CBSE Questions for Class 11 Engineering Maths Limits And Derivatives Quiz 13 - MCQExams.com

For the function, $$f(x) = (x - \frac{1}{x})^2$$, the first derivative with respect to x is 
  • $$2 (x - \frac{1}{x^3})$$
  • $$2 (x - \frac{1}{x})$$
  • $$2 (x + \frac{1}{x^2})$$
  • $$2 (x - \frac{1}{x^2})$$
$$\displaystyle \lim _{ \theta \rightarrow \pi /2 }{ \dfrac { 1-\sin  \theta  }{ (\pi /2-\theta )\cos { \theta  }  }  } $$ is equal to
  • $$1$$
  • $$-1$$
  • $$1/2$$
  • $$-1/2$$
If x + y = sin (x + y) then $$\dfrac{dy}{dx}$$ =
  • $$\dfrac{1}{2}$$
  • 0
  • -1
  • $$\dfrac{1}{3}$$
$$\lim_{n\rightarrow \infty}\dfrac{1}{n^{2}}\left[\sin^{3}\dfrac{\pi}{4n}+2\sin^{3}\dfrac{2\pi}{4n}+3\sin^{3}\dfrac{3\pi}{4n}+....+n\sin^{3}\dfrac{n\pi}{4n}\right]=$$
  • $$\dfrac{\sqrt{2}}{9\pi^{2}}\left(52-15\pi\right)$$
  • $$\dfrac{\sqrt{2}}{9\pi^{2}}\left(52+15\pi\right)$$
  • $$\dfrac{\sqrt{2}}{9\pi}\left(52-17\pi\right)$$
  • $$\dfrac{\sqrt{2}}{9\pi^{2}}\left(52+17\pi\right)$$
$$\lim- {x\to 0}$$ $$\dfrac{1- cos(1 - cos4x)}{x^4}$$ is equal to : 
  • 4
  • 16
  • 32
  • None of these
$$\underset { x\rightarrow 0 }{ lim } \left( \dfrac { \left( 1+x \right) ^{ \dfrac { 1 }{ x }  } }{ e }  \right) ^{ \dfrac { 1 }{ sinx }  }$$ is equal to 
  • $$\sqrt { e } $$
  • e
  • $$\dfrac { 1 }{ \sqrt { e } } $$
  • 1/e
$$ \underset { x\rightarrow 0 }{ lim } \cfrac { { \left( 25 \right)  }^{ x }-2\left( 15 \right)^ x+{ 9 }^{ x } }{ cos6x-cos2x }  $$ is equal to :
  • $$ log \left ( \cfrac {5} {3} \right ) $$
  • $$ \cfrac {1} {4} log 15 $$
  • $$ - \cfrac {1} {16} \left( \cfrac {5} {3} \right) ^2 $$
  • $$log \left ( \cfrac {3} {5} \right ) $$
Arrange the following limits in the ascending order :
(1)  $$\lim _ { x \rightarrow \infty } \left( \dfrac { 1 + x } { 2 + x } \right) ^ { x + 2 }$$

(2)  $$\lim _ { x \rightarrow 0 } ( 1 + 2 x ) ^ { 3 / x }$$

(3)  $$\lim _ { \theta \rightarrow 0 } \dfrac { \sin \theta } { 2 \theta }$$

(4)  $$\lim _ { x \rightarrow 0 } \dfrac { \log _ { e } ( 1 + x ) } { x }$$
  • $$1,2,3,4$$
  • $$1,3,4,2$$
  • $$1,4,3,2$$
  • $$3,4,1,2$$
If z = z(x) and $$(2cosx)\frac { dz }{ dx } +(sinx)z=sinx$$, z(0) = 3, then $$z(\frac { \pi  }{ 2 } )$$ equals :
  • 1
  • $$\frac { 3 }{ 2 } $$
  • $$\frac { 5 }{ 2 } $$
  • $$\frac { 1 }{ 2 } $$
$$\underset { x\rightarrow 0 }{ lim } \dfrac { x\tan { 2x } -2\tan { 2x }  }{ { \left( 1-cos2x \right)  } }$$ equals:
  • $$\dfrac{1}{4}$$
  • $$1$$
  • $$\dfrac{1}{2}$$
  • $$-\dfrac{1}{2}$$
If $$\mathop {\lim }\limits_{x \to 0} \frac{{x\left( {1 + a\cos x} \right) - b\sin x}}{{{x^3}}} = 1,$$ then
  • $$a = \frac{5}{2}$$
  • $$b = \frac{{ - 5}}{2}$$
  • $$a + b = 4$$
  • $$a + b = -4$$
$$lim_{n\to \infty} \Sigma^n_{r=1} \dfrac{\pi}{n} sin(\dfrac{\pi r}{n})$$ is equal to
  • $$1$$
  • $$2$$
  • $$3$$
  • $$4$$
Evaluate : $$\displaystyle\lim _{ x\rightarrow 0 }{ \left( \dfrac { { e }^{ x\ell n\left( { 3 }^{ x }-1 \right)  }-\left( { 3 }^{ x }-1 \right) ^{ x }\sin { x }  }{ { e }^{ x\ell nx } }  \right)  } $$ is equal to 
  • $$\dfrac{1}{e}\ell n3$$
  • $$e\ \ell n\ 3$$
  • $$3$$
  • $$\dfrac{1}{3}$$
The value of $$\displaystyle\lim_{x\to 0} |x|^{sinx}$$ equals 
  • $$0$$
  • $$-1$$
  • $$1$$
  • does not exist
If $$\displaystyle \lim _{ x\rightarrow 0 }{ \dfrac { \left( \sin { nx }  \right) \left[ (a-n)nx-tanx \right]  }{ { x }^{ 2 } }  } =0$$, then the value of $$a$$
  • $$\dfrac { 1 }{ n }$$
  • $$n-\dfrac { 1 }{ n } $$
  • $$n+\dfrac{1}{n}$$
  • $$None\ of\ these$$
$$\displaystyle \lim_{x\rightarrow0 }{\dfrac{(\cos\alpha)^{x}-(\sin\alpha)^{x}-\cos 2\alpha}{(x-4)}}, \alpha\in \left(0, \dfrac{\pi}{2}\right)$$ is equal to
  • $$\cos^{4}\alpha.\log(\cos\alpha)-\sin^{4}\alpha.\log(\sin\alpha)$$
  • $$\sin^{4}\alpha.\log(\cos\alpha)-\cos^{4}\alpha.\log(\sin\alpha)$$
  • $$\sin^{4}\alpha.\log(\cos\alpha)+\cos^{4}\alpha.\log(\sin\alpha)$$
  • $$None\ of\ the \ above$$
$$\underset { x\rightarrow 0 }{ Lt } \cfrac {tanx-x}{x^2tanx}$$ equals:
  • 1
  • 1/2
  • 1/3
  • None of these
evaluate$$ \underset { x\rightarrow 0 }{ lim } \frac { x-\int _{ 0 }^{ x }{ { cost }^{ 2 }dt }  }{ { x }^{ 3 }-6x } $$
  • $$3$$
  • $$-1$$
  • $$0$$
  • $$1$$
$$\displaystyle\lim_{x \to \pi/2} (sec x +tan x)$$ is equal to 
  • $$1$$
  • $$-1$$
  • $$\dfrac{1}{2}$$
  • $$0$$
The value f $$\lim_{x\rightarrow \pi/4}\dfrac{\sqrt{1-\sqrt{\sin 2x}}}{\pi-4x}=$$
  • $$-\dfrac{1}{4}$$
  • $$\dfrac{1}{4}$$
  • $$\dfrac{1}{2}$$
  • $$None\ of\ these$$
$$\displaystyle\lim_{x\rightarrow \infty}\left(\dfrac{x+1}{2x+1}\right)^{x^2}$$ equals?
  • $$0$$
  • e
  • $$1$$
  • $$\infty$$
$$\underset { x\rightarrow \pi/2 }{ lim } \left(\dfrac{cosec x-1}{cot^2x}\right)= $$
  • $$0$$
  • $$-\dfrac{1}{2}$$
  • $$\dfrac{1}{2}$$
  • $$1$$
If $$\displaystyle \lim_{x\rightarrow 0}\dfrac {ae^{-x}-b\cos x-\dfrac {1}{2}cx}{x\cos x}=2$$ then the value of $$a+b+c$$ is-
  • $$4$$
  • $$-4$$
  • $$2$$
  • $$-2$$
$$\underset{x \rightarrow 2}{lim} \dfrac{\sqrt[3]{60 + x^2} - 4}{\sin (x - 2)}$$ equals 
  • $$\dfrac{1}{4}$$
  • $$0$$
  • $$\dfrac{1}{12}$$
  • Does not exist
$$lim_{x\to \dfrac{\pi}{2}} tan^2x(\sqrt{2sin^2x + 3 sin x +4} - \sqrt{sin^2x + 6 sin x+2})$$ is equal to
  • $$\dfrac{3}{4}$$
  • $$\dfrac{1}{6}$$
  • $$\dfrac{1}{12}$$
  • $$\dfrac{5}{12}$$
$$\lim _ { x \rightarrow 0 } \frac { 1 - \cos x } { x \log ( 1 + x ) } =$$
  • $$1$$
  • $$0$$
  • $$-1$$
  • $$\frac { 1 } { 2 }$$
The value of $$\displaystyle\int^{\pi/2}_0ln|\tan x+\cot x|dx$$ is equal to?
  • $$\pi ln 2$$
  • $$-\pi ln 2$$
  • $$\dfrac{\pi}{2} ln 2$$
  • $$-\dfrac{\pi}{2} ln 2$$
$$\underset { x\rightarrow 0 }{ lim } (\cos  x+a\sin  b{ x) }^{ \frac { 1 }{ x }  }$$ is equal to 
  • $$e^a$$
  • $$e^{ab}$$
  • $$e^b$$
  • $$e^{a/b}$$
The values of $$\displaystyle\lim_{n\rightarrow \infty}\dfrac{\sqrt[4]{n^5+2}-\sqrt[3]{n^2+1}}{\sqrt[5]{n^4+2}-\sqrt[2]{n^3+1}}$$ is?
  • $$1$$
  • $$0$$
  • $$-1$$
  • $$\infty$$
Let $$f$$ be a differentiable function such that $$f'(x) = 7- \dfrac{3}{4}\dfrac{f(x)}{x}, (x > 0)$$ and $$f(1) \neq 4$$.
Then $$\underset{x\to 0^+}{\lim} xf \left(\dfrac{1}{x}\right) $$:
  • Exists and equals 4
  • Does not exist
  • Exist and equals
  • Exists and equals $$\dfrac{4}{7}$$
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