Download App

Higher Order Derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsHigher Order Derivatives5 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 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 $\hat{\text{a}}$ and $\hat{\text{b}}$ are unit vectors inclined at an angle $\theta$, prove that$\cos\frac{\theta}{2}=\frac{1}{2}\big|\hat{\text{a}}+\hat{\text{b}}\big|$
→
The mean and variance of a binomial variate with parameters n and p are 16 and 8, respectively. Find P(X = 0), P (X = 1) and P (X ≥ 2).
→
There are three coins. One is two-headed coin (having head on both faces), another is biased coin that comes up heads 75% of the times and third is also a biased coin that comes up tail 40% of the times. One of the three coins is chosen at random and tossed, and it shows heads. What is the probability that it was the two-headed coin?
→
Solve the following equation
$\text{x}\cos^2\text{y dx}=\text{y}\cos^2\text{x dx}$
→
Find the area of the region bounded by the curve $y^2 = 4x$ and the line $x = 3.$
→
Solve the matrix equations:
$\begin{bmatrix}\text{x}&1\end{bmatrix}\begin{bmatrix}1&0\\-2&-3\end{bmatrix}\begin{bmatrix}\text{x}\\5\end{bmatrix}=0$
→
Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}-\text{x}\text{e}^\text{x}-\frac{5}{2}+\cos^2\text{x}$
→
If $x^{13} y^7 = (x + y)^{20}$​​​​​​​, prove that $\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}$
→
If A and B are two independent events such that $\text{P}(\overline{\text{A}}\cap\text{B})=\frac{2}{15}$ and $\text{P}(\text{A}\cap\overline{\text{B}})=\frac{1}{6}$, then find P(B).
→
Solve the following LPP graphically:
Maximise Z = 105x + 90y
subject to the constraints
x + y $\leq$ 50
2x + y $\leq$80
x $\geq$ 0, y $\geq$ 0.
→