Download App

CONTINUITY AND DIFFERENTIABILITY — MATHS STD 12 Science — Question

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

Answer

$\text{a}(1-\cos\theta),\text{y}=\text{a}(\theta+\sin\theta),$
Differentiating w.r.t.$\theta$,
$\Rightarrow\frac{\text{dx}}{\text{d}\theta}=\text{a}(0+\sin\theta);\ ...\text{Eq}\ 1$
$\frac{\text{dy}}{\text{d}\theta}\ \text{a}(1+\cos\theta)\ ...\text{Eq}\ 2$
Dividing (2) by (1)
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\text{dy}}{\text{d}\theta}\times\frac{\text{d}\theta}{\text{dx}}=\frac{\text{a}(1+\cos\theta)}{\text{a}\sin\theta}$
Differentiating w.r.t.$\theta$,
$\Rightarrow\frac{\text{d}\Big(\frac{\text{dy}}{\text{dx}}\Big)}{\text{d}\theta}=\frac{\sin\theta(0-\sin\theta)-(1+\cos\theta)\cos\theta}{\sin^2\theta}...(3)$
$=-\frac{\sin^2\theta-\cos\theta-\cos^2\theta}{\sin^2\theta}$
$=-\frac{(1+\cos\theta)}{\sin^2\theta}...(4 )$
dividing (4) by (3)
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=-\frac{(1+\cos\theta)}{\sin^2\theta\times\text{a}\sin\theta}$
putting $\theta=\frac{\pi}{2}$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=-\frac{1}{\text{a}}$
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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Evaluate the following integrals:$\int\cos^{-1}\Big(\frac{1-\text{x}^2}{1+\text{x}^2}\Big)\text{dx}$
→
If $\text{A}=\begin{bmatrix} 3\\5\\2\end{bmatrix}$ and $\text{B}=\begin{bmatrix}1&0&4\end{bmatrix},$ verify that $(AB)^T = B^TA^T.$
→
 A fruit grower can use two types of fertilizer in his garden, brand $P$ and $Q$. The amounts $(in \ kg)$ of nitrogen, phosphoric acid, potash, and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least $240\ kg$ of phosphoric acid, at least $270\ kg$ of potash and at most $310\ kg$ of chlorine.
$Kg$ per bag
 
Brand $P$
Brand $P$
Nitrogen
 $32$ $3.5$
Phosphoric
 $1$ $2$
Potash
 $3$ $1.5$
Chlorine
 $1.5$ $2$
If the grower wants to minimize the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden?
→
Prove that:
  1. $\text{P}(\text{A})=\text{P}(\text{A}\cap\text{B})+\text{P}(\text{A}\cap\bar{\text{B}})$
  2. $\text{P}(\text{A}\cup\text{B})=\text{P}(\text{A}\cap\text{B})+\text{P}(\text{A}\cap\bar{\text{B}})+\text{P}(\bar{\text{A}}\cap\text{B})$
→
A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of first machine is 12 hours and that of second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines, and each unit of product B requires 2 hours on first machine and 1 hour on second machine. Each unit of product A is sold at ₹ 7 profit and that of B at a profit of ₹ 4. Find the production level per day for maximum profit graphically.
→
In each of the show that the given differential equation is homogeneous and solve each of them. $\text{x}^2\ \frac{\text{dy}}{\text{dx}}=\text{x}^2-2\text{y}^2+\text{xy}$ 
→
Evaluate: $\int\text{x sin}^{-1}\text{x dx}.$
→
Show the solution zone of the following inequalities on a graph paper:
$\text{x}+\text{y}\leq50$
$3\text{x}+\text{y}\geq90$
$\text{x},\text{y}\geq0$
→
Three shopkeepers $\text{A, B}$ and $C$ go to a store to buy stationery. A purchases $12$ dozen notebooks, $5$ dozen pens and $6$ dozen pencils. $B$ purchases $10$ dozen notebooks, $6$ dozen pens and $7$ dozen pencils. $C$ purchases $11$ dozen notebooks, $13$ dozen pens and $8$ dozen pencils. A notebook costs $40$ paise, a pen costs ₹ $1.25$ and a pencil costs $35$ paise. Use matrix multiplication to calculate each individual's bill.
→
Prove that:
$\begin{vmatrix}(\text{a}+1)(\text{a}+2)&\text{a}+2&1\\\text{a}+2)(\text{a}+3)&\text{a}+3&1\\\text{a}+3)(\text{a}+4)&\text{a}+4 &1\end{vmatrix}=-2$
→