JEE Mains · Maths · STD 12 - 2. inverse trigonometric function
Let \(\mathrm{S}=\left\{x: \cos ^{-1} x=\pi+\sin ^{-1} x+\sin ^{-1}(2 x+1)\right\}\). Then \(\sum_{x \in \mathrm{~S}}(2 x-1)^2\) is equal to ______.
- A 5
- B 10
- C 15
- D 20
Answer & Solution
Correct Answer
(A) 5
Step-by-step Solution
Detailed explanation
\begin{aligned} & \cos ^{-1} x=\pi+\sin ^{-1} x+\sin ^{-1}(2 x+1) \\ & 2 \cos ^{-1} x-\sin ^{-1}(2 x+1)=\frac{3 \pi}{2} \\ & 2 \alpha-\beta=\frac{3 \pi}{2} \text { where } \cos ^{-1} x=\alpha, \sin ^{-1}(2 x+1)=\beta \\ & 2 \alpha=\frac{3 \pi}{2}+\beta \\ & \cos 2 \alpha=\sin…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- If the domain of the function \(f(x)=\sec ^{-1}\left(\frac{2 x}{5 x+3}\right)\) is \([\alpha, \beta) \cup(\gamma, \delta]\), then \(|3 \alpha+10(\beta+\gamma)+21 \delta|\) is equal to \(.......\).JEE Mains 2023 Hard
- Let the solution curve \(y = y ( x )\) of the differential equation \(\quad \frac{d y}{d x}-\frac{3 x^5 \tan ^{-1}\left(x^3\right)}{\left(1+x^6\right)^{\frac{3}{2}}} y=2 x\) \(\exp \frac{x^3-\tan ^{-1} x^3}{\sqrt{(1+x)^6}}\) pass through the origin. Then \(y (1)\) is equal to:JEE Mains 2023 Hard
- Let f be a twice differentiable non-negative function such that \( (f(x))^{2}=25+\int_{0}^{x}((f(t))^{2}+(f'(t))^{2})dt \). Then the mean of \(f\left(\log _e(1)\right), f\left(\log _e(2)\right), \ldots \ldots, f\left(\log _e(625)\right)\) is equal to:JEE Mains 2026 Easy
- Consider the function \(f:(0, \infty) \rightarrow R\) defined by \(f(x)=e^{-\left|\log _e x\right|}\). If \(m\) and \(n\) be respectively the number of points at which \(f\) is not continuous and \(f\) is not differentiable, then \(\mathrm{m}+\mathrm{n}\) isJEE Mains 2024 Hard
- The integral \(\int \frac{\left(x^8-x^2\right) d x}{\left(x^{12}+3 x^6+1\right) \tan ^{-1}\left(x^3+\frac{1}{x^3}\right)}\) is equal to :JEE Mains 2024 Medium
- If \(\frac{\pi}{2} \leq x \leq \frac{3 \pi}{4}\), then \(\cos ^{-1}\left(\frac{12}{13} \cos x+\frac{5}{13} \sin x\right)\) is equal toJEE Mains 2025 Medium
More PYQs from JEE Mains
- Let the lines \(3 x-4 y-\alpha=0,8 x-11 y-33=0\), and \(2 x-3 y+\lambda=0\) be concurrent. If the image of the point
\((1,2)\) in the line \(2 x-3 y+\lambda=0\) is \(\left(\frac{57}{13}, \frac{-40}{13}\right)\), then \(|\alpha \lambda|\) is equal toJEE Mains 2025 Medium - Let the plane \(x+3 y-2 z+6=0\) meet the co-ordinate axes at the points \(A, B, C\). If the orthocentre of the triangle \(ABC\) is \(\left(\alpha, \beta, \frac{6}{7}\right)\), then \(98(\alpha+\beta)^2\) is equal to \(........\).JEE Mains 2023 Hard
- If \(\Delta=\left|\begin{array}{ccc}x-2 & 2 x-3 & 3 x-4 \\ 2 x-3 & 3 x-4 & 4 x-5 \\ 3 x-5 & 5 x-8 & 10 x-17\end{array}\right|=\) \(Ax ^{3}+ Bx ^{2}+ Cx + D ,\) then \(B + C\) is equal toJEE Mains 2020 Hard
- If for the complex numbers \(z\) satisfying \(|z-2-2 i| \leq 1\), the maximum value of \(|3 i z+6|\) is attained at \(\mathrm{a}+i \mathrm{~b}\), then \(\mathrm{a}+\mathrm{b}\) is equal to .... .JEE Mains 2021 Hard
- Let \(f(x) = e^x -x\) and \(g(x) = x^2 -x\), \(\forall \in R\). Then the set of all \(x \in R\), where the function \(h(x) = (fog)\, (x)\) is increasing isJEE Mains 2019 Hard
- The shortest distance between the lines \(\frac{x+2}{1}=\frac{y}{-2}=\frac{z-5}{2}\) and \(\frac{x-4}{1}=\frac{y-1}{2}=\frac{z+3}{0}\) is \(......\).JEE Mains 2023 Hard