CBSE Questions for Class 11 Engineering Maths Limits And Derivatives Quiz 7 - MCQExams.com

Differentiate the following w.r.t. $$x$$.
$$\tan^{3}x$$.
  • $$(3tan^2x)$$
  • $$(3tan^2x)(sec^2\,x)$$
  • $$(tan^2x)(sec^2\,x)$$
  • $$(tan^2x)(sin^2\,x)$$
Differentiate the following w.r.t. $$x$$.
$$\tan x^{2}$$.
  • $$sec^2(x^2)$$
  • $$sec^2(x^2)2x$$
  • $$sec^2(x^3)2x$$
  • $$tan^2(x^2)2x$$
Differentiate the following w.r.t. $$x$$.
$$\sin^{2} \sqrt {x}$$.
  • $$\dfrac{1}{2\sqrt{x}}\sin(3\sqrt{x})$$.
  • $$\dfrac{1}{\sqrt{x}}\sin(2\sqrt{x})$$.
  • $$\dfrac{1}{2\sqrt{x}}\sin(2\sqrt{x})$$.
  • $$\dfrac{1}{2\sqrt{x}}\sin(4\sqrt{x})$$.
$$\lim _{ x\rightarrow { 0 }^{ + } }{ \left( { \left( x\cos { x }  \right)  }^{ x }+{ \left( \cos { x }  \right)  }^{ \frac { 1 }{ \ln { x }  }  }+{ \left( x\sin { x }  \right)  }^{ x } \right)  } $$ is equal to
  • $$2$$
  • $$2+e$$
  • $$2+\dfrac { 1 }{ e }$$
  • $$3$$
If $$8f(x) + 6f\left( {{1 \over x}} \right) = x + 5$$ and $$y = {x^2}f(x)$$ ,then $${{dy} \over {dx}}$$ at x=-1 is equal to
  • 0
  • $${1 \over {14}}$$
  • $$ - {1 \over {14}}$$
  • None of these
If $$f:\mathrm{R}\to\mathrm{R}$$ is a differentiable function such that $$f'(x)\gt 2f(x)\forall x\in \mathrm{R}$$ and $$f(0)=1$$, then 
  • $$f(x)$$ is increasing in $$(0,\infty)$$
  • $$f(x)$$ is decreasing in $$(0,\infty)$$
  • $$f(x)\gt e^{2x} $$ in $$(0,\infty)$$
  • $$f(x)\lt e^{2x} $$ in $$(0,\infty)$$
$$\frac{d^n}{dx^n}[log(ax+b)]$$ is equal to:
  • $$\frac{(-1)^n n! a^n}{(ax+b)^{n+1}}$$
  • $$\frac{(-1)^{n-1} {(n-1)}! a^n}{(ax+b)^{n+1}}$$
  • $$\frac{(-1)^{n+1} ({n+1)}! a^{n-1}}{(ax+b)^{n+1}}$$
  • $$\frac{(-1)^{n-1} {(n-1)}! a^n}{(ax+b)^n}$$
$$\dfrac{\displaystyle \lim_{h \rightarrow 0}(h+1)^2}{\displaystyle \lim_{h\rightarrow 0}(1+h)^{2/h}}$$ is equal to
  • $$e^1$$
  • $$e^{-2}$$
  • $$e^2$$
  • $$e^{-1}$$
Function f(x)=$$\left| {x - 2} \right| - 2\left| {x - 4} \right|\,is$$ discontinous at:
  • $$x=2,4$$
  • $$x=2$$
  • No where
  • Except $$x=2$$
Let $${P_n} = \prod\limits_{k = 2}^n {\left( {1 - {1 \over {{}^{{}^{k + 1}}{C_2}}}} \right)} .$$ If $$\mathop {\lim }\limits_{x \to \infty } {P_n}$$ can be expressed as lowest rational in the form $$\dfrac { a }{ b } $$ , then value of $$(a+b)$$ is __________.
  • 4
  • 8
  • 10
  • 12
Given $$y=\dfrac {3}{x}, \dfrac {dy}{dx}=$$
  • $$3$$
  • $$\dfrac {3}{x^{2}}$$
  • $$\dfrac {-3}{x^{2}}$$
  • $$3x$$
Statement I: The function f(x) in the figure is differentiable at x = a
Statement II: The function f(x) continuous at x = a

1019713_fa884f00f0bf444a8a92598dbf0fe684.png
  • Both Statement I and Statement II are true and the Statement II is the correct explanation of the Statement I.
  • Both Statement I and Statement II are true and the Statement II is not the correct explanation of the Statement I.
  • Statement I is true but Statement II is false
  • Statement I is false but Statement II is true.
$$\mathop {\lim}\limits_{x \to \frac{\pi}{2}} \tan x = $$
  • $$\infty $$
  • $$ - \infty $$
  • $$0$$
  • does not exist
Suppose the function $$f(x)-f(2x)$$ has the derivative $$5$$ at $$x=1$$ and derivative $$7$$ at $$x=2$$. The derivative of the function $$f(x)-f(4x)$$ at $$x=1$$, has the value equal to?
  • $$19$$
  • $$9$$
  • $$17$$
  • $$14$$
$$\lim _{ x\rightarrow 0 }{ \log _{ \left( \tan ^{ 2 }{ x }  \right)  }{ \left( \tan ^{ 2 }{ 2x }  \right) = }  }$$
  • $$1$$
  • $$2$$
  • $$\dfrac {1}{2}$$
  • $$Does\ not\ exist$$
$$\displaystyle \lim_{n \rightarrow \infty} {^{n}C_{c}}\left(\dfrac {m}{n}\right)^{x}\left(1-\dfrac {m}{n}\right)^{n-x}$$ equal to
  • $$\dfrac {M^{x}}{x\ !}.e^{-m}$$
  • $$\dfrac {M^{x}}{x\ !}.e^{m}$$
  • $$0$$
  • $$\dfrac {m^{x+1}}{me^{m}x\ !}$$
If [.] denotes, GIF , then $$\underset{x \rightarrow 0}{lt} \left( \left[\dfrac{2018 sin^{-1} x}{x}\right] + \left[\dfrac{2020x}{tan^{-1} x}\right]\right)$$ = 
  • 4038
  • 4037
  • 4036
  • 4039
The shortest distance between line $$y-x=1$$ and curve $$x={y}^{2}$$ is
  • $$\dfrac {\sqrt {3}}{4}$$
  • $$\dfrac {3\sqrt {2}}{8}$$
  • $$\dfrac {8}{2\sqrt {2}}$$
  • $$\dfrac {4}{\sqrt {3}}$$
$${ x }_{ n }={ \left( 1-\cfrac { 1 }{ 3 }  \right)  }^{ 2 }{ \left( 1-\cfrac { 1 }{ 6 }  \right)  }^{ 2 }{ \left( 1-\cfrac { 1 }{ 10 }  \right)  }^{ 2 }...{ \left( 1-\cfrac { 1 }{ \cfrac { n(n+1) }{ 2 }  }  \right)  }^{ 2 },n\ge 2$$. Then the value of $$\displaystyle\lim _{ n\rightarrow \infty  }{ { x }_{ n } } $$
  • $$1/3$$
  • $$1/9$$
  • $$1/81$$
  • $$0$$ (zero)
The difference of slopes of lines represent by $${y^2} - 2xy{\sec ^2}\alpha  + \left( {3 + {{\tan }^2}\alpha } \right)\left( {{{\tan }^2}\alpha  - 1} \right){x^2} = 0$$ is
  • $$3$$
  • $$4$$
  • $$0$$
  • $$2$$
$$ \lim _{ x\rightarrow 1 }{ \dfrac { \sqrt { 1-\cos { 2\left( x-1 \right)  }  }  }{ x-1 }  }$$
  • Exists and it equals $$\sqrt {2}$$
  • Exists and it equals $$-\sqrt {2}$$
  • Does not exist because $$x-1\rightarrow 0$$
  • Does not exist because left hand limit is not equal to right hand limit
If $$f(x) = 3x^{10} - 7x^8 + 5x^6 - 21x^3 + 3x^2 - 7$$, then $$\underset{a \rightarrow 0}{\lim} \dfrac{f(1 - \alpha) - f(1)}{\alpha^3 + 3 \alpha}$$ is 
  • $$-\dfrac{53}{3}$$
  • $$\dfrac{53}{3}$$
  • $$-\dfrac{55}{3}$$
  • $$\dfrac{55}{3}$$
$$\displaystyle \lim_{n \rightarrow \infty}\dfrac {1^{2}+2^{2}+3^{2}+....+n^{2}}{n^{3}}$$ is equal to
  • $$1$$
  • $$1/2$$
  • $$1/3$$
  • $$0$$
If $$y=|\cos x|+|\sin x|$$, then $$\dfrac {dy}{dx}$$ at $$x=\dfrac {2\pi}{3}$$  is
  • $$\dfrac {1-\sqrt {3}}{2}$$
  • $$0$$
  • $$\dfrac {\sqrt {3}-1}{2}$$
  • $$\dfrac {\sqrt {3}+1}{2}$$
The value of $$\underset { x\longrightarrow \infty  }{ Lim } \dfrac{d}{dx}\overset { \sqrt { 3 }  }{ \underset { -\sqrt { 3 }  }{ \int }  } \dfrac{r^3}{(r+1)(r-1)}$$dr,is
  • $$0$$
  • $$1$$
  • $$\dfrac{1}{2}$$
  • non existent
Solve

$$\mathop {\lim }\limits_{x \to 0} \dfrac{{\sin 5x}}{{\tan 3x}}$$
  • $$-\dfrac{5}{3}$$
  • $$\dfrac{5}{3}$$
  • $$\dfrac{7}{3}$$
  • None of these
If $$a{x^2} + 2hxy + b{y^2} = 0$$ then $$\frac{{dy}}{{dx}}$$ is equal to
  • $$\frac{y}{x}$$
  • $$\frac{x}{y}$$
  • $$ - \frac{x}{y}$$
  • none of these
$$\underset{x \rightarrow \frac{\pi}{2}}{\lim} \dfrac{\cot x - \cos x}{\left(\dfrac{\pi}{2} -x \right)^3} = $$
  • $$\dfrac{-1}{2}$$
  • $$\dfrac{1}{2}$$
  • $$2$$
  • $$-2$$
Solve:
$$\displaystyle \int_{0}^{1}\dfrac{dx}{\sqrt{x+1}+\sqrt{x}}dx=$$
  • $$\dfrac{4}{3}(\sqrt{2}+1)$$
  • $$\dfrac{4}{3}(\sqrt{2}-1)$$
  • $$\dfrac{3}{4}(\sqrt{2}-1)$$
  • $$\dfrac{3}{4}(\sqrt{2}-2)$$
If $$y=a\ \sin\ x+b\ \cos\ x$$, then $$y^{2}+\left ( \dfrac{dy}{dx} \right )^{2}$$ is
  • function of $$x$$
  • function of $$y$$
  • function of $$x$$ and $$y$$
  • constant
0:0:1


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