CBSE Questions for Class 11 Commerce Applied Mathematics Differentiation Quiz 8 - MCQExams.com

If $$r=\left[2\phi +\cos^2\left(2\phi +\dfrac{\pi}4\right)\right]^{\tfrac12},$$ then what is the value of the derivative of $$\dfrac{dr}{d\phi}$$ at $$\phi=\dfrac{\pi}4?$$
  • $$2\left(\displaystyle\frac{1}{\pi+1}\right)^{\tfrac12}$$
  • $$2\left(\displaystyle\frac{2}{\pi+1}\right)^{2}$$
  • $$\left(\displaystyle\frac{2}{\pi+1}\right)^{\tfrac12}$$
  • $$2\left(\displaystyle\frac{2}{\pi+1}\right)^{\tfrac12}$$
$$\displaystyle \frac { d }{ dx } \left( { x }^{ x } \right) $$ is equal to:
  • $$\displaystyle { x }^{ x }\log { \left( \dfrac ex \right) } $$
  • $$\displaystyle { x }^{ x }\log { ex } $$
  • $$\displaystyle \log { ex } $$
  • $$\displaystyle { x }^{ x }\log { x } $$
If $$x^{2} + y^{2} = t + \dfrac {1}{t}$$ and $$x^{4} + y^{4} = t^{2} + \dfrac {1}{t^{2}}$$ then $$\dfrac {dy}{dx} =$$
  • $$-\dfrac {x}{y}$$
  • $$\dfrac {-y}{x}$$
  • $$\dfrac {x^{2}}{y^{2}}$$
  • $$\dfrac {y^{2}}{x^{2}}$$
$$f(x) = \log \left (e^{x} \left (\dfrac {x - 2}{x + 2}\right )^{\dfrac {3}{4}} \right ) \Rightarrow f'(0) =$$
  • $$\dfrac {1}{4}$$
  • $$4$$
  • $$\dfrac {-3}{4}$$
  • $$1$$
If $$xy \neq 0, x + y \neq 0$$ and $$x^{m}y^{n} = (x + y)^{m + n}$$ where $$m, n\in N$$ then $$\dfrac {dy}{dx} =$$
  • $$\dfrac {y}{x}$$
  • $$\dfrac {x + y}{xy}$$
  • $$xy$$
  • $$\dfrac {x}{y}$$
If $$sin^{-1} x + sin^{-1} y=\frac{\pi}{2}$$, then $$\dfrac{dy}{dx}$$ is equal to
  • $$\dfrac{x}{y}$$
  • $$-\dfrac{x}{y}$$
  • $$\dfrac{y}{x}$$
  • $$-\dfrac{y}{x}$$
If $$x=a\cos { \theta  } +a\log { \tan { \dfrac { \theta  }{ 2 }  }  } $$ and $$y=a\sin { \theta  } $$, then $$\dfrac { dy }{ dx } $$ is equal to
  • $$\cot { \theta } $$
  • $$\tan { \theta } $$
  • $$\sin { \theta } $$
  • $$\cos { \theta } $$
If $${ e }^{ x }\left( 1+x \right) \sec ^{ 2 }{ x{ e }^{ x } } dx =f\left( x \right) +$$ constant, then $$f\left( x \right) $$ is equal to
  • $$\cos { \left( x{ e }^{ x } \right) } $$
  • $$\sin { \left( x{ e }^{ x } \right) } $$
  • $$2\tan ^{ -1 }{ \left( x \right) } $$
  • $$\tan { \left( x{ e }^{ x } \right) } $$
State true or false, If $$y=\log x^2$$ then $$\dfrac{dy}{dx}=\dfrac2x$$
  • True
  • False
Let $$f(x)=cos^{-1}\left [ \frac{1}{\sqrt{13}}(2cos x-3 sin x) \right ]$$. Then $$f'(0.5)=$$____
  • $$0.5$$
  • $$1$$
  • $$0$$
  • $$-1$$
Let $$f(x)=(x-1) ^4(x-2) ^n, n\in N$$. Then $$f(x)$$ has
  • A maximum at $$x=1$$ if $$n$$ is odd.
  • A maximum at $$x=1$$ if $$n$$ is even.
  • A minimum at $$x=1$$ if $$n$$ is even.
  • A minimum at $$x=2$$ if $$n$$ is even.
If $${ x }^{ y }={ e }^{ x-y }$$ then $$\cfrac { dy }{ dx } $$ is equal to
  • $$\cfrac { \log { x } }{ \log { (x-y) } } $$
  • $$\cfrac { { e }^{ x } }{ { x }^{ x-y } } $$
  • $$\cfrac { \log { x } }{ { (1+\log { x } ) }^{ 2 } } $$
  • $$\cfrac { 1 }{ y } -\cfrac { 1 }{ x-y } $$
  • $$\dfrac{y(x -y)}{x^2}$$
If $$y = f(x^{2} + 2)$$ and $$f'(3) = 5$$, then $$\dfrac {dy}{dx}$$ at $$x = 1$$ is _____
  • $$5$$
  • $$25$$
  • $$15$$
  • $$10$$
If f(x) is a function such that $$f^{\prime \prime}(x)+f(x)=0$$ and $$g(x)=[f(x)]^2+[f'(x)]^2$$ and g(3)=8, then $$g(8)= $$_____
  • $$0$$
  • $$3$$
  • $$5$$
  • $$8$$
If $$y = \tan^{-1} \left (\dfrac {1}{1 + x + x^{2}}\right ) + \tan^{-1} \left (\dfrac {1}{x^{2} + 3x + 2}\right ) + \tan^{-1} \left (\dfrac {1}{x^{2} + 5x + 6}\right ) + .... +$$ upto $$n$$ terms then $$\dfrac {dy}{dx}$$ at $$x = 0$$ and $$n = 1$$ is equal to
  • $$\dfrac {1}{2}d$$
  • $$-\dfrac {1}{2}$$
  • $$0$$
  • $$\dfrac {1}{3}$$
If $$y={\cot}^{-1}\begin{bmatrix}\dfrac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\end{bmatrix}$$, where $$0 < x < \dfrac{\pi}{2}$$, then $$\dfrac{dy}{dx}$$ is equal to
  • $$-\dfrac{1}{2}$$
  • 2
  • $$\sin x + \cos x$$
  • $$\sin x - \cos x$$
If $$x^ay^b=(x-y)^{a+b}$$, then the value of $$\dfrac{dy}{dx}-\dfrac{y}{x}$$ is equal to
  • $$\dfrac{a}{b}$$
  • $$\dfrac{b}{a}$$
  • 1
  • 0
If $${ x }^{ m }+{ y }^{ m }=1$$ such that $$\cfrac { dy }{ dx } =-\cfrac { x }{ y } $$, then what should be the value of $$m$$?
  • $$0$$
  • $$1$$
  • $$2$$
  • None of the above
If $$\sec \left (\dfrac {x + y}{x - y}\right ) = a$$, then $$\dfrac {dy}{dx}$$ is.
  • $$\dfrac {x}{y}$$
  • $$\dfrac {y}{x}$$
  • $$y$$
  • $$x$$
If $$y=\sqrt{x+\sqrt{y+\sqrt{x+\sqrt{y+.....\infty}}}},$$ then $$\displaystyle\frac{dy}{dx}$$ is equal to?
  • $$\displaystyle\frac{y+x}{y^2-2x}$$
  • $$\displaystyle\frac{y^3-x}{2y^2-2xy-1}$$
  • $$\displaystyle\frac{y^3+x}{2y^2-x}$$
  • None of these
If $${ 2 }^{ x }+{ 2 }^{ y }={ 2 }^{ x+y }$$, then the value of $$\cfrac { dy }{ dx } $$ at $$(1,1)$$ is equal to
  • $$-2$$
  • $$-1$$
  • $$0$$
  • $$1$$
  • $$2$$
If $$y = x\tan y$$, then $$\dfrac {dy}{dx}$$ is equal to.
  • $$\dfrac {\tan y}{x - x^{2} - y^{2}}$$
  • $$\dfrac {y}{x - x^{2} - y^{2}}$$
  • $$\dfrac {\tan y}{y - x}$$
  • $$\dfrac {\tan x}{x - y^{2}}$$
If $$\log _{ 10 }{ \left( \cfrac { { x }^{ 2 }-{ y }^{ 2 } }{ { x }^{ 2 }+{ y }^{ 2 } }  \right)  } =2$$, then $$\cfrac { dy }{ dx } =$$............
  • $$-\cfrac { 99x }{ 101y } $$
  • $$\cfrac { 99x }{ 101y } $$
  • $$-\cfrac { 99y }{ 101x } $$
  • $$\cfrac { 99y }{ 101x } $$
If $$sin  x = \dfrac{2t}{1 + t^2}, tan y = \dfrac{2t}{1 - t^2}, $$ then $$\dfrac{dy}{dx}$$ is equal to
  • -1
  • 2
  • 0
  • 1
The slope of the tangent to the curve $$y^2 e^{xy} = 9e^{-3} x^2$$ at $$(-1, 3)$$ is
  • $$\dfrac{-15}{2}$$
  • $$\dfrac{-9}{2}$$
  • $$15$$
  • $$\dfrac{15}{2}$$
  • $$\dfrac{9}{2}$$
If $$x\sin (a + y) + \sin a\cos (a + y) = 0$$, then $$\dfrac {dy}{dx}$$ is equal to
  • $$\dfrac {\sin^{2}(a + y)}{\sin a}$$
  • $$\dfrac {\cos^{2}(a + y)}{\cos a}$$
  • $$\dfrac {\sin^{2}(a + y)}{\cos a}$$
  • $$\dfrac {\cos^{2}(a + y)}{\sin a}$$
If $$2^x + 2^y = 2^{x + y}$$, then $$\dfrac{dy}{dx}$$ is equal to
  • $$\dfrac{2^x + 2^y}{2^x - 2^y}$$
  • $$\dfrac{2^x + 2^y}{1 + 2^{x + y}}$$
  • $$2^{x - y} \left( \dfrac{2^y - 1}{1 - 2^x} \right )$$
  • $$\dfrac{2^{x + y} - 2^x}{2^y}$$
If $$x^{y} = e^{x - y}$$, then $$\dfrac {dy}{dx}$$ is equal to.
  • $$\dfrac {\log x}{1 + \log x}$$
  • $$\dfrac {\log x}{1 - \log x}$$
  • $$\dfrac {\log x}{(1 + \log x)^{2}}$$
  • $$\dfrac {y\log x}{x(1 + \log x)^{2}}$$
If $$y = \dfrac {1}{1 + x + x^{2}}$$, then $$\dfrac {dy}{dx}$$ is equal to
  • $$y^{2} (1 + 2x)$$
  • $$\dfrac {-(1 + 2x)}{y^{2}}$$
  • $$\dfrac {1 + 2x}{y^{2}}$$
  • $$-y (1 + 2x)$$
  • $$-y^{2} (1 + 2x)$$
If $$u = \tan^{-1}\left (\dfrac {\sqrt {1 - x^{2}} - 1}{x}\right )$$ and $$v= \sin^{-1} x$$, then $$\dfrac {du}{dv}$$ is equal to
  • $$\sqrt {1 - x^{2}}$$
  • $$-\dfrac {1}{2}$$
  • $$1$$
  • $$-x$$
  • $$-2$$
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