for continuity at $x=-1$
L.H.L. $=\frac{\pi}{4}-\frac{\pi}{4}=0$
R.H.L. $=0$
so, continuous at $x=-1$ for continuity at $x=1$
L.H.L. $=0$
R.H.L. $=\frac{\pi}{4}+\frac{\pi}{4}=\frac{\pi}{2}$
so, not continuous at $x=1$ For differentiability at $x =-1$
$L . H.D. =\frac{1}{1+1}=\frac{1}{2}$
$R.H.D. =-\frac{1}{2}$
so, non differentiable at $x=-1$
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$E=\left[\begin{array}{ccc}1 & 2 & 3 \\ 2 & 3 & 4 \\ 8 & 13 & 18\end{array}\right], P=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]$ and $F=\left[\begin{array}{ccc}1 & 3 & 2 \\ 8 & 18 & 13 \\ 2 & 4 & 3\end{array}\right]$
If $Q$ is a nonsingular matrix of order $3 \times 3$, then which of the following statements is (are) $TRUE$?
$(A)$F $=P E P$ and $P^2=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$
$(B)$ $\left| EQ + PFQ ^{-1}\right|=| EQ |+\left| PFQ ^{-1}\right|$
$(C)$ $\left|( EF )^3\right|>| EF |^2$
$(D)$ Sum of the diagonal entries of $P ^{-1} EP + F$ is equal to the sum of diagonal entries of $E + P ^{-1} FP$
$I.$ There can exist two triangles such that the sides of one triangle are all less than $1$ cm while the sides of the other triangle are all bigger than $10$ metres, but the area of the first triangle is larger than the area of second triangle.
$II$ .If $x, y, z$ are all different real numbers, then $\frac{1}{{{{(x - y)}^2}}} + \frac{1}{{{{(y - z)}^2}}} + \frac{1}{{{{(z - x)}^2}}}$ $=$ ${\left( {\frac{1}{{x - y}} + \frac{1}{{y - z}} + \frac{1}{{z - x}}} \right)^2}$
$III$. $log_3x · log_4x · log_5x = (log_3x · log_4x) $$+ (log_4x · log_5x) + (log_5x · log_3x)$ is true for exactly for one real value of $x.$
$IV$. $A$ matrix has $12$ elements. Number of possible orders it can have is six. Now indicate the correct alternatively.