Sample QuestionsSTD 12 - 5. continuity and differentiation questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If the function $f(x) = \left\{ \begin{array}{l}\frac{{k\cos x}}{{\pi - 2x}},{\rm{when }}x \ne \frac{\pi }{2}\\3,\;\;\;\;\;\;\;\;\;{\rm{when }}x = \frac{\pi }{2}\end{array} \right.$ be continuous at $x = \frac{\pi }{2}$, then $ k =$
Answer: B.
View full solution →In order that the function $f(x) = {(x + 1)^{1/x}}$ is continuous at $x = 0$, $f(0)$ must be defined as
- A
$f(0) = 0$
- ✓
$f(0) = e$
- C
$f(0) = 1/e$
- D
$f(0) = 1$
Answer: B.
View full solution →The value of $k$ so that the function $f(x) = \left\{ \begin{array}{l}k(2x - {x^2}),\;\;\;{\rm{when\,}}\,x < 0\\\,\,\,\,\,\,\,\,\,\cos x,\,\,\,\,\,\,{\rm{when\,}}\,x \ge {\rm{0}}\end{array} \right.$ is continuous at $x = 0$, is
Answer: D.
View full solution →If $f(x) = \left\{ \begin{array}{l}\frac{x}{{{e^{1/x}} + 1}},\,\,{\rm{when\,\,}}\,\,x \ne 0\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,0,\,\,{\rm{when \,\,}}x = 0\end{array} \right.$, then
- A
$\mathop {\lim }\limits_{x \to 0 + } f(x) = 1$
- B
$\mathop {\lim }\limits_{x \to 0 - } f(x) = 1$
- ✓
$f(x)$ is continuous at $x = 0$
- D
Answer: C.
View full solution →If $f(x) = \left\{ \begin{array}{l}{(1 + 2x)^{1/x}},\,{\rm{for\,\, }}x \ne 0\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{e^2},\,{\rm{for\,\, }}x = 0\,\,\,\end{array} \right.$, then
- A
$\mathop {\lim }\limits_{x \to 0 + } f(x) = e$
- ✓
$\mathop {\lim }\limits_{x \to 0 - } f(x) = {e^2}$
- C
$f(x)$ is discontinuous at $x = 0$
- D
Answer: B.
View full solution →