Download App

Higher Order Derivatives — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsHigher Order Derivatives5 Marks
Question
If $\text{x}=\text{a}(\theta+\sin\theta)\ \text{and}\ \text{y}=\text{a}(1+\cos\theta)$ prove that $\frac{\text{d}^2\text{y}}{\text{dx}^2}=-\frac{\text{a}}{\text{y}^2}.$
✓

Answer

Here
$\text{x}=\text{a}(\theta+\sin\theta)\ \text{and}\ \text{y}=\text{a}(1+\cos\theta)$
Differentiating w.r.t.$\theta$, we get
$\frac{\text{dx}}{\text{d}\theta}=\text{a}+\text{a}\cos\theta\ \text{and}\ \frac{\text{dy}}{\text{d}\theta}=-\text{a}\sin\theta$
$\therefore\frac{\text{dy}}{\text{dx}}=\frac{-\text{a}\sin\theta}{\text{a}+\text{a}\cos\theta}=\frac{-\sin\theta}{1+\cos\theta}$
Differentiating w.r.t.$\theta$, we get
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=-\Big\{\frac{(1+\cos\theta)\cos\theta+\sin^2\theta}{(1+\cos\theta)^2}\Big\}\frac{\text{d}\theta}{\text{dx}}$
$=\frac{-\cos\theta-\cos^2\theta-\sin^2\theta}{(1+\cos\theta)^2}\times\frac{1}{\text{a}+\text{a}\cos\theta}$
$=\frac{-(1+\cos\theta)}{\text{a}(1+\cos\theta)^3}$
$=\frac{-1}{\text{a}(1+\cos\theta)^2}$
$=\frac{-\text{a}}{\text{y}^2}\ [\because\text{y}=\text{a}(1+\cos\theta)]$
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 "Solve the Following Question.(5 Marks)" from Higher Order Derivatives→Higher Order Derivatives — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

If $\text{x}=\Big(\text{t}+\frac{1}{\text{t}}\Big)^\text{a},\text{y}=\text{a}^{\text{t}+\frac{1}{\text{t}}},$ find $\frac{\text{dy}}{\text{dx}}$
→
If $x = a \cos \theta, y = b \sin \theta$, show that $a^2\left[y \frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^2\right]+b^2=0$
→
Evaluate the following intregals:
$\int\frac{1}{(\text{x}^2+1)(\text{x}^2+2)}\ \text{dx}$
→
Prove that $\frac{\text{dy}}{\text{dx}}\Big\{\frac{\text{x}}{2}\sqrt{\text{a}^2-\text{x}^2}+\frac{\text{a}^2}{2}\sin^{-1}\frac{\text{x}}{\text{a}}\Big\}=\sqrt{\text{a}^2-\text{x}^2}$
→
Find the points of local maxima or local minima, if any, of the following functions, using the first derivatives test. Also, find the local maximum or local minimum values, as the case may be:
$\text{f}(\text{x})=\sin2\text{x},0\leq\text{x}\leq\pi$
→
If $\text{y}=(\sin^{-1}\text{x})^2,$ prove that $(1-\text{x}^2)\text{y}_2-\text{xy}_1-2=0$
→
Let $Z$ be the set of all integers and $Z_0$ be the set of all non-zero integers. Let a relation $R$ on $Z \times Z_0$ be defined as $(a, b)R(c, d) ⇔ ad = bc$ for all $(a, b), (c, d) \in Z \times Z_0,$ Prove that $R$ is an equivalence relation on $Z \times Z_0.$
→
Evaluate the following intregals:
$\int\frac{\text{x}^2}{(\text{x}^2+4)(\text{x}^2+9)}\ \text{dx}$
→
If $\text{f}\text{(x)}=\begin{cases}\frac{\text{x}-4}{\text{|x}-4|}+\text{a}, &\text{if x} <4\\\text{a}+\text{b},&\text{if x}=4\\\frac{\text{x}-4}{\text{|x}-4|}+\text{b}, & \text{if x}>4\end{cases}$ is continuous at x = 4. Find a, b.
→
On R − {1}, a binary operation * is defined by a * b = a + b − ab. Prove that * is commutative and associative. Find the identity element for * on R − {1}. Also, prove that every element of R − {1} is invertible.
→