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CBSE Questions for Class 12 Commerce Maths Continuity And Differentiability Quiz 2 - MCQExams.com

Find dydx if x=a(θsinθ) and y=a(1cosθ).
  • cot(θ2)
  • tan(θ2)
  • cos(θ2)
  • None of these
The derivative of f(\tan x) w.r.t. g (\sec x) at x=\displaystyle \frac{\pi }{4}, where {f}'(1)=2 and {g}'(\sqrt{2})=4, is
  • \displaystyle \frac{1}{\sqrt{2}}
  • \sqrt{2}
  • 1
  • none of these
 If r=\sqrt{x^{2}+y^{2}+z^{2}} and x=2\sin 3t,y=2\cos 3t,z=8t  then \displaystyle \frac{dr}{dt}=
  • \displaystyle \frac{32t}{\sqrt{1+16t^{2}}}
  • \displaystyle \frac{16t}{\sqrt{1+16t^{2}}}
  • \displaystyle \frac{t}{\sqrt{1+16t^{2}}}
  • \displaystyle \frac{4t}{\sqrt{1+16t^{2}}}
If y^x=x^y, then find \displaystyle\frac{dy}{dx}.
  • \displaystyle\frac{x(y\log{y}-y)}{y(x\log{x}-x)}
  • \displaystyle\frac{y(x\log{y}-y)}{x(y\log{x}-x)}
  • {y(x\log{y}-y)}
  • \displaystyle\frac{y(x\log{x}-y)}{x(y\log{y}-x)}
If \displaystyle x=\frac{2t}{1+t^2}, \displaystyle y=\frac{1-t^2}{1+t^2}, then \displaystyle\frac{dy}{dx} at t=2 is
  • \displaystyle\frac{4}{3}
  • \displaystyle -\frac{4}{3}
  • \displaystyle\frac{3}{4}
  • \displaystyle -\frac{3}{4}
\displaystyle\frac{dy}{dx} for y=x^x is
  • x^x(1-\log{x})
  • x^x(1-\log{y})
  • x^x(1+\log{y})
  • x^x(1+\log{x})
Let the function y=f(x) be given by x=t^{5}-5t^{3}-20t+7 and y=4t^{3}-3t^{2}-18t+3, where t\epsilon \left ( -2, 2 \right ). Then f^{'}(x) at t=1 is ?
  • \displaystyle \frac{5}{2}
  • \displaystyle \frac{2}{5}
  • \displaystyle \frac{7}{5}
  • none of these
  • Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
  • Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
  • Assertion is correct but Reason is incorrect
  • Both Assertion and Reason are incorrect
 \displaystyle\frac{dy}{dx} at \displaystyle t=\frac{\pi}{4} for \displaystyle x=a\left[\cos{t}+\frac{1}{2}\log{\tan^2{\frac{t}{2}}}\right] and y=a\sin{t} is
  • 1
  • 0
  • -1
  • \displaystyle \frac { 1 }{ 2 }
The relation between the parameter 't' and the angle \alpha between the tangent to the given curve and the x-axis is given by, 't' equals to
  • \displaystyle \dfrac {\pi}{2}-\alpha
  • \displaystyle \dfrac {\pi}{4}+\alpha
  • \alpha-\displaystyle \dfrac {\pi}{4}
  • \displaystyle \dfrac {\pi}{4}-\alpha
If y={(\tan{x})}^{\displaystyle{(\tan{x})}^{\displaystyle\tan{x}}}, then find \displaystyle\frac{dy}{dx} at \displaystyle x=\frac{\pi}{4}.
  • 0
  • 1
  • -1
  • 2
If x=a\sec^{3}{\theta} and y=a\tan^{3}{\theta}, then \displaystyle\frac{dy}{dx} at \theta=\displaystyle\frac{\pi}{3} is 
  • \displaystyle\frac{\sqrt{3}}{2}
  • \displaystyle\frac{\sqrt{5}}{2}
  • \displaystyle\frac{\sqrt{2}}{3}
  • \displaystyle\frac{\sqrt{4}}{3}
If   \displaystyle y^{x}=x^{\sin y} , find \cfrac{dy}{dx}.
  • \displaystyle \frac{y}{x}\left [ \frac{x \log y-\sin y}{y \:\log x\: \cos y-x} \right ]
  • \displaystyle \frac{y}{x}\left [ \frac{x \log y+\sin y}{y \:\log x\: \cos y+x} \right ]
  • \displaystyle \frac{-y}{x}\left [ \frac{x \log y-\sin y}{y \:\log x\: \cos y-x} \right ]
  • \displaystyle \frac{y}{x}\left [ \frac{x \log y-\sin y}{y \:\log x\: \cos y+x} \right ]
Discuss the applicability of Rolle's theorem to \displaystyle f(x)=\log \left[\frac{x^{2}+ab}{(a+b)x}\right], in the interval [a,b].
  • Yes Rolle's theorem is applicable and the stationary point is x=\sqrt { ab }
  • No Rolle's theorem is not applicable due to the discontinuity in the given interval
  • Yes Rolle's theorem is applicable and the stationary point is x= ab
  • none of these
Verify the Rolle's theorem for the function \displaystyle f(x)=x^{2}-3x+2 on the interval[1,2]
  • No Rolle's theorem is not applicable in the given interval
  • Yes Rolle's theorem is applicable in the given interval and the stationary point x=\frac { 5 }{ 4 }
  • Yes Rolle's theorem is applicable in the given interval and the stationary point x=\frac { 3 }{ 2 }
  • nnone of these
Differentiate \displaystyle x^{\sin^{-1}x} w.r.t. \displaystyle \sin ^{-1}x.
  • \displaystyle x^{\sin ^{-1}x}\left [ \log x+\sin ^{-1}x.\frac{\sqrt{\left ( 1-x^{2} \right )}}{x} \right ]
  • -\displaystyle x^{\sin ^{-1}x}\left [ \log x+\sin ^{-1}x\frac{\sqrt{\left ( 1-x^{2} \right )}}{x} \right ]
  • \displaystyle x^{\sin ^{-1}x}\left [ \log x+\sin ^{-1}x\frac{\sqrt{\left ( 1+x^{2} \right )}}{x} \right ]
  • -\displaystyle x^{\sin ^{-1}x}\left [ \log x+\sin ^{-1}x\frac{\sqrt{\left ( 1+x^{2} \right )}}{x} \right ]
\displaystyle y=(\cot x)^{\sin x}+(\tan  x)^{\cos x}.Find dy/dx 
  • \sin x(\cot x)^{ \sin x-1 }(-cosec ^{ 2 } x)+(\cot x)^{ \sin x }(log\cot x)\cos x+\\\cos { x } { (\tan { x } ) }^{ \cos { x-1 } }\sec ^{ 2 }{ x } +{ (\tan { x } ) }^{ cosx }(\log { tanx)(-sinx) }
  • \sin x(\cot x)^{ \sin x-1 }(-co\sec ^{ 2 } x)+\cos { x } { (\tan { x } ) }^{ \cos { x-1 } }\sec ^{ 2 }{ x }
  • \sin x(\cot x)^{ \sin x-1 }(-sec ^{ 2 } x)+(\cot x)^{ \sin x }(log\cot x)\cos x+\\\cos { x } { (\tan { x } ) }^{ \cos { x+1 } }co\sec ^{ 2 }{ x } +{ (\tan { x } ) }^{ cosx }(\log { tanx)(sinx) }
  • None of these
Verify Rolle's theorem for \displaystyle f(x)=x(x+3)e^{-x/2} in (-3,0)
  • Yes Rolle's theorem is applicable and the stationary point is x=-2
  • Yes Rolle's theorem is applicable and the stationary point is x=-1
  • No Rolle's theorem is not applicable in the given interval
  • Both A and B
Verify Rolle's theorem the function \displaystyle f(x)=x^{3}-4x   on  [-2,2]. If you think it is applicable in the given interval then find the stationary point ?
  • Yes Rolle's theorem is applicable and stationary point is x=\pm \dfrac{2}{\sqrt{3}}
  • No Rolle's theorem is not applicable
  • Yes Rolle's theorem is applicable and x=2\ or\  -2
  • none of these
Verify the Rolle's theorem for the function \displaystyle f(x)=x^{2} in (-1,1)
  • Yes Rolle's theorem is applicable and the stationary point is x=\frac { 1 }{ 2 }
  • Yes Rolle's theorem is applicable and the stationary point is x=0
  • No Rolle's theorem is not applicable in the given interval
  • Both A and B
Verify Rolle's theorem for the function \displaystyle f(x)=10x-x^{2} in the interval [0,10]
  • Yes Rolle's theorem is applicable and the stationary point is x=5
  • Yes Rolle's theorem is applicable and the stationary point is x=4
  • No Rolle's theorem is not applicable in the given interval
  • none of these
If y = \displaystyle (\tan x)^{\log x}, then \cfrac{dy}{dx} =
  • (\tan x)^{\log x} \left[\cfrac{\log \tan x}{x}+\cfrac{\log x}{\tan x}(\sec^2x) \right]
  • \frac { 1 }{ x } { tanx }^{ logx }log(tanx)+\frac { 1 }{ tanx } \sec ^{ 2 }{ x } logx
  • \frac { 1 }{ x } { tanx }^{ logx }log(tanx)+\frac { 1 }{ tanx } \sec ^{ 2 }{ x }
  • none of these
Verify Lagrange's mean value for the function f(x)=\displaystyle \sin x in \displaystyle \left [ 0,\frac{\pi }{2} \right ]
  • No Lagrange's theorem is not applicable
  • Yes Lagrange's theorem is applicable and \displaystyle c =\cos ^{-1}\left ( \frac{2}{\pi } \right )
  • Yes Lagrange's theorem is applicable and \displaystyle c =\sin ^{-1}\left ( \frac{4}{\pi } \right )
  • none of these
Find 'c' of the mean value theorem, if \displaystyle f(x)=x(x-1)(x-2);a=0, b=1/2
  • \displaystyle c=\frac{1+\sqrt{21}}{6}
  • \displaystyle c=\frac{1-\sqrt{21}}{6}
  • \displaystyle c=\frac{1+\sqrt{6}}{3}
  • \displaystyle c=\frac{1-\sqrt{6}}{3}
Discuss the applicapibility of Rolle's Theorem to the function \displaystyle f(x)=x^{2/3} in (-1,1).
  • Yes Rolle's theorem is applicable and the stationary point x=0
  • Yes Rolle's theorem is applicable and the stationary point x=\frac { 1 }{ 2 }
  • No Rolle's theorem is not applicable in the given interval
  • none of these
If \displaystyle x=2 \cos t-cos 2t, y=2\sin t-\sin 2t, find the value of dy/dx.
  • \tan\displaystyle \frac{3t}{2}
  • \tan\displaystyle \frac{-3t}{2}
  • \tan\displaystyle \frac{3t}{4}
  • \tan\displaystyle \frac{-3t}{4}
The function f is defined, where x=2t-\left| t \right|, t\in R. Draw the graph of f for the interval -1\le x\le 1. Also discuss its continuity and differentiability at x=0 
  • continuous and differentiable at x=0
  • not continuous but differentiable at x=0
  • neither continuous nor differentiable at x=0
  • differentiable but not continuous at x=0
The function f(x) = \displaystyle \sin ^{-1}(\cos x)is :-
  • discontinuous at x = 0
  • continuous at x = 0
  • differentiable at x = 0
  • none of these
Examine the origin for continuity and derivability in case of the function f defined by f(x)=x\tan ^{ -1 }{ (1/x) } , x\ne 0 and f(0)=0.
  • continuous but not differentiable at x=0
  • continuous and differentiable at x=0
  • not continuous and not differentiable at x=0
  • none of these
State true or false:
If x=t^2+3t-8, y=2t^2-2t-5, then \dfrac {dy}{dx} at (2, -1) is \dfrac {6}{7}.
  • True
  • False
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