CBSE Questions for Class 12 Commerce Maths Continuity And Differentiability Quiz 15 - MCQExams.com

if the function $$f(x)=\begin{cases} a+{\sin}^{-1}(x+b),\,x\geq 1 \\ x,\,x<1 \end{cases}$$ is differentiable at $$x=1$$, then $$\displaystyle \frac{a}{b}$$ is equal to 
  • $$1$$
  • $$0$$
  • $$-1$$
  • $$2$$
The function $$f(x) = \dfrac {x}{1 + |x|}$$ is differentiable at which of the following?
  • Every where
  • Everywhere except at $$x = 1$$
  • Everywhere except at $$x = 0$$
  • Everywhere except at $$x = 0$$ or $$1$$
If $$y=e^{\sin^{-1}(t^{2}-1)}$$ & $$x=e^{\sec^{-1}\left (\frac{1}{t^{2}-1}\right)}$$, then $$\dfrac{dy}{dx}$$ is equal to
  • $$\frac{x}{y}$$
  • $$\dfrac{-y}{x}$$
  • $$\dfrac{y}{x}$$
  • $$\dfrac{-x}{y}$$
If $$f(x)={ sin }^{ -1 }\left[ \dfrac { 2x }{ 1+{ x }^{ 2 } }  \right] $$,then $$f(x)$$ is differentiable on 
  • [-1,1]
  • R-{-1,1}
  • R-(-1,1)
  • None of these
f(X)=|x|+|x-1| is continuous at 
  • '0' only
  • 0,1 only
  • Every where
  • No where
If $$\displaystyle f_{n}(x) = e^{f_{n-1}(x)} $$ for all $$\displaystyle n \,\epsilon \,N $$ and $$ f_{0} (x) = x $$ then $$ \dfrac{d}{dx} \left \{f_{n} (x)\right \} $$ is equal to 
  • $$\displaystyle f_{n}(x) \,.\dfrac{d}{dx}\left \{f_{n - 1} (x)\right \} $$
  • $$\displaystyle f_{n}(x) \,.f_{n - 1}(x) $$
  • $$\displaystyle f_{n}(x) \,.f_{n - 1}(x) ..... f_{2}(x)\,.f_{1} (x) $$
  • $$\displaystyle \prod_{i = 1}^{n} f_{i}(x) $$
The order of the differential equation of all circles whose radius is $$4$$, is?
  • $$1$$
  • $$2$$
  • $$3$$
  • $$4$$
If $$f\left( x \right) = {\left| x \right|^{\left| {\sin x} \right|}}$$, then $${f'}\left( { - \dfrac{\pi }{4}} \right)$$ is equals
  • $${\left( {\dfrac{\pi }{4}} \right)^{1/\sqrt 2 }}\left( { - \dfrac{{\sqrt 2 }}{2}{\text{ln}}\dfrac{4}{\pi } - \dfrac{{2\sqrt 2 }}{\pi }} \right)$$
  • $${\left( {\dfrac{\pi }{4}} \right)^{1/\sqrt 2 }}\left( {\dfrac{{\sqrt 2 }}{2}{\text{ln}}\dfrac{4}{\pi } + \dfrac{{2\sqrt 2 }}{\pi }} \right)$$
  • $${\left( {\dfrac{\pi }{4}} \right)^{1/\sqrt 2 }}\left( {\dfrac{{\sqrt 2 }}{2}{\text{ln}}\dfrac{\pi }{4} + \dfrac{{2\sqrt 2 }}{\pi }} \right)$$
  • None of these.
Let $$f:(-1,1)\rightarrow R$$ be a differentiable function satisfying 
             $$(f'(x))^4=16(f(x))^2$$ for all $$x\in (-1,1)$$
   $$f(0)=0$$
The number of such functions is 
  • $$2$$
  • $$3$$
  • $$4$$
  • more than $$4$$
If $$\displaystyle y = \dfrac{\sqrt{(1 + t^{2})} - \sqrt{(1 - t^{2})}}{\sqrt{(1 + t^{2})} +\sqrt{(1 - t^{2})}} $$ and $$\displaystyle x = \sqrt {(1 - t^{4})} $$ , then $$ \dfrac{dy}{dx} $$
  • $$\displaystyle \dfrac{-1}{t^{2}\left \{1 +\sqrt{1 - t^{4}} \right \}} $$
  • $$\displaystyle \dfrac{\left \{\sqrt{(1 - t^{4})} - 1\right \}}{t^{6}} $$
  • $$\displaystyle \dfrac{1}{t^{2}\left \{1 + \sqrt{(1 - t^{4})}\right \}} $$
  • $$\displaystyle \dfrac{1 - \sqrt{( 1 - t^{4})}}{t^{6}} $$
f(x) is diffrentiable function and (f(x). g(x)) is differentiable a x=a , then 
  • g(x) must be differentiable at x=a
  • if g(x) is discontinuous , then f(a) =0
  • f(a) $$\neq $$ 0 then g(x) must be differentiable
  • nothing can be said
Give that f(x) =xg(x) /$$ \left | x \right | $$ , g(0) = 0 and f(x) is continous at x=Then the value of f' (0)
  • Does not exist
  • is -1
  • is 1
  • is 0
Let f(x)=
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  • f' is differentiable
  • f is differentiable
  • f' is continuous
  • f is continuous
If $$f(x) =  \left\{\begin{matrix} \dfrac{x\log \cos x}{\log(1+x^2)}, & x \neq 0\\ 0, & x=0\end{matrix}\right.$$ then
  • f(x) is not continuous at x=0.
  • f(x) is continuous at x=0.
  • f(x) is continuous at x=0 but not differentiable at x=0.
  • f(x) is differentiable at x=0.
Let f: $$ R\rightarrow R $$ be a function such that f(x+y)= f(x)+f(y),$$ \forall $$ x,y$$ \epsilon R $$. If f(x) is differentiable at x=0, then
  • f(x) is differentiable only in a finite interval containing zero
  • f(x) is continuous $$ \forall x\epsilon R $$
  • f(x) is constant $$ \forall x\epsilon R $$
  • f(x) is differentiable except at finitely many points
Let $$f(x)$$ be a function satisfying $$f(x+y)  = f(x)+f(y)$$ and $$f(x) = xg(x)$$ $$\forall x, ~y \in$$ R, where $$g(x)$$ is a continuous function then, which of the following is true?
  • $$f'(x)=g'(0)$$
  • $$f'(x)=g (0)$$
  • $$f(x)=g (0)$$
  • $$f(x)=g'(0)$$
Which of the following is differentiable at x= 0
  • cos $$ \left ( \left | x \right | \right )+\left | x \right | $$
  • cos $$ \left ( \left | x \right | \right )-\left | x \right | $$
  • sin$$ \left ( \left | x \right | \right )+\left | x \right | $$
  • $$sin \left ( \left | x \right | \right )-\left | x \right |$$
$$\cos |x|$$ is differentiable everywhere.
  • True
  • False
If the function $$f:[0,8] \rightarrow R$$ is differentiable, then for$$0<a, b<2, \int_{0}^{8} f(t) d t$$ is equal to
  • $$3\left[\alpha^{3} f\left(\alpha^{2}\right)+\beta^{2} f\left(\beta^{2}\right)\right]$$
  • $$3\left[\alpha^{3} f(\alpha)+\beta^{3} f(\beta)\right]$$
  • $${3}\left[\alpha^{2} f\left(\alpha^{3}\right)+\beta^{2} f\left(\beta^{3}\right)\right]$$
  • $$3\left[\alpha^{2} f\left(\alpha^{2}\right)+\beta^{2} f\left(\beta^{2}\right)\right]$$
Let f(x) and g(x) be differentiable for $$0\leq x\leq 1$$, such that f(0) such that f'(c)=2g'(c), then the value of g(1) must be 
  • 1
  • 3
  • -2
  • -1
$$ f(x)=x^{2}+x g^{\prime}(1)+g^{\prime \prime}(2) $$ and $$ g(x)=f(1) x^{2}+x f^{\prime}(x)+f^{\prime \prime}(x) $$

The value of $$ g(0) $$ is
  • 0
  • -3
  • 2
  • None of these
$$f(x)$$ is not invertible for
  • $$x \in\left[-\dfrac{\pi}{2}-\tan ^{-1} 2, \dfrac{\pi}{2}-\tan ^{-1} 2\right]$$
  • $$x \in\left[\tan ^{-1} \dfrac{1}{2}, \pi+\tan ^{-1} \dfrac{1}{2}\right]$$
  • $$x \in\left[\pi+\cot ^{-1} 2,2 \pi+\cot ^{-1} 2\right]$$
  • None of these
Let $$f(0,\infty)\rightarrow R$$ be a differentiable function such that $$f'(x)=2-\dfrac{f(x)}{x}$$ for all $$x\epsilon (0,\infty)$$ and $$f(1)\neq 1$$ Then
  • $$\underset{x\rightarrow 0+}{lim}f'(\dfrac{1}{x})=1$$
  • $$\underset{x\rightarrow 0+}{lim}xf'(\dfrac{1}{x})=2$$
  • $$\underset{x\rightarrow 0+}{lim}x^2f'(\dfrac{1}{x})=0$$
  • $$|f(x)|\le 2$$ for all $$x\epsilon(0,2)$$
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