfrac{\pi}{3}$$
Question 5
1 / -0

If $$\cos 3x=-1$$, where $$0^{o} \ge x \ge 360^{o}$$, then $$x$$

A
$$60^{o},180^{o},300^{o}$$

B
$$180^{o}$$

C
$$60^{o},180^{o}$$

D
$$180^{o},300^{o}$$

Solution

Given $$\cos 3 x=-1$$
$$\implies 3 x=2 n \pi\pm \pi=(2 n \pm 1)\pi$$ where $$n$$ is an integer
$$\implies x=\dfrac{(2 n\pm 1)\pi}{3}$$
$$\implies x=(2 n\pm 1)60^{\circ}=\pm 60^{\circ},\pm 180^{\circ},\pm 300^{\circ},\pm 420^{\circ},\cdots$$
As $$0^{\circ}\le x\le 360^{\circ},x=60^{\circ},180^{\circ},300^{\circ} $$
Question 6
1 / -0

Solution set of the equation $$\sin ^{ 2 }{ x } +\cos ^{ 2 }{ 3x }$$ =1$$ is given by

A
$$\left\{ \frac { n\pi }{ 4 } ,n\epsilon I \right\}$$

B
$$\left\{ n\pi ,n\epsilon I \right\}$$

C
$$\left\{ \frac { n\pi }{ 2 } ,n\epsilon I \right\}$$

D
$$none of these$$

Solution

According to question,

$$\sin^2x+\cos^23x=1$$

$$\implies cos^23x= 1-\sin^2x$$
$$\implies \cos^23x=\cos^2x$$

Solving through trigonometric eq,

$$3x=2n\pi+x$$
$$\implies 2x=2n\pi$$

=> $$x=n\pi$$ , $$n\epsilon I$$
Question 7
1 / -0

If $$\theta$$ is an acute angle such that $$\sec^2 \theta = 3$$. then the value of $$\dfrac{\tan^2 \theta - \text{cosec}^2 \theta}{\tan^2 \theta - \text{cosec}^2 \theta}$$ is

A
$$\dfrac{4}{7}$$

B
$$\dfrac{3}{7}$$

C
$$1$$

D
$$\dfrac{1}{7}$$

Solution

Given $$\text{sec}^2 \theta=3$$
for any value of $$\theta$$,$$\dfrac{\tan^2 \theta-\text{cosec}^2 \theta}{\tan^2 \theta-\text{cosec}^2 \theta}=1$$ $$(\because \text{the numerator and denominator is same})$$
Question 8
1 / -0

For $$x\ \in\ (0,\pi)$$, the equation $$\sin x+2\sin 2x-\sin 3x=3$$ has

A
infinitely solutions

B
three solution

C
one solution

D
no solution

Solution

$$\sin x+2\sin 2x-\sin 3x=3$$
$$\sin x+4\sin x\cos x=3+\sin 3x$$
$$\sin x (1+4\cos x)=3+\sin 3x$$
$$\sin x(1+4\cos x)=3+\sin x(3-4\sin^2 x)$$
$$\sin x(4\cos^2x-4\cos x-2)=-2$$
$$4\cos^2x-4\cos x-2=9$$
$$(2\cos x-1)^2-3 \ge -3$$
and on $$(0, \pi)$$ we have $$0 < \sin x \le 1$$ they
The $$LHS$$ of your last equation. should be $$\ge -3$$ on $$(0,\ \pi)$$
No Solution in $$(0,\ \pi)$$
Question 9
1 / -0

The number of value of $$x\in (0,2\pi)$$ satisfying $$\log_{\tan x}(2+4\cos^2x)=2$$

A
$$\dfrac{\pi}{4}$$

B
$$\dfrac{\pi}{2}$$

C
$$\dfrac{\pi}{3}$$

D
$$\dfrac{\pi}{6}$$

Solution
Question 10
1 / -0

The sum $$s=sin\theta+sin2\theta+.......+sin n \theta$$, equals

A
$$sin\dfrac{1}{2}(n+1)\theta sin \dfrac{1}{2}n \theta/ sin \dfrac{\theta}{2}$$

B
$$cos\dfrac{1}{2}(n+1)\theta sin \dfrac{1}{2}n \theta/ sin \dfrac{\theta}{2}$$

C
$$sin\dfrac{1}{2}(n+1)\theta sin \dfrac{1}{2}n \theta/ cos \dfrac{\theta}{2}$$

D
$$cos\dfrac{1}{2}(n+1)\theta sin \dfrac{1}{2}n \theta/ cos\dfrac{\theta}{2}$$

Solution

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Trigonometric Functions Test 18

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Trigonometric Functions Test 18

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SHARING IS CARING

Question 1

$$4$$

$$5$$

$$7$$

$$None\ of\ these$$

Solution

Question 2

$$1$$

$$-1$$

$$0$$

$$2$$

Solution

Question 3

$$0$$

$$1$$

$$2$$

infinite

Solution

Question 4

$$0$$

$$1$$

$$2$$

$$3$$

Solution

Question 5

$$60^{o},180^{o},300^{o}$$

$$180^{o}$$

$$60^{o},180^{o}$$

$$180^{o},300^{o}$$

Solution

Question 6

$$\left\{ \frac { n\pi }{ 4 } ,n\epsilon I \right\}$$

$$\left\{ n\pi ,n\epsilon I \right\}$$

$$\left\{ \frac { n\pi }{ 2 } ,n\epsilon I \right\}$$

$$none of these$$

Solution

Question 7

$$\dfrac{4}{7}$$

$$\dfrac{3}{7}$$

$$1$$

$$\dfrac{1}{7}$$

Solution

Question 8

infinitely solutions

three solution

one solution

no solution

Solution

Question 9

$$\dfrac{\pi}{4}$$

$$\dfrac{\pi}{2}$$

$$\dfrac{\pi}{3}$$

$$\dfrac{\pi}{6}$$

Solution

Question 10

$$sin\dfrac{1}{2}(n+1)\theta sin \dfrac{1}{2}n \theta/ sin \dfrac{\theta}{2}$$

$$cos\dfrac{1}{2}(n+1)\theta sin \dfrac{1}{2}n \theta/ sin \dfrac{\theta}{2}$$

$$sin\dfrac{1}{2}(n+1)\theta sin \dfrac{1}{2}n \theta/ cos \dfrac{\theta}{2}$$

$$cos\dfrac{1}{2}(n+1)\theta sin \dfrac{1}{2}n \theta/ cos\dfrac{\theta}{2}$$

Solution

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