Trigonometry · JEE Main previous year questions

2023 30 questions

1. Q282023 · 25 Jan 2023 · Shift 1· Numerical answer

   If the sum of all the solutions of ${\tan }^{-1}\left(\frac{2x}{1-x^2}\right)+{\cot }^{-1}\left(\frac{1-x^2}{2x}\right)=\frac{\pi }{3}$, $-1<x<1,x\ne 0$, is $\alpha -\frac{4}{\sqrt{3}}$, then $\alpha$ is equal to ____.

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   Answer: 2

2. Q292023 · 25 Jan 2023 · Shift 2· Numerical answer

   If m and n respectively are the numbers of positive and negative values of $\theta$ in the interval $[-\pi ,\pi ]$ that satisfy the equation $\cos 2\theta \cos \frac{\theta }{2}=\cos 3\theta \cos \frac{9\theta }{2}$, then mn is equal to ____.

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   Answer: 25

3. Q52023 · 24 Jan 2023 · Shift 1

   ${\tan }^{-1}\left(\frac{1+\sqrt{3}}{3+\sqrt{3}}\right)+{\sec }^{-1}\left(\sqrt{\frac{8+4\sqrt{3}}{6+3\sqrt{3}}}\right)$ is equal to :

   1. A.$\frac{\pi }{2}$
   2. B.$\frac{\pi }{3}$
   3. C.$\frac{\pi }{6}$
   4. D.$\frac{\pi }{4}$
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   Answer: B

4. Q122023 · 24 Jan 2023 · Shift 2

   The value of ${\left(\frac{1+\sin \frac{2\pi }{9}+i\cos \frac{2\pi }{9}}{1+\sin \frac{2\pi }{9}-i\cos \frac{2\pi }{9}}\right)}^3$ is

   1. A.$-\frac{1}{2}\left(1-i\sqrt{3}\right)$
   2. B.$-\frac{1}{2}\left(\sqrt{3}-i\right)$
   3. C.$\frac{1}{2}\left(1-i\sqrt{3}\right)$
   4. D.$\frac{1}{2}\left(\sqrt{3}+i\right)$
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   Answer: B

5. Q292023 · 24 Jan 2023 · Shift 2· Numerical answer

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

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   Answer: 2