Question
If x = a ( θ – sin θ ), y = a (1 + cos θ ), find $\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}$.
✓
Answer
$\therefore\frac{\text{dx}}{\text{d}\theta}=\text{a(1 - cos}\theta)\text{ and }\frac{\text{dy}}{d\theta}=-\text{a sin}\theta$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{-\sin\theta}{(1-\cos\theta)}$
$\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}=\frac{\text{(1-cos}\theta)\text{(-cos}\theta)+\sin\theta(\sin\theta)}{(1-\cos\theta)^{2}}\cdot\frac{\text{d}\theta}{\text{dx}}$
= $\frac{(\text{1-cos}\theta)}{\text{(1 - cos}\theta)^{2}}\cdot\frac{1}{\text{a(1 - cos}\theta)}$
$= \frac{1}{\text{a(1- cos}\theta)^{2}}\text{ or }\frac{1}{\text{4a}}\text{cosec}^{4}\frac{\theta}{2}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
$\int\frac{\text{x}^2+3\text{x}-1}{(\text{x}+1)^2}\text{dx}$
→Find the points of discontinuity, if any of the following function:
$\text{f(x)}=\begin{cases}\frac{\sin\text{x}}{\text{x}},&\text{if }\text{ x}<0\\2\text{x}+3,&\text{ x}\geq0\end{cases}$
→$\text{f(x)}=\begin{cases}\frac{\sin\text{x}}{\text{x}},&\text{if }\text{ x}<0\\2\text{x}+3,&\text{ x}\geq0\end{cases}$
If $\text{A}=\begin{bmatrix}1&0&2\\0&2&1\\2&0&3\end{bmatrix},$ then show that A is a root of the polynomial f(x) = x3 - 6x2 + 7x + 2.
→Show that the straight lines whose direction cosines are given by 2l + 2m – n = 0 and mn + nl + lm = 0 are at right angles.
Solve the equation
$\cos^{-1}\frac{\text{a}}{\text{x}}-\cos^{-1}\frac{\text{b}}{\text{x}}=\cos^{-1}\frac{1}{\text{b}}-\cos^{-1}\frac{1}{\text{b}}-\cos^{-1}\frac{1}{\text{a}}$
→$\cos^{-1}\frac{\text{a}}{\text{x}}-\cos^{-1}\frac{\text{b}}{\text{x}}=\cos^{-1}\frac{1}{\text{b}}-\cos^{-1}\frac{1}{\text{b}}-\cos^{-1}\frac{1}{\text{a}}$
Verify mean value theorem for the function:
$\text{f(x)}=\sin\text{x}-\sin2\text{x in }[0,\pi].$
→$\text{f(x)}=\sin\text{x}-\sin2\text{x in }[0,\pi].$
Show that the line $\frac{\text{x}+3}{-3}=\frac{\text{y}-1}{1}=\frac{\text{z}-5}{5}$ and $\frac{\text{x}+1}{-1}=\frac{\text{y}-2}{2}=\frac{\text{z}-5}{5}$ are coplanar. Hence, find the equation of the plane containing these lines.
→There are two types of fertilisers 'A' and 'B'. 'A' consists of 12 % nitrogen and 5 % phosphoric acid whereas 'B' consists of 4 % nitrogen and 5 % phosphoric acid. After testing the soil conditions, farmer finds that he needs at least 12 kg of nitrogen and 12 kg of phosphoric acid for his crops. If 'A' costs 10 per kg and 'B' cost 8 per kg, then graphically determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost.
A manufacturer produces three types of bolts, x, y and z which he sells in two markets. Annual sales (in ₹) are indicated below:
If unit sales prices of x, y and z are ₹ 2.50, ₹ 1.50 and ₹ 1.00 respectively, then answer the following questions using the concept of matrices.
→| Markets | Products | ||
| X | Y | Z | |
| I | 10000 | 2000 | 18000 |
| II | 6000 | 20000 | 8000 |
- Find the total revenue collected from the Market-I.
- ₹ 44000
- ₹ 48000
- ₹ 46000
- ₹ 53000
- Find the total revenue collected from the Market-II.
- ₹ 51000
- ₹ 53000
- ₹ 46000
- ₹ 49000
- If the unit costs of the above three commodities are ₹ 2.00, ₹ 1.00 and 50 paise respectively, then find the gross profit from both the markets.
- ₹ 53000
- ₹ 46000
- ₹ 34000
- ₹ 32000
- If matrix $\text{A}=[\text{a}_\text{ij}]_{2\times2},$ where $\text{a}_\text{ij}=1,$ if $\text{i}\neq\text{j}$ and $\text{a}_\text{ij}=0,$ if $\text{i}=\text{j}$ then A2 is equal to:
- I
- A
- OR
- None of these
- If A and B are matrices of same order, then (AB' - BA') is a.
- Skew-synunetric matrix.
- Null matrix.
- Symmetric matrix.
- Unit matrix.
If the points with position vectors $10\hat{\text{i}}+3\hat{\text{j}},\ 12\hat{\text{i}}-5\hat{\text{j}}$ and $\text{a}\hat{\text{i}}+11\hat{\text{j}}$ are collinear, find the value of a.
→