Download App

Higher Order Derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsHigher Order Derivatives5 Marks
Question
If $\text{x}=\text{a}(1-\cos^3\theta),\text{y}=\text{a}\sin^3\theta,$ Prove that $\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{32}{27\text{a}}\text{ at}\ \theta=\frac{\pi}{6}$
✓

Answer

Here,
$\text{x}=\text{a}(1-\cos^3\theta),\text{y}=\text{a}\sin^3\theta$
Differentiating w.r.t. x, we get
$\frac{\text{dx}}{\text{d}\theta}=3\text{a}\cos2\theta\sin\theta\text{ and }\frac{\text{dy}}{\text{d}\theta}=3\text{a}\sin2\theta\cos\theta$]
$\frac{\text{dy}}{\text{dx}}=\frac{3\text{a}\sin^2\theta\cos\theta}{3\text{a}\cos^2\theta\sin\theta}=\tan\theta$
Differentiating w.r.t. x, we get
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\sec^2\theta\frac{\text{d}\theta}{\text{dx}}$
$=\frac{\sec^2\theta}{3\text{a}\cos^2\theta\sin\theta}$
$=\frac{\sec^4\theta}{3\text{a}\sin\theta}$
$\therefore\frac{\text{d}^2\text{y}}{\text{dx}}\text{ at }\theta=\frac{\pi}{6}$
$\frac{\text{d}^2\text{y}}{\text{dx}}=\frac{\Big(\sec\frac{\text{x}}{6}\Big)^4}{3\text{a}\sin\frac{\text{x}}{6}}=\frac{32}{27\text{a}}$

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 Higher Order Derivatives→Higher Order Derivatives — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

→
If $\text{x}=\cos\text{t}+\log\tan\frac{\text{t}}{2},\text{y}=\sin\text{t},$ Then find the value of $\frac{\text{d}^2\text{y}}{\text{dt}^2}\ \text{and}\ \frac{\text{d}^2\text{y}}{\text{dx}^2}\ \text{at}\ \text{t}=\frac{\pi}{4}.$
→
Find the intervals in which the following functions are increasing or decreasing.
$f(x) = x^4 - 4x^3 - 4x^2 + 15$
→
Find the particular solution of the differential equation$(1-\text{y}^2)(1+\log\text{x})\text{dx}+2\text{xy dy}=0,$ given that $\text{y}=0$ when $\text{x}=1.$
→
Find the equation of the line passing through the points $\hat{\text{i}}+\hat{\text{j}}-3\hat{\text{k}}$ and perpendicular to the lines $\vec{\text{r}}=\hat{\text{i}}+\lambda\big(2\hat{\text{i}}+\hat{\text{j}}-3\hat{\text{k}}\big)$ and $\vec{\text{r}}=\big(2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}\big)+\mu\big(\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}\big).$
→
Solve the following systems of homogeneous linear equations by matrix method:
$x + y + z = 0$
$x - y - 5z = 0$
$x + 2y + 4z = 0$
→
Find the equation of a plane which passes through the point (3, 2, 0) and contains the line $\frac{\text{x}-3}{1}=\frac{\text{y}-6}{5}=\frac{\text{z}-4}{4}$
→
If $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ are non-zero, non-coplanar vectors, prove that the vector is coplanar:
$\vec{\text{a}}-2\vec{\text{b}}+3\vec{\text{c}},\ -3\vec{\text{b}}+5\vec{\text{c}}$ and $-2\vec{\text{a}}+3\vec{\text{b}}-4\vec{\text{c}}$
→
Using differentials, find the approximate values of the following:
$\Big(\frac{17}{81}\Big)^{\frac{1}{4}}$
→
It is given that the Rolle's theorem holds for the function $f(x) = x^3 + bx^2 + cx, \text{x}\in[1,2]$ at the point $\text{x}=\frac{4}{3},$ the values of b and c.
→