CBSE Questions for Class 12 Commerce Maths Continuity And Differentiability Quiz 3 - MCQExams.com

If $$f (x) =\displaystyle |\log _{10}x|$$ then at $$x = 1 $$
  • f is not continuous
  • f is continuous but not differentiable
  • f is differentiable
  • the derivative is 1
If $$\sqrt { x+y } +\sqrt { y-x } =c,$$ then $$\displaystyle \frac { { d }^{ 2 }x }{ d{ y }^{ 2 } } $$ is equal to
  • $$\displaystyle \frac { 2 }{ c } $$
  • $$\displaystyle-\frac { 2 }{ { c }^{ 2 } } $$
  • $$\displaystyle\frac { 2 }{ { c }^{ 2 } } $$
  • none of these
If $$f(x)=\begin{cases} a{ x }^{ 2 }-b,\quad if\quad \left| x \right| <-1 \\ -\cfrac { 1 }{ \left| x \right|  } if\quad \quad \left| x \right| \ge -1 \end{cases}$$ is differential at $$x=1$$. Find the values of $$a$$ and $$b$$
  • $$a=1/2$$; $$b=3/2$$
  • $$a=1/2$$; $$b=-3/2$$
  • $$a=-1/2$$; $$b=3/2$$
  • $$a=-1/2$$; $$b=-3/2$$
Find $$ \displaystyle  \frac{dy}{dx}$$ if $$ y=x^{x} $$
  • $$ \displaystyle x^{x}\left ( lnx+1 \right ) $$
  • $$ \displaystyle x^{x}\left ( lnx-1 \right ) $$
  • $$ \displaystyle x .x^{x-1}$$
  • $$ \displaystyle x^{x-1}\left ( lnx+1 \right ) $$

 Find $$ \displaystyle \frac { dy }{ dx }$$ at $$t =\displaystyle \frac { \pi  }{ 4 }$$ if $$y =\displaystyle  \cos ^{ 4 }{ t } $$  & $$x =\displaystyle \sin ^{ 4 }{ t } $$.


  • 1
  • 0
  • -1
  • 4

Differentiate $$ \displaystyle { x }^{ \ln x }$$ with respect to $$\ln x$$.


  • 2$$ \displaystyle \left( { x }^{ lnx } \right) \left( lnx \right)$$
  • $$ \displaystyle -\left( { lnx }^{ lnx } \right) \left( lnx \right)$$
  • $$ \displaystyle \left( { x }^{ lnx } \right) \left( lnx \right)/x$$
  • $$ \displaystyle -\left( { lnx }^{ lnx } \right) \left( lnx \right)/x$$
$$\displaystyle \frac{d}{dx}\left ( x^{\log x} \right )$$ is equal to
  • $$\displaystyle 2x^{\log x-1}\log x$$
  • $$\displaystyle x^{\log x-1}$$
  • $$\dfrac 23 (\log x)$$
  • $$\displaystyle x^{\log x-1}.\log x$$
If $$\displaystyle y= \frac {\sqrt[3]{1+3x}\sqrt[4]{1+4x}\sqrt[5]{1+5x}}{\sqrt[7]{1+7x}\sqrt[8]{1+8x}}$$, then $$y'(0)$$ is equal to
  • $$-1$$
  • $$1$$
  • $$2$$
  • Non existant
If $$f(x)$$ and $$g(x)$$ are both continuous at $$x= c$$ then which of the following is/are always continuous at $$x=c$$?
  • $$f(x) + g(x)$$
  • $$(f(x) - g(x))\times f(x)$$
  • $$g(x)\times f(x)$$
  • $$\dfrac{f(x) - g(x)}{g(x)}$$
Derivative of sin x w.r.t. cos x is
  • cos x
  • cot x
  • - cot x
  • tan x
If $$y=\tan ^{ -1 }{ \sqrt { \dfrac { 1-\sin { x }  }{ 1+\sin { x }  }  }  } $$, then the value of $$\dfrac { dy }{ dx } $$ at $$x=\dfrac { \pi  }{ 6 } $$ is
  • $$-\dfrac { 1 }{ 2 } $$
  • $$\dfrac { 1 }{ 2 } $$
  • $$1$$
  • $$-1$$
If $$x=a\, cos^3\theta$$ and $$y=a\, sin^3\theta$$, then $$\sqrt{1+\left(\dfrac{dy}{dx}\right)^2}$$ is equal to
  • $$|sec \,\theta |$$
  • $$sec^2\theta$$
  • $$sec\, \theta$$
  • $$| tan\, \theta|$$
If $$x=\cos^3\theta $$ and $$y=\sin^3\theta$$, then $$1+\left(\displaystyle\frac{dy}{dx}\right)^2$$ is equal to:
  • $$\tan^2\theta$$
  • $$\cot^2\theta$$
  • $$\sec^2\theta$$
  • $$\text{cosec}^2\theta$$
If $$y=1-\cos { \theta  } ,x=1-\sin { \theta  } $$, then $$\cfrac { dy }{ dx } $$ at $$\theta =\cfrac { \pi  }{ 4 } $$ is
  • $$-1$$
  • $$1$$
  • $$\cfrac{1}{2}$$
  • $$\cfrac{1}{\sqrt{2}}$$
If $$y = (1+x) (1+x^2)(1+x^4)......(1+x^{2n})$$ then the value of $$\begin{pmatrix}\dfrac{dy}{dx}\end{pmatrix}$$ at $$x=0$$ is
  • $$0$$
  • $$-1$$
  • $$1$$
  • $$2$$
The differential coefficient of $$\log _{ 10 }{ x } $$ with respect to $$\log _{ x }{ 10 } $$ is
  • $$1$$
  • $$-{ \left( \log _{ 10 }{ x } \right) }^{ 2 }\quad $$
  • $${ \left( \log _{ x }{ 10} \right) }^{ 2 }\quad $$
  • $$\cfrac { { x }^{ 2 } }{ 100 } $$
The function represented by  the following graph is.

572668_db4e79fb2abd478fa7332735dca6ad0f.png
  • Differentiable but not continuous $$x=1$$
  • Neither continuous nor Differentiable at $$x=1$$
  • Continuous but not Differentiable at $$x=1$$
  • Continuous but Differentiable at $$x=1$$
If $$x = a \cos^{3}\theta, y = a\sin^{3}\theta$$, then $$1 + \left (\dfrac {dy}{dx}\right )^{2}$$ is _____
  • $$\tan \theta$$
  • $$\tan^{2}\theta$$
  • $$\sec^{2}\theta$$
  • $$1$$
If $${ x }^{ y }={ e }^{ x-y }$$ then $$\cfrac { dy }{ dx } $$ is equal to
  • $$\cfrac { \log { x } }{ \log { (x-y) } } $$
  • $$\cfrac { { e }^{ x } }{ { x }^{ x-y } } $$
  • $$\cfrac { \log { x } }{ { (1+\log { x } ) }^{ 2 } } $$
  • $$\cfrac { 1 }{ y } -\cfrac { 1 }{ x-y } $$
  • $$\dfrac{y(x -y)}{x^2}$$
If $$x = ct$$ and $$y = \dfrac {c}{t}$$, find $$\dfrac {dy}{dx}$$ at $$t = 2$$
  • $$\dfrac {1}{4}$$
  • $$4$$
  • $$-\dfrac {1}{4}$$
  • $$0$$
Given a function $$f(x) = \left\{\begin{matrix}-1 & if & x \leq 0\\ ax + b & if & 0 < x < 1\\ 1 & if & x \geq 1\end{matrix}\right.$$ where $$a, b$$ are constants. The function is continuous everywhere.
What is the value of $$a$$?
  • $$-1$$
  • $$0$$
  • $$1$$
  • $$2$$
Consider the parametric equation $$x = \cfrac {a(1 - t^{2})}{1 + t^{2}}, y = \cfrac {2at}{1 + t^{2}}$$.
What is $$\cfrac {dy}{dx}$$ equal to?
  • $$\dfrac {y}{x}$$
  • $$-\dfrac {y}{x}$$
  • $$\dfrac {x}{y}$$
  • $$-\dfrac {x}{y}$$
Consider the function $$f(x) = \left\{\begin{matrix}\dfrac {\alpha \cos x}{\pi - 2x} & if & x\neq \dfrac {\pi}{2}\\ 3 & if & x = \dfrac {\pi}{2}\end{matrix}\right.$$
which is continuous at $$x = \dfrac {\pi}{2}$$, where $$\alpha$$ is a constant.
What is the value of $$\alpha$$?
  • $$6$$
  • $$3$$
  • $$2$$
  • $$1$$
If y = cos t and x = sin t, then what is $$\dfrac{dy}{dx}$$ equal to ?
  • $$xy$$
  • $$\dfrac{x}{y}$$
  • $$-\dfrac{y}{x}$$
  • $$-\dfrac{x}{y}$$
Consider the function
$$f(x)=\begin{cases}ax-2 & for & -2 < x < -1 \\ -1 & for & -1\le x\le 1 \\ a+2(x-1)^2 & for & 1 < x < 2\end{cases}$$
What is the value of a for which $$f(x)$$ is continuous at $$x=-1$$ and $$x=1$$?
  • -1
  • 1
  • 0
  • 2
Consider the function $$f(x)=\left\{\begin{matrix} x^2-5, & x\leq 3\\ \sqrt{x+13}, & x > 3\end{matrix}\right.$$.

Consider the following statements.
$$1$$. The function is discontinuous at $$x=3$$.
$$2$$. The function is not differentiable at $$x=0$$.
Which of the statements is$$/$$ are correct?
  • $$1$$ only
  • $$2$$ only
  • Both $$1$$ and $$2$$
  • Neither $$1$$ nor $$2$$
If $$y={\cot}^{-1}\begin{bmatrix}\dfrac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\end{bmatrix}$$, where $$0 < x < \dfrac{\pi}{2}$$, then $$\dfrac{dy}{dx}$$ is equal to
  • $$-\dfrac{1}{2}$$
  • 2
  • $$\sin x + \cos x$$
  • $$\sin x - \cos x$$
If $$x^ay^b=(x-y)^{a+b}$$, then the value of $$\dfrac{dy}{dx}-\dfrac{y}{x}$$ is equal to
  • $$\dfrac{a}{b}$$
  • $$\dfrac{b}{a}$$
  • 1
  • 0
Consider the following in respect of the function $$f(x) = | x- 3 |$$ :
$$f(x)$$ is continuous at $$x = 3$$
$$f(x)$$ is differentiable at $$x = 0$$.
Which of the above statements is/are correct ?
  • 1 only
  • 2 only
  • Both 1 and 2
  • Neither 1 nor 2
If $$f(x)=\sqrt{25-x^2}$$, then what is $$\displaystyle \lim_{ x\rightarrow 1 } \dfrac{f(x)-f(1)}{x-1}$$ equal to?
  • $$\dfrac{1}{5}$$
  • $$\dfrac{1}{24}$$
  • $$\sqrt{24}$$
  • $$-\dfrac{1}{\sqrt{24}}$$
0:0:1


Answered Not Answered Not Visited Correct : 0 Incorrect : 0

Practice Class 12 Commerce Maths Quiz Questions and Answers