CBSE Questions for Class 12 Commerce Maths Continuity And Differentiability Quiz 4 - MCQExams.com

Consider the function
$$f(x)=\begin{cases}-2\sin x & if & x\le -\dfrac{\pi}{2} \\ A\sin x+B & if & -\dfrac{\pi}{2} < x < \dfrac{\pi}{2} \\ \cos x & if & x \ge \dfrac{\pi}{2}\end{cases}$$
which is continuous everywhere.
The value of A is
  • 1
  • 0
  • -1
  • -2
If $${ x }^{ m }+{ y }^{ m }=1$$ such that $$\cfrac { dy }{ dx } =-\cfrac { x }{ y } $$, then what should be the value of $$m$$?
  • $$0$$
  • $$1$$
  • $$2$$
  • None of the above
If $$s=\sqrt{t^2+1}$$, then $$\dfrac{d^2s}{dt^2}$$ is equal to
  • $$\dfrac{1}{s}$$
  • $$\dfrac{1}{s^2}$$
  • $$\dfrac{1}{s^3}$$
  • $$\dfrac{1}{s^4}$$
Consider the function
$$f(x)=\begin{cases}-2\sin x & if & x\le -\dfrac{\pi}{2} \\ A\sin x+B & if & -\dfrac{\pi}{2} < x < \dfrac{\pi}{2} \\ \cos x & if & x \ge \dfrac{\pi}{2}\end{cases}$$
which is continuous everywhere.
The value of B is
  • 1
  • 0
  • -1
  • -2
If $$\sec \left (\dfrac {x + y}{x - y}\right ) = a$$, then $$\dfrac {dy}{dx}$$ is.
  • $$\dfrac {x}{y}$$
  • $$\dfrac {y}{x}$$
  • $$y$$
  • $$x$$
If $$g$$ is the inverse function of $$f$$ and $$f'(x) = \dfrac {1}{1 + x^{n}}$$, then $$g'(x)$$ is equal to.
  • $$1 + [g(x)]^n$$
  • $$1 - g(x)$$
  • $$1 + g(x)$$
  • $$-g(x)^{n}$$
If $$y = x\tan y$$, then $$\dfrac {dy}{dx}$$ is equal to.
  • $$\dfrac {\tan y}{x - x^{2} - y^{2}}$$
  • $$\dfrac {y}{x - x^{2} - y^{2}}$$
  • $$\dfrac {\tan y}{y - x}$$
  • $$\dfrac {\tan x}{x - y^{2}}$$
If $$\log _{ 10 }{ \left( \cfrac { { x }^{ 2 }-{ y }^{ 2 } }{ { x }^{ 2 }+{ y }^{ 2 } }  \right)  } =2$$, then $$\cfrac { dy }{ dx } =$$............
  • $$-\cfrac { 99x }{ 101y } $$
  • $$\cfrac { 99x }{ 101y } $$
  • $$-\cfrac { 99y }{ 101x } $$
  • $$\cfrac { 99y }{ 101x } $$
For what value of $$k$$, the function defined by $$f(x)=\cfrac { \log { (1+2x) } \sin { x }  }{ { x }^{ 2 } } $$ for $$x\ne 0$$
                                                                                     $$=k$$ for $$x=0$$
is continuous at $$x=0$$?
  • $$2$$
  • $$\cfrac { 1 }{ 2 } $$
  • $$\cfrac { \pi }{ 90 } $$
  • $$\cfrac { 90 }{ \pi } $$
If $$f(x) = be^{ax} + ae^{bx}$$, then $$f'' (0) =$$
  • $$0$$
  • $$2ab$$
  • $$ab(a + b)$$
  • $$ab$$
$$f(x) =\begin{cases}  2a - x \text{,  for  }-a < x < a \\ 3x-2a \text{,  for } x\ge a\end{cases}$$
Then which of the following is true?
  • $$f(x)$$ is discontinuous at $$x = a$$
  • $$f(x)$$ is not differentiable at $$x=a$$
  • $$f(x)$$ is differentiable at all $$x\geq a$$
  • $$f(x)$$ is continuous at all $$x < a$$
Derivative of $$\tan ^{ -1 }{ \left( \cfrac { x }{ \sqrt { 1-{ x }^{ 2 } }  }  \right)  } $$ with respect to $$\sin ^{ -1 }{ \left( 3x-4{ x }^{ 3 } \right)  } $$ is
  • $$\cfrac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } $$
  • $$\cfrac { 3 }{ \sqrt { 1-{ x }^{ 2 } } } $$
  • $$3$$
  • $$\cfrac { 1 }{ 3 } $$
If $$x^{y} = e^{x - y}$$, then $$\dfrac {dy}{dx}$$ is equal to.
  • $$\dfrac {\log x}{1 + \log x}$$
  • $$\dfrac {\log x}{1 - \log x}$$
  • $$\dfrac {\log x}{(1 + \log x)^{2}}$$
  • $$\dfrac {y\log x}{x(1 + \log x)^{2}}$$
If $$x=\sec { \theta  } -\cos { \theta  } $$ and $$y=\sec ^{ n }{ \theta  } -\cos ^{ n }{ \theta  } $$, then $${ \left( \dfrac { dy }{ dx }  \right)  }^{ 2 }$$ is
  • $$\dfrac { { n }^{ 2 }\left( { y }^{ 2 }+4 \right) }{ { x }^{ 2 }+4 } $$
  • $$\dfrac { { n }^{ 2 }\left( { y }^{ 2 }-4 \right) }{ { x }^{ 2 } } $$
  • $$n\dfrac { \left( { y }^{ 2 }-4 \right) }{ { x }^{ 2 }-4 } $$
  • $${ \left( \dfrac { ny }{ x } \right) }^{ 2 }-4$$
If $$f(x) =\begin{cases} x, & \text{for } x\leq 0 \\ 0, & \text{for } x>0\end{cases}$$ 
then $$f(x)$$ at $$x = 0$$ is
  • Continuous but not differentiable
  • Not continuous but differentiable
  • Continuous and differentiable
  • Not continuous and not differentiable
If $$f(x) =\begin{cases} \log (\sec^{2} x)^{\cot^{2}x}, & \text{for } x\neq 0 \\ K, & x=0\end{cases}$$ 
is continuous at $$x = 0$$ then $$K$$ is
  • $$e^{-1}$$
  • $$1$$
  • $$e$$
  • $$0$$
If $$f(x)=[x \sin \pi x]$$, then which of the following is incorrect?
  • $$f(x)$$ is continuous at $$x=0$$
  • $$f(x)$$ is continuous in $$(-1, 0)$$
  • $$f(x)$$ is differentiable at $$x=1$$
  • $$f(x)$$ is differentiable in $$(-1, 1)$$
Which of the following functions is differentiable at $$x = 0$$
  • $$\cos (|x|) + |x|$$
  • $$\cos (|x|) - |x|$$
  • $$\sin (|x|) + |x|$$
  • $$\sin (|x|) - |x|$$
If $$2^x + 2^y = 2^{x + y}$$, then $$\dfrac{dy}{dx}$$ is equal to
  • $$\dfrac{2^x + 2^y}{2^x - 2^y}$$
  • $$\dfrac{2^x + 2^y}{1 + 2^{x + y}}$$
  • $$2^{x - y} \left( \dfrac{2^y - 1}{1 - 2^x} \right )$$
  • $$\dfrac{2^{x + y} - 2^x}{2^y}$$
The value of $$k$$ for which the function $$f\left( x \right) =\begin{cases} \dfrac { 1-\cos { 4x }  }{ 8{ x }^{ 2 } } &,x\neq 0 \\ k &,x=0 \end{cases}$$ is continuous at $$x=0$$, is
  • $$k=0$$
  • $$k=1$$
  • $$k=-1$$
  • None of the above
Let $$f\left( x \right) =\left\{ \begin{matrix} { x }^{ n }\sin { \frac { 1 }{ x }  } ,x\neq 0 \\ \quad \quad 0,x=0 \end{matrix} \right\} $$. Then, $$f\left( x \right)$$ is continuous but not differentiable at $$x=0$$, if
  • $$n\in \left( 0,1 \right) $$
  • $$n\in \left[ 1,\infty \right) $$
  • $$n\in \left( -\infty ,0 \right) $$
  • $$n=0$$
If $$x = a \left (\cos t + \log \tan \dfrac {t}{2}\right ), y = a\sin t$$, then $$\dfrac {dy}{dx} =$$
  • $$\tan t$$
  • $$\cot t$$
  • $$-\cot t$$
  • $$-\tan t$$
If $$u = \tan^{-1}\left (\dfrac {\sqrt {1 - x^{2}} - 1}{x}\right )$$ and $$v= \sin^{-1} x$$, then $$\dfrac {du}{dv}$$ is equal to
  • $$\sqrt {1 - x^{2}}$$
  • $$-\dfrac {1}{2}$$
  • $$1$$
  • $$-x$$
  • $$-2$$
If $${ 2 }^{ x }+{ 2 }^{ y }={ 2 }^{ x+y }$$, then the value of $$\cfrac { dy }{ dx } $$ at $$(1,1)$$ is equal to
  • $$-2$$
  • $$-1$$
  • $$0$$
  • $$1$$
  • $$2$$
If $$x = a \cos^3 \theta$$ and $$y = a\sin^3 \theta$$, then $$1 + \left( \dfrac{dy}{dx} \right )^2$$ is
  • $$\tan \theta$$
  • $$\tan^2 \theta$$
  • $$1$$
  • $$\sec^2 \theta$$
  • $$\sec \theta$$
If $$y = 4x - 5$$ is a tangent to the curve $$y^{2} = px^{3} + q$$ at $$(2, 3)$$, then $$(p + q)$$ is equal to
  • $$-5$$
  • $$5$$
  • $$-9$$
  • $$9$$
  • $$0$$
If $$xe^{xy} + ye^{-xy} = \sin^{2}x$$, then $$\dfrac {dy}{dx}$$ at $$x = 0$$ is
  • $$2y^{2} - 1$$
  • $$2y$$
  • $$y^{2} - y$$
  • $$y^{2} + 1$$
  • $$y^{2} - 1$$
If $$ u = 2 (t - \sin t ) $$ and $$ v = 2 (1- \cos t), $$ then $$ \dfrac {dv}{du} $$ at $$ t = \dfrac {2 \pi}{3} $$ is equal to :
  • $$ \sqrt3$$
  • $$ - \sqrt3$$
  • $$ 2 \sqrt3$$
  • $$ \dfrac {2} {\sqrt3} $$
  • $$ \dfrac {1} {\sqrt3} $$
If $$x\sin (a + y) + \sin a\cos (a + y) = 0$$, then $$\dfrac {dy}{dx}$$ is equal to
  • $$\dfrac {\sin^{2}(a + y)}{\sin a}$$
  • $$\dfrac {\cos^{2}(a + y)}{\cos a}$$
  • $$\dfrac {\sin^{2}(a + y)}{\cos a}$$
  • $$\dfrac {\cos^{2}(a + y)}{\sin a}$$
If $$s=\sec ^{ -1 }{ \left( \cfrac { 1 }{ 2{ x }^{ 2 }-1 }  \right)  } $$ and $$t=\sqrt { 1-{ x }^{ 2 } } $$, then $$\cfrac { ds }{ dt } $$ at $$x=\cfrac { 1 }{ 2 }$$ is
  • $$1$$
  • $$2$$
  • $$-2$$
  • $$4$$
  • $$-4$$
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