MCQ
Maximum value of a second order determinant whose every element is either $0, 1$ or $2$ only is:
- A$0$
- B$1$
- C$2$
- ✓$4$
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$R _{1}=\left\{( a , b ) \in R ^{2}: a ^{2}+ b ^{2} \in Q \right\}$ and $R _{2}=\left\{( a , b ) \in R ^{2}: a ^{2}+ b ^{2} \notin Q \right\}$
where $Q$ is the set of all rational numbers. Then
$1.$ Which of the following is true for $0 < x < 1$ ?
$(A)$ $0 < $ f(x) $ < \infty$
$(B)$ $-\frac{1}{2} < f(x) < \frac{1}{2}$
$(C)$ $-\frac{1}{4} < f(x) < 1$
$(D)$ $-\infty < $ f $($ x $) < 0$
$2.$ If the function $e^{-x} f(x)$ assumes its minimum in the interval $[0,1]$ at $x=\frac{1}{4}$, which of the following is true?
$(A)$ $f^{\prime}(x)$
$(B)$ $f^{\prime}(x)>f(x), 0$
$(C)$ f $^{\prime}(x)$
$(D)$ $f^{\prime}(x)$
Give the answer question $1$ and $2.$
$f(x) =$ $\left\{ {\begin{array}{*{20}{c}} {\sin \,x}&{if\,\,\,x\,\, \leqslant \,\,c} \\ {ax\, + \,b}&{if\,\,\,x\,\, > \,\,c} \end{array}} \right.$ where $c$ is a known quantity.
If $f$ is derivable at $x = c$ , then the values of $'a'$ $and$ $'b’$ are _____ $and$______ respectively