Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
If $\text{y}=\cos^{-1}(2\text{x})+2\cos^{-1}\sqrt{1-4\text{x}^2}, -\frac{1}{2}<\text{x}<0,$ find $\frac{\text{dy}}{\text{dx}}.$
✓

Answer

Here, $\text{y}=\cos^{-1}(2\text{x})+2\cos^{-1}\sqrt{1-4\text{x}^2}$
Put $2\text{x}=\cos\theta, \text{So},$
$\text{y}=\cos^{-1}(\cos\theta)+2\cos^{-1}\sqrt{1-\cos^2\theta}$
$=\cos^2(\cos\theta)+2\cos^{-1}(\sin\theta)$
$\text{y}=\cos^{-1}(\cos\theta)+2\cos^{-1}\Big(\cos\Big(\frac{\pi}{2}-\theta\Big)\Big)\ .....(\text{i})$
Now, $-\frac{1}{2}<\text{x}<0$
$\Rightarrow -1<2\text{x}<0$
$\Rightarrow -1<\cos\theta<0$
$\Rightarrow \frac{\pi}{2}<\theta<\pi$
And
$\Rightarrow -\frac{\pi}{2}>-\theta>-\pi$
$\Rightarrow \Big(\frac{\pi}{2}-\frac{\pi}{2}\Big)>\Big(\frac{\pi}{2}-\theta\Big)>\Big(\frac{\pi}{2}-\pi\Big)$
$\Rightarrow 0>\Big(\frac{\pi}{2}-\theta\Big)>-\frac{\pi}{2}$
So, from equation (i),
$\text{y}=\theta+2\Big[-\Big(\frac{\pi}{2}-\theta\Big)\Big]$
$\begin{bmatrix} \text{Since}, \cos^{-1}\cos(\theta)=\theta, \text{if }\theta\in[0,\pi] \\ \cos^{-1}\cos(\theta)=-\theta, \text{if }\theta\in[-\pi,0] \end{bmatrix}$
$\text{y}=\theta-2\times\frac{\pi}{2}+2\theta$
$\text{y}=-\pi+3\theta$
$\text{y}=-\pi+3\cos^{-1}(2\text{x})\ \big[\text{Since}, 2\text{x}=\cos\theta\big]$
Differentiating it with respect to x using chain rule,
$\frac{\text{dy}}{\text{dx}}=0+3\Big(\frac{1}{\sqrt{1-(2\text{x})^2}}\Big)\frac{\text{d}}{\text{dx}}(2\text{x})$
$=\frac{-3}{\sqrt{1-4\text{x}^2}}(2)$
$\frac{\text{dy}}{\text{dx}}=-\frac{6}{\sqrt{1-4\text{x}^2}}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the equations of the tangent and the normal to the following curves at the indicated points.
$y = 2x^2 − 3x − 1$ at $(1, −2)$
→
Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\text{x}^{\text{x}}+\text{x}^\frac{1}{\text{x}}$
→
Solve the following differential equation
$(\text{x}^2+1)\frac{\text{dy}}{\text{dx}}=1$
→
Integrate the rational function $\frac{3 x+5}{x^{3}-x^{2}-x+1}$
→
Prove that:
$\begin{vmatrix}-\text{bc}&\text{b}^2+\text{bc}&\text{c}^2+\text{bc}\\\text{a}^2+\text{ac}&-\text{ac}&\text{c}^2+\text{ac}\\\text{a}^2+\text{ab}&\text{b}^2+\text{ab}&-\text{ab}\end{vmatrix}$
$=(\text{ab}+\text{bc}+\text{ca})^3$
→
Using properties of determinants, prove that,
$\begin{vmatrix}1& 1&1+3\text{x} \\1+3\text{y} & 1&1\\1&1+3\text{z}&1 \end{vmatrix}=9(3\text{xyz}+\text{xy}+\text{yz}+\text{zx})$
→
If $\text{A}=\begin{bmatrix}\cos\theta&\text{i}\sin\theta\\\text{i}\sin\theta&\cos\theta\end{bmatrix},$ then prove by principle of mathematical induction that $\text{A}^\text{n}=\begin{bmatrix}\cos\text{n}\theta&\text{i}\sin\text{n}\theta\\\text{i}\sin\text{n}\theta&\cos\text{n}\theta\end{bmatrix}$ for all $\text{n}\in\text{N}.$
→
Solve the following determinant equations:
$\begin{vmatrix}1&1&\text{x}\\\text{p}+1&\text{p}+1&\text{p}+\text{x}\\3&\text{x}+1&\text{x}+2\end{vmatrix}=0$
→
Find:
$\int\frac{x^{2}}{x^{4} + x^{2} - 2}dx$
→
Find the integrals of the functions in Exercises:
$\sin^{-1}(\cos\text{x})$
→