CBSE Questions for Class 11 Engineering Maths Limits And Derivatives Quiz 6 - MCQExams.com

If $$\displaystyle y=\frac{1}{1+x^{\beta -\alpha}+x^{\gamma -\alpha}}+\frac{1}{1+x^{\alpha-\beta}+x^{\gamma -\beta }}+\frac{1}{1+x^{\alpha -\gamma }+x^{\beta-\gamma }}$$
then $$\displaystyle \frac{dy}{dx}$$ is equal to-
  • $$0$$
  • $$1$$
  • $$\displaystyle

    (a+\beta +\gamma )X^{\alpha +\beta +\gamma -1}$$
  • None of these
If $$\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}$$
  • None of these
Evaluate $$\displaystyle \lim_{n\rightarrow \infty }\left [ \left ( 1+\frac{1}{n^{2}} \right )\left ( 1+\frac{2^{2}}{n^{2}} \right )\left ( 1+\frac{3^{2}}{n^{2}} \right )......\left ( 1+\frac{n^{2}}{n^{2}} \right ) \right ]^{1/n}$$
  • $$2e ^\left(\dfrac{\pi - 4}{2}\right)$$
  • $$2e ^\left(\dfrac{\pi - 2}{2}\right)$$
  • $$2e^ \left(\dfrac{\pi - 4}{4}\right)$$
  • $$2e^ \left(\dfrac{\pi - 4}{3}\right)$$
The limit of $$x\sin { \left( { e }^{ \frac { 1 }{ x }  } \right)  } $$ as $$x\rightarrow 0$$
  • is equal to $$0$$
  • is equal to $$1$$
  • is equal to $$\cfrac { e }{ 2 } $$
  • does not exist
If $$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}$$
The value of the constant $$\alpha$$ and $$\beta$$ such that $$\displaystyle \lim_{x\rightarrow \infty}\left(\displaystyle\frac{x^2+1}{x+1}-\alpha x-\beta\right)=0$$ are respectively.
  • $$(1, 1)$$
  • $$(-1, 1)$$
  • $$(1,-1)$$
  • $$(0, 1)$$
If the function $$f(x)$$ satisfies $$\displaystyle \lim_{x\rightarrow 1}\frac{f(x)-2}{x^2-1}=\pi$$, then $$\displaystyle \lim_{x\rightarrow 1}f(x)=$$
  • $$2$$
  • $$3$$
  • $$1$$
  • $$0$$
$$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$$
If $$f(x) = \sec (3x)$$, then $$f'\left (\dfrac {3\pi}{4}\right ) =$$
  • $$-3\sqrt {2}$$
  • $$-\dfrac {3\sqrt {2}}{2}$$
  • $$\dfrac {3}{2}$$
  • $$\dfrac {3\sqrt {2}}{2}$$
  • $$3\sqrt {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$$
If $$\displaystyle \lim_{x\rightarrow \infty}\dfrac{x^3+1}{x^2+1}-(ax+b)=2$$, then
  • $$a=2$$ and $$b=-1$$
  • $$a = 1$$ and $$b = 1$$
  • $$a = 1$$ and $$b = -1$$
  • $$a = 1$$ and $$b = -2$$
If $$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}$$
$$\displaystyle \lim_{x\rightarrow \pi/4} \dfrac {\tan x - 1}{\cos 2x}$$ is equal to
  • $$1$$
  • $$0$$
  • $$-2$$
  • $$-1$$
$$\displaystyle \lim_{x\rightarrow 3} = \dfrac {\sqrt {x} -\sqrt {3}}{\sqrt {x^{2} - 9}}$$ is equal to
  • $$1$$
  • $$3$$
  • $$\sqrt {3}$$
  • $$-\sqrt {3}$$
  • $$0$$
What is $$\displaystyle \lim_{x \rightarrow 0 }  x^2 \sin \left(\frac{1}{x}\right)$$ equal to ? 
  • 0
  • 1
  • 1/2
  • Limit does not exist.
The value of $$\displaystyle \lim _{ x\rightarrow \pi /6 }{ \cfrac { 2\sin ^{ 2 }{ x } +\sin { x } -1 }{ 2\sin ^{ 2 }{ x } -3\sin { x } -1 }  } $$ is
  • $$3$$
  • $$-3$$
  • $$6$$
  • $$0$$
If $$f\left( x \right) =\begin{vmatrix} \sin { x }  & \cos { x }  & \tan { x }  \\ { x }^{ 3 } & { x }^{ 2 } & x \\ 2x & 1 & 1 \end{vmatrix}$$, then $$\displaystyle\lim _{ x\rightarrow 0 }{ \dfrac { f\left( x \right)  }{ { x }^{ 2 } }  } $$ is
  • $$-1$$
  • $$3$$
  • $$1$$
  • Zero
If $$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)$$
If $$y = |\cos x| + |\sin x|$$, then $$\dfrac {dy}{dx}$$ at $$x = \dfrac {2\pi}{3}$$ is
  • $$\dfrac {1 - \sqrt {3}}{2}$$
  • $$0$$
  • $$\dfrac {1}{2}(\sqrt {3} - 1)$$
  • None of these
Which one of the following statements is correct?
  • $$\displaystyle \lim_{x \rightarrow 0} (fog) (x)$$ exists.
  • $$\displaystyle \lim_{x \rightarrow 0} (gof) (x)$$ exists.
  • $$\displaystyle \lim_{x \rightarrow 0-} (fog) (x) = \displaystyle \lim_{x \rightarrow 0-} (gof) (x)$$
  • $$\displaystyle \lim_{x \rightarrow 0+} (fog) (x) =\displaystyle \lim_{x \rightarrow 0-} (gof) (x)$$
Let   $$C(\theta)=\displaystyle\sum _{n=0}^{\infty}\dfrac{\cos(n\theta)}{n!}$$
Which of the following statements is FALSE? 
  • $$C(0).C(\pi)=1$$
  • $$C(0).C(\pi) > 2$$
  • $$C(\theta) > 0$$ for all $$\theta \in R$$
  • $$C(\theta) \neq 0$$ for all $$\theta \in R$$
$$\displaystyle\lim_{x\rightarrow\frac{\pi}{6}}\frac{\sin\left(x-\displaystyle\frac{\pi}{6}\right)}{\sqrt{3-2cos x}}$$ is equal to :
  • $$0$$
  • $$\displaystyle\frac{1}{(\sqrt{3}-2)}$$
  • $$1$$
  • $$\infty$$
If $$\underset{x\to 0}{\lim}\dfrac{x^a\sin^b x}{\sin(x^c)}, a, b, c, \in R \sim \{0\}$$ exists and has non-zero value, then 
  • $$a,b,c$$ are in A.P
  • $$a,b,c$$ are in G.P
  • $$a,b,c$$ are in H.P
  • none of these
If $$f(x)=\left| \log { \left| x \right|  }  \right| $$, then
  • $$f(x)$$ is continuous and differentiable for all $$x$$ in its domain
  • $$f(x)$$ is continuous for all $$x$$ in its domain but not differentiable at $$x=\pm 1$$
  • $$f(x)$$ is neither continuous nor differentiable at $$x=\pm 1$$
  • None of the above
If $$y=a\cos { \left( \sin { 2x }  \right)  } +b\sin { \left( \sin { 2x }  \right)  } $$, then $$y''+\left( 2\tan { 2x }  \right) y'=$$
  • $$0$$
  • $$4\left( \cos ^{ 2 }{ 2x } \right) y$$
  • $$-4\left( \cos ^{ 2 }{ 2x } \right) y$$
  • $$-\left( \cos ^{ 2 }{ 2x } \right) y$$
$$\lim _{ x\rightarrow 3 }{ \left( { x }^{ 3 }-4 \right) /\left( x+1 \right)  } =$$
  • $$(4/23)$$
  • $$(2/23)$$
  • $$(1/8)$$
  • $$(23/4)$$
If $$f(x) = \begin{vmatrix} \cos x& x & 1\\ 2\sin x & x^{2} & 2x\ \\ \tan x & x & 1\end{vmatrix}$$, then $$\displaystyle \lim_{x\rightarrow 0} \dfrac {f'(x)}{x}$$.
  • Exists and is equal to $$-2$$
  • Does not exist
  • Exist and is equal to $$0$$
  • Exists and is equal to $$2$$
$$\displaystyle\frac{d}{dx}\tan^{-1}\left(\displaystyle\frac{1-x}{1+x}\right)=$$ ____________.
  • $$\displaystyle\frac{2}{1+x^2}$$
  • $$\displaystyle\frac{-1}{1+x^2}$$
  • $$\displaystyle\frac{1}{1+x^2}$$
  • $$\displaystyle\frac{-2}{1+x^2}$$
Differentiate the following w.r.t. $$x$$.
$$\sin x\ log x$$.
  • $$\dfrac{\sin x}{x}-\cos x\,\, log \,x$$
  • $$\dfrac{\sin x}{x}+\cos x\,\, log \,x$$
  • $$\dfrac{\cos x}{x}+\cos x\,\, log \,x$$
  • $$\dfrac{\tan x}{x}+\cos x\,\, log \,x$$
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