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CBSE Questions for Class 12 Commerce Maths Continuity And Differentiability Quiz 11 - MCQExams.com

If y=x(logx)log(logx), then dydx is
  • yx((lnxx1)+2lnxln(lnx))
  • yx(logx)log(logx)(2log(logx)+1)
  • yxlnx[(lnx)2+2ln(lnx)]
  • yxlogylogx[2log(logx)+1]
If f(x)=|x||sinx|, then f(π4)=
  • (π5)12(22log5π22π)
  • (π4)12(22log4π22π)
  • (π3)12(22log3π33π)
  • None of these
If x=sin1t and y=log(1t2) , then d2ydx2|t=12=
  • 83
  • 83
  • 34
  • 34
If a function is represented parametrically by the equations x=1+logett2y=3+2logett, then which of the following statements are true?
  • y(x2xy)=y
  • yy=2x(y)2+1
  • xy=2y(y)2+2
  • y(y4xy)=(y)2
y=(1+1x)x+x1+1x.
Differentiate
  • (1+1x)x[log(1+1x)1x+1]+x1+1/x[x+1logxx2].
  • (1+1x)x[log(1+1x)1x+1]+x1+1/x[x+1+logxx2].
  • (1+1x)x[log(1+1x)1x+1]+x11/x[x+1logxx2].
  • (1+1x)x[log(1+1x)1x+1]+x11/x[x+1+logxx2].
If x=sint, y=sinkt then value of (1x2)y2xy1 is
  • k2y
  • k2y
  • ky2
  • ky2
If y=xlnxln(lnx), then dydx is equal to:
  • yx(lnxlnx1+2lnxln(lnx))
  • yxlnxln(lnx)(2ln(lnx)+1)
  • yxlnxlnx2+2ln(lnx)
  • ylnxxlnx(2ln(lnx)+1)
If f(x)=|x|+|cosx|, then
  • f(π2)=2
  • f(π2)=0
  • f(π2)=1
  • f(π2) does not exist
Given the parametric equations x=f(t),y=g(t). Then \displaystyle \frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } equals
  • \displaystyle \dfrac { \dfrac { { d }^{ 2 }y }{ d{ t }^{ 2 } } .\dfrac { dx }{ dt } -\dfrac { dy }{ dt } .\dfrac { { d }^{ 2 }x }{ d{ t }^{ 2 } }  }{ { \left( \dfrac { dx }{ dt }  \right)  }^{ 2 } }
  • \displaystyle \dfrac { \dfrac { { d }^{ 2 }y }{ d{ t }^{ 2 } } .\dfrac { dx }{ dt } -\dfrac { dy }{ dt } .\dfrac { { d }^{ 2 }x }{ d{ t }^{ 2 } }  }{ { \left( \dfrac { dx }{ dt }  \right)  }^{ 3 } }
  • \displaystyle \dfrac { \dfrac { { d }^{ 2 }y }{ d{ t }^{ 2 } }  }{ \dfrac { { d }^{ 2 }x }{ d{ t }^{ 2 } }  }
  • None of these
Derivative of {(x\cos{x})}^x with respect to x is
  • \displaystyle {(x\cos{x})}^x\left[(\log{x}+1)-\left\{\log{\cos{x}}+\frac{x}{\cos{x}}.(\sin{x})\right\}\right]
  • \displaystyle {(x\cos{x})}^x\left[(\log{x}+1)+\left\{\log{\cos{x}}+\frac{x}{\cos{x}}.(-\sin{x})\right\}\right]
  • \displaystyle {(x\cos{x})}^x\left[(\log{x}+1)+\left\{\log{\sin{x}}+\frac{x}{\cos{x}}.(\cos{x})\right\}\right]
  • None of these
The value of y''(1) if { x }^{ 3 }-2{ x }^{ 2 }{ y }^{ 2 }+5x+y-5=0 when y(1)>1, is equal to
  • \displaystyle \frac { 22 }{ 7 }
  • \displaystyle -7\frac { 21 }{ 28 }
  • 8
  • \displaystyle -8\frac { 22 }{ 27 }
The weight W of a certain stock of fish is given by W = nw, where n is the size of stock and w is the average weight of a fish. If n and w change with time t as n = 2t^2 + 3 and w = t^2 - t + 2, then the rate of change of W with respect to t at t = 1 is.
  • 1
  • 5
  • 8
  • 31
If x=log(1+t^2) and y=t-tan^{-1}t. Then, \dfrac{dy}{dx} is equal to 
  • e^x-1
  • t^2-1
  • \dfrac{\sqrt{e^x-1}}{2}
  • e^x-y
The function f(x)=\begin{cases} ax(x-1)+b\quad \quad when\quad x<1 \\ x-1\quad \quad \quad when\quad \quad 1\le x\le 3 \\ p{ x }^{ 2 }+qx+2\quad \quad when\quad x>3 \end{cases} Find the values of the constants a,b,p,q so that (i)f(x) is continuous for all x  (ii)f'(1) does not exist  (iii)f'(x) is continuous at x=3
  • a=1, b=0, p=1/3, q=-1
  • a\ne 1, b=0, p=1/3, q=-1
  • a\ne 1, b=0, p=1/3, q=1
  • a=1, b=0, p=1/3, q=1
Length of the subtangent at (x_l, y_l) on x^n y^m = a^{m+n}, m, n > 0,is
  • \dfrac{n}{m}x_l
  • \dfrac{m}{n}|x_l|
  • \dfrac{n}{m}|y_l|
  • \dfrac{n}{m}|x_l|
x12345
f(x)43713
The function f is continuous on the closed interval [1, 5] and values of the function are shown in the table above. If the values in the table are used to calculate a trapezoidal sum, the approximate value of \int_{1}^{5}f(x)dx is
  • 14
  • 14.5
  • 15
  • 29
The derivative of \displaystyle (\tan x)^{x} is equal to-
  • \displaystyle x(\tan x)^{x-1}
  • \displaystyle (\tan x)^{x}\left [ \sec x+\tan x \right ]
  • \displaystyle (\tan x)^{x}\left [ x\sec x \csc x +\log \tan x\right ]
  • \displaystyle(\tan x)^{x}\left [ \sec ^{2}x+x\tan x \right ]
Examine for continuity and differentiability at the points x=1, x=2, the function f defined by f(x)=\begin{cases} x\left[ x \right] ,\quad \quad \quad \quad 0\le x<2 \\ (x-1)\left[ x \right] ,\quad 2\le x\le 3 \end{cases} where \left[ x \right] = greatest integer less than or equal to x
  • discontinuous and not derivable at x=1,2
  • discontinuous and not derivable at x=1, continuous but not derivable at x=2
  • continuous and not derivable at x=1,2.
  • continuous and not derivable at x=1, discontinuous but not derivable at x=2.
If y=\sqrt { \left( a-x \right) \left( x-b \right)  } -\left( a-b \right) \tan ^{ -1 }{ \sqrt { \dfrac { a-x }{ x-b }  }  } , then \dfrac { dy }{ dx } is equal to
  • 1
  • \sqrt { \dfrac { a-x }{ x-b } }
  • \sqrt { \left( a-x \right) \left( x-b \right) }
  • \dfrac { 1 }{ \sqrt { \left( a-x \right) \left( b-x \right) } }
The derivative of y=x^{2^x} w.r.t x is :
  • x^{2^x}2^x\left(\displaystyle \frac{1}{x} + \ln x \ln 2 \right)
  • x^{2^x}\left(\displaystyle\frac{1}{x} \ln x \ln 2\right)
  • x^{2^x}2^x\left(\displaystyle\frac{1}{x} \ln x\right)
  • x^{2^x}2^x\left(\displaystyle\frac{1}{x}+\frac{\ln x}{\ln 2}\right)
Let f(x) = \left\{\begin{matrix}2a - x, &if\ -a < x < a \\ 3x - 2a, &if\ a \leq x \end{matrix}\right.. 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 x \geq a
  • f(x) is continuous at all x < a
If x = \sec \theta - \cos \theta and y = \sec^{n}\theta - \cos^{n}\theta, then \left (\dfrac {dy}{dx}\right )^{2} is equal to
  • \dfrac {n^{2}(y^{2} + 4)}{x^{2} + 4}
  • \dfrac {n^{2}(y^{2} - 4)}{x^{2}}
  • n\left (\dfrac {y^{2} - 4}{x^{2} - 4}\right )
  • \left (\dfrac {ny}{x}\right )^{2} - 4
If a curve is given by x=a\cos t+\displaystyle\frac{b}{2}\cos 2t and y=\sin t +\displaystyle\frac{b}{2}\sin 2t, then the points for which \displaystyle\frac{d^2y}{dx^2}=0, are given by.
  • \sin t=\displaystyle\frac{2a^2+b^2}{3ab}
  • \cos t=-\left [\displaystyle\frac{a^2+2b^2}{3ab}\right]
  • \tan t=a/b
  • None of the above
If \sin ^{ -1 }{ \left( \dfrac { { x }^{ 2 }-{ y }^{ 2 } }{ { x }^{ 2 }+{ y }^{ 2 } }  \right)  } =\log { a } , then \dfrac { { d }^{ 2 }y }{ d{ x }^{ 2 } } equals
  • \dfrac { x }{ y }
  • \dfrac { y }{ { x }^{ 2 } }
  • \dfrac { y }{ x }
  • 0
Find derivative of \tan^{-1}\dfrac{\cos x-\sin x}{\cos x+\sin x} w.r.t. x.
  • -1
  • 0
  • 1
  • 2
If y=\tan ^{ -1 }{ \left( \cfrac { 4x }{ 1+5{ x }^{ 2 } }  \right)  } +\tan ^{ -1 }{ \left( \cfrac { 2+3x }{ 2-3x }  \right)  } , then \cfrac { dy }{ dx } is
  • \cfrac { 6 }{ 1+4{ x }^{ 2 } }
  • \cfrac { 3 }{ 1+4{ x }^{ 2 } }
  • \cfrac { 5 }{ 1+{ 25x }^{ 2 } }
  • \dfrac{5}{(1+25x^{2})} - \dfrac{1}{(1+x^{2})}  - \dfrac{1.5}{(1+2.25x^{2})}
If x = A\cos 4t + B\sin 4t, then \dfrac {d^{2}x}{dt^{2}} is equal to
  • -16x
  • 16x
  • x
  • -x
Let f\left( x \right) and g\left( x \right) be defined and differentiable for x\ge { x }_{ 0 } and f\left( { x }_{ 0 } \right) =g\left( { x }_{ 0 } \right) , f^{ ' }\left( x \right) >g^{ ' }\left( x \right) for\ x>{ x }_{ 0 } then
  • f\left( x \right) < g\left( x \right) for some x>{ x }_{ 0 }
  • f\left( x \right) =g\left( x \right) for some x>{ x }_{ 0 }
  • f\left( x \right) >g\left( x \right) for all x>{ x }_{ 0 }
  • None of these
Find the derivative of \dfrac{\tan^{-1}x}{1+\tan^{-1}x} w.r.t. \tan^{-1}x.
  • \dfrac{1}{\sec^{-1}x}
  • \dfrac{1}{(\tan^{-1}x)^{2}}
  • \dfrac{1}{1+\tan^{2}x}
  • \dfrac{1}{(1+\tan^{-1}x)^{2}}
If the function f:\left[ 0,8 \right] \rightarrow R is differentiable, then for 0<a,b<2,\int _{ 0 }^{ 8 }{ f(t) } dt is equal to
  • $$2\left[ { \alpha }^{ 3 }f({ \alpha }^{ 2 })+{ \beta }^{ 2 }f({ \beta }^{ 2 }) \right] $
  • 3\left[ { \alpha }^{ 3 }f({ \alpha }^{ })+{ \beta }^{ 2 }f({ \beta }^{ }) \right]
  • 3\left[ { \alpha }^{ 2 }f({ \alpha }^{ 3 })+{ \beta }^{ 2 }f({ \beta }^{ 3 }) \right]
  • 3\left[ { \alpha }^{ 2 }f({ \alpha }^{ 2 })+{ \beta }^{ 2 }f({ \beta }^{ 2 }) \right]
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