Download App

CONTINUITY AND DIFFERENTIABILITY — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
Find $\frac{\text{dy}}{\text{dx}},$ when
$\text{x}=\text{ae}^{\theta}(\sin\theta-\cos\theta),\text{y}=\text{ae}^\theta(\sin\theta+\cos\theta)$
✓

Answer

We have, $\text{x}=\text{ae}^{\theta}(\sin\theta-\cos\theta)$ and $\text{y}=\text{ae}^{\theta}(\sin\theta+\cos\theta)$
$\Rightarrow\frac{\text{dx}}{\text{d}\theta}=\text{a}\Big[\text{e}^\theta\frac{\text{d}}{\text{d}\theta}(\sin\theta-\cos\theta)+(\sin\theta-\cos\theta)\frac{\text{d}}{\text{d}\theta}(\text{e}^\theta)\Big]$ and
$\frac{\text{dy}}{\text{d}\theta}=\text{a}\Big[\text{e}^\theta\frac{\text{d}}{\text{d}\theta}(\sin\theta+\cos\theta)+(\sin\theta+\cos\theta)\frac{\text{d}}{\text{d}\theta}(\text{e}^\theta)\Big]$
$\Rightarrow\frac{\text{dx}}{\text{d}\theta}=\text{a}\Big[\text{e}^\theta(\cos\theta+\sin\theta)+(\sin\theta-\cos\theta)(\text{e}^\theta)\Big]$ and $\frac{\text{dy}}{\text{d}\theta}=\text{a}\Big[\text{e}^\theta(\cos\theta-\sin\theta)+(\sin\theta+\cos\theta)(\text{e}^\theta)\Big]$
$\Rightarrow\frac{\text{dx}}{\text{d}\theta}=\text{a}\big[2\text{e}^\theta(\sin\theta)\big]$ and $\frac{\text{dy}}{\text{d}\theta}=\text{a}\big[2\text{e}^\theta(\cos\theta)\big]$
$\therefore\frac{\frac{\text{dy}}{\text{d}\theta}}{\frac{\text{dx}}{\text{d}\theta}}=\frac{\text{a}(2\text{e}^\theta\cos\theta)}{\text{a}(2\text{e}^\theta\sin\theta)}=\cot\theta$

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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

If $\text{y}=\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big)+\sec^{-1}\Big(\frac{1+\text{x}^2}{1-\text{x}^2}\Big), 0<\text{x}<1$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{4}{1+\text{x}^2}$
→
If $x^y + y^x = a^b$, then find $\frac{\text{dy}}{\text{dx}}.$
→
Find the shortest distance between the lines $\frac{\text{x}-2}{-1}=\frac{\text{y}-5}{2}=\frac{\text{z}-0}{3}$ and $\frac{\text{x}-0}{2}=\frac{\text{y}+5}{-1}=\frac{\text{z}-1}{2}.$
→
Find the equation of the lines joining the following pairs of vertices and then find the shortest distance between the lines(1) (0, 0, 0) and (1, 0, 2)
(2) (1, 3, 0) and (0, 3, 0)
→
Solve the following systems of homogeneous linear equations by matrix method: $x + y - z = 0 , x - 2y + z = 0a , 3x + 6y - 5z = 0$
→
Find the intervals in which the following functions are increasing or decreasing. $f(x) = 5 + 36x + 3x^2- 2x^3$
→
Differentiate the following functions with respect to x:
$\log\Big(\frac{\text{x}^2+\text{x}+1}{\text{x}^3-\text{x}+1}\Big)$
→
The mean and variance of a binomial variate with parameters n and p are 16 and 8, respectively. Find P(X = 0), P (X = 1) and P (X ≥ 2).
→
Integrate the function: $\frac{\sqrt{\tan x}}{\sin x \cos x}$
→
Solve the following differential equation : $\left(y+3 x^2\right) \frac{d x}{d y}=x$
→