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CBSE Questions for Class 12 Commerce Maths Continuity And Differentiability Quiz 4 - MCQExams.com

Consider the function
f(x)={2sinxifxπ2Asinx+Bifπ2<x<π2cosxifxπ2
which is continuous everywhere.
The value of A is
  • 1
  • 0
  • -1
  • -2
If xm+ym=1 such that dydx=xy, then what should be the value of m?
  • 0
  • 1
  • 2
  • None of the above
If s=\sqrt{t^2+1}, then \dfrac{d^2s}{dt^2} is equal to
  • \dfrac{1}{s}
  • \dfrac{1}{s^2}
  • \dfrac{1}{s^3}
  • \dfrac{1}{s^4}
Consider the function
f(x)=\begin{cases}-2\sin x & if & x\le -\dfrac{\pi}{2} \\ A\sin x+B & if & -\dfrac{\pi}{2} < x < \dfrac{\pi}{2} \\ \cos x & if & x \ge \dfrac{\pi}{2}\end{cases}
which is continuous everywhere.
The value of B is
  • 1
  • 0
  • -1
  • -2
If \sec \left (\dfrac {x + y}{x - y}\right ) = a, then \dfrac {dy}{dx} is.
  • \dfrac {x}{y}
  • \dfrac {y}{x}
  • y
  • x
If g is the inverse function of f and f'(x) = \dfrac {1}{1 + x^{n}}, then g'(x) is equal to.
  • 1 + [g(x)]^n
  • 1 - g(x)
  • 1 + g(x)
  • -g(x)^{n}
If y = x\tan y, then \dfrac {dy}{dx} is equal to.
  • \dfrac {\tan y}{x - x^{2} - y^{2}}
  • \dfrac {y}{x - x^{2} - y^{2}}
  • \dfrac {\tan y}{y - x}
  • \dfrac {\tan x}{x - y^{2}}
If \log _{ 10 }{ \left( \cfrac { { x }^{ 2 }-{ y }^{ 2 } }{ { x }^{ 2 }+{ y }^{ 2 } }  \right)  } =2, then \cfrac { dy }{ dx } =............
  • -\cfrac { 99x }{ 101y }
  • \cfrac { 99x }{ 101y }
  • -\cfrac { 99y }{ 101x }
  • \cfrac { 99y }{ 101x }
For what value of k, the function defined by f(x)=\cfrac { \log { (1+2x) } \sin { x }  }{ { x }^{ 2 } } for x\ne 0
                                                                                     =k for x=0
is continuous at x=0?
  • 2
  • \cfrac { 1 }{ 2 }
  • \cfrac { \pi }{ 90 }
  • \cfrac { 90 }{ \pi }
If f(x) = be^{ax} + ae^{bx}, then f'' (0) =
  • 0
  • 2ab
  • ab(a + b)
  • ab
f(x) =\begin{cases}  2a - x \text{,  for  }-a < x < a \\ 3x-2a \text{,  for } x\ge a\end{cases}
Then which of the following is true?
  • f(x) is discontinuous at x = a
  • f(x) is not differentiable at x=a
  • f(x) is differentiable at all x\geq a
  • f(x) is continuous at all x < a
Derivative of \tan ^{ -1 }{ \left( \cfrac { x }{ \sqrt { 1-{ x }^{ 2 } }  }  \right)  } with respect to \sin ^{ -1 }{ \left( 3x-4{ x }^{ 3 } \right)  } is
  • \cfrac { 1 }{ \sqrt { 1-{ x }^{ 2 } } }
  • \cfrac { 3 }{ \sqrt { 1-{ x }^{ 2 } } }
  • 3
  • \cfrac { 1 }{ 3 }
If x^{y} = e^{x - y}, then \dfrac {dy}{dx} is equal to.
  • \dfrac {\log x}{1 + \log x}
  • \dfrac {\log x}{1 - \log x}
  • \dfrac {\log x}{(1 + \log x)^{2}}
  • \dfrac {y\log x}{x(1 + \log x)^{2}}
If x=\sec { \theta  } -\cos { \theta  } and y=\sec ^{ n }{ \theta  } -\cos ^{ n }{ \theta  } , then { \left( \dfrac { dy }{ dx }  \right)  }^{ 2 } is
  • \dfrac { { n }^{ 2 }\left( { y }^{ 2 }+4 \right) }{ { x }^{ 2 }+4 }
  • \dfrac { { n }^{ 2 }\left( { y }^{ 2 }-4 \right) }{ { x }^{ 2 } }
  • n\dfrac { \left( { y }^{ 2 }-4 \right) }{ { x }^{ 2 }-4 }
  • { \left( \dfrac { ny }{ x } \right) }^{ 2 }-4
If f(x) =\begin{cases} x, & \text{for } x\leq 0 \\ 0, & \text{for } x>0\end{cases} 
then f(x) at x = 0 is
  • Continuous but not differentiable
  • Not continuous but differentiable
  • Continuous and differentiable
  • Not continuous and not differentiable
If f(x) =\begin{cases} \log (\sec^{2} x)^{\cot^{2}x}, & \text{for } x\neq 0 \\ K, & x=0\end{cases} 
is continuous at x = 0 then K is
  • e^{-1}
  • 1
  • e
  • 0
If f(x)=[x \sin \pi x], then which of the following is incorrect?
  • f(x) is continuous at x=0
  • f(x) is continuous in (-1, 0)
  • f(x) is differentiable at x=1
  • f(x) is differentiable in (-1, 1)
Which of the following functions is differentiable at x = 0
  • \cos (|x|) + |x|
  • \cos (|x|) - |x|
  • \sin (|x|) + |x|
  • \sin (|x|) - |x|
If 2^x + 2^y = 2^{x + y}, then \dfrac{dy}{dx} is equal to
  • \dfrac{2^x + 2^y}{2^x - 2^y}
  • \dfrac{2^x + 2^y}{1 + 2^{x + y}}
  • 2^{x - y} \left( \dfrac{2^y - 1}{1 - 2^x} \right )
  • \dfrac{2^{x + y} - 2^x}{2^y}
The value of k for which the function f\left( x \right) =\begin{cases} \dfrac { 1-\cos { 4x }  }{ 8{ x }^{ 2 } } &,x\neq 0 \\ k &,x=0 \end{cases} is continuous at x=0, is
  • k=0
  • k=1
  • k=-1
  • None of the above
Let f\left( x \right) =\left\{ \begin{matrix} { x }^{ n }\sin { \frac { 1 }{ x }  } ,x\neq 0 \\ \quad \quad 0,x=0 \end{matrix} \right\} . Then, f\left( x \right) is continuous but not differentiable at x=0, if
  • n\in \left( 0,1 \right)
  • n\in \left[ 1,\infty \right)
  • n\in \left( -\infty ,0 \right)
  • n=0
If x = a \left (\cos t + \log \tan \dfrac {t}{2}\right ), y = a\sin t, then \dfrac {dy}{dx} =
  • \tan t
  • \cot t
  • -\cot t
  • -\tan t
If u = \tan^{-1}\left (\dfrac {\sqrt {1 - x^{2}} - 1}{x}\right ) and v= \sin^{-1} x, then \dfrac {du}{dv} is equal to
  • \sqrt {1 - x^{2}}
  • -\dfrac {1}{2}
  • 1
  • -x
  • -2
If { 2 }^{ x }+{ 2 }^{ y }={ 2 }^{ x+y }, then the value of \cfrac { dy }{ dx } at (1,1) is equal to
  • -2
  • -1
  • 0
  • 1
  • 2
If x = a \cos^3 \theta and y = a\sin^3 \theta, then 1 + \left( \dfrac{dy}{dx} \right )^2 is
  • \tan \theta
  • \tan^2 \theta
  • 1
  • \sec^2 \theta
  • \sec \theta
If y = 4x - 5 is a tangent to the curve y^{2} = px^{3} + q at (2, 3), then (p + q) is equal to
  • -5
  • 5
  • -9
  • 9
  • 0
If xe^{xy} + ye^{-xy} = \sin^{2}x, then \dfrac {dy}{dx} at x = 0 is
  • 2y^{2} - 1
  • 2y
  • y^{2} - y
  • y^{2} + 1
  • y^{2} - 1
If u = 2 (t - \sin t ) and v = 2 (1- \cos t), then \dfrac {dv}{du} at t = \dfrac {2 \pi}{3} is equal to :
  • \sqrt3
  • - \sqrt3
  • 2 \sqrt3
  • \dfrac {2} {\sqrt3}
  • \dfrac {1} {\sqrt3}
If x\sin (a + y) + \sin a\cos (a + y) = 0, then \dfrac {dy}{dx} is equal to
  • \dfrac {\sin^{2}(a + y)}{\sin a}
  • \dfrac {\cos^{2}(a + y)}{\cos a}
  • \dfrac {\sin^{2}(a + y)}{\cos a}
  • \dfrac {\cos^{2}(a + y)}{\sin a}
If s=\sec ^{ -1 }{ \left( \cfrac { 1 }{ 2{ x }^{ 2 }-1 }  \right)  } and t=\sqrt { 1-{ x }^{ 2 } } , then \cfrac { ds }{ dt } at x=\cfrac { 1 }{ 2 } is
  • 1
  • 2
  • -2
  • 4
  • -4
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