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STD 12 - 5. continuity and differentiation — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 5. continuity and differentiation4 Marks
MCQ
Let $f :R \to R$ be a function defined as $f\left( x \right) = \left\{ \begin{array}{l}5,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,\,\,\,\,\,x \le 1\,\,\,\,\,\,\,\\ a + bx,\,\,\,\,if\,\,\,\,\,\,1 < x < 3\\ b + 5x,\,\,\,\,if\,\,\,\,\,\,3 \le x < 5\\ 30,\,\,\,\,\,\,\,\,\,\,if\,\,\,\,\,\,\,x \ge 5 \end{array} \right.\,\,\,\,$ Then $f$ is
  • A
    continuous if $a = 5$ and $b = 5$
  • B
    continuous if $a = 5$ and $b = 10$
  • C
    continuous if $a = 0$ and $b = 5$
  • D
    not continuous for any values of $a$ and $b$
✓

Answer

For $x=1 \ \text{R.H.L}=a+b$
$\text{L.H.L}=5$
So to be continuous at $x=1$
$a+b=5  .......... (i)$
for $x=3$
$\text{R.H.L}=b+15$
$\text{L.H.L}=a+3b$
$b+15=a+3b$
$a+2b=15  ........ (ii)$
for $x=5$
$\text{R.H.L}=30$
$\text{L.H.L}=b+25$
$b+25=30$
$b=5$.
From equation $(ii)$
$a=10$
but $a=10$ and $b=5$ does not satisfied equation $(i)$
So $f(x)$ is discontinuous for $a \in R$ and $b \in R$

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