CBSE Questions for Class 12 Commerce Maths Continuity And Differentiability Quiz 12 - MCQExams.com

Let f be a function which is continuous and differentiable for all real x. If $$f(2)=-4$$ and $$f'(x)\geq 6$$ for all $$x\epsilon [2, 4]$$, then.
  • $$f(4)<8$$
  • $$f(4)\geq 8$$
  • $$f(4) > 12$$
  • $$f(4) >8$$
Consider $$f(x)=\lim _{ n\rightarrow \infty  }{ \cfrac { { x }^{ n }-\sin { { x }^{ n } }  }{ { x }^{ n }+\sin { { x }^{ n } }  }  } $$ for $$x>0,x\neq 1,f(1)=0$$ then
  • $$f$$ is continuous at $$x=1$$
  • $$f$$ has a discontinuity at $$x=1$$
  • $$f$$ has an infinite or oscillatory discontinuity at $$x=1$$
  • $$f$$ has a removal type of discontinuity at $$x=1$$
A point where function $$f(x)$$ is not continuous where $$f(x)=\left[ \sin { \left[ x \right]  }  \right] $$ in $$\left( 0,2\pi  \right) $$; is ($$\left[ \ast  \right] $$ denotes greatest integer $$\le x$$)
  • $$(3,0)$$
  • $$(2,0)$$
  • $$(1,0)$$
  • $$(4,-1)$$
If $$u=f(x,y)$$ is a differentiable function of $$x$$ and $$y$$, where $$x$$ and $$y$$ are differentiable functions of $$t$$ then:
  • $$\cfrac { du }{ dt } =\cfrac { \partial f }{ \partial x } .\cfrac { \partial x }{ \partial t } +\cfrac { \partial f }{ \partial y } .\cfrac { \partial y }{ \partial t } $$
  • $$\cfrac { du }{ dt } =\cfrac { \partial f }{ \partial x } .\cfrac { dx }{ dt } -\cfrac { \partial f }{ \partial y } .\cfrac { \partial y }{ \partial t } $$
  • $$\cfrac { du }{ dt } =\cfrac { \partial f }{ \partial x } .\cfrac { dx }{ dt } +\cfrac { \partial f }{ \partial y } .\cfrac { dy }{ dt } $$
  • $$\cfrac { du }{ dt } =\cfrac { \partial f }{ \partial x } .\cfrac { \partial x }{ \partial t } -\cfrac { \partial f }{ \partial y } .\cfrac { \partial y }{ \partial t } $$
Let $$f:\left[ {0,2} \right] \to R$$ a function which is continuous on $$\left[ {0,2} \right]$$ and is differentiable on $$\left( {0,2} \right)$$ with $$f\left( 0 \right) = 1$$. Let $$F\left( x \right) = \int\limits_0^{{x^2}} {f\left( {\sqrt t } \right)} dt,$$ for $$x \in \left[ {0,2} \right]$$, if $$F'\left( x \right) = f'\left( x \right),\forall x \in \left( {0,2} \right),$$ then $$F(2)$$ equals to 
  • $${e^2} - 1$$
  • $${e^4} - 1$$
  • $$e-1$$
  • $${e^4}$$
If $$\int_{1}^{3} F'(x)dx=-12$$ and $$\int_{1}^{3} x^{3} F' '(x) dx=40$$, then the correct expression(s) is/are
  • $$9f'(3)+f'(1)-32=0$$
  • $$\int_{1}^{3} f(x)dx=12$$
  • $$9f' (3)-f'(1)+32=0$$
  • $$\int_{1}^{3} f(x) dx=-12$$
Which of the following function is differentiable at x=0
  • $$\cos \left( {|x|} \right) + |x|$$
  • $$\cos \left( {|x|} \right) - |x|$$
  • $$\sin \left( {|x|} \right) + |x|$$
  • $$\sin \left( {|x|} \right) - |x|$$
If $$f : R \rightarrow R$$ is defined by
$$f(x) = \left \{\begin{matrix} \dfrac{x + 2}{x^2 + 3x + 2} & if & x \in R - \{-1, -1\} \\ -1 & if & x = -2 \\ 0 & if  &x = -1\end{matrix} \right.$$ then $$f(x)$$ continuous on the set 
  • $$R$$
  • $$R - \{-2\}$$
  • $$R - \{-1, -2\}$$
  • $$R-\{-1\}$$
If $$ f\left( x+y \right) =f\left( x \right) +f\left( y \right)$$ then $$ f\left( x \right)$$ may be
  • $$x$$
  • $$x+1$$
  • $${ x }^{ 2 }+1$$
  • $$\log { x }$$
The set of points, where the function $$f(x) = x|x|$$, is differentiable, is given by
  • $$(-\infty, \infty)$$
  • $$(-\infty, 0) \cup (0, \infty)$$
  • $$(0, \infty)$$
  • $$[0, \infty)$$
The function $$\displaystyle f(x) = (x^2 - 1) |x^2 - 3x + 2| + cos (|x|)$$, is not differentiable at x=
  • 1
  • -1
  • 2
  • 0
The function $$f\left( x \right) = \,{\sin ^{ - 1}}\left( {\cos \,x} \right)\,is\,: - $$ 
  • discontinuous at $$=0$$
  • continuous at $$=0$$
  • differentiable $$=0$$
  • none of these
Given that $$f(x)$$ is a differentiable function of $$x$$ and that $$f\left( x \right).f\left( y \right) = f\left( x \right) + f\left( y \right) + f\left( {xy} \right) - 2$$ and that $$f\left( 2 \right) = 5.$$ Then $$f'\left( 3 \right)$$ is equal to
  • $$6$$
  • $$24$$
  • $$15$$
  • $$19$$
If $$y^{y^{y^{....{^\infty}}}} = \log_e(x+\log_e(x+....))$$, then $$\dfrac{dy}{dx}$$ at $$(x= e^2-2, y= \sqrt2)$$ is
  • $$\dfrac{\log\left(\dfrac{e}{2}\right)}{2\sqrt2(e^2-1)}$$
  • $$\dfrac{\log2}{2\sqrt2(e^2-1)}$$
  • $$\dfrac{\sqrt2 \log\dfrac{e}{2}}{(e^2-1)}$$
  • None of these
Let $$y=x^{x^x}$$, then differentiate $$y$$ w.r.t $$x$$.
  • $$x^{x^x}\left(\dfrac{1}{x}+\log x+(\log x)^2\right)$$
  • $$x^{x^x}(x^x)\left(\dfrac{1}{x}+\log x-(\log x)^2\right)$$
  • $$x^{x^x}(x^x)\left(\dfrac{1}{x}+\log x+(\log x)^2\right)$$
  • $$x^{x^x}(x^x)\left(\dfrac{1}{x}-\log x-(\log x)^2\right)$$
If $$x = 2\left( {\theta  + \sin \theta } \right)and\,y = 2\left( {1 - \cos \theta } \right),then\;value\;of\frac{{dy}}{{dx}}\;is\;$$
  • $$\tan \left( {\frac{\theta }{2}} \right)$$
  • $$\cot \left( {\frac{\theta }{2}} \right)$$
  • $$\sin \left( {\frac{\theta }{2}} \right)$$
  • $$\cos \left( {\frac{\theta }{2}} \right)$$
If $$f\left( x \right) = x + \left| x \right| + {\kern 1pt} \,\cos \left( {\left[ {{\pi ^2}} \right]x} \right)\,\,and\,\,g\left( x \right) = \,\sin \,x\,where\,\left[ . \right]$$ denotes the greatest integer function) then :-
$$(1)$$ $$f\left( x \right) + g\left( x \right)$$ is discontinuous
$$(2)$$$$f\left( x \right) + g\left( x \right)$$ is differentiable everywhere  
$$(3)$$ $$f\left( x \right) \times g\left( x \right)$$  is differentiable everywhere
 $$(4)$$ $$f\left( x \right) \times g\left( x \right)$$ is countimuos but not diffrentiable  at $$x=0$$ 
  • 1
  • 2
  • 3
  • 4
If $$f\left( x \right) = \left\{ \begin{array}{l}\frac{{1 - \left| x \right|}}{{1 + x}},{\rm{ }}x \ne  - 1\\1,{\rm{          }}x =  - 1{\rm{     }}\end{array} \right.$$   then $$f\left( {\left[ {2x} \right]} \right),$$ where $$\left[ {} \right]$$ represents the greatest integer function , is 
  • discontinuous at $$x = - 1$$
  • continuous at $$x = 0$$
  • continuous at $$x = \frac{1}{2}$$
  • continuous at $$x = 1$$
If $$x=a\cos^{3}\theta, y=a\sin^{3}\theta$$, then $$1+ {(\dfrac {dy}{dx})}^{2}$$ is
  • $$\sec^{2}\theta$$
  • $$\tan \theta$$
  • $$1$$
  • $$\tan^{2}\theta$$
Let f:R$$ \rightarrow  $$(0,1) be a continuous function.. Then, which of the following function(s) has (have) the value zero at some point in the interval (0,1)?
  • $$ e ^ { x } - \int _ { 0 } ^ { 1 } f ( t ) \sin t d t $$
  • $$ f ( x ) + \int _ { 0 } ^ { 1 } f ( t ) \sin t d t $$
  • $$ x - \int _ { 0 } ^ { \frac { \pi } { 2 } - x } f ( t ) \cos t d t $$
  • $$ x ^ { 3 } - f ( x ) $$
 If $$f:\left[ {0,1} \right] \to \left[ {0,1} \right]$$ be definded by f(x) =\begin{cases} x,\quad \quad \quad \quad \quad \quad if\quad x\quad is\quad rational\quad  \\ 1-x,\quad \quad \quad \quad if\quad x\quad is\quad irrational\quad \quad \quad \quad \quad  \end{cases} then $$\left( {f \circ f} \right)x$$ ______________.
  • constant
  • 1+x
  • x
  • None of these
If $$\left( \frac { 1-x }{ 1+x }  \right) =x$$ and $$g\left( x \right) =\int { f\left( x \right) } dx$$ then 
  • $$g\left( x \right)$$ is continuous in domain
  • $$g\left( x \right)$$ is discontinuous st two points in its domain
  • $$\lim _{ x\rightarrow \infty }{ g\left( x \right)=-1 }$$
  • $$\int { g\left( x \right) dx=-\frac { { x }^{ 2 } }{ 2 } +\left( 2x+1 \right) \lambda n\left( \frac { 1+x }{ e } \right) +C } $$
If $$f(x)=\begin{cases} \dfrac { 1-\sqrt { 2 } \sin { x }  }{ \pi -4x } ,\quad \quad ifx\neq \dfrac { \pi  }{ 4 }  \\ a\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ,\quad \quad ifx=\dfrac { \pi  }{ 4 }  \end{cases}$$ is continous at $$x=\dfrac {\pi}{4}$$ then $$a=$$
  • $$4$$
  • $$2$$
  • $$1$$
  • $$1/4$$
A function $$f$$ satisfies the relation $$f(x)=f'(x)+f"(x)+.....\infty$$ terms, where $$f(x)$$ is differentiable indefinitely, If $$f'(-1)=1$$ then $$f(-1)$$ is equal to
  • $$0$$
  • $$1$$
  • $$2$$
  • $$4$$
$$f(x)= x\sin\dfrac{1}{x} , \  for x\neq 0$$
       $$= 0,\  for x=0$$
Then.
  • $$f'(0^+) exit\ but \ f'(0^-)$$ does not exit
  • $$f'(0^+) \ and \ f'(0^-)$$ do not exit
  • $$f'(0^+) = f'(0^-)$$ 
  • none of these
$$f(x) = \left\{\begin{matrix}(3/x^{2})\sin 2x^{2} & if x M 0 \\\dfrac {x^{2} + 2x + c}{1 - 3x^{2}}  & if\ x \geq 0, x \neq \dfrac {1}{\sqrt {3}}\\ 0 & x = 1/ \sqrt {3}\end{matrix}\right.$$ then in order that $$f$$ be continuous at $$x = 0$$, the value of $$c$$ is
  • $$2$$
  • $$4$$
  • $$6$$
  • $$8$$
Let $$g(x)$$ be a continuous function for all $$x$$, and $$f(x)=f(\alpha)+(x-\alpha).g(x)\ \forall \ x\ =\epsilon \ R$$. Then;
  • $$f(x)$$ is necessarily differentiable at $$x=\alpha$$
  • $$f(x)$$ is not necessarily differentiable at $$x=\alpha$$
  • $$f(x)$$ is not necessarily continuous at $$x=\alpha$$
  • $$None\ of\ these$$
Let a function $$f: R\rightarrow R$$ be given by $$f(x+y)=f(x)f(y)$$ for all  $$x, y \in R$$ and $$f(x)\neq 0$$ for any $$x_{1}$$ function $$f(x)$$ is differentiable at $$x=0$$. Find $$f(x)    gi ven              f(0)=1$$.
  • $$e^{x}$$
  • $$x.{f'(0)}$$
  • $$\dfrac{x^{2}}{2}f'(x)$$
  • $$None\ of\ these$$
Define $$f\left( x \right) = \left\{ \begin{array}{l}{x^2} + bx + c\,\,\,\,\,,x < 1\\x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x \ge 1\end{array} \right.$$. If $$f(x)$$ is differentiable at $$x=1$$ then $$(b-c)=$$
  • $$-2$$
  • $$0$$
  • $$1$$
  • $$2$$
$$\dfrac { d }{ dx } \left[ { cos }^{ -1 }\left( x\sqrt { x } -\sqrt { \left( 1-x \right) \left( 1-{ x }^{ 2 } \right)  }  \right)  \right] =$$
  • $$\dfrac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } -\dfrac { 1 }{ 2\sqrt { { x-x }^{ 2 } } }$$
  • $$\dfrac { -1 }{ \sqrt { 1-{ x }^{ 2 } } } -\dfrac { 1 }{ 2\sqrt { { x-x }^{ 2 } } }$$
  • $$\dfrac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } +\dfrac { 1 }{ 2\sqrt { { x-x }^{ 2 } } }$$
  • $$\dfrac { 1 }{ \sqrt { 1-{ x }^{ 2 } } }$$
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