CBSE Questions for Class 12 Commerce Maths Continuity And Differentiability Quiz 10 - MCQExams.com

If $$y=\tan^{-1}\left(\dfrac{\sqrt{a+\sqrt{x}}}{1-\sqrt{ax}}\right)$$ then $$\dfrac{dy}{dx}=?$$
  • $$\dfrac{1}{(1+x)}$$
  • $$\dfrac{1}{\sqrt{x}(1+x)}$$
  • $$\dfrac{2}{\sqrt{x}(1+x)}$$
  • $$\dfrac{1}{2\sqrt{x}(1+x)}$$
If $$y=\sec^{-1}\left(\dfrac{x^{2}+1}{x^{2}-1}\right)$$ then $$\dfrac{dy}{dx}=?$$
  • $$\dfrac{-2}{(1+x^{2})}$$
  • $$\dfrac{2}{(1+x^{2})}$$
  • $$\dfrac{-1}{(1-x^{2})}$$
  • $$none\ of\ these$$
 If $$y=\tan ^{-1} \sqrt{\dfrac{x+1}{x-1}}, $$ then $$ \dfrac{d y}{d x} $$ is
  • $$\dfrac{-1}{2|x| \sqrt{x^{2}-1}} $$
  • $$ \dfrac{-1}{2 x \sqrt{x^{2}-1}} $$
  • $$ \dfrac{1}{2 x \sqrt{x^{2}-1}} $$
  • $$ None \ of \ these$$
$$ f(x)=x^{2}+x g^{\prime}(1)+g^{\prime \prime}(2) $$ and $$ g(x)=f(1) x^{2}+x f^{\prime}(x)+f^{\prime \prime}(x) $$
The value of $$ f(3) $$ is
  • 1
  • 0
  • -1
  • -2
The function given by y=$$ \left | x-1 \right | $$ is differentiable function and $$ f(1/n) $$ = 0 $$ \forall n\geq 1 $$ and n\epsilon I $$, then
  • f(x)=0, $$ x\epsilon (0,1] $$
  • f(0)=0, f'(0) =0
  • f(0)= 0=f'(0), $$ x\epsilon (0,1] $$
  • f(0)=0 and f'(0) need not to be zero
If $$P(1)=0$$ and $$\dfrac {dP(x)}{dx} > P(x)$$ for all $$x \ge 1$$, then
  • $$P(x) > 0\forall x > 1$$
  • $$P(x)$$ is a constant function
  • $$P(x) < 0\forall x > 1$$
  • None of these
The function $$ f(x)=\dfrac{x}{1+\left | x \right |} $$ is differentiable 
  • only at non-integer points
  • everywhere
  • only at $$x=0$$
  • none of these
If $$H(x_0)=0$$ for some $$x=x_0$$ and $$\dfrac {d}{dx}H(x) > 2cxH(x)$$ for all $$x \le x_0$$, where $$c > 0$$, then
  • $$H(x)=0$$ has root for $$x > x_0$$
  • $$H(x)=0$$ has no root for $$x > x_0$$
  • $$H(x)=0$$ is constant function
  • None of these
Given a function $$f:[0,4] \rightarrow \mathrm{R}$$ is differentiable, then for $$\displaystyle \operatorname{some} \alpha, \beta \in(0,2), \int_{0}^{4} f(t) d t$$ equals to
  • $$f\left(\alpha^{2}\right)+f\left(\beta^{2}\right)$$
  • $$2 \alpha f\left(\alpha^{2}\right)+2 \beta f\left(\beta^{2}\right)$$
  • $$\alpha f\left(b^{2}\right)+\beta f\left(\alpha^{2}\right)$$
  • $$f(\alpha) f(\beta)[f(\alpha)+f(\beta)]$$
If f(x)=$$[sin^{-1}(sin2x)]$$(where, [] denotes the greatest integer function), then 
  • $$\displaystyle \int_{0}^{\pi/2}f(x)dx=\dfrac{\pi}{2}-sin^{-1}(sin1)$$
  • f(x) is periodic with period
  • $$\displaystyle \lim_{x\dfrac{\pi}{2}}f(x)=-1$$
  • None of the above
If $$f:R \rightarrow (0, \infty)$$ be a differentiable function $$f(x)$$ satisfying $$f(x+ y) - f(x - y) = f(x) \{ f(y) - f(y) - y \}, \forall x, y \epsilon R, (f(y) \neq f(-y)$$ for all $$y \epsilon R)$$ and $$f' (0) = 2010$$.
Now answer the following questions

Which of the following is true for $$f(x)$$
  • $$f(x)$$ is one-one and into
  • $$\{ f(x) \}$$ is non-periodic, where {.} denotes fractional part of x.
  • $$f(x) = 4$$ has only two solutions.
  • $$f(x) = f^{-1} x$$ has only one solution
If f(x) = $$\displaystyle \int_{0}^{x}(f(t))^2 dt f:R→R$$ be differentiable function and f(g(x)) is differentiable function at x=a, then
  • g(x) must be differentiable at x=a
  • g(x) may be non-differentiable at x=a
  • g(x) must be discontinuous at x=a
  • None of the above
$$y=f(x)$$ is
  • injective but not surjective
  • surjective but not injective
  • bijective
  • neither injective nor surjective
The current statement(s) is/are
  • $$f'(1)<0$$
  • $$f(2)<0$$
  • $$f'(x)\neq 0$$ for any $$x \in (1,3)$$
  • $$f'(x) = 0$$ for some $$x \in (1,3)$$
If $$u=\sin^{-1}\Bigg(\dfrac{2x}{1+x^2}\Bigg)$$ and $$v=\tan^{-1}\Bigg(\dfrac{2x}{1-x^2}\Bigg)$$, then $$\dfrac{du}{dv}$$ is
  • $$\dfrac{1}{2}$$
  • $$x$$
  • $$\dfrac{1-x^2}{1+x^2}$$
  • $$1$$
The function $$f(x)=\Bigg\{\dfrac{\sin x}{x}+\cos x,if\,x\neq 0\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,k,if\,x=0$$  is continuous at $$x=0$$,then the value of k is
  • $$3$$
  • $$2$$
  • $$1$$
  • $$1.5$$
$$|\sin x|$$ is a differentiable function for every value of $$x$$.
  • True
  • False
The function $$f(x)=|x|+|x-1|$$ is
  • continuous at $$x = 0$$ as well as at $$x = 1$$.
  • continuous at $$x = 1$$ but not at $$x = 0$$.
  • discontinuous at $$x = 0$$ as well as at $$x = 1$$.
  • continuous at $$x = 0$$ but not at $$x = 1$$.
If $$f(x) = \sin^{-1} \left ( \frac{4^{x + \frac{1}{2}}}{1 + 2^{4x}} \right )$$, which of the following is not the derivative of f(x)?
  • $$\frac{2.4^{x} \cdot \log 4}{1 + 4^{2x}}$$
  • $$\frac{4^{x + 1} \cdot \log 2}{1 + 4^{2x}}$$
  • $$\frac{4^{x + 1} \cdot \log 4}{1 + 4^{4x}}$$
  • $$\frac{2^{2^(x + 1)} \cdot \log 2}{1 + 2^{4x}}$$
If $$y \tan^{-1} \left ( \sqrt{\frac{a - x}{a + x}} \right )$$, where -a < x < a, then $$\frac{dy}{dx} =$$.....
  • $$\frac{x}{\sqrt{a^2 - x^2}}$$
  • $$\frac{a}{\sqrt{a^2 - x^2}}$$
  • $$- \frac{1}{2 \sqrt{a^2 - x^2}}$$
  • $$\frac{1}{2 \sqrt{a^2 - x^2}}$$
If $$y = \tan^{-1} \left ( \frac{x}{1 + \sqrt{1 - x^2}} \right ) + \sin \left [ 2 \tan^{-1} \left ( \sqrt{\frac{1 - x}{1 + x}} \right )  \right ]$$ then $$\frac{dy}{dx} = $$...........
  • $$\frac{x}{\sqrt{1 - x^2}}$$
  • $$\frac{1 - 2x}{\sqrt{1 - x^2}}$$
  • $$\frac{1 - 2x}{2 \sqrt{1 - x^2}}$$
  • $$\frac{1 - 2x^2}{\sqrt{1 - x^2}}$$
If $$y = \sin(2 \sin^{-1} x)$$, then dx = .......
  • $$\frac{2 - 4x^2}{\sqrt{1 - x^2}}$$
  • $$\frac{2 + 4x^2}{\sqrt{1 - x^2}}$$
  • $$\frac{4x^2 - 1}{\sqrt{1 - x^2}}$$
  • $$\frac{1 - 2x^2}{\sqrt{1 - x^2}}$$
If function $$f(x)=\dfrac{x^2-9}{x-3}$$ is continuous at $$x=3$$, then value of $$(3)$$ will be:
  • $$6$$
  • $$3$$
  • $$1$$
  • $$0$$
If $$f(x)=\begin{cases} \begin{matrix} \dfrac{\log (1+mx)- \log (1-nx)}{x}; & x \ne 0 \end{matrix} \\ \begin{matrix} k; & x=0 \end{matrix} \\ \begin{matrix}  &  \end{matrix} \end{cases}$$
is continuous at $$x=0$$ then the value of $$k$$ will be:
  • $$0$$
  • $$m+n$$
  • $$m-n$$
  • $$m.n$$
If function $$f(x)=\begin{cases} \begin{matrix} \dfrac{\sin 3x}{x}; & x \ne 0 \end{matrix} \\ \begin{matrix} m; & x=0 \end{matrix} \\ \begin{matrix}  &  \end{matrix} \end{cases}$$
is continuous at $$x=2$$ then value of $$m$$ will be:
  • $$3$$
  • $$1/3$$
  • $$1$$
  • $$0$$
Let $$x={f}''(t) cost +{f}'(t) sint$$  and $$y={-f}''(t) sint+{f}'(t) cost.$$ Then $$\displaystyle \int \left [ \left(\frac{dx}{dt} \right)^2 + \left(\frac{dy}{dt} \right)^2 \right ]^{\frac{1}{2}} dt$$ equals
(Note : $$f(x), f'(x), f''(x) , f'''(x) >0$$ )
  • $${f}'(t)+{f}''(t)+c$$
  • $${f}''(t)+{f}'''(t)+c$$
  • $$f(t)+{f}''(t)+c$$
  • $${f}'(t)-{f}''(t)+c$$
The set of all points where the function $$f(\displaystyle \mathrm{x})=\frac{x}{1+|x|}$$ is differentiable is 
  • $$(-\infty, \infty)$$
  • $$(0,\infty)$$
  • $$(-\infty ,0)\cup (0,\infty )$$
  • $$(-\infty, 0)$$
If $$y=x^{\displaystyle x^{\displaystyle x^{\displaystyle \dots^{\displaystyle\infty}}}}$$, find $$\displaystyle\frac{dy}{dx}$$.
  • $$\displaystyle\frac{y^2}{x(1-y\log{x})}$$
  • $$\displaystyle\frac{y}{x(1-\log{x})}$$
  • $$\displaystyle\frac{y^2}{x(y-\log{x})}$$
  • None of these
Derivative of $$({\log{x}})^{\displaystyle\cos{x}}$$ with respect to $$x$$ is
  • $$\displaystyle({\log{x}})^{\displaystyle\cos{x}}\left[\frac{\cos{x}}{x\log{x}}-\sin{x}\log{(\log{x})}\right]$$
  • $$\displaystyle({\log{x}})^{\displaystyle\cos{x}}\left[\frac{\cos{x}}{x\log{x}}-\cos{x}\log{(\log{x})}\right]$$
  • $$\displaystyle({\log{x}})^{\displaystyle\sin{x}}\left[\frac{\sin{x}}{x\log{x}}-\cos{x}\log{(\log{x})}\right]$$
  • None of these
If $$y=x^{\left(x^{ x}\right)}$$, then $$\displaystyle\frac{dy}{dx}$$ is
  • $$y\left[x^{\displaystyle x}\left(\log{ex}\right)\log{x}+x^{\displaystyle x}\right]$$
  • $$y\left[x^{\displaystyle x}\left(\log{ex}\right)\log{x}+x\right]$$
  • $$y\left[x^{\displaystyle x}\left(\log{ex}\right)\log{x}+x^{\displaystyle x-1}\right]$$
  • $$y\left[x^{\displaystyle x}\left(\log_e{x}\right)\log{x}+x^{\displaystyle x-1}\right]$$
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