Download App

Differentiation — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDifferentiation5 Marks
Question
Find $\frac{\text{dy}}{\text{dx}},$ when
$\text{x}=\text{a}(\cos\theta+\theta\sin\theta)$ and $\text{y}=\text{a}(\sin\theta-\theta\sin\theta-\theta\cos\theta)$
✓

Answer

We have, $\text{x}=\text{a}(\cos\theta+\theta\sin\theta)$ and $\text{y}=\text{a}(\sin\theta-\theta\sin\theta-\theta\cos\theta)$
$\Rightarrow\frac{\text{dx}}{\text{d}\theta}=\text{a}\Big[\frac{\text{d}}{\text{d}\theta}\cos\theta+\frac{\text{d}}{\text{d}\theta}(\theta\sin\theta)\Big] $ and
$\frac{\text{dy}}{\text{d}\theta}=\text{a}\Big[\frac{\text{d}}{\text{d}\theta}\sin\theta-\frac{\text{d}}{\text{d}\theta}(\theta\cos\theta)\Big] $
$ \Rightarrow\frac{\text{dx}}{\text{d}\theta}=\text{a}\Big[-\sin\theta+\theta\frac{\text{d}}{\text{d}\theta}(\sin\theta)+\sin\theta\frac{\text{d}}{\text{d}\theta}(\theta)\Big]$ and
$\frac{\text{dy}}{\text{d}\theta}=\text{a}\Big[\cos\theta-\left\{\theta\frac{\text{d}}{\text{d}\theta}(\cos\theta)+\cos\theta\frac{\text{d}}{\text{d}\theta}(\theta)\right\}\Big]$
$ \Rightarrow\frac{\text{dx}}{\text{d}\theta}=\text{a}\big[-\sin\theta+\theta\cos\theta\big]$ and
$\frac{\text{dy}}{\text{d}\theta}=\text{a}\big[\cos\theta+\theta\sin\theta-\cos\theta\big]$
$ \Rightarrow\frac{\text{dx}}{\text{d}\theta}=\text{a}\theta\cos\theta$ and $\frac{\text{dy}}{\text{d}\theta}=\text{a}\theta\sin\theta$
$\therefore\frac{\text{dy}}{\text{dx}}=\frac{\frac{\text{dy}}{\text{d}\theta}}{\frac{\text{dx}}{\text{d}\theta}}=\frac{\text{a}\theta\sin\theta}{\text{a}\theta\cos\theta}=\tan\theta$

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 Differentiation→Differentiation — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Using direction ratios show that the points A(2, 3, -4) B(1, - 2, 3) and C(3, 8, -11) are collinear.
→
$\text{Evaluate:} \int\limits_{-a}^{a} \sqrt\frac{{a - x}}{a + x} {dx}$ 
→
Evaluate the following intregals:
$\int\frac{1}{\text{x}[6(\log\text{x})^2+7\log\text{x}+2]}\ \text{dx}$
→
Find the foot of the perpendicular from (1, 2, -3) on the line $\frac{\text{x}+1}{2}=\frac{\text{y}-3}{-2}=\frac{\text{z}}{-1}.$
→
If $\text{x}=\cos\text{t}(3-2\cos^2\text{t}),\text{y}\sin\text{t}(3-2\sin^2\text{t})$ find the value of $\frac{\text{dy}}{\text{dx}}\text{ at t}=\frac{\pi}{4}$
→
Evaluate the following integrals:
$\int_{0}^\limits{1}\text{x}\tan^{-1}\text{x}\text{ dx} $
→
Show that the points A, B and C with position vectors, $\vec{a}=3 \hat{i}-4 \hat{j}-4 \hat{k}$, $\vec{b}=2 \hat{i}-\hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-3 \hat{j}-5 \hat{k}$ respectively form the vertices of a right angled triangle.
→
Find the vector and cartesian equation of the line through the point (5, 2, -4) and which is parallel to the vector $3\hat{\text{i}}+2\hat{\text{j}}-8\hat{\text{k}}.$
→
Find the equation of the plane through the line of intersection of the planes $\vec{\text{r}}\cdot(\hat{\text{i}}+3\hat{\text{j}})+6=0$ and $\vec{\text{r}}\cdot(3\hat{\text{i}}-\hat{\text{j}}-4\hat{\text{k}})=0,$ which is at a unit distance from the origin.
→
A man takes a step forward with probability 0.4 and backward with probability 0.6. Find the probability that at the end of 5 steps, he is one step away from the starting point.
→