MCQ
Let $A$ be a square matrix such that $A A^T=I$. Then $\frac{1}{2} A\left[\left(A+A^T\right)^2+\left(A-A^T\right)^2\right]$ is equal to
- A$A^2+I$
- B$A^3+I$
- C$A^2+A^T$
- ✓$A^3+A^T$
On solving given expression, we get
$ \frac{1}{2} A\left[A^2+\left(A^T\right)^2+2 A A^T+A^2+\left(A^T\right)^2-2 A A^T\right] $
$ =A\left[A^2+\left(A^T\right)^2\right]=A^3+A^T$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$\mathrm{x}-2=-\mathrm{y}=\mathrm{z}-1,2(\mathrm{x}+1)=2(\mathrm{y}-1)=\mathrm{z}+1$
and be parallel to the line $\frac{x-2}{3}=\frac{y-1}{1}=\frac{z-2}{2}$.
Then which of the following points lies on $\mathrm{L}$ ?
$(i)$ $f _1(x)=\sin \left(\sqrt{1- e ^{-x^2}}\right)$
$(ii)$ $f_2(x)=\left\{\begin{array}{ll}\frac{|\sin x|}{\tan ^{-1} x} & \text { if } x \neq 0 \\ 1 & \text { if } x=0\end{array}\right.$, where the inverse trigonometric function of $\tan ^{-1} x$
assume values in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$,
$(iii)$ $f_3(x)=\left[\sin \left(\log _c(x+2)\right)\right]$, where, for $t \in R ,[t]$ denotes the greatest integer less than or equal to $t$,
(iv) $f_4(x)=\left\{\begin{array}{ll}x^2 \sin \left(\frac{1}{x}\right) & \text { if } x \neq 0 \\ 0 & \text { if } x=0\end{array}\right.$
| $LIST I$ | $LIST II$ |
| $P$ The function $f _1$ is | $1$ $NOT$ continuous at $x=0$ |
| $Q$ The function $f _2$ is | $2$ continuous at $x =0$ and $NOT$ differentiable at $x =0$ |
| $R$ The function $f_3$ is | $3$ differentiable at $x=0$ and its derivative is $NO$T continuous at $x =0$ |
| $S$ The function $f _4$ is | $4$ differentiable at $x =0$ and its derivative is continuous at $x =0$ |
The correct option is: