CBSE Questions for Class 12 Commerce Maths Continuity And Differentiability Quiz 11 - MCQExams.com

If $$y=x^{{\displaystyle(\log{x})}^{\displaystyle\log{(\log{x})}}}$$, then $$\displaystyle\frac{dy}{dx}$$ is
  • $$\displaystyle\frac{y}{x}((\ln{x^{\displaystyle x-1}})+2\ln{x}\ln{(\ln{x})})$$
  • $$\displaystyle\frac{y}{x}{(\log{x})}^{\displaystyle\log{(\log{x})}}(2\log{(\log{x})}+1)$$
  • $$\displaystyle\frac{y}{x\ln{x}}[{(ln{x})}^2+2\ln{(\ln{x})}]$$
  • $$\displaystyle\frac{y}{x}\frac{\log{y}}{\log{x}}[2\log{(\log{x})}+1]$$
If $$f(x)={|x|}^{|\sin{x}|}$$, then $$\displaystyle f^\prime\left(-\frac{\pi}{4}\right)=$$
  • $$\displaystyle {\left(\frac{\pi}{5}\right)}^{\tfrac{1}{\sqrt{2}}}\left(\frac{\sqrt{2}}{2}\log{\frac{5}{\pi}}-\frac{2\sqrt{2}}{\pi}\right)$$
  • $$\displaystyle {\left(\frac{\pi}{4}\right)}^{\tfrac{1}{\sqrt{2}}}\left(\frac{\sqrt{2}}{2}\log{\frac{4}{\pi}}-\frac{2\sqrt{2}}{\pi}\right)$$
  • $$\displaystyle {\left(\frac{\pi}{3}\right)}^{\tfrac{1}{\sqrt{2}}}\left(\frac{\sqrt{2}}{2}\log{\frac{3}{\pi}}-\frac{3\sqrt{3}}{\pi}\right)$$
  • None of these
If $$x=\sin^{-1}t$$ and $$y=\log(1-{t}^{2})$$ , then $$\displaystyle \left .\frac{{d}^{2}y}{{d}{x}^{2}}\right|_{{t}=\frac{1}{2}} = $$
  • $$-\dfrac{8}{3}$$
  • $$\dfrac{8}{3}$$
  • $$\dfrac{3}{4}$$
  • $$-\dfrac{3}{4}$$
If a function is represented parametrically by the equations $$\displaystyle x=\frac{1+\log_e{t}}{t^2}$$; $$\displaystyle y=\frac{3+2\log_e{t}}{t}$$, then which of the following statements are true?
  • $$y^{\prime\prime}(x-2xy^\prime)=y$$
  • $$yy^\prime=2x{(y^\prime)}^2+1$$
  • $$xy^\prime=2y{(y^\prime)}^2+2$$
  • $$y^{\prime\prime}(y-4xy^\prime)={(y^\prime)}^2$$
$$\displaystyle y=\left ( 1+\frac{1}{x} \right )^{x}+x^{1+\frac{1}{x}}.$$
Differentiate
  • $$\displaystyle \left ( 1+\frac{1}{x} \right )^{x}\left [ \log \left ( 1+\frac{1}{x} \right )-\frac{1}{x+1} \right ]+x^{1+1/x}\left [ \frac{x+1-\log x}{x^{2}} \right ].$$
  • $$\displaystyle \left ( -1+\frac{1}{x} \right )^{x}\left [ \log \left ( 1+\frac{1}{x} \right )-\frac{1}{x+1} \right ]+x^{1+1/x}\left [ \frac{x+1+\log x}{x^{2}} \right ].$$
  • $$\displaystyle \left ( 1+\frac{1}{x} \right )^{x}\left [ \log \left ( 1+\frac{1}{x} \right )-\frac{1}{x+1} \right ]+x^{1-1/x}\left [ \frac{x+1-\log x}{x^{2}} \right ].$$
  • $$\displaystyle \left ( -1+\frac{1}{x} \right )^{x}\left [ \log \left ( 1+\frac{1}{x} \right )-\frac{1}{x+1} \right ]+x^{1-1/x}\left [ \frac{x+1+\log x}{x^{2}} \right ].$$
If $$x=\sin t$$, $$y=\sin kt$$ then value of $$\left ( 1-x^{2} \right )y_{2}-xy_{1}$$ is
  • $$-k^{2}y$$
  • $$k^{2}y$$
  • $$ky^{2}$$
  • $$-ky^{2}$$
If $$\displaystyle y=x^{\ln x^{\ln \left ( \ln x \right )}}$$, then $$\displaystyle \frac{dy}{dx}$$ is equal to:
  • $$\displaystyle \frac{y}{x}\left ( \ln x^{\ln x-1}+2\ln x\ln (\ln x) \right )$$
  • $$\displaystyle \frac{y}{x}\ln x^{\ln \left ( \ln x \right )}(2\ln (\ln x)+1)$$
  • $$\displaystyle \frac{y}{x\ln x}\ln x^{2}+2\ln (\ln x)$$
  • $$\displaystyle \frac{y\ln x}{x\ln x}(2\ln (\ln x)+1)$$
If $$f(x)=\left | x \right |+\left | \cos x \right |$$, then
  • $$ f^{'}( \dfrac{\pi }{2} )=2$$
  • $$ f^{'}( \dfrac{\pi }{2} )=0$$
  • $$ f^{'}( \dfrac{\pi }{2} )=1$$
  • $$ f^{'}( \dfrac{\pi }{2} )$$ does not exist
Given the parametric equations $$x=f(t),y=g(t).$$ Then $$\displaystyle \frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } $$ equals
  • $$\displaystyle \dfrac { \dfrac { { d }^{ 2 }y }{ d{ t }^{ 2 } } .\dfrac { dx }{ dt } -\dfrac { dy }{ dt } .\dfrac { { d }^{ 2 }x }{ d{ t }^{ 2 } }  }{ { \left( \dfrac { dx }{ dt }  \right)  }^{ 2 } } $$
  • $$\displaystyle \dfrac { \dfrac { { d }^{ 2 }y }{ d{ t }^{ 2 } } .\dfrac { dx }{ dt } -\dfrac { dy }{ dt } .\dfrac { { d }^{ 2 }x }{ d{ t }^{ 2 } }  }{ { \left( \dfrac { dx }{ dt }  \right)  }^{ 3 } } $$
  • $$\displaystyle \dfrac { \dfrac { { d }^{ 2 }y }{ d{ t }^{ 2 } }  }{ \dfrac { { d }^{ 2 }x }{ d{ t }^{ 2 } }  } $$
  • None of these
Derivative of $${(x\cos{x})}^x$$ with respect to $$x$$ is
  • $$\displaystyle {(x\cos{x})}^x\left[(\log{x}+1)-\left\{\log{\cos{x}}+\frac{x}{\cos{x}}.(\sin{x})\right\}\right]$$
  • $$\displaystyle {(x\cos{x})}^x\left[(\log{x}+1)+\left\{\log{\cos{x}}+\frac{x}{\cos{x}}.(-\sin{x})\right\}\right]$$
  • $$\displaystyle {(x\cos{x})}^x\left[(\log{x}+1)+\left\{\log{\sin{x}}+\frac{x}{\cos{x}}.(\cos{x})\right\}\right]$$
  • None of these
The value of $$y''(1)$$ if $${ x }^{ 3 }-2{ x }^{ 2 }{ y }^{ 2 }+5x+y-5=0$$ when $$y(1)>1,$$ is equal to
  • $$\displaystyle \frac { 22 }{ 7 } $$
  • $$\displaystyle -7\frac { 21 }{ 28 } $$
  • $$8$$
  • $$\displaystyle -8\frac { 22 }{ 27 } $$
The weight $$W$$ of a certain stock of fish is given by $$W = nw$$, where n is the size of stock and $$w$$ is the average weight of a fish. If $$n$$ and $$w$$ change with time $$t$$ as $$n = 2t^2 + 3$$ and $$w = t^2 - t + 2$$, then the rate of change of $$W$$ with respect to $$t$$ at $$t = 1$$ is.
  • $$1$$
  • $$5$$
  • $$8$$
  • $$31$$
If $$x=log(1+t^2)$$ and $$y=t-tan^{-1}t$$. Then, $$\dfrac{dy}{dx}$$ is equal to 
  • $$e^x-1$$
  • $$t^2-1$$
  • $$\dfrac{\sqrt{e^x-1}}{2}$$
  • $$e^x-y$$
The function $$f(x)=\begin{cases} ax(x-1)+b\quad \quad when\quad x<1 \\ x-1\quad \quad \quad when\quad \quad 1\le x\le 3 \\ p{ x }^{ 2 }+qx+2\quad \quad when\quad x>3 \end{cases}$$ Find the values of the constants $$a,b,p,q$$ so that $$(i)f(x)$$ is continuous for all $$x$$  $$(ii)f'(1)$$ does not exist  $$(iii)f'(x)$$ is continuous at $$x=3$$
  • $$a=1, b=0, p=1/3, q=-1$$
  • $$a\ne 1, b=0, p=1/3, q=-1$$
  • $$a\ne 1, b=0, p=1/3, q=1$$
  • $$a=1, b=0, p=1/3, q=1$$
Length of the subtangent at $$(x_l, y_l)$$ on $$x^n y^m = a^{m+n}, m, n > 0,$$is
  • $$\dfrac{n}{m}x_l$$
  • $$\dfrac{m}{n}|x_l|$$
  • $$\dfrac{n}{m}|y_l|$$
  • $$\dfrac{n}{m}|x_l|$$
$$x$$$$1$$$$2$$$$3$$$$4$$$$5$$
$$f(x)$$$$4$$$$3$$$$7$$$$1$$$$3$$
The function f is continuous on the closed interval $$[1, 5]$$ and values of the function are shown in the table above. If the values in the table are used to calculate a trapezoidal sum, the approximate value of $$\int_{1}^{5}f(x)dx$$ is
  • $$14$$
  • $$14.5$$
  • $$15$$
  • $$29$$
The derivative of $$\displaystyle (\tan x)^{x}$$ is equal to-
  • $$\displaystyle x(\tan x)^{x-1}$$
  • $$\displaystyle (\tan x)^{x}\left [ \sec x+\tan x \right ]$$
  • $$\displaystyle (\tan x)^{x}\left [ x\sec x \csc x +\log \tan x\right ]$$
  • $$\displaystyle(\tan x)^{x}\left [ \sec ^{2}x+x\tan x \right ]$$
Examine for continuity and differentiability at the points $$x=1, x=2$$, the function $$f$$ defined by $$f(x)=\begin{cases} x\left[ x \right] ,\quad \quad \quad \quad 0\le x<2 \\ (x-1)\left[ x \right] ,\quad 2\le x\le 3 \end{cases}$$ where $$\left[ x \right] =$$ greatest integer less than or equal to $$x$$
  • discontinuous and not derivable at $$x=1,2$$
  • discontinuous and not derivable at $$x=1$$, continuous but not derivable at $$x=2$$
  • continuous and not derivable at $$x=1,2$$.
  • continuous and not derivable at $$x=1$$, discontinuous but not derivable at $$x=2$$.
If $$y=\sqrt { \left( a-x \right) \left( x-b \right)  } -\left( a-b \right) \tan ^{ -1 }{ \sqrt { \dfrac { a-x }{ x-b }  }  } $$, then $$\dfrac { dy }{ dx } $$ is equal to
  • $$1$$
  • $$\sqrt { \dfrac { a-x }{ x-b } } $$
  • $$\sqrt { \left( a-x \right) \left( x-b \right) } $$
  • $$\dfrac { 1 }{ \sqrt { \left( a-x \right) \left( b-x \right) } } $$
The derivative of $$y=x^{2^x}$$ w.r.t x is :
  • $$x^{2^x}2^x\left(\displaystyle \frac{1}{x} + \ln x \ln 2 \right)$$
  • $$x^{2^x}\left(\displaystyle\frac{1}{x} \ln x \ln 2\right)$$
  • $$x^{2^x}2^x\left(\displaystyle\frac{1}{x} \ln x\right)$$
  • $$x^{2^x}2^x\left(\displaystyle\frac{1}{x}+\frac{\ln x}{\ln 2}\right)$$
Let $$f(x) = \left\{\begin{matrix}2a - x, &if\ -a < x < a \\ 3x - 2a, &if\ a \leq x \end{matrix}\right.$$. 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 $$x \geq a$$
  • $$f(x)$$ is continuous at all $$x < a$$
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 a curve is given by $$x=a\cos t+\displaystyle\frac{b}{2}\cos 2t$$ and $$y=\sin t +\displaystyle\frac{b}{2}\sin 2t$$, then the points for which $$\displaystyle\frac{d^2y}{dx^2}=0$$, are given by.
  • $$\sin t=\displaystyle\frac{2a^2+b^2}{3ab}$$
  • $$\cos t=-\left [\displaystyle\frac{a^2+2b^2}{3ab}\right]$$
  • $$\tan t=a/b$$
  • None of the above
If $$\sin ^{ -1 }{ \left( \dfrac { { x }^{ 2 }-{ y }^{ 2 } }{ { x }^{ 2 }+{ y }^{ 2 } }  \right)  } =\log { a } $$, then $$\dfrac { { d }^{ 2 }y }{ d{ x }^{ 2 } } $$ equals
  • $$\dfrac { x }{ y } $$
  • $$\dfrac { y }{ { x }^{ 2 } } $$
  • $$\dfrac { y }{ x } $$
  • $$0$$
Find derivative of $$\tan^{-1}\dfrac{\cos x-\sin x}{\cos x+\sin x}$$ w.r.t. $$x$$.
  • $$-1$$
  • $$0$$
  • $$1$$
  • $$2$$
If $$y=\tan ^{ -1 }{ \left( \cfrac { 4x }{ 1+5{ x }^{ 2 } }  \right)  } +\tan ^{ -1 }{ \left( \cfrac { 2+3x }{ 2-3x }  \right)  } $$, then $$\cfrac { dy }{ dx } $$ is
  • $$\cfrac { 6 }{ 1+4{ x }^{ 2 } } $$
  • $$\cfrac { 3 }{ 1+4{ x }^{ 2 } } $$
  • $$\cfrac { 5 }{ 1+{ 25x }^{ 2 } } $$
  • $$\dfrac{5}{(1+25x^{2})} - \dfrac{1}{(1+x^{2})}  - \dfrac{1.5}{(1+2.25x^{2})}$$
If $$x = A\cos 4t + B\sin 4t$$, then $$\dfrac {d^{2}x}{dt^{2}}$$ is equal to
  • $$-16x$$
  • $$16x$$
  • $$x$$
  • $$-x$$
Let $$f\left( x \right)$$ and $$g\left( x \right)$$ be defined and differentiable for $$x\ge { x }_{ 0 }$$ and $$f\left( { x }_{ 0 } \right) =g\left( { x }_{ 0 } \right) , f^{ ' }\left( x \right) >g^{ ' }\left( x \right) for\ x>{ x }_{ 0 }$$ then
  • $$f\left( x \right) < g\left( x \right)$$ for some $$x>{ x }_{ 0 }$$
  • $$f\left( x \right) =g\left( x \right)$$ for some $$x>{ x }_{ 0 }$$
  • $$f\left( x \right) >g\left( x \right)$$ for all $$x>{ x }_{ 0 }$$
  • None of these
Find the derivative of $$\dfrac{\tan^{-1}x}{1+\tan^{-1}x}$$ w.r.t. $$\tan^{-1}x$$.
  • $$\dfrac{1}{\sec^{-1}x}$$
  • $$\dfrac{1}{(\tan^{-1}x)^{2}}$$
  • $$\dfrac{1}{1+\tan^{2}x}$$
  • $$\dfrac{1}{(1+\tan^{-1}x)^{2}}$$
If the function $$f:\left[ 0,8 \right] \rightarrow R$$ is differentiable, then for $$0<a,b<2,\int _{ 0 }^{ 8 }{ f(t) } dt$$ is equal to
  • $$2\left[ { \alpha }^{ 3 }f({ \alpha }^{ 2 })+{ \beta }^{ 2 }f({ \beta }^{ 2 }) \right] $
  • $$3\left[ { \alpha }^{ 3 }f({ \alpha }^{ })+{ \beta }^{ 2 }f({ \beta }^{ }) \right] $$
  • $$3\left[ { \alpha }^{ 2 }f({ \alpha }^{ 3 })+{ \beta }^{ 2 }f({ \beta }^{ 3 }) \right] $$
  • $$3\left[ { \alpha }^{ 2 }f({ \alpha }^{ 2 })+{ \beta }^{ 2 }f({ \beta }^{ 2 }) \right] $$
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