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

If y=11+xβα+xγα+11+xαβ+xγβ+11+xαγ+xβγ
then dydx is equal to-
  • 0
  • 1
  • (a+β+γ)Xα+β+γ1
  • None of these
If y=|cosx|+|sinx| then dydx at x=2π3 is:
  • 132
  • 0
  • 312
  • None of these
Evaluate limn[(1+1n2)(1+22n2)(1+32n2)......(1+n2n2)]1/n
  • 2e(π42)
  • 2e(π22)
  • 2e(π44)
  • 2e(π43)
The limit of xsin(e1x) as x0
  • is equal to 0
  • is equal to 1
  • is equal to e2
  • does not exist
If r=[2ϕ+cos2(2ϕ+π4)]12, then what is the value of the derivative of drdϕ at ϕ=π4?
  • 2(1π+1)12
  • 2(2π+1)2
  • (2π+1)12
  • 2(2π+1)12
The value of the constant α and β such that limx(x2+1x+1αxβ)=0 are respectively.
  • (1,1)
  • (1,1)
  • (1,1)
  • (0,1)
If the function f(x) satisfies limx1f(x)2x21=π, then limx1f(x)=
  • 2
  • 3
  • 1
  • 0
f(x)=log(ex(x2x+2)34)f(0)=
  • 14
  • 4
  • 34
  • 1
If f(x)=sec(3x), then f(3π4)=
  • 32
  • 322
  • 32
  • 322
  • 32
If y=f(x2+2) and f(3)=5, then dydx at x=1 is _____
  • 5
  • 25
  • 15
  • 10
If f(x) is a function such that f(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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