MCQ
$\mathop {\lim }\limits_{x \to 0} \frac{{\sin x + \log (1 - x)}}{{{x^2}}}$ is equal to
- A$0$
- B$\frac{1}{2}$
- ✓$ - \frac{1}{2}$
- DNone of these
$\mathop {\lim }\limits_{x \to 0} \,\frac{{\cos x - \frac{1}{{1 - x}}}}{{2x}} = \mathop {\lim }\limits_{x \to 0} \,\frac{{ - \sin x - \frac{1}{{{{(1 - x)}^2}}}}}{2} = - \frac{1}{2}$.
Aliter : $\mathop {\lim }\limits_{x \to 0} \,\frac{{\sin x + \log \,(1 - x)}}{{{x^2}}}$
$ = \mathop {\lim }\limits_{x \to 0} \,\,\frac{{\left( {x - \frac{{{x^3}}}{{3\,\,!}} + \frac{{{x^5}}}{{5\,\,!}} - ...} \right)}}{{{x^2}}} + \mathop {\lim }\limits_{x \to 0} \,\,\frac{{\left( { - x - \frac{{{x^2}}}{2} - \frac{{{x^3}}}{3} - \frac{{{x^4}}}{4} - ...} \right)}}{{{x^2}}}$
$\left( {\because \sin x = x - \frac{{{x^3}}}{{3\,!}} + \frac{{{x^5}}}{{5\,!}} - ..} \right.$
and $\left. {\log \,(1 - x) = - x - \frac{{{x^2}}}{2} - \frac{{{x^3}}}{3} - ..} \right)$
$ = \mathop {\lim }\limits_{x \to 0} \,\frac{{\frac{{ - {x^2}}}{2} - {x^3}\left( {\frac{1}{{3\,\,!}} + \frac{1}{3}} \right) - \frac{{{x^4}}}{4}...}}{{{x^2}}} = - \frac{1}{2}.$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
| List $I$ | List $II$ |
| $P.\quad$ Volume of parallelopiped determined by vectors $\vec{a}, \vec{b}$ and $\overrightarrow{ c }$ is $2$ . Then the volume of the parallelepiped determined by vectors $2(\vec{a} \times \vec{b}), 3(\vec{b} \times \vec{c})$ and $(\vec{c} \times \vec{a})$ is | $1.\quad$ $100$ |
| $Q.\quad$ Volume of parallelepiped determined by vectors $\vec{a}, \vec{b}$ and $\vec{c}$ is $5$ . Then the volume of the parallelepiped determined by vectors $3(\overrightarrow{ a }+\overrightarrow{ b }),(\overrightarrow{ b }+\overrightarrow{ c })$ and $2(\overrightarrow{ c }+\overrightarrow{ a })$ is | $2.\quad$ $30$ |
| $R.\quad$ Area of a triangle with adjacent sides determined by vectors $\vec{a}$ and $\vec{b}$ is $20$ . Then the area of the triangle with adjacent sides determined by vectors $(2 \vec{a}+3 \vec{b})$ and $(\vec{a}-\vec{b})$ is | $3.\quad$ $24$ |
| $S.\quad$ Area of a paralelogram with adjacent sides determined by vectors $\vec{a}$ and $\vec{b}$ is $30$ . Then the area of the parallelogram with adjacent sides determined by vectors $(\vec{a}+\vec{b})$ and $\vec{a}$ is | $4.\quad$ $60$ |
Codes: $ \quad P \quad Q \quad R \quad S $