JEE Mains · Maths · STD 12 - 2. inverse trigonometric function
Let \(( a , b ) \subset(0,2 \pi)\) be the largest interval for which \(\sin ^{-1}(\sin \theta)-\cos ^{-1}(\sin \theta) > 0, \theta \in(0,2 \pi)\) holds. If \(\alpha x^2+\beta x+\sin ^{-1}\left(x^2-6 x+10\right)+\cos ^{-1}\) \(\left(x^2-6 x+10\right)=0\) and \(\alpha-\beta=b-a\), then \(\alpha\) is equal to:
- A \(\frac{\pi}{48}\)
- B \(\frac{\pi}{16}\)
- C \(\frac{\pi}{8}\)
- D \(\frac{\pi}{12}\)
Answer & Solution
Correct Answer
(D) \(\frac{\pi}{12}\)
Step-by-step Solution
Detailed explanation
\(\sin ^{-1} \sin \theta-\left(\frac{\pi}{2}-\sin ^{-1} \sin \theta\right) > 0\) \(\Rightarrow \sin ^{-1} \sin \theta > \frac{\pi}{4}\) \(\Rightarrow \sin \theta > \frac{1}{\sqrt{2}}\) \(\text { So, } \theta \in\left(\frac{\pi}{4}, \frac{3 \pi}{4}\right)\)…
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
- Let an ellipse \(E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a^{2}>b^{2}\), passes through \(\left(\sqrt{\frac{3}{2}}, 1\right)\) and has ecentricity \(\frac{1}{\sqrt{3}} .\) If a circle, centered at focus \(\mathrm{F}(\alpha, 0), \alpha>0\), of \(\mathrm{E}\) and radius \(\frac{2}{\sqrt{3}}\), intersects \(\mathrm{E}\) at two points \(\mathrm{P}\) and \(\mathrm{Q}\), then \(\mathrm{PQ}^{2}\) is equal to:JEE Mains 2021 Hard
- If the function \(f:(-\infty,-1] \rightarrow(a, b]\) defined by \(f(x)=e^{x^3-3 x+1}\) is one-one and onto, then the distance of the point \(\mathrm{P}(2 \mathrm{~b}+4, \mathrm{a}+2)\) from the line \(x+e^{-3} y=4\) is :JEE Mains 2024 Hard
- If the system of equations \( 11 x+y+\lambda z=-5 \) \( 2 x+3 y+5 z=3 \) \( 8 x-19 y-39 z=\mu\) has infinitely many solutions, then \(\lambda^4-\mu\) is equal to :JEE Mains 2024 Hard
- If \(f(x)=\left|\begin{array}{ccc}x^3 & 2 x^2+1 & 1+3 x \\ 3 x^2+2 & 2 x & x^3+6 \\ x^3-x & 4 & x^2-2\end{array}\right|\) for all \(x \in \mathbb{R}\), then \(2 f(0)+f^{\prime}(0)\) is equal toJEE Mains 2024 Hard
- If the remainder when \(x\) is divided by \(4\) is \(3 ,\) then the remainder when \((2020+ x )^{2022}\) is divided by \(8\) is ....... .JEE Mains 2021 Hard
- A fair die is tossed repeatedly until a six is obtained. Let \(\mathrm{X}\) denote the number of tosses required and let \(\mathrm{a}=\mathrm{P}(\mathrm{X}=3), \mathrm{b}=\mathrm{P}(\mathrm{X} \geq 3)\) and \(\mathrm{c}=\) \(\mathrm{P}(\mathrm{X} \geq 6 \mid \mathrm{X}>3)\). Then \(\frac{\mathrm{b}+\mathrm{c}}{\mathrm{a}}\) is equal toJEE Mains 2024 Hard
More PYQs from JEE Mains
- Let the line \(\mathrm{L}\) be the projection of the line \(\frac{x-1}{2}=\frac{y-3}{1}=\frac{z-4}{2}\) in the plane \(x-2 y-z=3 .\) If \(d\) is the distance of the point \((0,0,6)\) from \(\mathrm{L}\), then \(\mathrm{d}^{2}\) is equal to .... .JEE Mains 2021 Hard
- Let the line \(\ell: x =\frac{1- y }{-2}=\frac{ z -3}{\lambda}, \lambda \in R\) meet the plane \(P : x +2 y +3 z =4\) at the point \((\alpha, \beta, \gamma)\). If the angle between the line \(\ell\) and the plane \(P\) is \(\cos ^{-1}\left(\sqrt{\frac{5}{14}}\right)\), then \(\alpha+2 \beta+6 \gamma\) is equal toJEE Mains 2023 Hard
- The largest value of \(a,\) for which the perpendicular distance of the plane containing the lines \(\vec{r}=(\hat{i}+\hat{j})+\lambda(\hat{i}+a \hat{j}-\hat{k})\) and \(\vec{r}=(\hat{i}+\hat{j})+\mu(-\hat{i}+\hat{j}-a \hat{k})\) from the point \((2,1,4)\) is \(\sqrt{3}\), is\(...\)JEE Mains 2022 Hard
- Let \( a_{1}=1 \) and for \( n\ge1 \), \( a_{n+1}\)
= \(\frac{1}{2}a_{n}+\frac{n^{2}-2n-1}{n^{2}(n+1)^{2}} \). Then \( |\sum_{n=1}^{\infty}(a_{n}-\frac{2}{n^{2}})| \) is equal to ........... .JEE Mains 2026 Easy - The number of distinct real roots of the equation \(3 x^{4}+4 x^{3}-12 x^{2}+4=0\) is ..... .JEE Mains 2021 Hard
- Two sides of a parallelogram are along the lines \(4 x+5 y=0\) and \(7 x+2 y=0\). If the equation of one of the diagonals of the parallelogram is \(11 \mathrm{x}+7 \mathrm{y}=9\), then other diagonal passes through the point:JEE Mains 2021 Hard