CBSE Questions for Class 12 Commerce Maths Continuity And Differentiability Quiz 1 - MCQExams.com

If $$\mathrm{f}(\mathrm{x})$$ is a differentiable function and $$\mathrm{g}(\mathrm{x})$$ is a double differentiable function such that $$|\mathrm{f}(\mathrm{x})|\leq 1$$ and $$\mathrm{f}'(\mathrm{x})=\mathrm{g}(\mathrm{x})$$. If $$\mathrm{f}^{2}(0)+\mathrm{g}^{2}(0)=9$$such that there exists some $$\mathrm{c}\in(-3, 3)$$ such that $$\mathrm{g}(\mathrm{c}).\ \mathrm{g}''(\mathrm{c})<0$$, True or false
  • True
  • False
Let f be a polynomial function such that $$f(3x)=f'(x)\cdot f''(x)$$, for all $$x\in R$$. Then.
  • $$f''(2)-f'(2)=0$$
  • $$f(2)+f'(2)=28$$
  • $$f''(2)-f(2)=4$$
  • $$f(2)-f'(2)+f''(2)=10$$
If $$y=\left[\displaystyle x+\sqrt{x^2-1}\right]^{15}+\left[\displaystyle x-\sqrt{x^2-1}\right]^{15}$$, then $$(x^2-1)\displaystyle\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}$$ is equal to.
  • $$125y$$
  • $$225y^2$$
  • $$225y$$
  • $$224y^2$$
Let $$f : R \rightarrow R$$ and $$g : R \rightarrow R$$ be functions satisfying $$f(x + y) = f(x) + f(y) + f(x)f(y)$$ and $$f(x) = xg(x)$$ for all $$x, y \in R$$. If $$\underset{x \rightarrow 0}{\lim} g(x) = 1$$, then which of the following statements is/are TRUE?
  • f is differentiable at every $$x \in R$$
  • If $$g(0) = 1$$, then g is differentiable at every $$x \in R$$
  • The derivative $$f'\left( 1 \right) $$ is equal to $$ 1$$
  • The derivative $${ f }^{ \prime  }\left( 0 \right) $$ is equal to $$1$$
For the curve $$x = t^2 - 1, y = t^2 - t$$, tangent is parallel to $$x$$ - axis where,
  • $$t = 0$$
  • $$t=\dfrac{1}{\sqrt{3}}$$
  • $$t=\dfrac{1}{2}$$
  • $$t=-\dfrac{1}{\sqrt{3}}$$
$$\displaystyle \frac{d}{dx}(\tan ^{-1}x)$$
  • $$ \displaystyle \frac{1}{1+x^{2}}.$$
  • $$\displaystyle \frac{-1}{1+x^{2}}.$$
  • $$\displaystyle \frac{-1}{1-x^{2}}.$$
  • $$\displaystyle \frac{1}{1-x^{2}}.$$
Say true or false.
Every continuous function is always differentiable.
  • True
  • False
$$\displaystyle \frac{d(sin^{-1}x)}{dx}$$
  • $$\displaystyle \frac{1}{\sqrt{\left ( 1-x^{2} \right )}}$$
  • $$\displaystyle \frac{1}{\sqrt{\left ( 1-x^{4} \right )}}$$
  • $$\displaystyle \frac{1}{\sqrt{\left ( 1-x^{3} \right )}}$$
  • $$\displaystyle \frac{1}{\sqrt{\left ( 1+x^{2} \right )}}$$
If $$y$$ is expressed in terms of a variable $$x$$ as $$y = f(x)$$, then $$y$$ is called
  • Explicit function
  • Implicit function
  • Linear function
  • Identity function
The function $$f(x)=\dfrac{1-\sin x+\cos x}{1+\sin x+\cos x}$$ is not defined at $$x=\pi$$. The value of $$f(\pi)$$ so that $$f(x)$$ is continuous at $$x=\pi$$ is
  • $$-\dfrac{1}{2}$$
  • $$\dfrac{1}{2}$$
  • $$-1$$
  • $$1$$
If the tangent to the curve $$x=a \, (\theta + \sin \, \theta), y=a (1+ \cos \,\theta)$$ at $$ \theta=\dfrac{\pi}{3}$$ makes an angle $$\alpha (0 \leq\alpha < \pi)$$ with x-axis, then $$\alpha$$ =
  • $$\dfrac{\pi}{3}$$
  • $$\dfrac{2 \pi}{3}$$
  • $$\dfrac{\pi}{6}$$
  • $$\dfrac{5 \pi}{6}$$
If $$f(x)=\begin{cases}
ax &a<1&\\
ax^2+bx+2 &a\ge 1&.
\end{cases}$$
Then the values of $$a$$, $$b$$ for which $$f(x)$$ is differentiable, are
  • $$a=\large{\frac{3}{4}}$$, $$b=\large{\frac{1}{4}}$$
  • $$a=2$$, $$b=-2$$
  • $$a=\large{\frac{3}{2}}$$, $$b=\large{\frac{1}{4}}$$
  • $$a=\large{\frac{3}{4}}$$, $$b=-2$$
If $$f(x)=\dfrac{1}{2}\left [ \left | \sin x \right |+\sin x \right ],\ 0 < x \leq 2\pi$$, then $$f$$ is
  • Increasing in $$\left ( \dfrac{\pi}{2},\dfrac{3\pi}{2} \right )$$
  • Decreasing in $$\left ( 0,\dfrac{\pi}{2} \right )$$ and increasing in $$\left ( \dfrac{\pi}{2},\pi \right )$$
  • Increasing in $$\left ( 0,\dfrac{\pi}{2} \right )$$ and decreasing in $$\left ( \dfrac{\pi}{2},\pi \right )$$
  • Increasing in $$\left ( 0,\dfrac{\pi}{4} \right )$$ and decreasing in $$\left ( \dfrac{\pi}{4},\pi \right )$$
If $$x=a\cos ^{ 4 }{ t } ,y=b\ cosec^{ 4 }{ t } $$, then $$\cfrac { dx }{ dy } $$ at $$t=\cfrac { 3\pi  }{ 4 } $$
  • $$\cfrac{-b}{a}$$
  • $$\cfrac{b}{a}$$
  • $$\cfrac{a}{b}$$
  • $$\cfrac{-a}{16b}$$
Let $$y=\sqrt{(\sin x+\sin 2x+\sin 3x)^2+(\cos x+\cos 2x+\cos 3x)^2}$$ then which of  the following(s) is correct?
  • $$\dfrac{dy}{dx}$$ when $$x=\dfrac{\pi}{2}$$ is $$-2$$
  • Value of y when $$x=\dfrac{\pi}{5}$$ is $$\dfrac{3+\sqrt{5}}{2}$$
  • Value of y when $$x=\dfrac{\pi}{12}$$ is $$\dfrac{\sqrt{1}+\sqrt{2}+\sqrt{3}}{2}$$
  • y simplifies to $$(1+2\cos x)$$ in $$[0, \pi]$$
If $$y=\tan^{-1}\left(\dfrac{2^{x+1}}{1+2^{2x}}\right) $$ then $$\dfrac{dy}{dx}$$ at $$x=0$$ is
  • $$\dfrac{1}{10}log2$$
  • $$\dfrac{1}{5}log2$$
  • $$-\dfrac{1}{10}log2$$
  • $$log2$$
$$\displaystyle \int _0^1 \dfrac {e^x}{1+e^{2x}}dx$$
  • $$\tan ^{-1}e-\dfrac \pi 4$$
  • $$\tan ^{-1}e+\dfrac \pi 4$$
  • $$\tan e-\dfrac \pi 4$$
  • None of these
If $$y = \tan ^ { - 1 } \left( \cot \left( \dfrac { \pi } { 2 } - x \right) \right) ,$$ then $$\dfrac { d y } { d x } =$$
  • $$1$$
  • $$-1$$
  • $$0$$
  • $$\dfrac { 1 } { 2 }$$
If $$\displaystyle\int \dfrac{f(x)dx}{\log(\sin x)}$$ $$=\log (\log(\sin x))$$ then $$f(x) =$$
  • $$sinx$$
  • $$cosx$$
  • $$\log(sinx)$$
  • $$cotx$$
The set of points of continuity of the following $$\sqrt { \dfrac { 1 }{ 2 } -cos^{ 2 }x } $$ contains in the interval 
  • $$\left[ \dfrac { \pi }{ 4 } ,\dfrac { 3\pi }{ 4 } \right] $$
  • $$\left[ \dfrac { 5\pi }{ 4 } ,\dfrac { 7\pi }{ 4 } \right] $$
  • $$\left[ \dfrac { 21\pi }{ 4 } ,\dfrac { 23\pi }{ 4 } \right] $$
  • All above the
If $$y=\log \sin x$$ find $$x$$ if $$y=0$$
  • $$\dfrac {\pi}2$$
  • $$\pi $$
  • $${\dfrac {\pi}3}$$
  • $$\dfrac {-\pi}2$$
If $$f ( x ) =  \tan x$$ and $$f$$ is inverse of $$g ,$$ then $$g ^ { \prime } ( x )$$is equal to
  • $$\cot x$$
  • $$\dfrac{1}{1+x^2}$$
  • $$\dfrac{1}{1-x^2}$$ 
  • $$\tan x$$
Find the values of a and b so that the function $$f(x)=\left\{\begin{matrix} x^2+3x+a, & if & x\leq 1\\ bx+2, & if & x > 1\end{matrix}\right.$$ is differentiable at each $$x\in R$$.
  • $$a=1,b=3$$
  • $$a=5,b=3$$
  • $$a=3,b=5$$
  • $$a=3,b=1$$
The set of points where the functions f given by $$f(x)=|x-3|\cos x$$ differentiable is
  • R
  • R-{$$3$$}
  • $$(0,\infty)$$
  • None of these
Let $$f(x)$$ be differentiable function such that $$f\left (\displaystyle \frac{x+y}{1-xy}\right)=f(x)+f(y)\forall x$$ and $$y$$. lf $$ \displaystyle \lim_{x \rightarrow 0}\frac{f(x)}{x}=\frac{1}{3}$$, then $$f(1)$$ equals
  • $$\displaystyle \frac{\pi}{4}$$
  • $$\displaystyle \frac{\pi}{12}$$
  • $$\displaystyle \frac{\pi}{6}$$
  • $$\displaystyle \frac{\pi}{3}$$

$$\displaystyle \mathrm{A}:\dfrac{d}{dx}(\sin x)\ at\ x=\frac{\pi}{2}$$

$$\displaystyle \mathrm{B}:\dfrac{d}{dx}(\tan^{-1}{x})$$ at $$ {x}=1$$

$$\displaystyle \mathrm{C}:\dfrac{d}{dx}(\mathrm{e}^{x})$$ at $${x}=0$$

$$\mathrm{D}:\dfrac{d}{dx}(x^{x})\ at\ x=e$$

Arrangement of the above values in the increasing order of the magnitude
  • B, C, A, D
  • D, A, B, C
  • D, B, C, A
  • A, B, C, D
$${A} :$$ If $$x = ct, y = \dfrac{c}{t}$$, then at $$t=1,  \dfrac{dy}{dx} =$$
$$B:$$ If $$x=3\cos\theta -\cos^{3}\theta, y=3\sin\theta-\sin^{3}\theta$$, then at $$\theta = \dfrac{\pi}{3}, \dfrac{dy}{dx} = $$
$$C:$$ If $$x = a\left(t+ \dfrac{1}{t}\right), y = a\left(t-\dfrac{1}{t}\right)$$, then at $$t =2, \dfrac{dy}{dx} = $$
$$D:$$ Derivative of $$\log(\sec x)$$ with respect to $$\tan x$$ at $$x = \dfrac{\pi}{4}$$ is$$\\$$
Arrangement of the above values in the increasing order is
  • $$C, D, A, B$$
  • $$C, A, D, B$$
  • $$A, B, D, C$$
  • $$B, D, A, C$$
If $$t\left( 1+{ x }^{ 2 } \right) =x$$ and $${ x }^{ 2 }+{ t }^{ 2 }=y,$$ then at $$x=2,$$ the value of $$\displaystyle\frac{dy}{dx}$$
  • $$\displaystyle\frac{488}{125}$$
  • $$\displaystyle\frac{88}{125}$$
  • $$\displaystyle\frac{101}{125}$$
  • None of these
If $$\mathrm{x}=\mathrm{a}\mathrm{t}^{2},\ \mathrm{y}=2\mathrm{a}\mathrm{t}$$, then $$\displaystyle \frac{\mathrm{d}^{2}\mathrm{y}}{\mathrm{d}\mathrm{x}^{2}}$$ is
  • $$-\dfrac{1}{t^{2}}$$
  • $$-\dfrac{1}{2at^{2}}$$
  • $$-\dfrac{1}{t^{3}}$$
  • $$-\dfrac{1}{2at^{3}}$$

 $$\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{x}}[l \mathrm{o}\mathrm{g}(\mathrm{a}\mathrm{x})^{\mathrm{x}}]$$, where $$a$$ is a constant, is equal to
  • $$1$$
  • $$\log {ax}$$
  • $$1/a$$
  • $$\log {(ax)+1}$$
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