Trigonometric Functions Test 32

Result

Trigonometric Functions Test 32

Score
-
out of -
Rank
-
out of -

TIME Taken - -

SHARING IS CARING

If our Website helped you a little, then kindly spread our voice using Social Networks. Spread our word to your readers, friends, teachers, students & all those close ones who deserve to know what you know now.

Weekly Quiz Competition

Question 1
1 / -0

If $$cos\alpha+cos\beta+cos\gamma= 0= sin\alpha+sin\beta+sin\gamma$$ then $$cos2^n\alpha+cos2^n\beta+cos2^n\gamma=?$$

A
$$0$$

B
$$1$$

C
$$cos 2(\alpha+\beta+gamma)$$

D
$$3cos(\alpha+\beta+\gamma)$$
Question 2
1 / -0

If $$\cos 3x+\sin\left(2x-\dfrac{7\pi}{6}\right)=-2$$, then $$x$$ is equal to $$\left(where\ m\in Z\right)$$

A
$$\dfrac{\pi}{3}\left(6m-1\right)$$

B
$$\dfrac{\pi}{3}\left(6m+1\right)$$

C
$$\dfrac{\pi}{3}\left(2m+1\right)$$

D
$$none\ of\ these$$

Solution
Question 3
1 / -0

If $${ tan }^{ 2 }\theta +{ cot }^{ 2 }\theta =\frac { a }{ b-cos4\theta } +c$$ (where defined) is satisfied for all $$\theta $$, then the value of (a + b + c) is equal to $$(a,b,c\in I)$$

A
6

B
7

C
8

D
9

Solution
Question 4
1 / -0

If $$sin({ cot }^{ -1 }(x+1)=cos({ tan }^{ -1 }x)$$, then x is equal to

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

B
$$\dfrac { 1 }{ 2 } $$

C
Zero

D
$$\dfrac { 9 }{ 4 } $$

Solution
Question 5
1 / -0

The value of $$\sin{\left(\alpha+\beta\right)}$$ is

A
$$\dfrac{24}{25}$$

B
$$\dfrac{12}{13}$$

C
$$\dfrac{1}{25}$$

D
$$None\ of\ these$$

Solution
Question 6
1 / -0

If $$cos(\alpha +\beta )sin(\gamma +\delta )=cos(\alpha -\beta )sin(\gamma -\delta )$$ then

A
$$cot\alpha cot\beta cot\delta =cot\gamma $$

B
$$cot\beta cot\delta cot\gamma =cot\alpha $$

C
$$cot\alpha cot\gamma cot\delta =cot\beta $$

D
$$cot\alpha cot\beta cot\gamma =cot\delta $$

Solution

$$(\cos \alpha \cos \beta-\sin \alpha \sin \beta)(\sin y \cos \delta+\cos y \sin \delta)=$$
$$(\cos \alpha \cos \beta+\sin \alpha \sin \beta)(\sin y \cos \delta-\cos y \sin \delta)$$
$$\Rightarrow \quad 2 \cos \alpha \cos \beta \cos y \sin \delta=2$$ sin $$\alpha$$ sin $$\beta$$ siny cos $$\delta$$
$$=\dfrac{\cos \alpha}{\sin \alpha} \times \dfrac{\cos \beta}{\sin \beta} \times \dfrac{\cos y}{\sin y}=\dfrac{\cos \delta}{\sin \delta}$$
$$=\cot \alpha \cot \beta \cot y=\cot \delta$$
$$\therefore$$ option $$D$$ is correct.
Question 7
1 / -0

In a $$\triangle ABC, \sum a \cos A=4 \sin A \sin B \sin C$$, then value of $$\left(\dfrac{\sum \sin A}{\sum a}\right)^{2}$$ is-

A
$$\dfrac{1}{4}$$

B
$$\dfrac{1}{2}$$

C
$$1$$

D
$$\dfrac{1}{3}$$

Solution
Question 8
1 / -0

$$sin\left( { 4tan }^{ -1 }{ \cfrac { 1 }{ 3 } } \right) =$$

A
$$\frac { 12 }{ 25 } $$

B
$$\frac { 24 }{ 25 } $$

C
$$\frac { 1 }{ 5 } $$

D
none of these

Solution
Question 9
1 / -0

If $$0 < \phi < \dfrac {\pi}{2}$$ and if $$\displaystyle x=\sum^{\alpha}_{n=0}\cos^{2}\phi:y=\sum^{\alpha}_{n=0}\sin^{2n}\phi$$ then

A
$$x^{2}+y^{2}=1$$

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

C
$$x+y=xy$$

D
$$\dfrac {1}{x}+\dfrac {1}{y}=1$$

Solution
Question 10
1 / -0

The value of $$\cos^2 \theta + \cos^2 \left(\dfrac{2\pi}{2} - \theta\right) + \cos^2 \left(\dfrac{2\pi}{3} + \theta \right)$$ is

A
$$-\dfarc{3}{2}$$

B
$$\dfrac{3}{2}$$

C
$$\dfarc{1}{2}$$

D
$$1$$

Solution

$$ \begin{aligned} & \cos ^{2} \alpha+\cos ^{2}\left(\frac{2 \pi}{3}+\alpha\right)+\cos ^{2}\left(\frac{2 \pi}{3}-\alpha\right) \\ =& \cos ^{2} \alpha+\left(\cos \left(\frac{2 \pi}{3}\right) \cos \alpha-\sin \left(\frac{2 \pi}{3}\right) \sin \alpha\right)^{2} \\ &+\left(\cos \left(\frac{2 \pi}{3}\right) \cos \alpha+\sin \left(\frac{2 \pi}{3}\right) \sin \alpha\right)^{2} \\ &=\cos ^{2} \alpha+\left(\frac{-1}{2} \cos \alpha-\frac{\sqrt{3}}{2} \sin \alpha\right)^{2}+\left(\frac{-1}{2} \cos \alpha\right.\\ &\left.+\frac{\sqrt{3}}{2} \sin \alpha\right)^{2} \\ &=\cos ^{2} \alpha+\frac{1}{4} \cos ^{2} \alpha+\frac{3}{4} \sin ^{2} \alpha+\frac{\sqrt{3}}{2} \cos \alpha \sin \alpha \\ &+\frac{1}{4} \cos ^{2} \alpha+\frac{3}{4} \sin ^{2} \alpha-\frac{\sqrt{3}}{2} \cos \alpha \sin \alpha \\ &=\frac{3}{2} \cos ^{2} \alpha+\frac{3}{2} \sin ^{2} \alpha \\ &=\frac{3}{2}\left(\cos ^{2} \alpha+\sin ^{2} \alpha\right) \\ &=\frac{3}{2} \\ & \text { option } B \text { is correct } \end{aligned} $$