7. Trigonometric Equations(11th)

Trigonometric Equations

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

(A) 5

(B) 7

(D) 9

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

(A) 4

(B) 3

(D) 5

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3. The number of solutions of the equation $2x + 3 \tan x = \pi, x \in [-2\pi, 2\pi] - \{\pm \frac{\pi}{2}, \pm \frac{3\pi}{2}\}$ is:

(A) 4

(B) 5

(D) 6

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4. If $\theta \in [-\frac{7\pi}{6}, \frac{4\pi}{3}]$ , then the number of solutions of $\sqrt{3} \csc^2 \theta - 2(\sqrt{3} - 1) \csc \theta - 4 = 0$ is equal to:

(A) 7

(B) 10

(D) 8

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5. If $\theta \in [-2\pi, 2\pi]$ , then the number of solutions of $2\sqrt{2} \cos^2 \theta + (2 - \sqrt{6}) \cos \theta - \sqrt{3} = 0$ is equal to:

(A) 8

(B) 6

(D) 12

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6. The sum of all values of $\theta \in [0, 2\pi]$ satisfying $2 \sin^2 \theta = \cos 2\theta$ and $2 \cos^2 \theta = 3 \sin \theta$ is:

(A) $\pi$

(B) $\frac{5\pi}{6}$

(D) $4\pi$

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7. Let $|\cos \theta \cos(60 - \theta) \cos(60 + \theta)| \leq \frac{1}{8}, \theta \in [0, 2\pi]$ . Then, the sum of all $\theta \in [0, 2\pi]$ where $\cos 3\theta$ attains its maximum value, is:

(A) $6\pi$

(B) $9\pi$

(D) $15\pi$

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8. The number of solutions of the equation $4 \sin^2 x - 4 \cos^3 x + 9 - 4 \cos x = 0; x \in [-2\pi, 2\pi]$ is:

(A) 0

(B) 3

(D) 2

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9. If $2 \sin^3 x + \sin 2x \cos x + 4 \sin x - 4 = 0$ has exactly 3 solutions in the interval $[0, \frac{n\pi}{2}]$ , $n \in \mathbb{N}$ , then the roots of the equation $x^2 + nx + (n - 3) = 0$ belong to:

(A) $(0, \infty)$

(B) $\mathbb{Z}$

(D) $(-\infty, 0)$

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10. The sum of the solutions $x \in \mathbb{R}$ of the equation $\frac{3 \cos 2x - \cos^3 2x}{\cos^2 x - \sin^2 x} = x^3 - x^2 + 6$ is:

(A) 3

(B) 1

(D) -1

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11. If $\alpha, -\frac{\pi}{2} < \alpha < \frac{\pi}{2}$ is the solution of $4 \cos \theta + 5 \sin \theta = 1$ , then the value of $\tan \alpha$ is:

(A) $\frac{10 - \sqrt{10}}{12}$

(B) $\frac{\sqrt{10 - 10}}{6}$

(D) $\frac{10 - \sqrt{10}}{6}$

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12. If $2 \tan^2 \theta - 5 \sec \theta = 1$ has exactly 7 solutions in the interval $[0, \frac{n\pi}{2}]$ , for the least value of $n \in \mathbb{N}$ , then $\sum_{k=1}^{n} \frac{k}{2^k}$ is equal to:

(A) $\frac{1}{2^{14}} (2^{15} - 15)$

(B) $1 - \frac{15}{2^{18}}$

(D) $\frac{1}{2^{13}} (2^{14} - 15)$

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13. The number of elements in the set $S = \{\theta \in [0, 2\pi) : 3 \cos^4 \theta - 5 \cos^2 \theta - 2 \sin^4 \theta + 2 = 0\}$ is:

(A) 9

(B) 8

(D) 10

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14. Let $S = \{x \in (-\frac{\pi}{2}, \frac{\pi}{2}) : 9^{1- \tan^2 x} + 9^{\tan^2 x} = 10\}$ and $\beta = \sum_{x \in S} \tan^2 (\frac{x}{3})$ , then $\frac{1}{6} (\beta - 14)^2$ is equal to:

(A) 16

(B) 32

(D) 64

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15. The number of elements in the set $S = \{x \in \mathbb{R} : 2 \cos (\frac{x^2 + x}{6}) = 4^x + 4^{-x}\}$ is:

(A) 1

(B) 3

(D) infinite

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16. Let $S = \{\theta \in (0, \pi) : \sum_{m=1}^9 \sec (\theta + (m-1) \frac{\pi}{6}) \sec (\theta + \frac{m\pi}{6}) = -\frac{8}{\sqrt{3}}\}$ . Then:

(A) $S = \{\frac{\pi}{12}\}$

(B) $S = \{\frac{2\pi}{3}\}$

(D) $\sum_{\theta \in S} \theta = \frac{3\pi}{4}$

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17. Let $S = \{\theta \in [0, 2\pi) : 8^{2\sin^2 \theta} + 8^{2 \cos^2 \theta} = 16\}$ . Then $\frac{1}{8} + \sum_{\theta \in S} \left(\sec (\frac{\pi}{4} + 2\theta) \csc (\frac{\pi}{4} + 2\theta)\right)$ is equal to:

(A) 0

(B) -2

(D) 12

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18. The number of solutions of $|\cos x| = \sin x$ , such that $-4\pi \leq x \leq 4\pi$ is:

(A) 4

(B) 6

(D) 12

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19. Let for some real numbers $\alpha$ and $\beta$ , $\alpha = a - i\beta$ . If the system of equations $4i\alpha x + (1 + i)y = 0$ and $8 \left(\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3}\right) x + \alpha y = 0$ has more than one solution, then $\frac{\alpha}{\beta}$ is equal to: