MCQ
If $\frac{{{d^2}y}}{{d{x^2}}} = 0,$ then
- ✓$y = ax + b$
- B${y^2} = ax + b$
- C$y = \log x$
- D$y = {e^x} + c$
Integrating $(i)$ with respect to $x$, $\frac{{dy}}{{dx}} = a$…..$(ii)$
where $a$ is an arbitrary constant
Again integrating $(ii)$ with respect to $x$
$\int {\frac{{dy}}{{dx}}dx} = \int {adx + b} $ or $y = ax + b$,
where $b$ is another arbitrary constant.
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| column-$I$ | column-$II$ |
| $(A)$ A line from the origin meets the lines $\frac{x-2}{1}=\frac{y-1}{-2}=\frac{z+1}{1}$ and $\frac{x-\frac{8}{3}}{2}=\frac{y+3}{-1}=\frac{z-1}{1}$ at $P$ and $Q$ respectively. If length $P Q=d$, then $d^2$ is | $(p)$ $-4$ |
| $(B)$ The values of $x$ satisfying $\tan ^{-1}(x+3)-\tan ^{-1}(x-3)=\sin ^{-1}\left(\frac{3}{5}\right)$ are | $(q)$ $0$ |
| $(C)$ Non-zero vectors $\vec{a}, \vec{b}$ and $\overrightarrow{\mathrm{c}}$ satisfy $\vec{a} \cdot \vec{b}=0$, $(\vec{b}-\vec{a}) \cdot(\vec{b}+\vec{c})=0$ and $2|\vec{b}+\vec{c}|=|\vec{b}-\vec{a}|$. If $\vec{a}=\mu \vec{b}+4 \vec{c}$, then the possible values of $\mu$ are | $(r)$ $4$ |
| $(D)$ Let $f$ be the function on $[-\pi, \pi]$ given by $f(0)=9$ and $f(x)=$ $\sin \left(\frac{9 x}{2}\right) / \sin \left(\frac{x}{2}\right)$ for $x \neq 0$. The value of $\frac{2}{\pi} \int_{-\pi}^\pi f(x) d x$ is | $(s)$ $5$ |
| $(t)$ $6$ |