CBSE Questions for Class 11 Engineering Maths Limits And Derivatives Quiz 5 - MCQExams.com

$$\displaystyle \lim_{x\rightarrow \dfrac{\pi}{2}}\dfrac{\left ( 1-\tan \dfrac{x}{2} \right )\left ( 1-\sin x \right )}{\left ( 1+\tan \dfrac{x}{2} \right )\left ( \pi -2x \right )^{3}}$$ is
  • $$0$$
  • $$\displaystyle \frac{1}{32}$$
  • $$\infty $$
  • $$\displaystyle \frac{1}{8}$$
If P(x) is a polynomial such that $$P\left ( x^{2}+1 \right )=\left \{ P\left ( x^{2} \right ) \right \}^{2}+1$$ and $$P(0)=0$$ then $$P^{'}(0)$$ is equal to
  • $$1$$
  • $$0$$
  • $$-$$1
  • none of these
Let $$f\left ( x \right )=\begin{cases}\sin x, x\neq n\pi 
                   \\ 2,  x=n\pi \end{cases}$$, where $$n\epsilon \mathbb{Z}$$ and
$$g\left ( x \right )=\begin{cases}x^{2}+1, x\neq 2 \\
              3, x=2 \end{cases}$$.
Then $$\displaystyle \lim_{x\to 0}g\left ( f\left ( x \right ) \right )$$ is
  • $$0$$
  • $$1$$
  • $$3$$
  • none of these
$$\displaystyle \frac{d}{dx}(\log_{e}\left ( \frac{1+x}{1-x} \right )^{1/4}-\frac{1}{2}\tan^{-1}x.)$$
  • $$\displaystyle \frac{x^{2}}{1-x^{4}}.$$
  • $$\displaystyle \frac{x^{3}}{1-x^{4}}.$$
  • $$\displaystyle \frac{x^{4}}{1-x^{4}}.$$
  • $$-\displaystyle \frac{x^{2}}{1-x^{4}}.$$
Differentiate $$\displaystyle \tan x^{n}+\tan ^{n}x-\tan ^{-1}\frac{a+x^{n}}{1-ax^{n}}.$$
  • $$ \left ( \sec^{2}x^{n} \right ).nx^{n-1}+n\tan^{n-1}x.\sec^{2}x-\left [\dfrac{ 1}{\left ( 1-x^{2n} \right ) }\right ]nx^{n-1}$$
  • $$ \left ( \sec^{2}x^{n} \right ).nx^{n-1}+n\tan^{n-1}x.\sec^{2}x-\left [ \dfrac{1}{\left ( 1+x^{2n} \right )} \right ]nx^{n}$$
  • $$ \left ( \sec^{2}x^{n} \right ).nx^{n-1}+n\tan^{n}x.\sec^{2}x-\left [\dfrac{ 1}{\left ( 1+x^{2n} \right )} \right ]nx^{n-1}$$
  • $$ \left ( \sec^{2}x^{n} \right ).nx^{n-1}+n\tan^{n-1}x.\sec^{2}x-\left [ \dfrac{1}{\left ( 1+x^{2n} \right )} \right ]nx^{n-1}$$
$$\displaystyle \frac{d}{dx}(\tan ^{-1}\frac{\sin x+\cos x}{\cos x-\sin x})$$
  • $$-1$$
  • $$-2$$
  • $$1$$
  • $$2$$
$$\displaystyle \frac{d}{dx}\tan ^{-1}\left(\frac{a \cos x-b\sin x}{b\cos x+a\sin x}\right)$$
  • $$-1$$
  • $$-2$$
  • $$1$$
  • $$2$$
Differentiate the following: $$\displaystyle \cot ^{-1}\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{\left ( 1+\sin x \right )}-\sqrt{\left ( 1-\sin x \right )}}$$
  • $$ \displaystyle \frac{1}{2}.$$
  • $$\displaystyle \frac{-1}{2}.$$
  • $$\displaystyle \frac{1}{4}.$$
  • $$\displaystyle \frac{-1}{4}.$$
$$\displaystyle \dfrac{d}{dx}\tan ^{-1}\left(\dfrac{\cos x}{1+\sin x}\right)$$
  • $$-\displaystyle \dfrac{1}{2}$$
  • $$-\displaystyle \dfrac{1}{4}$$
  • $$-\displaystyle \dfrac{1}{8}$$
  • $$\displaystyle \dfrac{1}{2}$$
Let $$ \displaystyle f\left ( x \right ) $$ be defined by $$ \displaystyle f\left ( x \right )=\left\{\begin{matrix}\sin 2x & \text{if } 0< x\leq \dfrac{\pi}6\\ ax+b& \text{if } \dfrac{\pi}6< x\leq 1\end{matrix}\right. $$. The values of $$a$$ and $$b$$ such that $$ \displaystyle f $$ and $$ \displaystyle {f}' $$ are continuous, are
  • $$ \displaystyle a=1,b=\dfrac1{\sqrt{2}}+\dfrac{\pi}6 $$
  • $$ \displaystyle a=\dfrac1{\sqrt{2}},b=\dfrac1{\sqrt{2}} $$
  • $$ \displaystyle a=1,b=\dfrac{\sqrt{3}}2-\dfrac{\pi}6 $$
  • None of these
A polynomial $$f(x)$$ leaves remainder $$15$$ when divided by $$(x-3)$$ and $$(2x+1)$$ when divided by $$(x-1)^2$$. When $$f$$ is divided by $$(x-3)(x-1)^2,$$ the remainder is
  • $$2x^2+2x+3$$
  • $$2x^2-2x-3$$
  • $$2x^2-2x+3$$
  • none of these
If $$f\left( x \right) $$ is a polynomial of degree $$n(>2)$$ and $$f\left( x \right) =f\left( k-x \right) ,($$ where $$k$$ is a fixed real number$$),$$ then degree of $$f'(x)$$ is
  • $$n$$
  • $$n-1$$
  • $$n-2$$
  • None of these
If $$2f\left( \sin { x }  \right) +f\left( \cos { x }  \right) =x$$, then $$\displaystyle \frac { d }{ dx } f\left( x \right)$$ is
  • $$\sin{x}+\cos{x}$$
  • $$2$$
  • $$\displaystyle \frac { 1 }{ \sqrt { 1-{ x }^{ 2 } }  } $$
  • 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 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 $$ \displaystyle {f}'\left ( x \right )=g\left ( x \right ) $$ and $$ \displaystyle {g}'\left ( x \right )=-f\left ( x \right ) $$ and $$ \displaystyle f\left ( 2 \right )=4={f}'\left ( 2 \right ) $$ then $$ \displaystyle f^{2}\left ( 16 \right )+g^{2}\left ( 16 \right ) $$ is
  • 16
  • 32
  • 64
  • None of these
Let $$f\left( x \right)=\sqrt { x-1 } +\sqrt { x+24-10\sqrt { x-1 }  } ;1<x<26$$ be a real valued function. Then $$f'(x)$$ for $$1<x<26$$ is
  • $$0$$
  • $$\displaystyle \frac { 1 }{ \sqrt { x-1 }  } $$
  • $$2\sqrt { x-1 } -5$$
  • none of these
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
  • $$0$$
  • $$ \displaystyle x+1 $$
  • $$1$$
  • $$-1$$
A curve passing through the point $$(1,1)$$ is such that the intercept made by a tangent to it on x-axis is three times the x co-ordinate of the point of tangency, then the equation of the curve is:
  • $$\displaystyle y=\frac{1}{x^{2}}$$
  • $$\displaystyle y=\sqrt{x}$$
  • $$\displaystyle y=\frac{1}{\sqrt{x}}$$
  • none
If $$\displaystyle \lim_{x\rightarrow 0}(f(x)\:g(x))$$ exists for any functions $$f$$ and $$g$$ then
  • $$\displaystyle \lim_{x\rightarrow a}f(x)$$ and $$\displaystyle \lim_{x\rightarrow a}g(x)$$ exist
  • $$\displaystyle \lim_{x\rightarrow a}f(x)$$ exist but $$\displaystyle \lim_{x\rightarrow a}g(x)$$ may not exist
  • $$\displaystyle \lim_{x\rightarrow a}f(x)$$ may not exist but $$\displaystyle \lim_{x\rightarrow a}g(x)$$ exist
  • $$\displaystyle \lim_{x\rightarrow a}f(x)$$ and $$\displaystyle \lim_{x\rightarrow a}g(x)$$ may not exist
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 $${ S }_{ n }$$ 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
Let $$y = \sqrt { x + \sqrt { x + \sqrt { x + ......\infty  }  }  } $$ then $$\displaystyle\frac { dy }{ dx } $$
  • $$\displaystyle\frac { 1 }{ 2y-1 } $$
  • $$\displaystyle\frac { x }{ x-2y } $$
  • $$\displaystyle\frac { 1 }{ \sqrt { 1+4x } } $$
  • $$\displaystyle\frac { y }{ 2x+y } $$
$$\underset{x\rightarrow0}{lim}\displaystyle\frac{1-cos^{3}x+sin^{3}x+\ell n(1+x^{3})+\ell n(1+cos\,\,x)}{x^{2}-1+2\,cos^{2}x+tan^{4}x+sin^{3}x}$$ is equal to -
  • $$\,\,\displaystyle\frac{3}{4}$$
  • $$\,\,ln2$$
  • $$\,\,\displaystyle\frac{ln2}{4}$$
  • $$\,\,3/2$$

$$ \displaystyle f^{ ' }\left( x \right) =g\left( x \right) $$ 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
$$\displaystyle \frac{d}{dx}\left ( \tan ^{-1}\left ( \frac{\sqrt{x}-x}{1+x^{3/2}} \right ) \right )$$ equals $$\displaystyle ($$for $$x\geq 0)$$
  • $$\displaystyle \frac{1}{2\sqrt{x}(1+x)}-\frac{1}{1+x^{2}}$$
  • $$\displaystyle \frac{1}{2\sqrt{x}(1+x)}+\frac{1}{1+x^{2}}$$
  • $$\displaystyle \frac{1}{1+x}-\frac{1}{1+x^{2}}$$
  • None of these
Evaluate $$\displaystyle \lim_{n\rightarrow \infty }\left [ \frac{n!}{n^{n}} \right ]^{1/n}$$.
  • $$\displaystyle\frac{1}{e}$$
  • $$\displaystyle\frac{1}{t}$$
  • $$\displaystyle\frac{1}{n}$$
  • $$none\ of\ above$$
$$\displaystyle \frac{d}{dx}\left ( \tan ^{-1}\left ( \frac{a-x}{1+ax} \right ) \right )$$ equals if ax > -1
  • $$\displaystyle \frac{a}{1+x^{2}}$$
  • $$\displaystyle \frac{1}{1+x^{2}}$$
  • $$\displaystyle- \frac{a}{1+x^{2}}$$
  • $$\displaystyle -\frac{1}{1+x^{2}}$$
If $$f(x) = \displaystyle \left | \cos x-\sin x \right |$$ then $$\displaystyle f'\left ( \dfrac{\pi}4 \right )$$ is equal to-
  • $$\displaystyle \sqrt{2}$$
  • $$\displaystyle -\sqrt{2}$$
  • $$0$$
  • $$Does\ not\ exist$$
If $$y = sec  x^0$$ then $$\displaystyle \frac{dy}{dx} = $$
  • $$sec x tan x$$
  • $$sec x^0 tan x^0$$
  • $$\displaystyle \frac{\pi}{180} sec x^0 tan x^0$$
  • $$\displaystyle \frac{180}{\pi} sec x^0 tan x^0$$
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