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5. continuity and differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
MCQ
Let $f: R \rightarrow R$ be defined as

$f(x)=\left[\begin{array}{ll}{\left[e^{x}\right],} \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,x<0 \\ a e^{x}+[x-1], \,\,\,\,\,\,\,\,\,0 \leq x<1 \\ b+[\sin (\pi x)], \,\,\,\,\,\,\,\,\,\,\,\,1 \leq x<2 \\ {\left[e^{-x}\right]-c,} \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,x \geq 2\end{array}\right.$

where a,b,c $\in R$ and $[t]$ denotes greatest integer less than or equal to $t.$ Then, which of the following statements is true $?$

  • A
    There exists $a,b,c$ $\in R$ such that $f$ is continuous of $R$.
  • B
    If $f$ is discontinuous at exactly one point, then $a+b+c=1$
  • ✓
    If $f$ is discontinuous at exactly one point, then $a+b+c \neq 1$
  • D
    $f$ is discontinuous at atleast two points, for any values of $a , b$ and $c$.
✓

Answer

Correct option: C.
If $f$ is discontinuous at exactly one point, then $a+b+c \neq 1$
c
$f ( x )$ is discontinuous at $x =1$

For continuous at $x =0 ; a =1$

For continuous at $x =2 ; b + c =1$

$a+b+c=2$

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