CBSE Questions for Class 12 Commerce Maths Continuity And Differentiability Quiz 7 - MCQExams.com

If $$x=\sqrt { { a }^{ \sin ^{ -1 }{ t }  } } ;\quad y=\sqrt { { a }^{ \cos ^{ -1 }{ t }  } } $$, then $$\cfrac { dy }{ dx } =\cfrac { -y }{ x } $$
  • True
  • False

$$f(x)=\dfrac {1-\cos(1-\cos x)}{x^4}$$ is continuous at $$ x= 0  $$, then $$f(0)=$$

  • $$\dfrac{1}{2}$$
  • $$\dfrac{1}{4}$$
  • $$\dfrac{1}{6}$$
  • $$\dfrac{1}{8}$$
Let $$f(x)$$ be a function defined on$$(-a,a)$$ with $$a > 0$$. Amuse that $$f(x)$$ is continuous at $$x=0$$ and $$\underset{x\to 0}{\lim}\dfrac {f(x)-f(kx)}{x}=\alpha$$, where $$k \in (0,1)$$ then compute $$f'(0^{+})$$ and $$f'(0^{-})$$, and comment upon the differentiablity of $$f$$ at $$x=0$$? Denote $$\alpha$$.
  • $$\underset{x\to o}{\lim}\dfrac{f(x) - f(0)}{x} \cdot \dfrac{f(kx) - f(0)}{kx} \times k = \alpha$$
  • $$\underset{x\to o}{\lim}\dfrac{f(kx) - f(0)}{kx} \cdot \dfrac{f(kx) - f(0)}{kx} \times k = \alpha$$
  • $$\underset{x\to 0}{\lim}\dfrac{f(x) - f(kx)}{x} \cdot \dfrac{f(kx) - f(0)}{kx} \times k = \alpha$$
  • $$\underset{x\to o}{\lim}\dfrac{f(kx) - f(0)}{x} \cdot \dfrac{f(kx) - f(0)}{kx} \times k = \alpha$$
If $$x=a\left(\cos t+\log \tan\dfrac{t}{2}\right), y=a\sin t$$, then evaluate $$\dfrac{d^2y}{dx^2}$$ at $$t=\dfrac{\pi}{3}$$.
  •  $$\cfrac{4\sqrt3}{a}$$
  •  $$\cfrac{\sqrt3}{a}$$
  •  $$\cfrac{8\sqrt3}{a}$$
  •  $$\cfrac{8\sqrt3}{5a}$$
Let $$f(x)$$ be a continuous funct and $$g(x)$$ be a discontinuous $$funct^{n}$$, then   $$f(x) + g(x)$$ is continuous $$funct^{n}$$. ? it is .  
  • True
  • False
If $$x=\cos ec \theta-\sin \theta,y=\cos ec^{n}\theta-\sin^{n}\theta$$ then $$(x^{2}+4)\left(\dfrac {dy} {dx}\right)^{2}-n^{2}y^{2}=$$ 
  • $$n^{2}$$
  • $$2n^{2}$$
  • $$3n^{2}$$
  • $$4n^{2}$$
If $$x=a\sec^{2}\theta$$, $$y=a\tan^{3}\theta$$ then $$\dfrac {d^{3}y}{dx^{3}}$$
  • $$\dfrac {-3}{8a^{2}}\cot^{3}\theta$$
  • `$$\dfrac {3}{8a^{2}}\cot^{3}\theta$$
  • $$3\sec^{2}\theta \tan \theta$$
  • $$\dfrac {3}{4a^{2}}\cot^{3}\theta$$
The value of k which makes $$f(x)=\left\{\begin{matrix} \sin\dfrac{1}{x}, x\neq 0\\ k, x=0\end{matrix}\right.$$ continuous at $$x=0$$ is?
  • $$8$$
  • $$1$$
  • $$-1$$
  • None
Let $$f(x) = \underset{0}{\overset{x}{\int}} |2t - 3| dt$$, then $$f$$ is 
  • continuous at $$x = \dfrac{3}{2}$$
  • continuous at $$x = 3$$
  • differentiable at $$x = \dfrac{3}{2}$$
  • differentiable at $$x = 0$$
Let $$f(x) = \left\{\begin{matrix}x &x < 1 \\ 2 - x & 1 \leq x \leq 2\\ -2 + 3x - x^{2} & x > 2\end{matrix}\right.$$ then $$f(x)$$ is
  • Differentiable at $$x = 0$$
  • Differentiable at $$x = 2$$
  • Differentiable at $$x = 1$$ and $$x = 2$$
  • Not differentiable at $$x = 0$$
Range of function $$f(x)-g(x)$$ in [-$$\pi, \pi$$] is-
  • $$[0, e^{\pi}+1]$$
  • $$[0, e^{\pi}-1]$$
  • $$e^{\pi}-1, e^{\pi}+1]$$
  • $$[e^{-\pi}, e^{\pi}]$$
Which of the following is NOT CORRECT-
  • $$b=d$$
  • $$a=d$$
  • $$a\neq b$$
  • $$c\neq d$$
If $$f\left( x \right) = \left( {x - a} \right)g\left( x \right)$$ and $$g\left( x \right)$$ is continuous $$x = a$$ then $${f^{'}}\left( 1 \right) = $$
  • $$g\left( 1 \right)$$
  • $${g^{'}}\left( 1 \right)$$
  • $$ - {g^{'}}\left( 1 \right)$$
  • $${g^{'}}\left( { - 1} \right)$$
Function $$f(x)-g(x)$$ is-
  • continuous & differentiable at $$x=0$$
  • continuous but not differentiable at $$x=0$$
  • discontinuous at $$x=0$$
  • non differentiable because it is discontinuous at $$x=0$$
Differentiate $$\log(1+x^{2})$$ with respect $$\tan^{-1}x$$.
  • $$2x$$
  • $$x^2$$
  • $$x$$
  • $$x^3$$
An angle $$\theta$$ through which a pulley turns with time $$'t'$$ is completed by $$\theta = t^{2} + 3t - 5\ sq. cms/ min$$. Then the angular velocity for $$t = 5\ sec$$.
  • $$5^{c}/ sec$$
  • $$13^{c}/ sec$$
  • $$23^{c}/ sec$$
  • $$35^{c}/ sec$$
If the following function is continuous at $$x=0$$, find the value of $$k$$:
$$f\left( x \right) = \left\{ {\begin{array}{ccccccccccccccc}{\dfrac{{\sin \frac{{3x}}{2}}}{x}}&{,x \ne 0}\\k&{,x = 0}\end{array}} \right.$$
  • $$\dfrac{1}{3}$$
  • $$\dfrac{2}{3}$$
  • $$\dfrac{4}{3}$$
  • $$\dfrac{3}{2}$$
Let $$g\left(x\right)=\dfrac{f\left(x\right)}{x+1}$$ where $$f\left(x\right) $$ is differentiable on  $$\left[0,5\right]$$ such that $$f\left(0\right)=4,f\left(5\right)=-1$$. There exists $$c\in \left(0,5\right)$$ such that $$g^{ ' }\left( c \right) $$ is ?
  • $$-\dfrac{1}{6}$$
  • $$\dfrac{1}{6}$$
  • $$-\dfrac{5}{6}$$
  • $$-1$$
Let $$f(x)=3x^{10}-7x^8+5x^6-21x^3+3x^2-7$$.
The value of $$\lim _{ h\rightarrow 0 }{ \dfrac { f\left( 1-h \right) -f\left( 1 \right)  }{ { h }^{ 3 }+3h }  } $$ is
  • $$\dfrac{50}{3}$$
  • $$\dfrac{22}{3}$$
  • $$13$$
  • None of these
If $$f$$ is differentiable on $$(a,b)$$ and if $$f(a)=f(b)=0$$ then for any $$\alpha $$ there is an $$x\in (a,b)$$ such that $$\alpha f(x)+f'(x)-1=0$$
  • True
  • False
A function $$f:R\rightarrow R$$ is such that $$f(1)=3$$ and $$f'(1)=6$$. Then $$\rightarrow { \displaystyle \lim _{ x\rightarrow 0 }{ \left[ \dfrac { f\left( 1+x \right)  }{ f\left( 1 \right)  }  \right]  }  }^{ 1/x }=$$ ?
  • $$1$$
  • $$e^{2}$$
  • $$e^{1/2}$$
  • $$e^{3}$$
If $$y=y(x)$$ and it follows the relation $$e^{xy^{2}}+y\cos(x^{2})=5$$ then $$y'(0)$$ is equal to
  • $$4$$
  • $$-16$$
  • $$-4$$
  • $$16$$
If $$f$$ is a real valued differentiable function satisfying $$\left| f\left( x \right) -f\left( y \right)  \right| \le { \left( x-y \right)  }^{ 2 },x,y\epsilon R\ and f\left( 0 \right) ,\ then f\left( 1 \right)$$ equals ?
  • $$-1$$
  • $$0$$
  • $$2$$
  • $$1$$
If $$f (x)$$ is differentiable everywhere, then:
  • $$| f |$$ is differentiable everywhere
  • $$|  { f } | ^ { 2 }$$ is differentiable everywhere
  • $$ f | f |$$ is not differentiable at some point
  • $$ f + | f |$$ is differentiable everywhere
Let $$f$$ differentiable at $$x=0$$ and $$f'\left( 0 \right) = 1$$. Then $$\mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( h \right) - f\left( { - 2h} \right)}}{h} = $$=
  • 3
  • 2
  • 1
  • 6
Let $$f(x)$$ be differentiable function such that $$f\left(\dfrac{x+y}{1-xy}\right)=f(x)+f(y) \forall x$$ and $$y$$. If $$\underset { x\rightarrow 0 }{ lt } \dfrac { f\left( x \right)  }{ x } =\dfrac { 1 }{ 3 }$$ then $$f(1)$$ equals 
  • $$\dfrac{1}{4}$$
  • $$\dfrac{1}{6}$$
  • $$\dfrac{1}{12}$$
  • $$\dfrac{1}{8}$$
If $$f(x)=x^{n}\sin \dfrac {1}{x},f(0)=0$$
  • $$f$$ is continuous differentiable at $$x=0$$ for any $$n< 2, n\in N$$
  • $$f$$ is continuous differentiable for all $$x$$ for any $$n\ge 2, n\in N$$
  • $$f$$ is discountinuous at $$x=0$$ if $$n=0$$
  • $$f'(x)$$ is discountinuous at $$x=0$$ if $$n=0$$
If $$x=\dfrac{e^t+e^{-t}}{2}, y=\dfrac{e^t-e^{-t}}{2}$$, then $$\dfrac{dx}{dy}=$$
  • $$-\dfrac{x}{y}$$
  • $$\dfrac{y}{x}$$
  • $$\dfrac{x}{y}$$
  • $$-\dfrac{y}{x}$$
If $$y=a^{{a}^{{x}}}$$, then $$\dfrac {dy}{dx}=$$

  • $$y.a^{x}{(\log a)}^{2}$$
  • $$y.a^{x}.\log a$$
  • $${(y.a^{x})}^{2}$$
  • $$(y.a^{x})$$
Solve:
$$\dfrac { d } { d x } \tan ^ { - 1 } ( \sec x + \tan x )$$
  • $$1$$
  • $$1 / 2$$
  • $$\cos x$$
  • sec $$x$$
0:0:1


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