MHT CET · Maths · Differentiation
If \(y=\sin ^2\left(\cot ^{-1} \sqrt{\frac{1+x}{1-x}}\right)\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) has the value
- A \(\frac{-1}{2}\)
- B \(\frac{1}{2}\)
- C -1
- D 1
Answer & Solution
Correct Answer
(A) \(\frac{-1}{2}\)
Step-by-step Solution
Detailed explanation
\(y=\sin ^2\left(\cot ^{-1} \sqrt{\frac{1+x}{1-x}}\right)\)
Let \(\theta=\cot ^{-1} \sqrt{\frac{1+x}{1-x}}\)
\(\begin{array}{ll}
\therefore & \cot ^2 \theta=\frac{1+x}{1-x} \\
\therefore & 1+\cot ^2 \theta=\frac{2}{1-x} \\
\therefore & \sin ^2 \theta=\frac{1-x}{2} \\
\therefore & \theta=\sin ^{-1} \sqrt{\frac{1-x}{2}} \\
\therefore & y=\left[\sin \left(\sin ^{-1} \sqrt{\frac{1-x}{2}}\right)\right]^2 \\
\therefore & y=\frac{1-x}{2} \\
\therefore & \frac{\mathrm{~d} y}{\mathrm{~d} x}=\frac{-1}{2}
\end{array}\)
Let \(\theta=\cot ^{-1} \sqrt{\frac{1+x}{1-x}}\)
\(\begin{array}{ll}
\therefore & \cot ^2 \theta=\frac{1+x}{1-x} \\
\therefore & 1+\cot ^2 \theta=\frac{2}{1-x} \\
\therefore & \sin ^2 \theta=\frac{1-x}{2} \\
\therefore & \theta=\sin ^{-1} \sqrt{\frac{1-x}{2}} \\
\therefore & y=\left[\sin \left(\sin ^{-1} \sqrt{\frac{1-x}{2}}\right)\right]^2 \\
\therefore & y=\frac{1-x}{2} \\
\therefore & \frac{\mathrm{~d} y}{\mathrm{~d} x}=\frac{-1}{2}
\end{array}\)
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
- The area bounded by the curve \(y^2=2 x+1\) and the line \(x-y=1\) isMHT CET 2022 Easy
- Let \(\mathrm{f}(x)=\frac{1-\tan x}{4 x-\pi}, x \neq \frac{\pi}{4}, x \in\left[0, \frac{1}{2}\right], \quad \mathrm{f}(x)\) is continuous in \(\left[0, \frac{\pi}{2}\right]\), then \(f\left(\frac{\pi}{4}\right)\) isMHT CET 2024 Medium
- The foot of the perpendicular from the point \((1,2,3)\) on the line \(\overline{\mathrm{r}}=(6 \hat{\mathrm{i}}+7 \hat{\mathrm{j}}+7 \hat{\mathrm{k}})+\lambda(3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})\) has the co-ordinatesMHT CET 2023 Easy
- If \(\sin ^{-1}(4 x)+\sin ^{-1}(4 \sqrt{3} x)=-\frac{\pi}{2}\), then the absolute value of \(x\) isMHT CET 2025 Medium
- If the vectors \(\hat{i}+2 \hat{\jmath}+x \hat{k}\) and \(y t+6 \hat{k}+4 \hat{k}\) are collinear, then the values of \(x\) and
\(y\) are respectively,MHT CET 2020 Easy - Let \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) be a function such that \(\mathrm{f}(x)=x^3+x^2 \mathrm{f}^{\prime}(1)+x \mathrm{f}^{\prime \prime}(2)+6, x \in \mathrm{R}\), then \(\mathrm{f}(2)\) isMHT CET 2023 Hard
More PYQs from MHT CET
- \(\int_{2}^{3} \frac{x}{x^{2}-1} d x=\)MHT CET 2020 Easy
- India shares about __________% of total biodiversity wealth on earth.MHT CET 2021 Hard
- At certain temperature, rod A and \(\operatorname{rod} \mathrm{B}\) of different materials have lengths \(L_A\) and \(L_B\) respectively. Their co-efficients of linear expansion are \(\alpha_A\) and \(\alpha_B\) respectively. It is observed that the difference between their lengths remain constant at all temperatures. The ratio \(L_A / L_B\) is given byMHT CET 2024 Medium
- Identify the reagent ' \(R\) ' used in the following reaction.
Ketone \(\xrightarrow{\mathrm{R}}\) SemicarbazoneMHT CET 2025 Medium - Following are avian characters found in Archaeopteryx lithegraphica, EXCЕРТ ____________ .MHT CET 2023 Medium
- In the following circuit, the current flowing through zener diode is
MHT CET 2024 Hard