MCQ
$\int_{}^{} {x\cos x\;dx = } $
- ✓$x\sin x + \cos x$
- B$x\sin x - \cos x$
- C$x\cos x + \sin x$
- D$x\cos x - \sin x$
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$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)
$(A)$ $\left(\frac{7}{3}, \frac{7}{3}, \frac{5}{3}\right)$ $(B)$ $(-1,,-1,0)$ $(C)$ $(1,1,1)$ $(D)$ $\left(\frac{7}{9}, \frac{7}{9}, \frac{8}{9}\right)$
$1.$ The probability that $x_1+x_2+x_3$ is odd, is $x _1+ x _2+ x _3$
$(A)$ $\frac{29}{105}$ $(B)$ $\frac{53}{105}$ $(C)$ $\frac{57}{105}$ $(D)$ $\frac{1}{2}$
$2.$ The probability that $x_1, x_2, x_3$ are in an arithmetic progression, is
$(A)$ $\frac{9}{105}$ $(B)$ $\frac{10}{105}$ $(C)$ $\frac{11}{105}$ $(D)$ $\frac{7}{105}$
Give the answer question $1$ and $2.$
be continuous for some $a, b, c \in R$ and $f ^{\prime}(0)+ f ^{\prime}(2)= e ,$ then the value of of $a$ is
$f(x)= \begin{cases}\frac{1-\cos 2 x}{x^2} & , x<0 \\ \alpha & , x=0, \text { where } \alpha, \beta \in R \text {. If } \\ \frac{\beta \sqrt{1-\cos x}}{x} & , x>0\end{cases}$
$f$ is continuous at $\mathrm{x}=0$, then $\alpha^2+\beta^2$ is equal to :