Trigonometry-JEE-Main-PYQ’s

Mathematics Questions

The value of $\cot \left( \frac{\pi}{24} \right)$ is:

(A) $\sqrt{2} + \sqrt{3} + 2 - \sqrt{6}$

(B) $\sqrt{2} + \sqrt{3} + 2 + \sqrt{6}$

(D) $3\sqrt{2} - \sqrt{3} - \sqrt{6}$

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If $15\sin^4\alpha + 10\cos^4\alpha = 6$ , for some $\alpha \in \mathbb{R}$ , then the value of $27\sec^6\alpha + 8\csc^6\alpha$ is equal to:

(A) 500

(B) 400

(D) 350

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If for $x \in \left(0, \frac{\pi}{2}\right)$ , $\log_{10} \sin x + \log_{10} \cos x = -1$ and $\log_{10} (\sin x + \cos x) = \frac{1}{2} (\log_{10} n - 1), \quad n > 0,$ then the value of $n$ is equal to:

(A) 16

(B) 9

(D) 20

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If $0 < x, y < \pi$ and $\cos x + \cos y - \cos(x + y) = \frac{3}{2},$ then $\sin x + \cos y$ is equal to:

(A) $\frac{1 + \sqrt{3}}{2}$

(B) $\frac{1}{2}$

(D) $\frac{1 - \sqrt{3}}{2}$

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If $e^{(\cos^2 x + \cos^4 x + \cos^6 x + \ldots \infty)\log_e 2}$ satisfies the equation $t^2 - 9t + 8 = 0,$ then the value of $\frac{2\sin x}{\sin x + \sqrt{3} \cos x} \quad \left(0 < x < \frac{\pi}{2}\right)$ is:

(A) $\sqrt{3}$

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

(D) $\frac{1}{2}$

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If $L = \sin^2\left(\frac{\pi}{16}\right) - \sin^2\left(\frac{\pi}{8}\right) \quad \text{and} \quad M = \cos^2\left(\frac{\pi}{16}\right) - \sin^2\left(\frac{\pi}{8}\right),$ then:

(A) $L = -\frac{1}{2\sqrt{2}} + \frac{1}{2} \cos \frac{\pi}{8}$

(B) $M = \frac{1}{2\sqrt{2}} + \frac{1}{2} \cos \frac{\pi}{8}$

(D) $L = \frac{1}{4\sqrt{2}} - \frac{1}{4} \cos \frac{\pi}{8}$

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If the equation $\cos^4 \theta + \sin^4 \theta + \lambda = 0$ has real solutions for $\theta$ , then $\lambda$ lies in the interval:

(A) $\left[ -\frac{3}{2}, -\frac{5}{4} \right]$

(B) $\left( -\frac{1}{2}, -\frac{1}{4} \right]$

(D) $[-1, -\frac{1}{2}]$

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If $x = \sum_{n=0}^{\infty} (-1)^n \tan^{2n} \theta \quad \text{and} \quad y = \sum_{n=0}^{\infty} \cos^{2n} \theta$ for $0 < \theta < \frac{\pi}{4}$ , then: