CBSE Questions for Class 12 Commerce Maths Continuity And Differentiability Quiz 5 - MCQExams.com

Let $$y=\tan ^{ -1 }{ \left( \sec { x } +\tan { x }  \right)  } $$. Then, $$\cfrac { dy }{ dx } = $$
  • $$\cfrac { 1 }{ 4 } $$
  • $$\cfrac { 1 }{ 2 } $$
  • $$\cfrac { 1 }{ \sec { x } +\tan { x } } $$
  • $$\cfrac { 1 }{ \sec ^{ 2 }{ x } } $$
  • $$\cfrac { 1 }{ \tan { x } } $$
If $$x = \sin t$$ and $$y = \tan t$$, then $$\dfrac{dy}{dx}$$ is equal to
  • $$\cos^3 t$$
  • $$\dfrac{1}{\cos^3 t}$$
  • $$\dfrac{1}{\cos^2 t}$$
  • $$\sin^2 t$$
  • $$\dfrac{1}{\sin^2 t}$$
If $$f(x)=\cos ^{ -1 }{ \left\{ \cfrac { 1-{ \left( \log _{ e }{ x }  \right)  }^{ 2 } }{ 1+{ \left( \log _{ e }{ x }  \right)  }^{ 2 } }  \right\}  } $$, then $$f'(e)$$
  • Does not exist
  • Is equal to $$\cfrac { 2 }{ e } $$
  • Is equal to $$\cfrac { 1 }{ e } $$
  • Is equal to $$1$$
If $$y=\tan ^{ -1 }{ \left( \cfrac { a\cos { x } -b\sin { x }  }{ b\cos { x } +a\sin { x }  }  \right)  } $$, then $$\cfrac { dy }{ dx } $$ is equal to
  • $$2$$
  • $$-1$$
  • $$\cfrac { a }{ b } $$
  • $$0$$
If $$x=\cos { \theta  } ,y=\sin { 5\theta  } $$, then
$$(1-{ x }^{ 2 })\cfrac { { d }^{ 2 }y }{ d{ x }^{ 2 } } -x\cfrac { dy }{ dx } =$$
  • $$-5y$$
  • $$5y$$
  • $$25y$$
  • $$-25y$$
If $$y=x\log { \left( \cfrac { x }{ a+bx }  \right)  } $$, then $${ x }^{ 3 }\cfrac { { d }^{ 2 }y }{ d{ x }^{ 2 } } $$ is equal to
  • $$x\cfrac { dy }{ dx } -y$$
  • $${ \left( x\cfrac { dy }{ dx } -y \right) }^{ 2 }$$
  • $$y\cfrac { dy }{ dx } -x$$
  • $${ \left( y\cfrac { dy }{ dx } -x \right) }^{ 2 }$$
Let $$f\left( x \right) $$ be a non-negative continuous function such that the area bounded by the curve $$y=f\left( x \right) $$, $$x$$-axis and the ordinates $$x=\dfrac { \pi  }{ 4 } $$ and $$x=\beta > \dfrac { \pi  }{ 4 } $$ is $$\left( \beta \sin { \beta  } +\dfrac { \pi  }{ 4 } \cos { \beta  } +\sqrt { 2 } \beta  \right) $$. Then, $$f\left( \dfrac { \pi  }{ 2 }  \right) $$ is
  • $$\left( 1-\dfrac { \pi }{ 4 } +\sqrt { 2 } \right) $$
  • $$\left( 1-\dfrac { \pi }{ 4 } -\sqrt { 2 } \right) $$
  • $$\left( \dfrac { \pi }{ 4 } -\sqrt { 2 } +1 \right) $$
  • $$\left( \dfrac { \pi }{ 4 } +\sqrt { 2 } -1 \right) $$
Let $$f(x)=2\tan^{-1}x+\sin^{-1}\left(\displaystyle\frac{2x}{1+x^2}\right)$$. Then
  • $$f'(2)=f'(3)$$
  • $$f'(2)=0$$
  • $$f'(1/2)=16/5$$
  • All of these
The value of a for which the function $$f(x)=\left\{\begin{matrix} \tan^{-1} a-3x^2, & 0<x<1 \\ -6x, & x\geq 1\end{matrix}\right.$$ has a maximum at $$x=1$$, is?
  • $$0$$
  • $$1$$
  • $$2$$
  • None of these
If $$ y^x = 2^x , $$ then $$ \dfrac {dy}{dx} $$ is equal to :
  • $$ \dfrac {y}{x} \log \left( \dfrac {2}{y} \right) $$
  • $$ \dfrac {x}{y} \log \left( \dfrac {2}{y} \right) $$
  • $$ \dfrac {y}{x} \log \left( \dfrac {y}{2} \right) $$
  • $$ \dfrac {x}{y} \log \left( \dfrac {y}{2} \right) $$
  • $$ \dfrac {y}{x} \log \left( 2y \right) $$
If $$y = x - x^{2}$$, then the derivative of $$y^{2} w.r.t. x^{2}$$ is
  • $$2x^{2} + 3x - 1$$
  • $$2x^{2} - 3x + 1$$
  • $$2x^{2} + 3x + 1$$
  • None of these
Find the derivative of $$\sin(2\sin^{-1}x)$$.
  • $$\dfrac{2\cos(2\sin^{-1}x)}{\sqrt{1-x^{2}}}$$
  • $$\dfrac{\cos(2\sin^{-1}x)}{\sqrt{1-x^{2}}}$$
  • $$\dfrac{2\cos(2\cos^{-1}x)}{\sqrt{1-x^{2}}}$$
  • $$-\dfrac{\cos(2\cos^{-1}x)}{\sqrt{1-x^{2}}}$$
Let $$y=\sin^{-1}x$$, then find $$(1-x^{2})y_{2}-xy_{1}$$.
Where $$y_{1}$$ and $$y_{2}$$ denote the first and second order derivatives respectively.
  • $$1$$
  • $$-1$$
  • $$0$$
  • $$2$$
Differentiate $$\cos^{-1}(4x^{3}-3x)$$ w.r.t $$x$$.
  • $$\dfrac{3}{\sqrt{1-x^{2}}}$$
  • $$-\dfrac{3}{\sqrt{1-x^{2}}}$$
  • $$\dfrac{1}{\sqrt{1-x^{2}}}$$
  • $$-\dfrac{1}{\sqrt{1-x}}$$
Find derivative of $$\tan^{-1}\dfrac{\cos x}{1+\sin x}$$.
  • $$\dfrac{1}{2}$$
  • $$-\dfrac{1}{2}$$
  • $$\dfrac{3}{2}$$
  • $$-\dfrac{3}{2}$$
If $$y=\dfrac{1}{2}(\sin^{-1}x)^{2}$$, then find $$(1-x^{2})y_{2}-xy_{1}$$. 
Where $$y_{1}$$ and $$y_{2}$$ denote first and second derivatives of $$y$$ respectively.
  • $$-1$$
  • $$0$$
  • $$1$$
  • $$2$$
The derivative of $$\sin^{-1}x$$ with respect to $$\cos^{-1}\sqrt{1-x^2}$$ is?
  • $$\displaystyle\frac{1}{\sqrt{1-x^2}}$$
  • $$\cos^{-1}x$$
  • $$1$$
  • $$0$$
If $$y=\sec^{-1}\dfrac{\sqrt{x}+1}{\sqrt{x}-1}+\sin^{-1}\dfrac{\sqrt{x}-1}{\sqrt{x}+1}$$, then $$\dfrac{dy}{dx}=$$
  • $$0$$
  • $$1$$
  • $$2$$
  • $$3$$
If $$y=\sin^{-1}(x^{2})$$ then find $$\dfrac{dy}{dx}$$ using first principle.
  • $$\dfrac{2x}{\sqrt{1-x^{4}}}$$
  • $$\dfrac{2}{\sqrt{1-x^{2}}}$$
  • $$\dfrac{x}{\sqrt{1-x^{4}}}$$
  • $$-\dfrac{1}{\sqrt{1-x^{4}}}$$
If $$y=\cos^{-1}(\sqrt{x})$$, then find $$\dfrac{dy}{dx}$$ using first principle.
  • $$-\dfrac{1}{\sqrt{1-x}}$$
  • $$\dfrac{1}{\sqrt{1-x}}$$
  • $$-\dfrac{1}{2\sqrt{x}\sqrt{1-x}}$$
  • $$\dfrac{1}{2\sqrt{x}\sqrt{1-x}}$$
Find derivative of $$\sin^{-1}(x^{2})$$ using first principle.
  • $$\dfrac{2x}{\sqrt{1-x^{2}}}$$
  • $$\dfrac{2x}{\sqrt{1-x}}$$
  • $$\dfrac{2x}{\sqrt{1-x^{4}}}$$
  • $$\dfrac{x}{\sqrt{1-x^{4}}}$$
If $$(f(x))^{g(y)} = e^{f(x) - g(y)}$$ then $$\dfrac {dy}{dx} =$$.
  • $$\dfrac {f^{1}(x)\log f(x)}{g^{1}(y) (1 + \log f(x))^{2}}$$
  • $$\dfrac {f^{1}(x)\log f(x)}{g^{1}(y) (1 + \log f(x))^{3}}$$
  • $$\dfrac {f^{1}(x).\log f(x)}{g^{-1}(y) (1 - \log f(x))^{2}}$$
  • $$\dfrac {f^{1}(x)\log f(x)}{g(y) (1 + \log f(x))^{2}}$$
If $$x + y = \tan^{-1}y$$ and $$y'' =f(y) y'$$ then $$f(y) =$$
  • $$\dfrac {1}{y(1+y^2)}$$
  • $$\dfrac {3}{y(1+y^2)}$$
  • $$\dfrac {2}{y(1+y^2)}$$
  • $$\dfrac {-2}{y(1+y^2)}$$
Let $$f$$ be differentiable $$(x\epsilon R)$$, if $$f(2) = -2$$ and $$f'(x) \geq 2$$ for $$x\epsilon [1, 6]$$, then
  • $$f(6) < 6$$
  • $$f(6) \geq 6$$
  • $$f(6) = 5$$
  • $$f(6) \leq 5$$
Given $$\quad f(x)=\begin{cases} \log _{ a }{ { \left( a\left| \left[ x \right] +\left[ -x \right]  \right|  \right)  }^{ x } } \left( \cfrac { { a }^{ \cfrac { 2 }{ \left( \cfrac { \left[ x \right] +\left[ -x \right]  }{ \left| x \right|  }  \right)  } -5 } }{ 3+{ a }^{ \cfrac { 1 }{ \left|x\right| }  } }  \right) \quad \text{for}\quad \left| x \right| \neq 0;a>1 \\ 0\quad \quad \quad \quad \quad \quad \quad \text{for}\quad x=0\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  \end{cases}$$,
where $$\left[ . \right] $$ represents the integral part function, then;
  • $$f$$ is continuous but not differentiable at $$x=0$$
  • $$f$$ is continuous and differentiable at $$x=0$$
  • The differentiability of $$f$$ at $$x=0$$ depends on the value of $$a$$
  • $$f$$ is continuous and differentiable at $$x=0$$ and for $$a=e$$ only
If $$y = \sin^{-1}\dfrac {1}{2}(\sqrt {1 + x} + \sqrt {1 - x})$$ then $$y' =$$
  • $$\dfrac {1}{2\sqrt {1 - x^{2}}}$$
  • $$\dfrac {-1}{2\sqrt {1 - x^{2}}}$$
  • $$\dfrac {1}{2\sqrt {1 + x^{2}}}$$
  • $$\dfrac {-1}{2\sqrt {1 + x^{2}}}$$
Let $$g: [1, 6]\rightarrow [0, \infty]$$ be a real valued differentiable function satisfying $$g'(x) = \dfrac {2}{x + g(x)}$$ and $$g(1) = 0$$, the maximum value of $$g$$ cannot exceed.
  • $$ln\ 2$$
  • $$ln\ 6$$
  • $$6\ ln\ 2$$
  • $$2\ ln\ 6$$
Let $$f:(-1,1)\rightarrow R$$ be the differentiable function with $$f(0)=-1$$ and $$f'(0)=1$$. 

If $$g(x)={ \left( f(2f(x)+2 \right)  }^{ 2 }$$, then $$g'(0)=$$
  • $$0$$
  • $$-2$$
  • $$4$$
  • $$-4$$
Determine the value of k for which the following function is continuous at $$x=3$$.
$$f(x)=\dfrac{x^2-9}{x-3}, x \neq 3$$

$$f(x)=k, x=3$$
  • 2
  • 4
  • 6
  • 8
Let $$f(x)=4$$ and $$f'(x)=4$$, then $$\displaystyle \lim _{ x\rightarrow 2 }{ \cfrac { xf(2)-2f(x) }{ x-2 }  } $$
  • $$2$$
  • $$-2$$
  • $$-4$$
  • $$4$$
0:0:1


Answered Not Answered Not Visited Correct : 0 Incorrect : 0

Practice Class 12 Commerce Maths Quiz Questions and Answers