CBSE Questions for Class 12 Commerce Maths Application Of Derivatives Quiz 10 - MCQExams.com

The equation of the curves through the point $$(1,0)$$ and whose slope is $$ \dfrac{y -1}{x^{2} + x} $$ is
  • $$ (y - 1)(x + 1) + 2x = 0 $$
  • $$ 2x(y - 1) + x + 1 = 0 $$
  • $$ x(y - 1)(x + 1) + 2 = 0 $$
  • None of these
The function f(x) is 
  • increasing for all x
  • non-monotonic
  • decreasing for all x
  • None of these
If $$ f(x) = \displaystyle \int_{1}^{x} e^{t^2/2}(1-t^2)dt, $$ then $$\dfrac{d}{dx} f(x) $$ at x=1 is 
  • $$0$$
  • $$1$$
  • $$2$$
  • $$-1$$
The curve for which the ratio of the length of the segment by any tangent on the $$Y-$$axis to the length of the radius vector is constant $$(K)$$, is
  • $$(y+\sqrt {x^2 -y^2})x^{k-1}=c$$
  • $$(y+\sqrt {x^2 +y^2})x^{k-1}=c$$
  • $$(y-\sqrt {x^2 -y^2})x^{k-1}=c$$
  • $$(y+\sqrt {x^2 +y^2})x^{k-1}=c$$
Number of critical point for $$y=f(x)$$ for $$x \in [0,2]$$
  • $$0$$
  • $$1$$
  • $$2$$
  • $$3$$
The point of the curve $$ y^2 = x$$ where the tangent makes an angle of $$ \frac { \pi}{4} $$ with x-axis is 
  • $$ ( \frac {1}{2}, \frac {1}{4} ) $$
  • $$ ( \frac {1}{4}, \frac {1}{2} ) $$
  • $$ (4,2 ) $$
  • $$ (1,1) $$
The abscissa of the point on the curve $$ 3y=6x- 5x^3 $$ the normal at which passes through origin is :
  • $$ 1 $$
  • $$ \frac {1}{3} $$
  • $$ 2 $$
  • $$ \frac {1}{2} $$
The curve $$ y=x^{\frac{1}{5}} $$ has at $$ (0,0) $$
  • a vertical tangent (parallel to y-axis)
  • a horizontal tangent (parallel to x-axis)
  • an oblique tangent
  • no tangent
The equation of the curve satisfying the differential equation $$y_2(x^2 + 1) = 2xy_1$$ passing through the point $$(0, 1)$$ and having slope of tangent at $$x = 0$$ as $$3$$ (where $$y_2$$ and $$y_1$$ represents 2nd and 1st order derivative), then
  • $$y=f(x)$$ is a strictly increasing function
  • $$y=f(x)$$ is non-monotomic finction
  • $$y=f(x)$$ has three distinct real roots
  • $$y=f(x)$$ has only one negative root
The tangent to the curve $$ y=e^{2x} $$ at the point $$ (0,1) $$ meets x-axis at:
  • $$ (0,1) $$
  • $$ \left( -\frac { 1 }{ 2 } ,0 \right) $$
  • $$ (2,0) $$
  • $$ (0,2) $$
The slope of tangent to the curve $$ x=t^2+3t-8,y=2t^2-2t-5 $$ at the point $$ (2,-1) $$ is:
  • $$ \frac{22}{7} $$
  • $$ \frac{6}{7} $$
  • $$ \frac{-6}{7} $$
  • $$ -6 $$
The curve for which the slope of the tangent at any point is equal to the ratio of the abscissa to the ordinate of the point is :
  • an ellipse
  • parabola
  • circle
  • rectangular hyperbola
The line $$5x-2y+4k=0$$ is tangent to $$4x^{2}-y^{2}=36$$, then k is:
  • $$\dfrac{9}{4}$$
  • $$\dfrac{81}{16}$$
  • $$\dfrac{4}{9}$$
  • $$\dfrac{2}{3}$$
The function $$ f\left( x \right) =tan x-x $$
  • always increases
  • always decreases
  • never increases
  • sometime increase and sometimes decreases
Which of the following function is decreasing on $$\left( 0,\frac { \pi  }{ 2 }  \right) $$
  • $$ sin 2 x $$
  • $$ tan x $$
  • $$ cos x $$
  • $$ cos 3 x $$
If the tangent at $$(1,1)$$ on $$y^{2}=x(2-x)^{2}$$ meets the curve again at $$P$$, then $$P$$ is
  • $$(4,4)$$
  • $$(-1,2)$$
  • $$(3,6)$$
  • $$\left(\dfrac{9}{4}, \dfrac{3}{8}\right)$$
The slope of the tangent to the curve $$x = t^{2} + 3 t - 8, y = 2t^{2} - 2t - 5$$ at the point $$(2, -1)$$ is
  • $$\dfrac{22}{7}$$
  • $$\dfrac{6}{7}$$
  • $$\dfrac{7}{6}$$
  • $$\dfrac{-6}{7}$$
The line $$y = mx + 1$$ is a tangent to the curve $$y^{2} = 4x$$ if the value of m is .......
  • $$1$$
  • $$2$$
  • $$3$$
  • $$\dfrac{1}{2}$$
The normal at the point $$(1, 1)$$ on the curve $$2y + x^{2} - 3$$ is .............
  • $$x + y = 0$$
  • $$x - y = 0$$
  • $$x + y = 1$$
  • $$x - y = 1$$
The slope of the normal to the curve $$ y = 2x ^{2} + 3 \sin x $$ at $$ x = 0 $$ is 
  • $$3$$
  • $$1/3$$
  • $$-3$$
  • $$-1 /3$$
The normal to the curve $$x^{2} = 4y$$ passing $$(1, 2)$$ is
  • $$x + y = 3$$
  • $$x - y = 3$$
  • $$x + y = 1$$
  • $$x - y = 1$$
The line $$ y = x + 1 $$ is a tangent to the curve $$y^{2} = 4 x $$ at the point 
  • $$ ( 1 , 2 ) $$
  • $$ ( 2 , 1 ) $$
  • $$ ( 1 , -2 ) $$
  • $$ ( -1 , 2 )$$
The points on the curve $$9 y^{2} = x^{3}$$, where the normal to the curve makes equal intercepts with the axes are ...........
  • $$\left ( 4, \pm \dfrac{8}{3} \right )$$
  • $$\left ( 4, \dfrac{-8}{3} \right )$$
  • $$\left ( 4, + \dfrac{8}{3} \right )$$
  • $$\left (\pm 4, \dfrac{8}{3} \right )$$
For $$a\in[\pi,2\pi]$$ and $$n\in I$$, the critical points of $$\displaystyle f(x)=\frac{1}{3}\sin a\tan^{3}x+(\sin a - 1 ) \tan x +\sqrt{\frac{a-2}{8-a}}$$ is
  • $$x=n\pi$$
  • $$x=2n\pi$$
  • $$x=(2n+1)\pi$$
  • no critical points
Let $$\mathrm{f}(\mathrm{x})=\mathrm{a}\mathrm{x}^{3}+\mathrm{b}\mathrm{x}^{2}+ cx + \mathrm{d}$$, where $$a,b,c,d $$ are real and $$3\mathrm{b}^{2}<\mathrm{c}^{2}$$, is an increasing function and $$\mathrm{g}(\mathrm{x})=\mathrm{a}\mathrm{f}'(\mathrm{x})+\mathrm{b}\mathrm{f}''(\mathrm{x})+\mathrm{c}^{2}$$. lf $$\displaystyle \mathrm{G}(\mathrm{x})=\int_{\alpha}^{\mathrm{x}}\mathrm{g}(\mathrm{t})\mathrm{d}\mathrm{t},\alpha \in \mathrm{R}$$, then for $$ \alpha < x < \alpha +1 $$,
  • G(x) is a decreasing function
  • G(x) is an increasing function
  • G(x) is neither increasing nor decreasing
  • G(x) is a one-one function
Let $$f'\left( \sin { x }  \right) <0$$ and $$\displaystyle f''\left( \sin { x }  \right) >0,\quad \forall \quad x\in \left( 0,\frac { \pi  }{ 2 }  \right) $$ and $$g\left( x \right)=f\left( \sin { x }  \right) +f\left( \cos { x }  \right) ,$$ then $$g(x)$$ is decreasing in
  • $$\displaystyle \left( \frac { \pi  }{ 4 } ,\frac { \pi  }{ 2 }  \right) $$
  • $$\displaystyle \left( 0,\frac { \pi  }{ 4 }  \right) $$
  • $$\displaystyle \left( 0,\frac { \pi  }{ 2 }  \right) $$
  • $$\displaystyle \left( \frac { \pi  }{ 6 } ,\frac { \pi  }{ 2 }  \right) $$
The point of contact of vertical tangent to the curve given by the equations $$\mathrm{x}=3-2\cos\theta, \mathrm{y}=2+3\sin\theta$$ is
  • (1, 5)
  • (1, 2)
  • (5, 2)
  • (2, 5)
The value of a for which the function $$\displaystyle \mathrm{f}(\mathrm{x})=(4\mathrm{a}-3)(\mathrm{x}+\log 5)+2(\mathrm{a}-7)\cot\frac{\mathrm{x}}{2}\sin^{2}\frac{\mathrm{x}}{2}$$ does not possess critical points is
  • $$(-\displaystyle \infty,-\frac{4}{3})\mathrm{\cup}(2,\infty)$$
  • $$(-\infty, -1)$$
  • $$[1, \infty)$$
  • $$(-2,\infty)$$
The greatest inclination between the tangents is
  • $$\displaystyle \tan^{-1} \left ( \dfrac{\mathrm{a}+\mathrm{b}}{2\sqrt{\mathrm{a}\mathrm{b}}}\right )$$
  • $$\displaystyle \tan^{-1} \left (\dfrac{\mathrm{a}-\mathrm{b}}{2\sqrt{\mathrm{a}\mathrm{b}}}\right )$$
  • $$\tan^{-1}\sqrt{\dfrac{\mathrm{a}}{\mathrm{b}}}$$
  • $$\tan^{-1}\sqrt{\dfrac{\mathrm{b}}{\mathrm{a}}}$$
A function $$y=f(x)$$ has a second order derivative $$f''(x)=6(x-1)$$ .
If its graph passes through the point $$(2,1)$$ and at that point the tangent to the graph is $$y=3x-5$$, then the function is
  • $$(x-1)^{2}$$
  • $$(x+1)^{2}$$
  • $$(x+1)^{3}$$
  • $$(x-1)^{3}$$
If $$ f(x) = \displaystyle \frac{x}{{\sin x}}$$ and $$g(x) = \displaystyle \frac{x}{{\tan x}}$$  where $$0 < x \leq 1$$ then in the interval
  • Both f(x) and g(x) are increasing functions
  • Both f(x) and g(x) are decreasing functions
  • f(x) is an increasing function
  • g(x) is an increasing function
A function $$y = f (x)$$ is given by $$x = \cos^2\theta$$ & $$y =\dfrac{\cot\,  \theta }{\sec^2\, \theta }$$ for all $$\theta >0$$, then $$f$$ is :
  • increasing in $$x \in \left (0, \dfrac {3}{2}\right)$$ & decreasing in $$x \in \left ( \dfrac {3}{2}, \infty\right)$$
  • increasing in $$x \in (0, 1)$$
  • increasing in $$x \in (0, 2)$$
  • decreasing in $$x \in ( 2, \infty)$$
Suppose $$a,b,c$$ are such that the curve $$y = ax^2 + bx + c$$ is tangent to $$y = 3x -3$$ at $$(1, 0)$$ and is also tangent to $$y = x + 1$$ at $$(3, 4)$$ then the value of $$(2a -b -4c)$$ equals
  • $$7$$
  • $$8$$
  • $$9$$
  • $$10$$
For the curve $$y=3  \sin \theta  \cos  \theta,  x= e^{\theta} \sin \theta,  0  \leq \theta  \leq  \pi$$, the tangent is parallel to x-axis when $$\theta$$ is :
  • $$\displaystyle \frac{\pi}{4}$$
  • $$\displaystyle \frac{\pi}{2}$$
  • $$\displaystyle \frac{3\pi}{4}$$
  • $$\displaystyle \frac{\pi}{6}$$
If $$f(x) = x^{2/3}$$ then
  • $$(0,0)$$ is a point of maxima
  • $$(0,0)$$ is a point of minima
  • $$(0,0)$$ is a critical point
  • There is no critical point
If $$f(x)=\left\{\begin{matrix}x^2+2 & x<0\\ 3 & x = 0\\ x+2 & x>0\end{matrix}\right.$$, then which of the following statement(s) is/are false ?
  • $$f(x)$$ has a local maximum at $$x=0$$
  • $$f(x)$$ is strictly decreasing on the left of $$x=0$$
  • $$f'(x)$$ is strictly increasing on the left of $$x=0$$
  • $$f'(x)$$ is strictly increasing on the right of $$x=0$$
For the curve represented parametrically by the equations, $$x = 2 ln \cot( t) + 1$$ & $$y = \tan( t) + \cot( t)$$
  • tangent at $$t = \pi/4$$ is parallel to x - axis
  • normal at $$t = \pi/4$$ is parallel to y - axis
  • tangent at $$t = \pi/4$$ is parallel to the line $$y = x$$
  • tangent and normal intersect at the point $$(2, 1)$$
A curve passes through $$(2, 0)$$ and the slope of the tangent at any point $$(x, y)$$ is $$x^2 -2x$$ for all values of $$x$$. The point of minimum ordinate on the curve where $$x > 0$$ is $$(a, b)$$'
Then find the value of $$a + 6b$$.
  • $$2$$
  • $$4$$
  • $$-2$$
  • $$-4$$
The value of $$x$$ at which tangent to the curve $$y=x^3-6x^2+9x+4,   0\leq x \leq 5$$ has maximum slope is
  • $$0$$
  • $$2$$
  • $$\dfrac{5}{2}$$
  • $$5$$
The point on the curve $$y^{2} = x ,$$ the tangent at which makes an angle of $$45^{0}$$ with positive direction of $$x -$$ axis will be given by
  • $$\left (\displaystyle \frac{1}{2},\displaystyle \frac{1}{4} \right )$$
  • $$\left ( \displaystyle \frac{1}{2}, \displaystyle \frac{1}{2} \right )$$
  • $$(2,4)$$
  • $$\left ( \displaystyle \frac{1}{4}, \displaystyle \frac{1}{2} \right )$$
A function $$y=f(x)$$ has a second-order derivative $$f''(x)=6(x-1)$$. If its graph passes through the point $$(2,1)$$ and at the point tangent to the graph is $$y=3x-5$$, then the value of $$f(0)$$ is 
  • $$1$$
  • $$-1$$
  • $$2$$
  • $$0$$
The period of oscillation $$T$$ of a pendulum of length $$l$$ at a place of acceleration due to gravity $$g$$ is given by $$T=2\pi \sqrt {\dfrac {l}{g}}$$. If the calculated length is $$0.992$$ times the actual length and if the value assumed for $$g$$ is $$1.002$$ times its actual value, the relative error in the computed value of $$T$$ is
  • $$0.005$$
  • $$-0.005$$
  • $$0.003$$
  • $$-0.003$$
The focal length of a mirror is given by $$\dfrac {1}{v}-\dfrac {1}{u}=\dfrac {2}{f}$$. If equal errors ($$\alpha$$) are made in measuring $$u$$ and $$v$$, then the relative error in $$f$$ is
  • $$\dfrac {2}{\alpha}$$
  • $$\alpha \left (\dfrac {1}{u}+\dfrac {1}{v}\right )$$
  • $$\alpha \left (\dfrac {1}{u}-\dfrac {1}{v}\right )$$
  • none of these
The tangent of the acute angle between the curves $$y=|x^2-1| $$ and $$y=\sqrt {7-x^2}$$ at their points of intersection is
  • $$\displaystyle \frac {5\sqrt 3}{2}$$
  • $$\displaystyle \frac {3\sqrt 5}{2}$$
  • $$\displaystyle \frac {5\sqrt 3}{4}$$
  • $$\displaystyle \frac {3\sqrt 5}{4}$$
The angle made by the tangent of the curve $$x=a (t+\sin t \cos t)$$, $$y=a(1+sint)^2$$ with the $$x- axis$$ at any point on it is
  • $$\displaystyle \frac {1}{4}(\pi +2t)$$
  • $$\displaystyle \frac {1-\sin t}{\cos t}$$
  • $$\displaystyle \frac {1}{4}(2t-\pi)$$
  • $$\displaystyle \frac {1+\sin t}{\cos 2t}$$
Consider the function $$f(x)= \begin{cases} x \sin \displaystyle \frac {\pi}{x}, for  \ x>0\\ 0,                   for \   x=0 \end{cases}$$. Then, the number of points in $$(0,1)$$ where the derivative $$f'(x)$$ vanishes is
  • $$0$$
  • $$1$$
  • $$2$$
  • infinite
The abscissas of points $$P$$ and $$Q$$ on the curve $$y=e^x+e^{-x}$$ such that tangents at $$P$$ and $$Q$$ make $$60^{\circ}$$ with the $$x$$-axis are
  • $$\ln \left (\displaystyle \frac {\sqrt 3+\sqrt 7} {7}\right )$$ and $$\ln \left (\displaystyle \frac {\sqrt 3+\sqrt 5} {2}\right )$$
  • $$\ln \left (\displaystyle \frac {\sqrt 3+\sqrt 7} {2}\right )$$
  • $$\ln \left (\displaystyle \frac {\sqrt 7-\sqrt 3} {7}\right )$$
  • $$\pm\ln \left (\displaystyle \frac {\sqrt 3+\sqrt 7} {2}\right )$$
The graphs $$y=2x^3-4x+2$$ and $$y=x^3+2x-1$$ intersect at exactly 3 distinct points. The slope of the line passing through two of these points
  • is equal to 4
  • is equal to 6
  • is equal to 8
  • is not unique
If the curve represented parametrically by the equations $$x=2 \ln\cot t+1$$ and $$y=\tan t+ \cot t$$
  • tangent and normal intersect at the point $$(2,1)$$
  • normal at $$t=\displaystyle \frac{\pi}{4}$$ is parallel to the $$y$$ axis
  • tangent at $$t=\displaystyle \frac{\pi}{4}$$ is parallel to the line $$y=x$$
  • tangent at $$t=\displaystyle \frac{\pi}{4}$$ is parallel to the $$x$$ axis
Let $$S$$ be a square with sides of length $$x$$. If we approximate the change in size of the area of $$S$$ by $$\displaystyle h.\frac{dA}{dx}|_{x=x_0}$$, when the sides are changed from $$x_0$$ to $$x_o+h$$, then the absolute value of the error in our approximation, is
  • $$h^2$$
  • $$2hx_0$$
  • $$x_0^2$$
  • $$h$$
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