Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
If $\text{x}=\text{a}\sec\theta,\text{y}=b\tan\theta$ prove that $\frac{\text{d}^2\text{y}}{\text{dx}^2}=-\frac{\text{b}^4}{\text{a}^2\text{y}^3}$
✓

Answer

Here,
$\text{x}=\text{a}\sec\theta,\text{y}=b\tan\theta$
Differentiating w.r.t.x, we get
$\frac{\text{dx}}{\text{d}\theta}=\text{a}\sec\theta\tan\theta\ \text{and}\ \frac{\text{dy}}{\text{d}\theta}=b\sec^2\theta$
$\therefore\frac{\text{dy}}{\text{dx}}=\frac{\text{dy}}{\text{d}\theta}\times\frac{\text{d}\theta}{\text{dx}}=\frac{\text{b}\sec^2\theta}{\text{a}\sec\theta\tan\theta}=\frac{\text{b}\ \text{cosec}\theta}{\text{a}}$
Differentiating w.r.t.x, we get
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{\text{b}}{\text{a}}\times-\text{cosec}\theta\cot\theta\times\frac{\text{d}\theta}{\text{dx}} $
$=-\frac{\text{b}}{\text{a}}\times\text{cosec}\theta\cot\theta\times\frac{1}{\text{a}\sec\theta\tan\theta}$
$=\frac{-\text{b}}{\text{a}^2}\times\frac{1}{\tan^3\theta}$
$\therefore\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{-\text{b}}{\frac{\text{a}^2\text{y}^3}{\text{b}^3}}=-\frac{\text{b}^4}{\text{a}^2\text{y}^3}$
Hence proved

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 one-parameter families of solution curves of the following differential equation: (or solve the following differential equation)$(\text{x}\log\text{x})\frac{\text{dy}}{\text{dx}}+\text{y}=\log\text{x}$
→
Integrate the rational function $\frac{\left(x^{2}+1\right)\left(x^{2}+2\right)}{\left(x^{2}+3\right)\left(x^{2}+4\right)}$
→
Find one-parameter families of solution curves of the following differential equation: (or solve the following differential equation)$\frac{\text{dy}}{\text{dx}}+3\text{y}=\text{e}^{\text{mx}},$ m is given real number.
→
Discuss the continuity of the following functions at the indicated point:
$\text{f(x)}=\begin{cases}\frac{\text{x}^\text{x}-1}{\log(1+2\text{x})}&,\text{ if x}\neq0&\\ \ &&\text{at x}=0\\7&,\text{ if x}=0\end{cases}$
→
Show that the binary operation $\ast \text{ on A = R - {-1}}$ defined as a $\text{a} \ast \text{b} = \text{a + b + ab}$ for all $\text{a, b}\in \text{A}$ is communicative and associative on A. Also find the identity element of $\ast$ in A and prove that every element of a is invertible.
→
Let $\text{F}(\alpha)=\begin{bmatrix}\cos\alpha & -\sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1\end{bmatrix}$ and
$\text{G}(\beta)=\begin{bmatrix} \cos\beta & 0 & \sin\beta \\ 0 & 1 & 0 \\ -\sin\beta & 0 & \cos\beta \end{bmatrix}$
Show that
$\big[\text{F}(\alpha)\big]^{-1}=\text{F}(-\alpha)$
→
Verify mean value theorem for the function:
$\text{f(x)}=\frac{1}{4\text{x}-1}\text{ in }[1,4].$
→
Evaluate the following integrals:
$\int\frac{\text{dx}}{(\text{x}^2+1)(\text{x}^2+4)}$
→
Consider $f: R_+\rightarrow [– 5, \infty )$ given by $f(x) = 9x^2 + 6x – 5.$ Show that f is invertible with $f^{-1}(\text{y})=\left(\frac{\Big(\sqrt{\text{y}+6}\Big)-1}{3}\right).$
→
Evaluate the following definite integrals:
$\int\limits_{0}^{\frac{\pi}{2}}\sin^3\text{x}\text{ dx}$
→