CBSE Questions for Class 11 Commerce Applied Mathematics Differentiation Quiz 16 - MCQExams.com

Given a function '$$g$$' whcih has a derivative $$g'(x)$$ for every real '$$x$$' and which satisfy $$g'(0)=2$$ and $$g(x+y)={e}^{y}. g(x)+{e}^{x}.g(y)$$ for all $$x, y$$. Find $$g(x)$$.
  • $$2x{e}^{x}$$
  • $$xe^x$$
  • $$x+e^x$$
  • $$x-e^x$$
Let $$f$$ be a differential function such that $$f(x)=f(4-x)$$ and $$g(x)=f(2+x)$$ for all $$x\in R,$$ then
  • graph of $$f(x)$$ is symmetric about the line $$x=2$$
  • $$f'(2)=0$$
  • graph of $$g(x)$$ is symmetric about x-axis
  • $$g'(0)=0$$
Length of the subtangent at $$(x_l, y_l)$$ on $$x^n y^m = a^{m+n}, m, n > 0,$$is
  • $$\dfrac{n}{m}x_l$$
  • $$\dfrac{m}{n}|x_l|$$
  • $$\dfrac{n}{m}|y_l|$$
  • $$\dfrac{n}{m}|x_l|$$

The slope(s) of common tangent(s) to the curves $$ \displaystyle y={ e }^{ -x }$$ and $$ \displaystyle y={ e }^{ -x }\sin { x } $$ can be -
  • $$ \displaystyle -{ e }^{ -\tfrac{\pi}2 }$$
  • $$ \displaystyle -{ e }^{ -\pi }$$
  • $$ \displaystyle \frac { \pi }{ 2 } $$
  • $$1$$
The derivative of $$\displaystyle (\tan x)^{x}$$ is equal to-
  • $$\displaystyle x(\tan x)^{x-1}$$
  • $$\displaystyle (\tan x)^{x}\left [ \sec x+\tan x \right ]$$
  • $$\displaystyle (\tan x)^{x}\left [ x\sec x \csc x +\log \tan x\right ]$$
  • $$\displaystyle(\tan x)^{x}\left [ \sec ^{2}x+x\tan x \right ]$$
If $$x=\dfrac { 1-{ t }^{ 2 } }{ 1+{ t }^{ 2 } } $$ and $$y=\dfrac { 2at }{ 1+{ t }^{ 2 } } $$, then $$\dfrac { dy }{ dx } $$ is equal to:
  • $$\dfrac { a\left( 1-{ t }^{ 2 } \right) }{ 2t } $$
  • $$\dfrac { a\left( { t }^{ 2 }-1 \right) }{ 2t } $$
  • $$\dfrac { a\left( { t }^{ 2 }+1 \right) }{ 2t } $$
  • $$\dfrac { a\left( { t }^{ 2 }-1 \right) }{ t } $$
If $$\displaystyle e^{\sin (x^{2}+y^{2})}=\tan \frac{y^{2}}{4}+\sin ^{-1}x$$, then $$y' (0)$$ can be- 
  • $$\displaystyle \frac{1}{3\sqrt{\pi }}$$
  • $$\displaystyle -\frac{1}{3\sqrt{\pi }}$$
  • $$\displaystyle -\frac{1}{5\sqrt{\pi }}$$
  • $$\displaystyle -\frac{1}{3\sqrt{5\pi }}$$
The solution of differential equation $$\displaystyle ydx+\left( x-{ y }^{ 2 } \right) dy=0$$
  • $$\displaystyle { e }^{ \frac { y }{ x } }=\sin { x } +c$$
  • $$\displaystyle y=cx\log { x } $$
  • $$\displaystyle x=\frac { { y }^{ 2 } }{ 3 } +\frac { c }{ y } $$
  • $$\displaystyle \cos { \left( \frac { y-2 }{ x } \right) } =a$$
The derivative of $$y=x^{2^x}$$ w.r.t x is :
  • $$x^{2^x}2^x\left(\displaystyle \frac{1}{x} + \ln x \ln 2 \right)$$
  • $$x^{2^x}\left(\displaystyle\frac{1}{x} \ln x \ln 2\right)$$
  • $$x^{2^x}2^x\left(\displaystyle\frac{1}{x} \ln x\right)$$
  • $$x^{2^x}2^x\left(\displaystyle\frac{1}{x}+\frac{\ln x}{\ln 2}\right)$$
If $$x=\sqrt{a^{\sin^{-1}t}}$$, $$y=\sqrt{a^{\cos^{-1}t}}$$, then $$\dfrac{dy}{dx}=$$ ________.
  • $$\dfrac{-y}{x}$$
  • $$\dfrac{x}{y}$$
  • $$\dfrac{y}{x}$$
  • $$\dfrac{-x}{y}$$
If $$x = \sec \theta - \cos \theta$$ and $$y = \sec^{n}\theta - \cos^{n}\theta$$, then $$\left (\dfrac {dy}{dx}\right )^{2}$$ is equal to
  • $$\dfrac {n^{2}(y^{2} + 4)}{x^{2} + 4}$$
  • $$\dfrac {n^{2}(y^{2} - 4)}{x^{2}}$$
  • $$n\left (\dfrac {y^{2} - 4}{x^{2} - 4}\right )$$
  • $$\left (\dfrac {ny}{x}\right )^{2} - 4$$
If $$y = x^{x^{x^{x^{.^{.^{\infty}}}}}}$$, then $$y' =$$
  • $$\dfrac {-y^{2}}{x(1 - y\log x)}$$
  • $$\dfrac {y^{2}}{1 - y\log x}$$
  • $$\dfrac {y^{2}}{x(1 - y\log x)}$$
  • $$\dfrac {-y^{2}}{1 - y\log x}$$
If $$\int_{0}^{x}(t^{2}+2t+2)$$ dt, $$2\leq x\leq 4$$
  • The maximum value of f(x) is $$\dfrac{136}{3}$$
  • The minimum value of f(x) is 10
  • The maximum value of f(x) is 26
  • None of these
If $$x\sqrt {1 + y}  + y\sqrt {1 + x}  = 0$$, then $$\dfrac {dy}{dx}$$ is equal to
  • $$\cfrac{1}{(1+x)^{2}}$$
  • $$\cfrac{-1}{(1+x)^{2}}$$
  • $$\cfrac{1}{(1-x)^{2}}$$
  • None of these
If $$y^{y^{y^{....{^\infty}}}} = \log_e(x+\log_e(x+....))$$, then $$\dfrac{dy}{dx}$$ at $$(x= e^2-2, y= \sqrt2)$$ is
  • $$\dfrac{\log\left(\dfrac{e}{2}\right)}{2\sqrt2(e^2-1)}$$
  • $$\dfrac{\log2}{2\sqrt2(e^2-1)}$$
  • $$\dfrac{\sqrt2 \log\dfrac{e}{2}}{(e^2-1)}$$
  • None of these
Let $$y=x^{x^x}$$, then differentiate $$y$$ w.r.t $$x$$.
  • $$x^{x^x}\left(\dfrac{1}{x}+\log x+(\log x)^2\right)$$
  • $$x^{x^x}(x^x)\left(\dfrac{1}{x}+\log x-(\log x)^2\right)$$
  • $$x^{x^x}(x^x)\left(\dfrac{1}{x}+\log x+(\log x)^2\right)$$
  • $$x^{x^x}(x^x)\left(\dfrac{1}{x}-\log x-(\log x)^2\right)$$
Let $$y=f(x)$$ be a differentiable curve satisfying $$\int _{ 2 }^{ x }{ f\left( t \right) dt+2=\dfrac { { x }^{ 2 } }{ 2 } + } \int _{ x }^{ 2 }{ { t }^{ 2 }f\left( t \right) dt }$$,then $$\displaystyle \int _{ -\pi /4 }^{ \pi /4 }{ \dfrac { f\left( x \right) +{ x }^{ 9 }-{ x }^{ 3 }+x+1 }{ \cos ^{ 2 }{ x }  } dx }$$ equals-
  • $$0$$
  • $$1$$
  • $$2$$
  • $$4$$
if $$\int \dfrac {\cos 4x+1}{\cos x-\tan x}dx=A \cos 4x+B;$$ where $$A$$ & $$B$$ are constants, then 
  • $$A=-1/4$$ & $$B$$ may have any value
  • $$A=-1/8$$ & $$B$$ may have any value
  • $$A=-1/2$$ & $$B=-1/4$$
  • $$A=B=1/2$$
Consider the following statements:
$$S_1: \lim_\limits{x \to 0} \dfrac{[x]}{x}$$ is an indeterminate form (where [.] denotes greatest integer function).
$$S_2: \lim_\limits{x\to\infty}\dfrac{sin(3^x)}{3^x}=0$$
$$S_3: \lim_\limits{x \to \infty}\sqrt{\dfrac{x- sinx}{x+cos^2x}}$$ does not exist.
$$S_4:  \lim_\limits{n\to \infty}\dfrac{(n+2)!+(n+1)!}{(n+3)! }(n \in N=0$$
State, in order, whether $$S_1, S_2, S_3, S_4$$ are true or false
  • FTFT
  • FTTT
  • FTFF
  • TTFT
If $$x^{2} + y^{2} = a^{2}$$ and $$k=1/a$$, then $$k$$ is equal to ?
  • $$\dfrac { y\prime \prime }{ \sqrt { 1+y\prime } }$$
  • $$\dfrac { \left| y\prime \prime \right| }{ \sqrt { { \left( 1+{ y\prime }^{ 2 } \right) }^{ 3 } } }$$
  • $$\dfrac {2 y\prime \prime }{ \sqrt { 1+y\prime } }$$
  • $$\dfrac { y\prime \prime }{ 2\sqrt { { \left( 1+{ y\prime }^{ 2 } \right) }^{ 3 } } } $$
If $$x ^ { y } + y ^ { x } = a ^ { b }$$  then find that $$\dfrac { d y } { d x }$$ 
  • $$ -\dfrac{yx^{x-1}+y^x\log y}{x^y\log x+xy^{x-1}}$$
  • $$ \dfrac{yx^{x-1}+y^x\log y}{x^y\log x+xy^{x-1}}$$
  • $$ -\dfrac{yx^{x-1}+y^x\log y}{x^y\log x+xy^{x}}$$
  • None of these
$$\frac{d}{{dx}}\left( {\frac{{1 + {x^2} + {x^4}}}{{1 + x + {x^2}}}} \right) = ax + b$$, then $$\left( {a,b} \right) = $$
  • $$\left( { - 1,2} \right)$$
  • $$\left( { - 2,1} \right)$$
  • $$\left( {2, - 1} \right)$$
  • $$\left( {1,2} \right)$$
If $$f(x+y)=f(x)f(y)\forall\ x,y\ \in\ R$$ and$$f'(0)=5,f(2)=6$$, then $$f'(2)$$ 
  • $$20$$
  • $$10$$
  • $$30$$
  • $$40$$
If $$x \sin y=\sin (y+a)$$ and $$\dfrac{dy}{dx}=\dfrac{A}{1+x^{2}-2 x \cos a}$$ then the value of $$A$$ is
  • $$2$$
  • $$\cos a$$
  • $$-\sin a $$
  • $$-2$$
Let $$g(x)$$ be a continuous function for all $$x$$, and $$f(x)=f(\alpha)+(x-\alpha).g(x)\ \forall \ x\ =\epsilon \ R$$. Then;
  • $$f(x)$$ is necessarily differentiable at $$x=\alpha$$
  • $$f(x)$$ is not necessarily differentiable at $$x=\alpha$$
  • $$f(x)$$ is not necessarily continuous at $$x=\alpha$$
  • $$None\ of\ these$$
For the function, $$f(x) = (x - \frac{1}{x})^2$$, the first derivative with respect to x is 
  • $$2 (x - \frac{1}{x^3})$$
  • $$2 (x - \frac{1}{x})$$
  • $$2 (x + \frac{1}{x^2})$$
  • $$2 (x - \frac{1}{x^2})$$
If  $$y = y ( x )$$  is an implicit function of  $$x$$  given by  $$y \cos x + x \cos y = \pi,$$  the  $$y ^ { \prime \prime } ( 0 )$$  is equal to
  • $$\pi$$
  • $$- \pi$$
  • $$0$$
  • $$2 \pi$$
If $$x^{m}y^{n}=(x+y)^{m+n}$$, then $$xy^{3}=y$$, if $$\dfrac {x}{y} \neq \dfrac {m}{n}$$
  • True
  • False
The value of $$\int _{ 0 }^{ \left[ x \right]  }{ \left( x-\left[ x \right]  \right) dx\quad is\left( \left[ . \right] denotes\quad greatest\quad integer\quad function \right)  } $$
  • $$\left[ x \right] $$
  • $$2\left[ x \right] $$
  • $$\dfrac { \left[ x \right] }{ 2 } $$
  • $$3\left[ x \right] $$
Let $$f ' (x)= -f(x)$$, where f(x) is a continous double differentiable function and $$g(x)=f '(x)$$. If $$F(x)=\left[ f\left( \dfrac { x }{ 2 }  \right)  \right] ^{ 2 }+\left[ g\left( \dfrac { x }{ 2 }  \right)  \right] ^{ 2 }$$ and F(5)=5,then F(10) =
  • 0
  • 5
  • 10
  • 25
0:0:1


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Practice Class 11 Commerce Applied Mathematics Quiz Questions and Answers