7. Trigonometric Equations(11th)

Trigonometric Equations

1. Let $S = \{\theta \in [-\pi, \pi] - \{\pm \frac{\pi}{2}\} : \sin \theta \tan \theta + \tan \theta = \sin 2\theta\}$ . If $T = \sum_{\theta \in S} \cos 2\theta$ , then $T + n(S)$ is equal to:

(A) $7 + \sqrt{3}$

(B) 9

(D) 10

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2. If $n$ is the number of solutions of the equation $2 \cos x (4 \sin (\frac{\pi}{4} + x) \sin (\frac{\pi}{4} - x) - 1) = 1, x \in [0, \pi]$ and $S$ is the sum of all these solutions, then the ordered pair $(n, S)$ is:

(A) $(3, \frac{13\pi}{9})$

(B) $(2, \frac{2\pi}{3})$

(D) $(3, \frac{5\pi}{3})$

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3. The number of solutions of the equation $32^{ \tan^2 x} + 32^{ \sec^2 x} = 81, 0 \leq x \leq \frac{\pi}{4}$ is:

(A) 3

(B) 1

(D) 2

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4. The sum of solutions of the equation $\frac{\cos x}{1+\sin x} = |\tan 2x|, x \in (-\frac{\pi}{2}, \frac{\pi}{2}) - \{\frac{\pi}{4}, -\frac{\pi}{4}\}$ is:

(A) $-\frac{11\pi}{30}$

(B) $\frac{\pi}{10}$

(D) $-\frac{\pi}{15}$

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5. The sum of all values of $x$ in $[0, 2\pi]$ for which $\sin x + \sin 2x + \sin 3x + \sin 4x = 0$ , is equal to:

(A) $8\pi$

(B) $11\pi$

(D) $9\pi$

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6. The number of solutions of $\sin^7 x + \cos^7 x = 1$ , $x \in [0, 4\pi]$ is equal to:

(A) 11

(B) 7

(D) 9

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7. The number of solutions of the equation $x + 2 \tan x = \frac{\pi}{2}$ in the interval $[0, 2\pi]$ is:

(A) 4

(B) 3

(D) 5

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8. The number of roots of the equation $81^{\sin^2 x} + 81^{\cos^2 x} = 30$ in the interval $[0, \pi]$ is equal to:

(A) 2

(B) 3

(D) 8

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9. All possible values of $\theta \in [0, 2\pi]$ for which $\sin 2\theta + \tan 2\theta > 0$ lie in:

(A) $(0, \frac{\pi}{4}) \cup (\frac{\pi}{2}, \frac{3\pi}{4}) \cup (\frac{3\pi}{2}, \frac{11\pi}{6})$

(B) $(0, \frac{\pi}{2}) \cup (\pi, \frac{3\pi}{2})$

(D) $(0, \frac{\pi}{4}) \cup (\frac{\pi}{2}, \frac{3\pi}{4}) \cup (\pi, \frac{5\pi}{4}) \cup (\frac{3\pi}{2}, \frac{7\pi}{4})$

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10. If $[x]$ denotes the greatest integer $\leq x$ , then the system of linear equations $\sin \theta |x| + [-\cos \theta] y = 0$ and $\cot \theta |x| + y = 0$

(A) has a unique solution if $\theta \in (\frac{\pi}{2}, \frac{2\pi}{3})$ and have infinitely many solutions if $\theta \in (\pi, \frac{7\pi}{6})$

(B) have infinitely many solutions if $\theta \in (\frac{\pi}{2}, \frac{2\pi}{3})$ and has a unique solution if $\theta \in (\pi, \frac{7\pi}{6})$

(D) has a unique solution if $\theta \in (\frac{\pi}{2}, \frac{2\pi}{3}) \cup (\pi, \frac{7\pi}{6})$

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11. Let $S$ be the set of all $\alpha \in \mathbb{R}$ such that the equation $\cos 2x + \alpha \sin x = 2\alpha - 7$ has a solution. Then $S$ is equal to:

(A) $[2, 6]$

(B) $[3, 7]$

(D) $\mathbb{R}$

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12. The number of solutions of the equation $1 + \sin^4 x = \cos^2 3x$ , $x \in [-\frac{5\pi}{2}, \frac{5\pi}{2}]$ is:

(A) 5

(B) 3

(D) 4

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13. Let $S = \{\theta \in [-2\pi, 2\pi] : 2 \cos^2 \theta + 3 \sin \theta = 0\}$ . Then the sum of the elements of $S$ is:

(A) $\pi$

(B) $2\pi$

(D) $\frac{5\pi}{3}$

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14. The sum of all values of $\theta \in (0, \frac{\pi}{2})$ satisfying $\sin^2 2\theta + \cos^4 2\theta = \frac{3}{4}$ is:

(A) $\frac{5\pi}{4}$

(B) $\frac{\pi}{2}$

(D) $\frac{3\pi}{8}$

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15. If $0 \leq x \leq \frac{\pi}{2}$ , then the number of values of $x$ for which $\sin x - \sin 2x + \sin 3x = 0$ , is:

(A) 3

(B) 1

(D) 2

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16. If sum of all solutions of the equation $8 \cos x \cdot (\cos (\frac{\pi}{6} + x) \cdot \cos (\frac{\pi}{6} - x) - \frac{1}{2}) = 1$ in $[0, \pi]$ is $k\pi$ , then $k$ is equal to: