CBSE Questions for Class 11 Engineering Maths Limits And Derivatives Quiz 10 - MCQExams.com

Differentiate the following function with respect to x.
$$(2x^2-3)\sin x$$.
  • $$4x\sin x+(2x^2+3)\cos x$$.
  • $$4x\sin x+(2x^2-3)\sin x$$.
  • $$4x\sin x+(2x^2-3)\cos x$$.
  • $$4x\cos x+(2x^2-3)\cos x$$.
If $$f$$ is differentiable at $$x = 1$$ and $$\underset{h \rightarrow 0}{\lim} \dfrac{1}{h} f (1 + h) = 5, f'(1) = $$
  • $$0$$
  • $$1$$
  • $$3$$
  • $$4$$
  • $$5$$
If $$y=\sqrt{\sin x+\sqrt{\sin x+\sqrt{\sin x+}}}.....\infty$$ then $$\dfrac{dy}{dx}=?$$
  • $$\dfrac{\sin x}{(2y-1)}$$
  • $$\dfrac{\cos x}{(y-1)}$$
  • $$\dfrac{\cos x}{(2y-1)}$$
  • $$none\ of\ these$$
If $$y=(\tan x)^{\cot x}$$ then $$\dfrac{dy}{dx}=?$$
  • $$\cot x.(\tan x)^{\cot x-1}.\sec^2x$$
  • $$-(\tan x)^{\cot x}.\csc^2x$$
  • $$(\tan x)^{\cot x}.\csc^2 x(1-\log \tan x)$$
  • $$none\ of\ these$$
If $$y=\sqrt{x\sin x}$$ then $$\dfrac{dy}{dx}=?$$
  • $$\dfrac{(x\cos x+\sin x)}{2\sqrt{x\sin x}}$$
  • $$\dfrac{1}{2}(x\cos x+\sin x).\sqrt{x\sin x}$$
  • $$\dfrac{1}{2\sqrt{x\sin x}}$$
  • $$none\ of\ these$$
If $$x=a(\cos\theta+\theta\sin\theta)$$ and $$y=a(\sin\theta-\theta\cos\theta)$$ then $$\dfrac{dy}{dx}=?$$
  • $$\cot\theta$$
  • $$\tan\theta$$
  • $$a\cot\theta$$
  • $$a\tan\theta$$
If $$x=a\sec\theta, y=b\tan\theta$$ then $$\dfrac{dy}{dx}=$$
  • $$\dfrac{b}{a}\sec\theta$$
  • $$\dfrac{b}{a} \ cosec\theta$$
  • $$\dfrac{b}{a}\cot\theta$$
  • $$none\ of\ these$$
If $$y=\sqrt{\dfrac{1+\sin x}{1-\sin x}}$$ then $$\dfrac{dy}{dx}=?$$
  • $$\dfrac{1}{2}\sec^2\left(\dfrac{\pi}{4}-\dfrac{x}{2}\right)$$
  • $$\dfrac{1}{2}\csc^2\left(\dfrac{\pi}{4}-\dfrac{x}{2}\right)$$
  • $$\dfrac{1}{2}\csc \left(\dfrac{\pi}{4}-\dfrac{x}{2}\right)\cot \left(\dfrac{\pi}{4}-\dfrac{x}{2}\right)$$
  • $$none\ of\ these$$
If $$\lim _{ x\rightarrow 0 }{ \cfrac { x\left( 1+a\cos { x }  \right) -b\sin { x }  }{ { x }^{ 3 } }  } =1$$ then
  • $$a=-5/2$$, $$b=-1/2$$
  • $$a=-3/2$$, $$b=-1/2$$
  • $$a=-3/2$$, $$b=-5/2$$
  • $$a=-5/2$$, $$b=-3/2$$
The value of $$ \displaystyle \lim _{x \rightarrow \pi} \dfrac{1+\cos ^{3} x}{\sin ^{2} x} $$ is
  • 1/3
  • 2/3
  • -1/4
  • 3/2
If $$\displaystyle {f}'(x) = sin\,x + sin\,4x .\, cos \,x $$ then $$\displaystyle {f}'(x) \left (2x^{2} + \dfrac{\pi}{2} \right ) $$ at $$ x = \sqrt{\dfrac{\pi}{2}} $$ is equal to 
  • $$ -1 $$
  • $$ 0 $$
  • $$\displaystyle -2\sqrt{2\pi} $$
  • None of these
If $$\displaystyle sin\, y = x\, sin ( a + y) $$ and
$$\displaystyle \dfrac{dy}{dx} = \dfrac{A}{ 1 + x^{2} - 2x \, cos a } $$ then the value of $$ A $$ is
  • $$ 2 $$
  • $$ cos\, a $$
  • $$ sin\,a $$
  • None of these
$$ \displaystyle \lim _{x \rightarrow 0} \dfrac{\sin x^{n}}{(\sin x)^{m}},(m<n) $$ is equal to
  • 1
  • 0
  • n/m
  • None of these
 $$ The \ value \ of  \displaystyle \lim _{x \rightarrow 1}(2-x)^{\tan \dfrac{\pi x}{2}} $$ is
  • $$e^{-2 \pi} $$
  • $$ e^{1 / \pi} $$
  • $$ e^{2 /\pi} $$
  • $$ e^{-1 / \pi} $$
$$\displaystyle \lim _{x \rightarrow 0} \dfrac{x\left(e^{x}-1\right)}{1-\cos x} $$ is equal to
  • 0
  • $$\infty$$
  • -2
  • 2
$$ \displaystyle \lim _{x \to \pi / 2}\left[x \tan x-\left(\dfrac{\pi}{2}\right) \sec x\right] $$ is equal to 
  • 1
  • -1
  • 0
  • $$None \ of \ these$$
$$ \displaystyle \lim _{x \rightarrow-\infty} \dfrac{x^{2} \tan \dfrac{1}{x}}{\sqrt{8 x^{2}+7 x+1}} $$ is equal to
  • $$ -\dfrac{1}{2 \sqrt{2}} $$
  • $$ \dfrac{1}{2 \sqrt{2}} $$
  • $$ \dfrac{1}{\sqrt{2}} $$
  • Does not exist
 $$\displaystyle \lim _{x \rightarrow 0} \dfrac{x^{4}\left(\cot ^{4} x-\cot ^{2} x+1\right)}{\left(\tan ^{4} x-\tan ^{2} x+1\right)} $$ is equal to
  • 1
  • 0
  • 2
  • None of these
If $$ f(x)=\dfrac{\cos x}{(1-\sin x)^{1 / 3}}, $$ then
  • $$\lim _{ x\rightarrow \dfrac { \pi^- }{2 } }{ f(x)=-\infty } $$
  • $$\lim _{ x\rightarrow \dfrac { \pi^+ }{2 } }{ f(x)=\infty } $$
  • $$\lim _{ x\rightarrow \dfrac { \pi }{2 } }{ f(x)=\infty } $$
  • none of these
$$ \displaystyle \lim _{x \rightarrow 0} \dfrac{x \tan 2 x-2 x \tan x}{(1-\cos 2 x)^{2}} $$ is equal to
  • 2
  • -2
  • 1/2
  • -1/2
$$\displaystyle  \lim _{x \rightarrow 1} \dfrac{1+\sin \pi\left(\dfrac{3 x}{1+x^{2}}\right)}{1+\cos \pi x} $$ is equal to
  • 0
  • 1
  • 2
  • 4
$$\displaystyle \lim _{x \rightarrow 1} \dfrac{1-x^{2}}{\sin 2 \pi x} \text { is equal to }$$
  • $$ \dfrac{1}{2 \pi} $$
  • $$ \dfrac{-1}{\pi} $$
  • $$ \dfrac{-2}{\pi} $$
  • None of these
Let $$f(x)= \sin x+ax+b$$, then which of the following is/are true.
  • $$f(x)=0$$ has on;ly one root which is possitive if $$a>1, b<0$$
  • $$f(x)=0$$ has on;ly one root which is negative if $$a>1, b<0$$
  • $$f(x)=0$$ has on;ly one root which is negaitive if $$a < -1, b<0$$
  • None of these
The value of $$ \displaystyle \lim _{x \rightarrow a} \sqrt{a^{2}-x^{2}} \cot \dfrac{\pi}{2} \sqrt{\dfrac{a-x}{a+x}} $$ is
  • $$\dfrac{2 a}{\pi} $$
  • $$-\dfrac{2 a}{\pi} $$
  • $$ \dfrac{4 a}{\pi} $$
  • $$-\dfrac{4 a}{\pi} $$
Which of the following is not true about $$y=f(x)$$?
  • It is an increasing function
  • It is a monotonic function
  • It has infinite points of inflections
  • None of these
$$ \displaystyle \lim _{n \rightarrow \infty} \sum_{x=1}^{20} \cos ^{2 n}(x-10) $$ is equal to
  • 0
  • 1
  • 19
  • 20
$$\displaystyle  \lim _{x \rightarrow 0} \dfrac{\sin \left(x^{2}\right)}{\ln \left(\cos \left(2 x^{2}-x\right)\right)} $$ is equal to
  • 2
  • -2
  • 1
  • -1
Differential coefficient of $$\sec (\tan^{-1} x)$$ w.r.t. x is
  • $$\dfrac{x}{\sqrt{1+x^2}}$$
  • $$\dfrac{x}{1+x^2}$$
  • $$x\sqrt{1+x^2}$$
  • $$\dfrac{1}{\sqrt{1+x^2}}$$
The value of $$\displaystyle \lim_{x\rightarrow \infty} \dfrac {\sin x}{x}$$ is
  • $$0$$
  • $$\infty$$
  • $$1$$
  • $$-1$$
The value of $$\displaystyle \lim_{x\rightarrow 0} \dfrac {1 - \cos x}{x^{2}}$$ is
  • $$0$$
  • $$1/2$$
  • $$-1/2$$
  • $$-1$$
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