Trigonometric Functions Test 33

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Trigonometric Functions Test 33

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Question 1
1 / -0

If
$$x=sin\left(2\tan^{-1}2\right), y=\sin\left(\dfrac{1}{2}\tan^{-1}\dfrac{4}{3}\right)$$. Then

A
$$x=y^2$$

B
$$y^2=1-x$$

C
$$x^2\dfrac{y}{2}$$

D
$$y^2=1+x$$

Solution
Question 2
1 / -0

The smallest positive number p for which the equation cos (p sin x) = sin (p cos x) has a solution in $$[0, 2 \pi]$$ is

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

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

C
$$\dfrac{\pi}{4 \sqrt{2}}$$

D
$$\dfrac{\pi}{2 \sqrt{2}}$$

Solution

$$ \begin{aligned} &\begin{aligned} &-\cos (p \sin x)=\sin (p \cos x) \\ \Rightarrow & \cos (p \sin x)=\cos \left(\frac{\pi}{2}-p \cos x\right) \end{aligned}\\ &2 sin x \pm\left(\frac{\pi}{2}-p \cos x\right) \quad(n \geqslant 0)\\ &\Rightarrow p \sin x+p \cos x=2 n \pi+\frac{\pi}{2}\\ &4\\ &p \sin x-p \cos x=2 n \pi-\frac{\pi}{2}\\ &\Rightarrow p \sqrt{2}\left(\sin \left(x+\frac{\pi}{4}\right)\right)=\frac{(4 n+1)}{2} \pi \end{aligned} $$
or $$\frac{\sqrt{2}\left(\sin \left(x-\frac{\pi}{4}\right)\right)}=\frac{(4 n-1) n}{2}$$
As $$-1 \leq \sin \left(x+\frac{\pi}{4}\right) \leqslant 1$$
$$\Rightarrow-p \sqrt{2} \leq p \sqrt{2} \sin \left(x+\frac{\pi}{4}\right) \leqslant p \sqrt{2}$$
$$\Rightarrow-p \sqrt{2} \leq\left(\frac{4 n+1}{2}\right) \pi \leq p \sqrt{2} \quad$$... (i)
And similarly,
$$\Rightarrow-p \sqrt{2} \leq\left(\frac{4 n-1}{2}\right)^{n} \leq p \sqrt{2}$$
(ii) is subset of (i),
$$\therefore$$ We can consider only (i).
If $$\begin{array}{ll}n \geqslant 0, & \left(\frac{4 n+1}{2}\right) \pi \Rightarrow\left(\frac{4 n+1}{2}\right) n \leqslant \sqrt{2} p\end{array}$$
$$\Rightarrow \sqrt{2} p \geqslant \frac{\pi}{2}$$
$$\Rightarrow p \geqslant \frac{\pi}{2 \sqrt{2}} \Rightarrow \quad p \geqslant \frac{\pi \sqrt{2}}{4}$$
$$\therefore$$ option $$D$$ is correct.
Question 3
1 / -0

If $$sin \alpha + sin \beta = \dfrac{1}{2} $$ and $$cos \alpha + cos \beta = \dfrac{\sqrt{3}}{2}$$ then $$3 \beta + \alpha = $$

A
$$0^o$$

B
$$60^o$$

C
$$120^o$$

D
$$90^o$$

Solution
Question 4
1 / -0

In $$\Delta ABC, \sum \dfrac{b^2 - c^2}{a^2} sin \, 2A$$ =

A
$$0$$

B
$$a^2 + b^2$$

C
$$b^2 + c^2$$

D
$$c^2 + a^2$$

Solution
Question 5
1 / -0

$$[3(\sin(\cos\ 1)+\cos(\cos\ 1))]$$ is equal to (where[.] is denotes the greatest integer function)

A
$$3$$

B
$$4$$

C
$$5$$

D
$$6$$

Solution
Question 6
1 / -0

If $$sin\alpha sin \beta - cos \alpha \beta - 1, = 0$$ then the value of $$cot \alpha tan \beta $$ is

A
-1

B
0

C
1

D
None

Solution
Question 7
1 / -0

If $$x=\sin\alpha+\sin\beta$$ and $$y=\cos\alpha+\cos\beta$$
then $$x=\tan\alpha+\tan\beta=$$

A
$$\dfrac { 8xy }{ 2\left( { y }^{ 2 }-{ x }^{ 2 } \right) +\left( { x }^{ 2 }+{ y }^{ 2 } \right) .\left( { x }^{ 2 }+{ y }^{ 2 }-2 \right) }$$

B
$$\dfrac { 4xy }{ 2\left( { y }^{ 2 }-{ x }^{ 2 } \right) +\left( { x }^{ 2 }+{ y }^{ 2 } \right) .\left( { x }^{ 2 }+{ y }^{ 2 }-2 \right) }$$

C
$$\dfrac { 8xy }{ \left( { x }^{ 2 }+{ y }^{ 2 } \right) .\left( { x }^{ 2 }+{ y }^{ 2 }-2 \right) }$$

D
$$4xy$$

Solution
Question 8
1 / -0

The value of $$\csc \dfrac{\pi}{13}-\sqrt{3} \sec \dfrac{\pi}{18}$$ is a

A
Surd

B
Rational which is not integral

C
Negative integer

D
Natural number

Solution
Question 9
1 / -0

If $$\cos^{2} \theta + 3\cos^{2}\theta + 4\sin^{2}\theta + .... (200) terms = 10025$$, where $$\theta$$ is an acute angle, then the value of $$\sin \theta - \cos \theta$$ is

A
$$\dfrac {1-\sqrt {3}}{2}$$

B
$$\dfrac {1+\sqrt {3}}{2}$$

C
$$\dfrac {\sqrt {3}-1}{2}$$

D
$$0$$

Solution
Question 10
1 / -0

The number of solution of the equation $${x^3} + 2{x^2} + 5x + 2\cos x = 0\,in\,[0,2\pi ]$$ is

A
$$0$$

B
$$1$$

C
$$2$$

D
$$3$$

Solution