Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
If $\text{x}=\text{a}\sin\text{t}\ \text{and}\ \text{y}=\text{a}(\cos\text{t}+\log\tan\frac{\text{t}}{2}),$ find $\frac{\text{d}^2\text{y}}{\text{dx}^2}$
✓

Answer

$\text{x}=\text{a}\sin\text{t}\ \text{and}\ \text{y}=\text{a}(\cos\text{t}+\log\tan\frac{\text{t}}{2}),$
$\frac{\text{dx}}{\text{dt}}=\text{a}\cos\text{t}$
$\frac{\text{d}^2\text{x}}{\text{dt}^2}=-\text{a}\sin\text{t}$
$\frac{\text{dy}}{\text{dt}}=-\text{a}\sin\text{t}+\text{a}\frac{1}{\tan\frac{\text{t}}{2}}\times\sec^2\frac{\text{t}}{2}\times\frac{1}{2}$
$=-\text{a}\sin\text{t}+\text{a}\frac{1}{2\sin\frac{\text{t}}{2}\cos\frac{\text{t}}{2}}$
$=-\text{a}\sin\text{t}+\text{a}\ \text{cosec}\ \text{t}$
$\frac{\text{d}^2\text{y}}{\text{dt}^2}=-\text{a}\cos\text{t}-\text{a cosec t}\cot\text{t}$
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{\frac{\text{dx}}{\text{dt}}\frac{\text{d}^2\text{y}}{\text{dt}^2}-\frac{\text{dy}}{\text{dt}}\frac{\text{d}^2\text{x}}{\text{dt}^2}}{\Big(\frac{\text{dx}}{\text{dt}}\Big)^3}$
$=\frac{\text{a}\cos\text{t}(-\text{a}\cos\text{t}-\text{a}\ \text{cosec t}\cot\text{t})-(-\text{a}\sin\text{t}+\text{a}\text{cosec t})(-\text{a}\sin\text{t})}{(\text{a}\cos\text{t})^3}$
$=\frac{-\text{a}^2\cos^2\text{t}-\text{a}^2\cot^2\text{t}-\text{a}^2\sin^2\text{t}+\text{a}^2}{\text{a}^3\cos^3\text{t}}$
$=\frac{-\text{a}^2\cos^2\text{t}-\text{a}^2\sin^2\text{t}-\text{a}^2\cot^2\text{t}+\text{a}^2}{\text{a}^3\cos^3\text{t}}$
$=\frac{-\text{a}^2(\cos^2\text{t}+\sin^2\text{t})-\text{a}^2\cot^2\text{t}+\text{a}^2}{\text{a}^3\cos^3\text{t}}$
$=-\frac{1}{\text{a}\sin^2\text{t}\cos\text{t}}$

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

Similar questions

Solve the following differential equations $\frac{\text{dy}}{\text{dx}}=\frac{2\text{x}(\log\text{x}+1)}{\sin\text{y+y}\cos\text{y}},$ given that $\text{y}=0,$ when $\text{x}=1.$
→
Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}\sin\alpha&\cos\alpha&\cos(\alpha+\delta)\\\sin\beta&\cos\beta&\cos(\beta+\delta)\\\sin\gamma&\cos\gamma&\cos(\gamma+\delta)\end{vmatrix}$
→
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius $r$ is $\frac{4\text{r}}{3}.$ Also show that the maximum volume of the cone is$\frac{8}{27}$of the volume of the sphere.
→
Draw a rough sketch of the region $\{(x, y) : y^2 < 5x, 5x^2 + < 36\}$ and find the area by the region using mwthod of integration.
→
Find the shortest distance between the following pairs of lines whose vector equation are:
$\vec{\text{r}}=(8+3\lambda)\hat{\text{i}}-(9+16\lambda)\hat{\text{j}}+(10+7\lambda)\hat{\text{k}}$ and $\vec{\text{r}}=15\hat{\text{i}}+29\hat{\text{j}}+5\hat{\text{k}}+\mu\big(3\hat{\text{i}}+8\hat{\text{j}}-5\hat{\text{k}}\big)$
→
Maximize $Z = -x_1 + 2x_2$
Subject to
$-\text{x}_1+3\text{x}_2\leq10$
$\text{x}_1+\text{x}_2\leq6$
$\text{x}_1+\text{x}_2\leq2$
$\text{x}_1,\text{x}_2\geq0$
→
Find the absolute maximum and the absolute minimum value of the following functions in the given intervals:
$\text{f}(\text{x})=4\text{x}-\frac{\text{x}^{2}}{2}\ \text{in}\ [2,4,5]$
→
Find the vector equation of the plane through the line of intersection of the planes $x + y + z = 1$ and $2x + 3y + 4z = 5$ which is perpendicular to the plane $x - y + z = 0$.
→
Evaluate the following integrals:
$\int\text{e}^{2\text{x}}\Big(\frac{1-\sin2\text{x}}{1-\cos2\text{x}}\Big)\text{dx}$
→
Form the differential equation corresponding to $\text{y}^2-2\text{ay}+\text{x}^2=\text{a}^2$ by eliminating a.
→