7. Trigonometric Equations(11th)

Trigonometric Equations

1. The number of values of $x$ in the interval $[0, 3\pi]$ satisfying the equation $2 \sin^2 x + 5 \sin x - 3 = 0$ is:

(A) 4

(B) 6

(D) 2

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2. The number of solution of $\tan x + \sec x = 2 \cos x$ in $[0, 2\pi]$ is:

(A) 2

(B) 3

(D) 1

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3. The number of solutions of $\sin^2 x + (2 + 2x - x^2) \sin x - 3(x - 1)^2 = 0$ , where $-\pi \leq x \leq \pi$ , is:

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4. Let $S = \{\sin^2 2\theta : (\sin^4 \theta + \cos^4 \theta) x^2 + (\sin 2\theta) x + (\sin^6 \theta + \cos^6 \theta) = 0$ has real roots $\}$ . If $\alpha$ and $\beta$ be the smallest and largest elements of the set $S$ , respectively, then $3((\alpha - 2)^2 + (\beta - 1)^2)$ equals:

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5. If $m$ and $n$ respectively are the numbers of positive and negative values of $\theta$ in the interval $[- \pi, \pi]$ that satisfy the equation $\cos 2\theta \cos \frac{\theta}{2} = \cos 3\theta \cos \frac{\theta}{2}$ , then $mn$ is equal to:

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6. Let $S = \{\theta \in [0, 2\pi) : \tan(\pi \cos \theta) + \tan(\pi \sin \theta) = 0\}$ . Then $\sum_{\theta \in S} \sin^2 (\theta + \frac{\pi}{4})$ is equal to:

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7. Let $S = \{\theta \in (0, 2\pi) : 7 \cos^2 \theta - 3 \sin^2 \theta - 2 \cos^2 2\theta = 2\}$ . Then, the sum of roots of all the equations $x^2 - 2(\tan^2 \theta + \cot^2 \theta) x + 6 \sin^2 \theta = 0$ , $\theta \in S$ , is:

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8. Let $S = [-\pi, \frac{\pi}{2}) \cup \{-\frac{\pi}{2}, -\frac{\pi}{4}, -\frac{3\pi}{4}, \frac{\pi}{4}, \frac{\pi}{2}\}$ . Then the number of elements in the set $A = \{\theta \in S : \tan \theta (1 + \sqrt{5} \tan(2\theta)) = \sqrt{5} - \tan(2\theta)\}$ is:

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9. If the sum of solutions of the system of equations $2 \sin^2 \theta - \cos 2\theta = 0$ and $2 \cos^2 \theta + 3 \sin \theta = 0$ in the interval $[0, 2\pi]$ is $k\pi$ , then $k$ is equal to:

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10. Let $S_1 = \{x \in [0, 12\pi] : \sin^5 x + \cos^5 x = 1\}$ and $S_2 = \{x \in [0, 8\pi] : \sin^7 x + \cos^7 x = 1\}$ . Then $n(S_1) - n(S_2)$ is equal to:

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11. The number of solutions of the equation $\sin x = \cos^2 x$ in the interval $(0, 10)$ is:

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12. The number of elements in the set $S = \{\theta \in [-4\pi, 4\pi] : 3 \cos^2 2\theta + 6 \cos 2\theta - 10 \cos^2 \theta + 5 = 0\}$ is:

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13. The number of solutions of the equation $2\theta - \cos^2 \theta + \sqrt{2} = 0$ in $\mathbb{R}$ is equal to:

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14. The number of values of $x$ in the interval $(\frac{\pi}{4}, \frac{7\pi}{4})$ for which $14 \csc^2 x - 2 \sin^2 x = 21 - 4 \cos^2 x$ holds, is:

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15. Let $S$ be the sum of all solutions (in radians) of the equation $\sin^4 \theta + \cos^4 \theta - \sin \theta \cos \theta = 0$ in $[0, 4\pi]$ . Then $\frac{8S}{\pi}$ is equal to:

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16. The number of solutions of the equation $|\cot x| = \cot x + \frac{1}{\sin x}$ in the interval $[0, 2\pi]$ is:

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17. If $\sqrt{3}(\cos^2 x) = (\sqrt{3} - 1) \cos x + 1$ , the number of solutions of the given equation when $x \in [0, \frac{\pi}{2}]$ is:

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