Loading [MathJax]/jax/element/mml/optable/MathOperators.js

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)6 for all xϵ[2,4], then.
  • f(4)<8
  • f(4)8
  • f(4)>12
  • f(4)>8
Consider f(x)=limnxnsinxnxn+sinxn for x>0,x1,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)=[sin[x]] in (0,2π); is ([] denotes greatest integer 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:
  • dudt=fx.xt+fy.yt
  • dudt=fx.dxdtfy.yt
  • dudt=fx.dxdt+fy.dydt
  • dudt=fx.xtfy.yt
Let f:[0,2]R a function which is continuous on [0,2] and is differentiable on (0,2) with f(0)=1. Let F(x)=x20f(t)dt, for x[0,2], if F(x)=f(x),x(0,2), then F(2) equals to 
  • e21
  • e41
  • e1
  • e4
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 } } }
0:0:1


Answered Not Answered Not Visited Correct : 0 Incorrect : 0

Practice Class 12 Commerce Maths Quiz Questions and Answers