MCQ
Matrix $\left[ {\begin{array}{*{20}{c}}0&{ - 4}&1\\4&0&{ - 5}\\{ - 1}&5&0\end{array}} \right]$is
- AOrthogonal
- BIdempotent
- ✓Skew- symmetric
- DSymmetric
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$f(x)=\left\{\begin{array}{rc}x^5+5 x^4+10 x^3+10 x^2+3 x+1, & x<0 \\ x^2-x+1, & 0 \leq x<1 \\ \frac{2}{3} x^3-4 x^2+7 x-\frac{8}{3}, & 1 \leq x<3 \\ (x-2) \log _e(x-2)-x+\frac{10}{3}, & x \geq 3\end{array}\right.$
Then which of the following options is/are correct?
$(1)$ $f^{\prime}$ has a local maximum at $x =1$ $(2)$ $f$ is onto
$(3)$ $f$ is increasing on $(-\infty, 0)$ $(4)$ $f^{\prime}$ is $NOT$ differentiable at $x =1$
$l_1: \overrightarrow{ r }=(\hat{ i }-11 \hat{ j }-7 \hat{ k })+\lambda(\hat{ i }+2 \hat{ j }+3 \hat{ k }), \lambda \in R$
and $l_2: \overrightarrow{ r }=(-\hat{ i }+\hat{ k })+\mu(2 \hat{ i }+2 \hat{ j }+\hat{ k }), \mu \in R$.
If $P$ is the point of intersection of $l$ and $l_1$, and $Q (\alpha$ $, \beta, \gamma)$ is the foot of perpendicular from $P$ on $l_2$, then $9(\alpha+\beta+\gamma)$ is equal to $..........$.