MCQ
$\int_{}^{} {\frac{{dx}}{{1 + 3{{\sin }^2}x}} = } $
- A$\frac{1}{3}{\tan ^{ - 1}}(3{\tan ^2}x) + c$
- ✓$\frac{1}{2}{\tan ^{ - 1}}(2\tan x) + c$
- C${\tan ^{ - 1}}(\tan x) + c$
- DNone of these
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$a_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}, n \geq 1$
$b_1=1 \text { and } b_n=a_{n-1}+a_{n+1}, n \geq 2.$
Then which of the following options is/are correct?
$(1)$ $a_1+a_2+a_3+\ldots . .+a_n=a_{n+2}-1$ for all $n \geq 1$
$(2)$ $\sum_{n=1}^{\infty} \frac{ a _{ n }}{10^{ n }}=\frac{10}{89}$
$(3)$ $\sum_{n=1}^{\infty} \frac{b_n}{10^n}=\frac{8}{89}$
$(4)$ $b=\alpha^n+\beta^n$ for all $n>1$
| Red | Blue | Green | |
| Bag I | 3 | 2 | 5 |
| Bag II | 4 | 3 | 3 |
| Bag III | 5 | 1 | 4 |
$l_1:(3+ t ) \hat{ i }+(-1+2 t ) \hat{ j }+(4+2 t ) \hat{ k },-\infty< t <\infty $
$l_2:(3+2 t ) \hat{ i }+(3+2 t ) \hat{ j }+(2+ s ) \hat{ k },-\infty< s <\infty$
Then, the coordinate$(s)$ of the point$(s)$ on $l_2$ at a distance of $\sqrt{17}$ from the point of intersection of $l$ and $l_1$ is(are)
$(A)$ $\left(\frac{7}{3}, \frac{7}{3}, \frac{5}{3}\right)$ $(B)$ $(-1,,-1,0)$ $(C)$ $(1,1,1)$ $(D)$ $\left(\frac{7}{9}, \frac{7}{9}, \frac{8}{9}\right)$
If the volume of the parallelopiped, whose adjacent sides are represented by the vectors $\overrightarrow{ u }, \overrightarrow{ v }$ and $\overrightarrow{ w }$ , is $\sqrt{2}$, then the value of $|3 \vec{u}+5 \vec{v}|$ is. . . . .