CBSE Questions for Class 11 Engineering Maths Limits And Derivatives Quiz 16 - MCQExams.com

$$\lim _ { x \rightarrow 0 } \int _ { 0 } ^ { x } \dfrac { \left( \tan ^ { - 1 } t \right) ^ { 2 } } { \sqrt { 1 + x ^ { 2 } } } d t$$  is equal to
  • $$\pi ^ { 2 }$$
  • $$\dfrac { \pi ^ { 2 } } { 2 }$$
  • $$\dfrac { \pi ^ { 2 } } { 4 }$$
  • None of these
If $$x = 3\cos \theta - 2\cos^{3} \theta$$ and $$y = 3\sin \theta - 2\sin^{3}\theta$$, then $$\dfrac {dy}{dx} =$$
  • $$\sin \theta$$
  • $$\cos \theta$$
  • $$\tan \theta$$
  • $$\cot \theta$$
the value of $$\underset { x\longrightarrow \infty  }{ lim } \frac { { X }^{ 4 }sin\left( \frac { 1 }{ x }  \right) +{ x }^{ 3 } }{ 1+\left| x \right| ^{ 3 } } $$
  • 1
  • -1
  • 2
  • does not exist
If $$y=e^{\sin^{2}x +\sin^{4}x + \sin ^{6}x +....+\infty} $$ , then $$\dfrac {dy}{dx} =?$$
  • $$2~\tan x~\sec^2x~e^{\tan^2~x}$$
  • $$2~\sec^2x~e^{\tan^2~x}$$
  • $$\sec^2x~e^{\tan^2~x}$$
  • $$\tan x~\sec x~e^{\tan^2~x}$$
$$\underset { x\rightarrow 1 }{ lim } { \left[ cosec { \dfrac { \pi x }{ 2 }  }  \right]  }^{ { 1 }/{ \left( 1-x \right)  } }$$ (where $$[.]$$ represents the greatest integer function) is equal to
  • $$0$$
  • $$1$$
  • $$\infty$$
  • $$Does \ not \ exist$$
$$\lim _{ x\rightarrow 0 }{ \dfrac { 1-\cos { x }  }{ { { x\log { (1+x) }  } } }  } $$ =
  • 1
  • 0
  • -1
  • 1/2
If $$\alpha$$ and $$\beta$$ be the roots of the equation $$ax^{2} + bx + c = 0$$ then $$\displaystyle \lim_{x\rightarrow \dfrac {1}{\alpha}} \sqrt {\dfrac {1 - \cos^{2} (cx^{2} + bx + a)}{4(1 - \alpha x)^{2}}}$$
  • Does not exist
  • Equals $$\left |\dfrac {c}{2\alpha} \left (\dfrac {1}{\alpha} +\dfrac {1}{\beta}\right )\right |$$
  • Equals $$\left |\dfrac {c}{2\alpha} \left (\dfrac {1}{\alpha} - \dfrac {1}{\beta}\right )\right |$$
  • Equals $$\left |\dfrac {c}{2} \left (\dfrac {1}{\alpha} +\dfrac {1}{\beta}\right )\right |$$
The value of $$\begin{matrix} lim \\ x\rightarrow \frac { 1 }{ \sqrt { 2 }  }  \end{matrix}\dfrac { x-cos\left( { sin }^{ -1 }x \right)  }{ 1-tan\left( { sin }^{ -1 }x \right)  } is$$
  • $$-\dfrac { 1 }{ \sqrt { 2 } } $$
  • $$\dfrac { 1 }{ \sqrt { 2 } } $$
  • $$\sqrt { 2 } $$
  • $$-\sqrt { 2 } $$
$$\overset {lim}{x \rightarrow \pi/2} \dfrac{\sin(x \ cos x)}{cos(x\, \ sin x)}$$ is equal to
  • $$1$$
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{2}{\pi}$$
  • does not exist
$$\underset { x\rightarrow 0 }{ lim } \dfrac { sec4x-sec2x }{ sec3x-secx } =$$
  • 3/2
  • 2/3
  • 1/3
  • 3/4
$$\cfrac { d }{ dx } \left( 3\cos { \left( \cfrac { \pi  }{ 6 } +{ x }^{ 0 } \right)  } -4\cos ^{ 3 }{ \left( \cfrac { \pi  }{ 6 } +{ x }^{ 0 } \right)  }  \right) =....\quad $$
  • $$\cos { \left( 3{ x }^{ 0 } \right) } $$
  • $$\cfrac { \pi }{ 60 } \sin { \left( 3{ x }^{ 0 } \right) } $$
  • $$\cfrac { \pi }{ 60 } \cos { \left( 3{ x }^{ 0 } \right) } $$
  • $$-\cfrac { \pi }{ 60 } \sin { \left( 3{ x }^{ 0 } \right) } $$
Value of $$L=\displaystyle\lim_{n\rightarrow \infty n}\dfrac{1}{4}\left[1.\left(\displaystyle\sum_{k=1}^{n}k\right)+2.\left(\sum_{k=1}^{n-1}k\right)+3.\left(\sum_{k=1}^{n-2}k\right)+...+n.1\right]$$ is
  • $$1/24$$
  • $$1/12$$
  • $$1/6$$
  • $$1/3$$
If $$y=4^{log2sinx}+9^{log3cosx}$$  then  $$\frac{dy}{dx}$$ = 
  • 0
  • 1
  • -1
  • 2
Let $$\displaystyle x^{cos\,y} + y^{cox\,x} = 5 $$ , Then 
  • $$ at x = 0 , y =0 ,{y}' =0 $$
  • $$ at x = 0 , y = 1 , {y}' = 0 $$
  • $$ at x = y , y = 1 , {y}' = -1 $$
  • $$ at x = 1 , y = 0 , {y}' = 1 $$
The value of $$\displaystyle \lim_{n\infty}\dfrac{1}{n^2}\left\{ sin^3\dfrac{\pi}{4n}+2sin^3\dfrac{2\pi}{4n} + ... + nsin^3\dfrac{n\pi}{4n}\right\}$$ is equal to 
  • $$\dfrac{\sqrt{2}}{9\pi^2}(52-15 \pi )$$
  • $$\dfrac{2}{9\pi^2}(52-15n)$$
  • $$\dfrac{1}{9\pi^2}(15n-15)$$
  • None of these
If $$ \displaystyle \lim _{x \rightarrow 0} \dfrac{x^{n}-\sin x^{n}}{x-\sin ^{n} x} $$ is non-zero finite, then $$ n $$ must be equal
  • 4
  • 1
  • 2
  • 3
If $$ L=\displaystyle \lim _{x \rightarrow 0} \dfrac{\sin x+a e^{x}+b e^{-x}+c \ln (1+x)}{x^{3}}=\infty $$

Equation $$ a x^{2}+b x+c=0 $$ has
  • real and equal roots
  • complex roots
  • unequal positive real roots
  • unequal roots
$$\displaystyle \lim _{x \rightarrow \infty} \dfrac{2+2 x+\sin 2 x}{(2 x+\sin 2 x) e^{\sin x}} $$ is equal to
  • 0
  • 1
  • -1
  • Does not exists
$$\displaystyle  \lim _{x \rightarrow \pi / 2} \dfrac{\sin (x \cos x)}{\cos (x \sin x)} $$ is equal to
  • 0
  • p/2
  • p
  • 2p
The value of $$ \displaystyle \lim _{x \rightarrow 0} \dfrac{\sqrt{\dfrac{1}{2}(1-\cos 2 x)}}{x} $$ is
  • 1
  • -1
  • 0
  • None of these
Solution of the equation $$ \dfrac {dy }{dx}  + \dfrac {1}{x} \tan y = \dfrac {1}{x^2}  \tan y \sin y$$ is 
  • $$ 2x = sin y ( x ) $$
  • $$ 2x =sin y ( 1 +cx^2) $$
  • $$ 2x +sin y ( 1 +cx^2 ) $$
  • None of these
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