CBSE Questions for Class 11 Commerce Applied Mathematics Differentiation Quiz 6 - MCQExams.com

If for a non-zero $$x,$$ the function $$f(x)$$ satisfies the equation $$\displaystyle af\left( x \right)+bf\left( \frac { 1 }{ x }  \right) =\frac { 1 }{ x } -5\left( a\neq b \right) $$ then $$f'(x)$$ is equal to
  • $$\displaystyle \frac { 1 }{ { b }^{ 2 }-{ a }^{ 2 } } \left( \frac { a }{ { x }^{ 2 } } +b \right) $$
  • $$\displaystyle \frac { 1 }{ { a }^{ 2 }-{ b }^{ 2 } } \left( \frac { a }{ { x }^{ 2 } } +b \right) $$
  • $$\displaystyle \frac { 1 }{ { a }^{ 2 }-{ b }^{ 2 } } \left( \frac { a }{ { x }^{ 2 } } -b \right) $$
  • none of these
If for all $$x,y$$ the function $$f$$ is defined by $$f\left( x \right)+f\left( y \right)+f\left( x \right).f\left( y \right)=1$$ and $$f\left( x \right)>0$$, then
  • $$f'\left( x \right) $$ does not exist
  • $$f'\left( x \right) =0$$ for all $$x$$
  • $$f'\left( 0 \right) <f'\left( 1 \right) $$
  • None of these
If $$ \displaystyle {f}'\left ( x \right )=f\left ( x \right ) $$ for all x and $$ \displaystyle {f}'\left ( 0\right )=4 $$ then $$ \displaystyle f\left ( x \right ) $$ is equal to
  • $$ \displaystyle 2e^{2x} $$
  • $$ \displaystyle e^{4x} $$
  • $$ \displaystyle x^{4}+4x^{2}+4x $$
  • $$ \displaystyle 4e^{x} $$
If the functions $$ \displaystyle f\left ( x \right )=\sin \left ( x+a \right ) $$ and $$ \displaystyle g\left ( x \right )=b\sin x+c\cos x $$ satisfy $$ \displaystyle f\left ( 0 \right )=g\left ( 0 \right ) $$ and $$ \displaystyle {f}'\left ( 0 \right )={g}'\left ( 0 \right ) $$ then
  • $$ \displaystyle b=\dfrac{\pi}2 $$
  • $$ \displaystyle b=\cos a $$
  • $$ \displaystyle c=\sin a $$
  • $$ \displaystyle c=\cos a $$
If $$ \displaystyle y=\sec ^{ -1 }{ \left( \frac { x+1 }{ x-1 }  \right)  } +\sin ^{ -1 }{ \left( \frac { x-1 }{ x+1 }  \right)  }  $$ then $$ \displaystyle \frac{dy}{dx} $$ is equal to
  • $$0$$
  • $$ \displaystyle x+1 $$
  • $$1$$
  • $$-1$$
If $$ \displaystyle \sqrt{1-x^{6}} +\sqrt{1-y^{6}}=a\left ( x^{3}-y^{3} \right ) $$ and $$ \displaystyle \frac{dy}{dx}=f\left ( x,y \right )\sqrt{\frac{1-y^{6}}{1-x^{6}}} $$ then
  • $$ \displaystyle f\left ( x,y \right )=\dfrac{y}{x}$$
  • $$ \displaystyle f\left ( x,y \right )=\dfrac{x^{2}}{y^{2}} $$
  • $$ \displaystyle f\left ( x,y \right )=2\dfrac{y^{2}}{x^{2}}$$
  • $$ \displaystyle f\left ( x,y \right )=\dfrac{y^{2}}{x^{2}}$$
If $$f(x+y)=f(x)f(y) \forall  x,y$$ and $$f(5)=2, f'(0)=3$$; then $$f'(5)$$ is equal to-
  • $$2$$
  • $$4$$
  • $$6$$
  • $$8$$
Consider the function: $$ f(-\infty,\infty)\rightarrow (-\infty,\infty)$$ defined by $$\displaystyle f(x)=\frac{x^{2}-ax+1}{x^{2}+ax+1},0<a<2$$
Which of the following is true?
  • $$(2+a)^{2}f''(1)+(2-a)^{2}f'(-1)=0$$
  • $$(2-a)^{2}f''(1)-(2+a)^{2}f'(-1)=0$$
  • $$ f(1)+f(-1)=(2-a)^{2}$$
  • $$ f(1)f(-1)=-(2-a)^{2}$$
Differentiation of $$x \sin x$$ with respect to $$x$$ is
  • $$\displaystyle x\cos x+\sin x$$
  • $$\displaystyle x\sin x + \cos x$$
  • $$ \displaystyle x\cos x$$
  • $$\displaystyle x\cos x - \sin x$$
Let $$y = \sqrt { x + \sqrt { x + \sqrt { x + ......\infty  }  }  } $$ then $$\displaystyle\frac { dy }{ dx } $$
  • $$\displaystyle\frac { 1 }{ 2y-1 } $$
  • $$\displaystyle\frac { x }{ x-2y } $$
  • $$\displaystyle\frac { 1 }{ \sqrt { 1+4x } } $$
  • $$\displaystyle\frac { y }{ 2x+y } $$
If $$x=\sqrt { { a }^{ \sin ^{ -1 }{ t }  } } $$ and $$y=\sqrt { { a }^{ \cos ^{ -1 }{ t }  } } $$ then $$\displaystyle\frac{dy}{dx}=$$
  • $$\displaystyle\frac{y}{x}$$
  • $$\displaystyle\frac{x}{y}$$
  • $$\displaystyle-\frac{x}{y}$$
  • $$\displaystyle-\frac{y}{x}$$
Let $$f(x+y)=f(x)f(y)$$ for all $$x$$ and $$y$$. If $$f(7)=2$$ and $${f}'\left ( 0 \right )=3$$, then $${f}'\left ( 7 \right )$$ is equal to
  • $$5$$
  • $$6$$
  • $$0$$
  • none of these

$$ \displaystyle f^{ ' }\left( x \right) =g\left( x \right) $$ and $$ \displaystyle g^{ ' }\left( x \right) =-f\left( x \right)$$ for all real x and $$ \displaystyle f\left( 5 \right) =2=f^{ ' }\left( 5 \right) $$ then $$ \displaystyle f^{ 2 }\left( 10 \right) +g^{ 2 }\left( 10 \right) $$ is -
  • $$2$$
  • $$4$$
  • $$8$$
  • None of these
If $$\sqrt { y + x } + \sqrt { y - x } = c$$, then $$\displaystyle\frac { dy }{ dx } $$ is equal to
  • $$\displaystyle\frac { 2x }{ { c }^{ 2 } } $$
  • $$\displaystyle\frac { x }{ y+\sqrt { { y }^{ 2 }-{ x }^{ 2 } } } $$
  • $$\displaystyle\frac { y-\sqrt { { y }^{ 2 }-{ x }^{ 2 } } }{ x } $$
  • $$\displaystyle\frac { { c }^{ 2 } }{ 2y } $$
If $$g$$ is the inverse of $$f$$ and $$f'(x) = \displaystyle \frac{1}{1+x^{3}}$$ then $$g'(x)$$ is equal to-
  • $$\displaystyle 1+\left [ g\left ( x \right ) \right ]^{3}$$
  • $$\displaystyle \frac{-1}{2\left ( 1+x^{2} \right )}$$
  • $$\displaystyle \frac{1}{2\left ( 1+x^{2} \right )}$$
  • None of these

If $$ \displaystyle x\sqrt { \left( 1+y \right)  } +y\sqrt { \left( 1+x \right)  } =0$$. then $$ \displaystyle \frac{dy}{dx}$$ equals -
  • $$ \displaystyle \frac { 1 }{ \left( 1+x \right) ^{ 2 } } $$
  • -$$ \displaystyle \frac { 1 }{ \left( 1+x \right) ^{ 2 } } $$
  • $$ \displaystyle -\frac { 1 }{ \left( 1+x \right) } +\frac { 1 }{ \left( 1+x \right) ^{ 2 } } $$
  • None of these

If $$ \displaystyle y=\dfrac { x }{ a+\dfrac { x }{ b+y }  } $$, then $$ \displaystyle \frac{dy}{dx}$$ is
  • $$ \displaystyle \frac{a}{ab+2ay}$$
  • $$ \displaystyle \frac{b}{ab+2by} $$
  • $$ \displaystyle \frac{a}{ab+2by} $$
  • $$ \displaystyle \frac{b}{ab+2ay}$$
If $${ S }_{ n }$$ denotes the sum of $$n$$ terms of $$g.p$$. whose common ratio is $$r$$, then $$\displaystyle \left( r-1 \right) \frac { d{ S }_{ n } }{ dr } $$ is equal to
  • $$\left( n-1 \right) { S }_{ n }+n{ S }_{ n-1 }$$
  • $$\left( n-1 \right) { S }_{ n }-n{ S }_{ n-1 }$$
  • $$\left( n-1 \right) { S }_{ n }$$
  • None of these
The values of $$f'(1)$$ is
  • $$0$$
  • $$1$$
  • $$2$$
  • $$3$$
If $${ y }^{ 2 } + { b }^{ 2 } = 2xy$$, then $$\displaystyle\frac { dy }{ dx } $$ equals
  • $$\displaystyle\frac { 1 }{ xy-{ b }^{ 2 } } $$
  • $$\displaystyle\frac { y }{ y-x } $$
  • $$\displaystyle\frac { xy-{ b }^{ 2 } }{ { \left( y-x \right) }^{ 2 } } $$
  • $$\displaystyle\frac { xy-{ b }^{ 2 } }{ y } $$
Find $$ \displaystyle  \frac{dy}{dx}$$ if $$ y=x^{x} $$
  • $$ \displaystyle x^{x}\left ( lnx+1 \right ) $$
  • $$ \displaystyle x^{x}\left ( lnx-1 \right ) $$
  • $$ \displaystyle x .x^{x-1}$$
  • $$ \displaystyle x^{x-1}\left ( lnx+1 \right ) $$

Find $$ \displaystyle \frac { dy }{ dx } $$,   if $$x + y =\displaystyle \sin { \left( x-y \right)  } $$
  • $$ \displaystyle \frac { \cos { \left( x-y \right) -1 } }{ \cos { \left( x-y \right) +1 } }$$
  • $$ \displaystyle \frac { \cos { \left( x-y \right) +1 } }{ \cos { \left( x-y \right) -1 } }$$
  • $$ \displaystyle \frac { \cos { \left( x+y \right) +1 } }{ \cos { \left( x-y \right) -1 } }$$
  • $$ \displaystyle \frac { \cos { \left( x+y \right) -1 } }{ \cos { \left( x-y \right)+1 } }$$
A balloon which always remains spherical, has a variable diameter $$\displaystyle \frac {3} {2}(2x+3)$$. The rate of change of volume with respect to $$x$$ will be
  • $$\displaystyle \frac {27 \pi}{8} (2x-3)^2$$
  • $$\displaystyle \frac {27 \pi}{8} (2x+3)^2$$
  • $$\displaystyle \frac {27 \pi}{8} (3x-2)^2$$
  • $$\displaystyle \frac {8} {27 \pi} (2x+3)^2$$
The surface area of a cube is increasing at the rate of $$2 cm^2/sec$$. When its edge is 90 cm, the volume is increasing at the rate of.
  • $$1620 cm^3/sec$$
  • $$810 cm^3/sec$$
  • $$405 cm^3/sec$$
  • $$45 cm^3/sec$$
If $$f(x) = \displaystyle \left | \cos x-\sin x \right |$$ then $$\displaystyle f'\left ( \dfrac{\pi}4 \right )$$ is equal to-
  • $$\displaystyle \sqrt{2}$$
  • $$\displaystyle -\sqrt{2}$$
  • $$0$$
  • $$Does\ not\ exist$$
The surface area of a spherical bubble is increasing at the rate of $$2 cm^2/s$$. When the radius of the bubble is 6 cm, then the rate by which the volume of the bubble increasing is.
  • $$6 cm^3/sec$$
  • $$9 cm^3/sec$$
  • $$3 cm^3/sec$$
  • $$13 cm^3/sec$$
The radius of a sphere is changing at the rate of 0.1 cm/sec. The rate of change of its surface area when the radius is 200 cm, is.
  • $$8\pi cm^2/sec$$
  • $$12\pi cm^2/sec$$
  • $$160\pi cm^2/sec$$
  • $$200\pi cm^2/sec$$
If $$\displaystyle y=\left | \cos x \right |+\left | \sin x \right |$$ then $$\displaystyle \frac{dy}{dx}$$ at $$x=\dfrac{2\pi }{3}$$ is:
  • $$\displaystyle \frac{1-\sqrt{3}}{2}$$
  • $$0$$
  • $$\displaystyle \frac{\sqrt{3}-1}{2}$$
  • None of these
The surface area of a sphere when its volume is increasing at the same rate as its radius, is.
  • 1 sq. units
  • $$\dfrac {1} {2 \sqrt {\pi}}$$ sq. units
  • $$4 \pi$$ sq. units
  • $$\dfrac {4 \pi} {3 }$$ sq. units
If $$f(x) = \displaystyle \left | \left ( x-4 \right )\left ( x-5 \right ) \right |$$ then $$f'(x)$$ is equal to-
  • $$-2x + 9$$, for all $$\displaystyle x\: \: \epsilon \: \: R $$
  • $$2x - 9$$, if $$4 < x < 5$$
  • $$-2x + 9$$, if $$4 < x < 5$$
  • None of these
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Practice Class 11 Commerce Applied Mathematics Quiz Questions and Answers