Trigonometric functions — practice questions

Question 1 · difficulty L2 · understanding

Let the range of the function $f(x)=6+16 \cos x \cdot \cos \left(\frac{\pi}{3}-x\right) \cdot \cos \left(\frac{\pi}{3}+x\right) \cdot \sin 3 x \cdot \cos 6 x, x \in \mathbf{R}$ be $[\alpha, \beta]$. Then the distance of the point $(\alpha, \beta)$ from the line $3 x+4 y+12=0$ is :

• A. 11
• B. 10
• C. 8
• D. 9

Question 2 · difficulty L3 · understanding

If the value of $$\frac{3 \cos 36^{\circ}+5 \sin 18^{\circ}}{5 \cos 36^{\circ}-3 \sin 18^{\circ}}$$ is $$\frac{a \sqrt{5}-b}{c}$$, where $$a, b, c$$ are natural numbers and $$\operatorname{gcd}(a, c)=1$$, then $$a+b+c$$ is equal to :

• A. 54
• B. 52
• C. 50
• D. 40

Question 3 · difficulty L3 · understanding

The number of solutions, of the equation $$e^{\sin x}-2 e^{-\sin x}=2$$, is :

• A. 0
• B. 1
• C. 2
• D. more than 2

Question 4 · difficulty L3 · understanding

The value of $$36\left(4 \cos ^{2} 9^{\circ}-1\right)\left(4 \cos ^{2} 27^{\circ}-1\right)\left(4 \cos ^{2} 81^{\circ}-1\right)\left(4 \cos ^{2} 243^{\circ}-1\right)$$ is :

• A. 18
• B. 36
• C. 54
• D. 27

Question 5 · difficulty L3 · understanding

If $$\tan 15^\circ + {1 \over {\tan 75^\circ }} + {1 \over {\tan 105^\circ }} + \tan 195^\circ = 2a$$, then the value of $$\left( {a + {1 \over a}} \right)$$ is :

• A. $$5 - {3 \over 2}\sqrt 3$$
• B. $$4 - 2\sqrt 3$$
• C. 2
• D. 4

Question 6 · difficulty L3 · understanding

The set of all values of $$\lambda$$ for which the equation $${\cos ^2}2x - 2{\sin ^4}x - 2{\cos ^2}x = \lambda$$ has a real solution $$x$$, is :

• A. $$\left[ { - 2, - 1} \right]$$
• B. $$\left[ { - {3 \over 2}, - 1} \right]$$
• C. $$\left[ { - 2, - {3 \over 2}} \right]$$
• D. $$\left[ { - 1, - {1 \over 2}} \right]$$

Question 7 · difficulty L3 · understanding

$$\alpha = \sin 36^\circ$$ is a root of which of the following equation?

• A. $$16{x^4} - 10{x^2} - 5 = 0$$
• B. $$16{x^4} + 20{x^2} - 5 = 0$$
• C. $$16{x^4} - 20{x^2} + 5 = 0$$
• D. $$4{x^4} - 10{x^2} + 5 = 0$$

Question 8 · difficulty L3 · understanding

The value of 2sin (12$$^\circ$$) $$-$$ sin (72$$^\circ$$) is :

• A. $${{\sqrt 5 (1 - \sqrt 3 )} \over 4}$$
• B. $${{1 - \sqrt 5 } \over 8}$$
• C. $${{\sqrt 3 (1 - \sqrt 5 )} \over 2}$$
• D. $${{\sqrt 3 (1 - \sqrt 5 )} \over 4}$$

Question 9 · difficulty L3 · understanding

The value of $$\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ}$$ is __________.

Question 10 · difficulty L3 · understanding

The number of integral values of 'k' for which the equation $$3\sin x + 4\cos x = k + 1$$ has a solution, k$$\in$$R is ___________.