CBSE Questions for Class 11 Engineering Maths Limits And Derivatives Quiz 12 - MCQExams.com

If $$y=\left | \cos x \right |+\left | \sin x \right |$$ then $$\frac{dy}{dx}$$ at $$x=\frac{2\pi }{3}$$ is
  • $$\frac{1-\sqrt{3}}{2}$$
  • $$0$$
  • $$\frac{1}{2}\left ( \sqrt{3}-1 \right )$$
  • none of these
Let $$\tan \alpha .x+\sin \alpha .y=\alpha $$ and $$\alpha \ \text{cosec} \alpha .x+\cos \alpha .y=1$$ be two variable straight line, $$\alpha $$ being the parameter. Let $$P$$ be the point of intersection of the lines. In the limiting position when $$\alpha \rightarrow 0$$, the point $$P$$ lies on the line
  • $$x=2$$
  • $$x=-1$$
  • $$y+1=0$$
  • $$y=2$$
If $$\displaystyle y=\sum _{ r=1 }^{ x }{ \tan ^{ -1 }{ \frac { 1 }{ 1+r+{ r }^{ 2 } }  }  } $$ then $$\displaystyle \frac { dy }{ dx } $$ is equal to
  • $$\displaystyle \frac { 1 }{ 1+{ x }^{ 2 } } $$
  • $$\displaystyle \frac { 1 }{ 1+{ \left( 1+x \right)  }^{ 2 } } $$
  • $$0$$
  • None of these
Let f and g be differentiable function such that $${f}'\left ( x \right )=2g\left ( x \right )$$ and $${g}'\left ( x \right )=-f\left ( x \right )$$, and let $$T\left ( x \right )=\left ( f\left ( x \right ) \right )^{2}-\left ( g\left ( x \right ) \right )^{2}$$. Then $${T}'\left ( x \right )$$ is equal to
  • $$T(x)$$
  • $$0$$
  • $$2f(x)g(x)$$
  • $$6f(x)g(x)$$
The value of $$\displaystyle \underset { n\rightarrow \infty  }{ lim } \left( \frac { 1 }{ n+1 } +\frac { 1 }{ n+2 } +...+\frac { 1 }{ 6n }  \right) $$ is
  • $$\displaystyle \log { 2 } $$
  • $$\displaystyle \log { 6 } $$
  • $$\displaystyle 1$$
  • $$\displaystyle \log { 3 } $$
If the prime sign (') represents differentiation w.r.t. $$x$$ and $$f^{'}=\sin x+\sin 4x.\cos x$$, then $$f^{'}\left ( 2x^{2}+\cfrac{\pi }{2} \right )$$ at $$x=\sqrt{\dfrac{\pi }{2}}$$ is equal to
  • $$0$$
  • $$-1$$
  • $$-2\sqrt{2\pi }$$
  • none of these
For $$n\epsilon N$$, let $$f\left ( x \right )=min\:\left \{ 1-\tan ^{n}x, 1-\sin ^{n}x, 1-x^{n} \right \}$$, $$x\epsilon \left ( -\cfrac {\pi}{2}, \cfrac {\pi}{2} \right )$$. The left hand derivative of $$f$$ at $$x=\cfrac {\pi}{4}$$ is
  • $$-2n$$
  • $$-2(n+1)$$
  • $$\displaystyle -n\frac{\pi }{4}$$
  • $$\displaystyle -n\left ( \frac{\pi }{4} \right )^{n-1}$$
If $$f '$$ (0) = 0 and f(x) is a differentiable and increasing function,then lim $$ x \rightarrow 0$$  $$\frac {x.f ' (x^2)}{f ' (x)}$$
  • is always equal to zero
  • may not exist as left hand limit may not exist
  • may not exist as left hand limit may not exist
  • right hand limit is always zero
The left-handed derivative of $$f\left( x \right) =\left| x \right| \sin { \left( \pi x \right)  }$$ at $$x=K$$ where $$K$$ is an integer, is :
  • $${ \left( -1 \right) }^{ K }\left( K-1 \right) \pi$$
  • $${ \left( -1 \right) }^{ K-1 }\left( K-1 \right) \pi$$
  • $${ \left( -1 \right) }^{ K }\ K\pi$$
  • $${ \left( -1 \right) }^{ K-1 }\ K\pi$$
If $$\sum _{ r=1 }^{ k }{ \cos ^{ -1 }{ \beta  }  } =\cfrac { k\pi  }{ 2 } $$ for any $$k\ge 1$$ and $$A=\sum _{ r=1 }^{ k }{ { \left( { \beta  }_{ r } \right)  }^{ r } } $$, then $$\lim _{ x\leftarrow A }{ \cfrac { { \left( 1+x \right)  }^{ 1/3 }-{ \left( 1-2x \right)  }^{ 1/4 } }{ x+{ x }^{ 2 } }  } $$ is equal to
  • $$0$$
  • $$\cfrac{1}{2}$$
  • $$\cfrac { \pi }{ 2 } $$
  • $$\cfrac { 5 }{ 6 } $$
The value of $$\lim _{ x\rightarrow 0 }{ \left( { \left( \sin { x }  \right)  }^{ 1/x }+{ \left( 1+x \right)  }^{ \sin { x }  } \right)  } $$ whre $$x> 0$$ is
  • $$0$$
  • $$-1$$
  • $$1$$
  • 2
The value of $$\displaystyle \lim_{x \rightarrow 1^{-}}\dfrac {1 - \sqrt {x}}{(\cos^{-1}x)^{2}}$$
  • $$\dfrac{-1}{4}$$
  • $$1/2$$
  • $$2$$
  • None of these
$$\displaystyle \lim_{x \rightarrow 0} \frac{ae^x + b cos x + c. e^{-x}}{sin^2 x} = 4$$ then b =
  • 2
  • 4
  • -2
  • -4
If $$\int_{0}^{x}(t^{2}+2t+2)$$ dt, $$2\leq x\leq 4$$
  • The maximum value of f(x) is $$\dfrac{136}{3}$$
  • The minimum value of f(x) is 10
  • The maximum value of f(x) is 26
  • None of these
If $$x\sqrt {1 + y}  + y\sqrt {1 + x}  = 0$$, then $$\dfrac {dy}{dx}$$ is equal to
  • $$\cfrac{1}{(1+x)^{2}}$$
  • $$\cfrac{-1}{(1+x)^{2}}$$
  • $$\cfrac{1}{(1-x)^{2}}$$
  • None of these
$$\mathop {\lim }\limits_{x \to {a^ + }} {{\left\{ x \right\}\sin \left( {x - a} \right)} \over {{{\left( {x - a} \right)}^2}}}$$ 

is equal to (where {.} denotes the fraction
part of x and $$a \in N$$
  • 0
  • 1
  • does not exist
  • none of thes
Consider the following statements:
$$S_1: \lim_\limits{x \to 0} \dfrac{[x]}{x}$$ is an indeterminate form (where [.] denotes greatest integer function).
$$S_2: \lim_\limits{x\to\infty}\dfrac{sin(3^x)}{3^x}=0$$
$$S_3: \lim_\limits{x \to \infty}\sqrt{\dfrac{x- sinx}{x+cos^2x}}$$ does not exist.
$$S_4:  \lim_\limits{n\to \infty}\dfrac{(n+2)!+(n+1)!}{(n+3)! }(n \in N=0$$
State, in order, whether $$S_1, S_2, S_3, S_4$$ are true or false
  • FTFT
  • FTTT
  • FTFF
  • TTFT
$$If {A_i} = \frac{{x - {a_i}}}{{\left| {x - {a_i}} \right|}}, \,i = 1,2,3,.....n$$ and $${a_1}< {a_2}< {a_3}....< {a_{n,}} \, then$$
$$\mathop {\lim }\limits_{x \to {a_m}} \left( {{A_1}{A_2}......{A_n}} \right), 1 \le m \le n$$
  • is equal to $${\left( { - 1} \right)^m}$$
  • is equal to $${\left( { - 1} \right)^{m + 1}}$$
  • is equal to $${\left( { - 1} \right)^{n-m}}$$
  • is equal to $${\left( { - 1} \right)^{n-m-1}}$$
If $$l=\lim\limits_{n\to 3}\dfrac{x^2-9}{\sqrt{x^2+7}-4}$$ and $$m=\lim\limits_{n\to -3}\dfrac{x^2-9}{\sqrt{x^2+7}-4}$$, then
  • $$l\ne m$$
  • $$l=2m$$
  • $$l=-m$$
  • $$l=m$$
If $$\mathop {\lim }\limits_{x \to \infty } \left( {\frac{{{x^2} + x + 1}}{{x + 1}} - ax - b} \right)\, = 4$$,then
  • $$a=1,b=4$$
  • $$a=1,b=-4$$
  • $$a=2,b=-3$$
  • $$a=2,b=3$$
Let$$f(\theta) = \dfrac{1}{tan^{9}\theta} {(1+tan\theta)^{10}+(2+tan\theta)^{10}+....+(20+tan\theta)^{10}}-20tan\theta$$. The left hand limit of $$f(\theta)$$ as $$\theta \rightarrow \dfrac{\pi}{2}$$ is:
  • 1900
  • 2000
  • 2100
  • 2200
if $$f\left( x \right) = \left\{ {\matrix{   {\cos \left[ x \right],} & {x \ge 0}  \cr    {\left| x \right| + a,} & {x < 0}  \cr 
 } } \right\}$$ Find
the value of a , given that $$\mathop {\lim }\limits_{x \to 0} f\left( x \right)$$  exists,
where[.]  denotes
  • -1
  • 2
  • 1
  • 0
$$\displaystyle \lim _{ x\rightarrow \frac { \pi  }{ 4 }  }{ { \left( \sin { 2x }  \right)  }^{ \sec ^{ 2 }{ 2x }  } }$$ is equal to 
  • $$-\dfrac {1}{2}$$
  • $$\dfrac {1}{2}$$
  • $$e^{-\dfrac {1}{2}}$$
  • $$e^{\dfrac {1}{2}}$$
$$ \underset { x\rightarrow \cfrac { \pi  }{ 2 }  }{ lim } \cfrac { cot \,  x-cos\, x }{ \left( \pi -{ 2x } \right)^ 3 } $$ equals
  • $$ \cfrac {1} {24} $$
  • $$ \cfrac {1} {16} $$
  • $$ \cfrac {1} {8} $$
  • $$ \cfrac {1} {4} $$
$$\displaystyle\underset{x\rightarrow 0}{Lt}\left(cosec x-\dfrac{1}{x}\right)=?$$
  • $$0$$
  • $$1/2$$
  • $$1$$
  • Does not exits
The value of $$\displaystyle \lim_{x\rightarrow 0}\dfrac {1-\cos^{3}x}{x\sin x\cos x}$$ is
  • $$\dfrac {2}{5}$$
  • $$\dfrac {3}{5}$$
  • $$\dfrac {3}{2}$$
  • $$\dfrac {3}{4}$$
$$\displaystyle \lim_{x \rightarrow 0}\dfrac {1}{x\sqrt {x}}\left(a\ arc\ tan \dfrac {\sqrt {x}}{a}-b\ arc\ \tan \dfrac {\sqrt {x}}{b}\right)$$ has the value equal to
  • $$\dfrac {a-b}{3}$$
  • $$0$$
  • $$\dfrac {(a^{2}-b^{2})}{6a^{2}b^{2}}$$
  • $$\dfrac {a^{2}-b^{2}}{3a^{2}b^{2}}$$
Integrate:
 $$lim_{x\rightarrow 0}\dfrac{(1-\cos{2x})^{2}}{2x\tan{x}-x\tan{2x}}$$
  • $$2$$
  • $$\dfrac{-1}{2}$$
  • $$-2$$
  • $$\dfrac{1}{2}$$
$$\displaystyle\lim_{n\rightarrow \infty}\left(\tan\theta +\dfrac{1}{2}\tan \dfrac{\theta}{2}+\dfrac{1}{2^2}\tan \dfrac{\theta}{2^2}+...+\dfrac{1}{2^n}\tan\dfrac{\theta}{2^n}\right)$$ equals?
  • $$\dfrac{1}{\theta}$$
  • $$\dfrac{1}{\theta}-2\cot 2\theta$$
  • $$2\cot 2\theta$$
  • None of these
$$\displaystyle \lim_{x\rightarrow \infty}{x^{2}\sin\left(\log_{e}\sqrt{\cos\dfrac{\pi}{x}}\right)}$$
  • $$0$$
  • $$-\dfrac{\pi^{2}}{2}$$
  • $$-\dfrac{\pi^{2}}{4}$$
  • $$-\dfrac{\pi^{2}}{8}$$
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