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CBSE Questions for Class 12 Commerce Maths Continuity And Differentiability Quiz 10 - MCQExams.com

If y=tan1(a+x1ax) then dydx=?
  • 1(1+x)
  • 1x(1+x)
  • 2x(1+x)
  • 12x(1+x)
If y=sec1(x2+1x21) then dydx=?
  • 2(1+x2)
  • 2(1+x2)
  • 1(1x2)
  • none of these
 If y=tan1x+1x1, then dydx is
  • 12|x|x21
  • 12xx21
  • 12xx21
  • None of these
f(x)=x2+xg(1)+g(2) and g(x)=f(1)x2+xf(x)+f(x)
The value of f(3) is
  • 1
  • 0
  • -1
  • -2
The function given by y=|x1| is differentiable function and f(1/n) = 0 n1 and n\epsilon I $$, then
  • f(x)=0, xϵ(0,1]
  • f(0)=0, f'(0) =0
  • f(0)= 0=f'(0), xϵ(0,1]
  • f(0)=0 and f'(0) need not to be zero
If P(1)=0 and dP(x)dx>P(x) for all x1, then
  • P(x)>0x>1
  • P(x) is a constant function
  • P(x)<0x>1
  • None of these
The function f(x)=x1+|x| is differentiable 
  • only at non-integer points
  • everywhere
  • only at x=0
  • none of these
If H(x0)=0 for some x=x0 and ddxH(x)>2cxH(x) for all xx0, where c>0, then
  • H(x)=0 has root for x>x0
  • H(x)=0 has no root for x>x0
  • H(x)=0 is constant function
  • None of these
Given a function f:[0,4]R is differentiable, then for someα,β(0,2),40f(t)dt equals to
  • f(α2)+f(β2)
  • 2αf(α2)+2βf(β2)
  • αf(b2)+βf(α2)
  • f(α)f(β)[f(α)+f(β)]
If f(x)=[sin1(sin2x)](where, [] denotes the greatest integer function), then 
  • π/20f(x)dx=π2sin1(sin1)
  • f(x) is periodic with period
  • lim
  • None of the above
If f:R \rightarrow (0, \infty) be a differentiable function f(x) satisfying f(x+ y) - f(x - y) = f(x) \{ f(y) - f(y) - y \}, \forall x, y \epsilon R, (f(y) \neq f(-y) for all y \epsilon R) and f' (0) = 2010.
Now answer the following questions

Which of the following is true for f(x)
  • f(x) is one-one and into
  • \{ f(x) \} is non-periodic, where {.} denotes fractional part of x.
  • f(x) = 4 has only two solutions.
  • f(x) = f^{-1} x has only one solution
If f(x) = \displaystyle \int_{0}^{x}(f(t))^2 dt f:R→R be differentiable function and f(g(x)) is differentiable function at x=a, then
  • g(x) must be differentiable at x=a
  • g(x) may be non-differentiable at x=a
  • g(x) must be discontinuous at x=a
  • None of the above
y=f(x) is
  • injective but not surjective
  • surjective but not injective
  • bijective
  • neither injective nor surjective
The current statement(s) is/are
  • f'(1)<0
  • f(2)<0
  • f'(x)\neq 0 for any x \in (1,3)
  • f'(x) = 0 for some x \in (1,3)
If u=\sin^{-1}\Bigg(\dfrac{2x}{1+x^2}\Bigg) and v=\tan^{-1}\Bigg(\dfrac{2x}{1-x^2}\Bigg), then \dfrac{du}{dv} is
  • \dfrac{1}{2}
  • x
  • \dfrac{1-x^2}{1+x^2}
  • 1
The function f(x)=\Bigg\{\dfrac{\sin x}{x}+\cos x,if\,x\neq 0\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,k,if\,x=0  is continuous at x=0,then the value of k is
  • 3
  • 2
  • 1
  • 1.5
|\sin x| is a differentiable function for every value of x.
  • True
  • False
The function f(x)=|x|+|x-1| is
  • continuous at x = 0 as well as at x = 1.
  • continuous at x = 1 but not at x = 0.
  • discontinuous at x = 0 as well as at x = 1.
  • continuous at x = 0 but not at x = 1.
If f(x) = \sin^{-1} \left ( \frac{4^{x + \frac{1}{2}}}{1 + 2^{4x}} \right ), which of the following is not the derivative of f(x)?
  • \frac{2.4^{x} \cdot \log 4}{1 + 4^{2x}}
  • \frac{4^{x + 1} \cdot \log 2}{1 + 4^{2x}}
  • \frac{4^{x + 1} \cdot \log 4}{1 + 4^{4x}}
  • \frac{2^{2^(x + 1)} \cdot \log 2}{1 + 2^{4x}}
If y \tan^{-1} \left ( \sqrt{\frac{a - x}{a + x}} \right ), where -a < x < a, then \frac{dy}{dx} =.....
  • \frac{x}{\sqrt{a^2 - x^2}}
  • \frac{a}{\sqrt{a^2 - x^2}}
  • - \frac{1}{2 \sqrt{a^2 - x^2}}
  • \frac{1}{2 \sqrt{a^2 - x^2}}
If y = \tan^{-1} \left ( \frac{x}{1 + \sqrt{1 - x^2}} \right ) + \sin \left [ 2 \tan^{-1} \left ( \sqrt{\frac{1 - x}{1 + x}} \right )  \right ] then \frac{dy}{dx} = ...........
  • \frac{x}{\sqrt{1 - x^2}}
  • \frac{1 - 2x}{\sqrt{1 - x^2}}
  • \frac{1 - 2x}{2 \sqrt{1 - x^2}}
  • \frac{1 - 2x^2}{\sqrt{1 - x^2}}
If y = \sin(2 \sin^{-1} x), then dx = .......
  • \frac{2 - 4x^2}{\sqrt{1 - x^2}}
  • \frac{2 + 4x^2}{\sqrt{1 - x^2}}
  • \frac{4x^2 - 1}{\sqrt{1 - x^2}}
  • \frac{1 - 2x^2}{\sqrt{1 - x^2}}
If function f(x)=\dfrac{x^2-9}{x-3} is continuous at x=3, then value of (3) will be:
  • 6
  • 3
  • 1
  • 0
If f(x)=\begin{cases} \begin{matrix} \dfrac{\log (1+mx)- \log (1-nx)}{x}; & x \ne 0 \end{matrix} \\ \begin{matrix} k; & x=0 \end{matrix} \\ \begin{matrix}  &  \end{matrix} \end{cases}
is continuous at x=0 then the value of k will be:
  • 0
  • m+n
  • m-n
  • m.n
If function f(x)=\begin{cases} \begin{matrix} \dfrac{\sin 3x}{x}; & x \ne 0 \end{matrix} \\ \begin{matrix} m; & x=0 \end{matrix} \\ \begin{matrix}  &  \end{matrix} \end{cases}
is continuous at x=2 then value of m will be:
  • 3
  • 1/3
  • 1
  • 0
Let x={f}''(t) cost +{f}'(t) sint  and y={-f}''(t) sint+{f}'(t) cost. Then \displaystyle \int \left [ \left(\frac{dx}{dt} \right)^2 + \left(\frac{dy}{dt} \right)^2 \right ]^{\frac{1}{2}} dt equals
(Note : f(x), f'(x), f''(x) , f'''(x) >0 )
  • {f}'(t)+{f}''(t)+c
  • {f}''(t)+{f}'''(t)+c
  • f(t)+{f}''(t)+c
  • {f}'(t)-{f}''(t)+c
The set of all points where the function f(\displaystyle \mathrm{x})=\frac{x}{1+|x|} is differentiable is 
  • (-\infty, \infty)
  • (0,\infty)
  • (-\infty ,0)\cup (0,\infty )
  • (-\infty, 0)
If y=x^{\displaystyle x^{\displaystyle x^{\displaystyle \dots^{\displaystyle\infty}}}}, find \displaystyle\frac{dy}{dx}.
  • \displaystyle\frac{y^2}{x(1-y\log{x})}
  • \displaystyle\frac{y}{x(1-\log{x})}
  • \displaystyle\frac{y^2}{x(y-\log{x})}
  • None of these
Derivative of ({\log{x}})^{\displaystyle\cos{x}} with respect to x is
  • \displaystyle({\log{x}})^{\displaystyle\cos{x}}\left[\frac{\cos{x}}{x\log{x}}-\sin{x}\log{(\log{x})}\right]
  • \displaystyle({\log{x}})^{\displaystyle\cos{x}}\left[\frac{\cos{x}}{x\log{x}}-\cos{x}\log{(\log{x})}\right]
  • \displaystyle({\log{x}})^{\displaystyle\sin{x}}\left[\frac{\sin{x}}{x\log{x}}-\cos{x}\log{(\log{x})}\right]
  • None of these
If y=x^{\left(x^{ x}\right)}, then \displaystyle\frac{dy}{dx} is
  • y\left[x^{\displaystyle x}\left(\log{ex}\right)\log{x}+x^{\displaystyle x}\right]
  • y\left[x^{\displaystyle x}\left(\log{ex}\right)\log{x}+x\right]
  • y\left[x^{\displaystyle x}\left(\log{ex}\right)\log{x}+x^{\displaystyle x-1}\right]
  • y\left[x^{\displaystyle x}\left(\log_e{x}\right)\log{x}+x^{\displaystyle x-1}\right]
0:0:1


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