Download App

CONTINUITY AND DIFFERENTIABILITY — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
If $\text{x}=\text{a}(\theta-\sin\theta),\text{y}=\text{a}(1+\cos\theta)$ find $\frac{\text{d}^2\text{y}}{\text{dx}^2}$
✓

Answer

$\text{x} =\text{a} (\theta -\sin\theta)\dots\text{ eq. } 1$
$\text{y}=\text{a}(1+\cos\theta)\dots\text{ eq. 2}$
To find: $\frac{\text{d}^2\text{y}}{\text{dx}^2}$
As, $\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{\text{d}}{\text{dx}}\Big(\frac{\text{dy}}{\text{dx}}\Big)$
So, lets first find $\frac{\text{dy}}{\text{dx}}$ using parametric form and differentiate it again.
$\frac{\text{dx}\text{}}{\text{d}\theta}=\frac{\text{d}}{\text{d}\theta}\text{a}(\theta-\sin\theta)=\text{a}(1-\cos\theta)\dots\ \text{eq. 3}$
Similarly,
$\frac{\text{dy}}{\text{d}\theta}=\frac{\text{d}}{\text{d}\theta}\text{a}(1+\cos\theta)=-\text{a}\sin\theta\dots\text{ eq. }4$
$\Big[\because\frac{\text{d}}{\text{dx}}\cos\text{x}=-\sin\text{x},\frac{\text{d}}{\text{dx}}\sin\text{x}=\cos\text{x}\Big]$
$\therefore\frac{\text{dy}}{\text{dx}}=\frac{\frac{\text{dy}}{\text{d}\theta}}{\frac{\text{dx}}{\text{d}\theta}}=\frac{-\text{a}\sin\theta}{\text{a}(1-\cos\theta)}=\frac{-\sin\theta}{(1-\cos\theta)}\dots\ \text{eq. }5$
Differentiating again w.r.t. x:
$\frac{\text{d}}{\text{dx}}\Big(\frac{\text{dy}}{\text{dx}}\Big)=-\frac{\text{d}}{\text{dx}}\Big(\frac{\sin\theta}{1-\cos\theta}\Big)$
Using product rule and chain rule of differentiation together:
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\Big\{-\frac{1}{1-\cos\theta}\frac{\text{d}}{\text{d}\theta}\sin\theta-\sin\theta\frac{\text{d}}{\text{d}\theta}\frac{1}{(1-\cos\theta)}\Big\}\frac{\text{d}\theta}{\text{dx}}$
Apply chain rule to determine $\frac{\text{d}}{\text{d}\theta}\frac{1}{(1-\cos\theta)}$
$\frac{\text{d}^2\text{y}}{\text{dx}}=\Big\{\frac{-\cos\theta}{1-\cos\theta}+\frac{\sin^2\theta}{(1-\cos\theta)^2}\Big\}\frac{1}{\text{a}(1-\cos\theta)}$ [using eq.3]
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\bigg\{\frac{-\cos\theta(1-\cos\theta)+\sin^2\theta}{(1-\cos\theta)^2}\bigg\}\frac{1}{\text{a}(1-\cos\theta)}$
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\bigg\{\frac{-\cos\theta+\cos^2\theta+\sin^2\theta}{(1-\cos\theta)^2}\bigg\}\frac{1}{\text{a}(1-\cos\theta)}$
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\Big\{\frac{1-\cos\theta}{(1-\cos\theta)^2}\Big\}\frac{1}{\text{a}(1-\cos\theta)}[\because\cos^2\theta+\sin^2\theta=1]$
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{1}{\text{a(1}-\cos\theta)^2}$
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{1}{\text{a}\Big(2\sin^2\frac{\theta}{2}\Big)^2}\big[\because-\cos\theta=2\sin^2\frac{\theta}{2}\big]$
$\therefore\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{1}{\text{4a}}\text{cosec}^4\frac{\theta}{2}$

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

Similar questions

A man 180cm tall walks at a rate of 2m/ sec. away, from a source of light that is 9m above the ground. How fast is the length of his shadow increasing when he is 3m away from the base of light?
→
Find the absolute maximum and the absolute minimum value of the following functions in the given intervals:
$\text{f}(\text{x})=4\text{x}-\frac{\text{x}^{2}}{2}\ \text{in}\ [2,4,5]$
→
If $\text{y}=\text{cosec}^{-1}\text{x},\text{x}>1$ prove that $\text{x}(\text{x}^2-1)\frac{\text{d}^2\text{y}}{\text{dx}^2}+(2\text{x}^2-1)\frac{\text{dy}}{\text{dx}}=0.$
→
If $\text{A} = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix} \text{and B = } \text{A} = \begin{bmatrix} 2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5 \end{bmatrix}$ are two square matrices, find AB and hence solve the system of linear
equations x – y = 3, 2x + 3y + 4z = 17 and y + 2z = 7.
→
A die is thrown 5 times. Find the probability that an odd number will come up exactly three times.
→
If $y = 2 \cos (\log x) + 3 \sin (\log x),$ Prove that $x^{2} \frac{d^{2}y}{dx}^{2} + x\frac{dy}{dx} + y = 0.$
→
There are two types of fertilisers 'A' and 'B'. 'A' consists of 12% nitrogen and 5% phosphoric acid whereas 'B' consists of 4% nitrogen and 5% phosphoric acid. After testing the soil conditions, farmer finds that he needs at least 12kg of nitrogen and 12kg of phosphoric acid for his crops. If 'A' costs Rs. 10 per kg and 'B' cost Rs. 8 per kg, then graphically determine how much of each type of fertiliser should be used so that nutrient requiremnets are met at a minimum cost.
→
Suppose $N$ is set of natural numbers and $R$, is defined in $N \times N$ such that :
$(a, b) R (c, d) \Leftrightarrow a d(b+c)=b c(a+d)$. Prove that $R$ in equivalence in $N \times N$.
→
Examine the continuity of the function
 $\text{f}\text{(x)}=\begin{cases}3\text{x}-2 &, \text{ if x} \leq 0\\\text{x}+1 &, \text{ if x} > 0\end{cases}\text{at x}=0$ 
 Also sketch the graph of this function.
→
If $\tan^{-1}\Big(\frac{\text{x}^2-\text{y}^2}{\text{x}^2+\text{y}^2}\Big)=\text{a}$ prove that $\frac{\text{dx}}{\text{dx}}=\frac{\text{y}}{\text{x}}\frac{(1-\tan\text{a})}{(1+\tan\text{a})}$
→