CBSE Questions for Class 12 Commerce Maths Application Of Derivatives Quiz 11 - MCQExams.com

The number of points on the curve $$x^{3/2}+y^{3/2}=a^{3/2}$$, where the tangents are equally inclined to the axes, is
  • $$2$$
  • $$1$$
  • $$0$$
  • $$4$$
Let $$\displaystyle f(x) = ln \: mx (m > 0)$$ and $$g(x) = px$$. Then the equation $$\displaystyle |f(x)| = g(x)$$ has only one solution for
  • $$\displaystyle 0 < p < \frac {m}{e}$$
  • $$\displaystyle p < \frac {e}{m}$$
  • $$\displaystyle 0 < p < \frac {e}{m}$$
  • $$\displaystyle p > \frac {m}{e}$$
Let the equation of a curve be $$x=a\left ( \theta +\sin \theta  \right )$$, $$y=a\left ( 1-\cos \theta  \right )$$. If $$\theta $$ changes at a constant rate $$k$$ then the rate of change of slope of the tangent to the curve at $$\displaystyle \theta =\frac{\pi }{2}$$ is
  • $$\displaystyle \frac{2k}{\sqrt{3}}$$
  • $$\displaystyle \frac{k}{\sqrt{3}}$$
  • $$k$$
  • none of these
The real number $$\displaystyle '\alpha'$$ such that the curve $$\displaystyle f(x) = e^x$$ is tangent to the curve $$\displaystyle g(x) = \alpha x^2$$.
  • $$\displaystyle \frac{e^2}{4}$$
  • $$\displaystyle \frac{e^2}{2}$$
  • $$\displaystyle \frac{e}{4}$$
  • $$\displaystyle \frac{e}{2}$$
The area of a triangle is computed using the formula $$S=\dfrac {1}{2}$$ bc sin A. If the relative errors made in measuring b, c and calculating S are respectively $$0.02$$, $$0.01$$ and $$0.13$$ the approximate error in A when $$A=\pi /6$$ is
  • $$0.05$$ radians
  • $$0.01$$ radians
  • $$0.05$$ degree
  • $$0.01$$ degree
If the circle $$x^2+y^2+2gx+2fy+c=0$$ is touched by $$y=x$$ at P such that OP = $$6\sqrt{2}$$
then the value of c is
  • 36
  • 144
  • 72
  • None of these
The angle made by the tangent of the curve $$\displaystyle x = a(t + \sin t \cos t); y = a (1 + \sin t)^2$$ with the x-axis at any point on it is
  • $$\displaystyle \frac {1}{4} (\pi + 2t)$$
  • $$\displaystyle \frac {1 - \sin t}{ \cos t}$$
  • $$\displaystyle \frac {1}{4} (2t - \pi)$$
  • $$\displaystyle \frac {1 + \sin t}{\cos 2 t}$$
Let $$\displaystyle \:f : R\rightarrow R$$ be a function such that $$\displaystyle \:f \left ( x \right )= ax+3\sin x+4\cos x.$$ Then $$\displaystyle \:f \left ( x \right )$$ is invertible if
  • $$\displaystyle \:a \epsilon \left ( -5, 5 \right )$$
  • $$\displaystyle \:a \epsilon \left ( -\infty , 5 \right )$$
  • $$\displaystyle \:a \epsilon \left ( -5, +\infty \right )$$
  • none of these
For the curve represented parametrically by the equations, $$\displaystyle x = \displaystyle\frac{2}{cot\:t}  + 1$$ & $$\displaystyle y = tan \: t + cot \: t$$
  • tangent at $$\displaystyle t = \displaystyle\frac{\pi}{4}$$ is parallel to x - axis
  • normal at $$\displaystyle t =\displaystyle\frac{\pi}{4}$$ is parallel to y - axis
  • tangent at $$\displaystyle t =\displaystyle\frac{\pi}{4}$$ is parallel to the line $$y = x$$
  • tangent and normal intersect at the point $$(2, 1)$$
Consider the curve represented parametrically by the equation
$$\displaystyle x = t^3 - 4t^2 - 3t$$ and $$\displaystyle y = 2t^2 + 3t - 5$$ where $$\displaystyle t \: \epsilon \: R$$.
If $$H$$ denotes the number of point on the curve where the tangent is horizontal and $$V$$ the number of point where the tangent is vertical then
  • $$H = 2$$ and $$V = 1$$
  • $$H = 1$$ and $$V = 2$$
  • $$H = 2$$ and $$V = 2$$
  • $$H = 1$$ and $$V = 1$$
If the line $$ax+by+c=0$$ is a normal to the rectangular hyperbola $$xy=1$$ then
  • $$a>0, b>0$$
  • $$a>0, b<0$$
  • $$a<0, b>0$$
  • $$a<0, b<0$$
The point (s) on the curve $$\displaystyle y^{3}+3x^{2}= 12y,$$ where the tangent is vertical (i.e., parallel to the y-axis),  is / true
  • $$\displaystyle \left ( \pm \frac{4}{\sqrt{3}},-2 \right )$$
  • $$\displaystyle \left ( \pm \frac{\sqrt{11}}{3},1 \right )$$
  • $$\displaystyle \left ( 0, 0 \right )$$
  • $$\displaystyle \left ( \pm \frac{4}{\sqrt{3}},2 \right )$$
For a $$\displaystyle a\epsilon \left [ \pi , 2\pi  \right ],$$ the function $$\displaystyle f\left ( x \right )= \frac{1}{3}\sin a \tan ^{3}x+\left ( \sin a-1 \right )\tan x+ \frac{\sqrt{a-2}}{\sqrt{8-a}}$$
  • $$\displaystyle x= n\pi \left ( n\epsilon I \right )$$ as critical points
  • no critical points
  • $$\displaystyle x= 2n\pi \left ( n\epsilon I \right )$$ as critical points
  • $$\displaystyle x= \left ( 2n+1 \right )\pi \left ( n\epsilon I \right )$$ as critical points.
The point(s) on the curve $$y^{3}+3x^{2}=12y$$ the tangent is vertical is (are)
  • $$\left ( \pm 4/\sqrt{3}\: -2 \right )$$
  • $$\left ( \pm \sqrt{11/3},1 \right )$$
  • $$\left ( 0,0 \right )$$
  • $$\left ( \pm 4/\sqrt{3}\: ,2 \right )$$
The set of values of $$\displaystyle \lambda $$ for which the function $$\displaystyle f\left ( x \right )= \left ( 4\lambda -3 \right )\left ( x+\log 5 \right )+2\left ( \lambda -7 \right ). \cot \dfrac{x}{2}\sin ^{2}\dfrac{x}{2}.$$ does not posses critical point is:
  • $$\displaystyle \left ( 1,\infty \right )$$
  • $$\displaystyle \left ( 2,\infty \right )$$
  • $$\displaystyle \left ( -\infty ,-4/3 \right )$$
  • $$\displaystyle \left ( -\infty ,-1 \right )$$
The positive value of $$k$$ for which $$ke^x-x=0$$ has only one real solution is 
  • $$\dfrac{1}{e}$$
  • 1
  • e
  • $$\log_{e}2$$
The coordinates of the point $$P$$ on the curve $$y^{2}= 2x^{3}$$ the tangent at which is perpendicular to the line $$4x-3y + 2 = 0$$, are given by
  • $$\left ( 2,4 \right )$$
  • $$\left ( 1,\sqrt{2} \right )$$
  • $$\left ( 1/2,-1/2 \right )$$
  • $$\left ( 1/8,-1/16 \right )$$
Find the co-ordinates of the point (s) on the curve $$\displaystyle y= \frac{x^{2}-1}{x^{2}+1}, x> 0$$ such that tangent at these point (s)have the greatest slope.
  • $$\displaystyle \left ( \frac{1}{\sqrt{3}},-\frac{1}{\sqrt{2}} \right ).$$
  • $$\displaystyle \left ( \frac{1}{\sqrt{3}},-\frac{1}{{2}} \right ).$$
  • $$\displaystyle \left ( \frac{1}{{3}},-\frac{4}{{5}} \right ).$$
  • $$\displaystyle \left ( {\sqrt{3}},\frac{1}{{2}} \right ).$$
The equation of the tangents to $$\displaystyle 4x^{2}-9y^{2}=36$$ which are perpendicular to the straight line $$\displaystyle 2y+5x= 10$$ are
  • $$\displaystyle 5\left ( y-3 \right )=2 \left ( x-\sqrt{\frac{117}{4}} \right )$$
  • $$\displaystyle 5\left ( y-2 \right )=2 \left ( x-\sqrt{-18} \right )$$
  • $$\displaystyle 5\left ( y+2 \right )=2 \left ( x-\sqrt{-18} \right )$$
  • none of these
For the curve $${x}^{2}+4xy+8{y}^{2}=64$$ the tangents are parallel to the $$x$$-axis only at the points
  • $$(0,2\sqrt { 2 } )$$ and $$(0,-2\sqrt { 2 } )$$
  • $$(8,-4)$$ and $$(-8,4)$$
  • $$(8\sqrt { 2 } ,-2\sqrt { 2 } )$$ and $$(-8\sqrt { 2 } ,2\sqrt { 2 } )$$
  • $$(9,0)$$ and $$(-8,0)$$
Given function $$f(x)=\left(\displaystyle\frac{e^{2x}-1}{e^{2x}+1}\right)$$ is.
  • Increasing
  • Decreasing
  • Even
  • None of these
If the line $$ax + by + c = 0$$ is a normal to the curve $$xy = 1$$. Then
  • $$a> 0, b> 0$$
  • $$a> 0, b < 0$$
  • $$a < 0, b > 0$$
  • $$a < 0, b < 0$$
The coordinates of the points(s) at which the tangents to the curve $$\displaystyle y=x^{3}-3x^{2}-7x+6$$ cut the positive semi axis OX a line segment half that on the negative semi axis OY is/are given by
  • $$(-1, 9)$$
  • $$(3, -15)$$
  • $$(1, -3)$$
  • none
The tangent to the curve $$x= a\sqrt{\cos 2\theta }\cos \theta $$, $$y= a\sqrt{\cos 2\theta }\sin \theta
$$ at the point corresponding to $$\theta = \pi /6$$ is
  • parallel to the $$x$$-axis
  • parallel to the $$y$$-axis
  • parallel to line $$y = x$$
  • none of these
The tangent to the curve $$y=e^{x}$$ drawn at the point $$\left ( c,e^{c} \right )$$ intersects the line joining the points $$(c -1,e^{c-1})$$ and $$(c +1,e^{c+1}) $$
  • on the left of $$x = c$$
  • on the right of $$x = c$$
  • at no paint
  • at all points
If $$f(x)=e^x(x-2)^2$$ then
  • f is increasing in $$(-\infty, 0)$$ and $$(2, \infty)$$ deceasing in $$(0, 2)$$
  • f is increasing in $$(-\infty, 0)$$ and deceasing in $$(0, \infty)$$
  • f is increasing in $$(2, \infty)$$ and deceasing in $$(-\infty, 0)$$
  • f is increasing in $$(0, 2)$$ and deceasing in $$(-\infty, 0)$$ and $$(2, \infty)$$
The value of $$a$$ for which the function $$f(x)=(4a-3)(x+\log 5)+2(a-7)\cot \dfrac x2\sin ^2\dfrac x2$$ does not possess critical points is
  • $$(-\infty,-\dfrac 43)$$
  • $$(-\infty,-1)$$
  • $$[1,\infty)$$
  • $$(2,\infty)$$
All the critical points of $$f(x)=\dfrac {|2-x|}{x^2}$$ is/are:
  • $$x=0,2$$
  • $$x=2,4$$
  • $$x=2,-4$$
  • None of the above.
If the slope of the curve $$y=\cfrac { ax }{ b-x } $$ at the point $$(1,1)$$ is $$2$$, then the values of $$a$$ and $$b$$ are respectively
  • $$1,-2$$
  • $$-1,2$$
  • $$1,2$$
  • None of these
If $$f:[1, 10]\rightarrow[1,10]$$ is a non-decreasing function and $$g:[1,10] \rightarrow [1,10]$$ is a non-increasing function, Let $$h(x) = f(g(x))$$ with $$h(1)=1$$. then, $$h(2)$$
  • less than $$1$$
  • is more than two
  • is equal to $$2$$
  • is not defined
If $$y = 4x - 5$$ is a tangent to the curve $$y^{2} = px^{3} + q$$ at $$(2, 3)$$, then
  • $$p = 2, q = -7$$
  • $$p = -2, q = 7$$
  • $$p = -2, q = -7$$
  • $$p = 2, q = 7$$
The curve given by $$x+y={ e }^{ xy }$$ has a tangent parallel to the y-axis at the point
  • $$(0,1)$$
  • $$(1,0)$$
  • $$(1,1)$$
  • $$(-1,-1)$$
Abscissa of $$p_{1}, p_{2}, p_{3} .... p_{n}$$ are in
  • A.P.
  • G.P.
  • H.P
  • None
If $$g(x)$$ is continuous function at $$x = a$$, such that $$g(a) > 0$$ and $$f'(x) (g(x))(x^{2} - ax + a^{2}) \forall x\epsilon R$$, then $$f(x)$$ is
  • Increasing in the neighbourhood of $$x = a$$
  • Decreasing in the neighbourhood of $$x = a$$
  • Constant in the neighbourhood of $$x = a$$
  • Maximum at $$x = a$$
Let $$f'(\sin { x } )<0$$ and $$f''(\sin { x } )>0>\forall x\in \left( 0,\cfrac { \pi  }{ 2 }  \right) $$ and $$g(x)=f'(\sin { x } )+f'(\cos { x } ) $$, then $$g(x)$$ is decreasing in
  • $$\left( \cfrac { \pi }{ 4 } ,\cfrac { \pi }{ 2 } \right) $$
  • $$\left( 0,\cfrac { \pi }{ 4 } \right) $$
  • $$\left( 0,\cfrac { \pi }{ 2 } \right) $$
  • $$\left( \cfrac { \pi }{ 6 } ,\cfrac { \pi }{ 2 } \right) $$
$$f(x)=e^{3x}-sinx+x^{2}$$, Find $$f '(x)$$
  • $$3e^{3x}-cosx+2x$$
  • $$e^{3x}+cosx+2x$$
  • $$3e^{3x}+cosx+x$$
  • $$3e^{3x}+sinx+2x$$
Given that f(x) is a differentiable function of x and that $$f(x).f(y)=f(x)-4-f(y)+f(xy)-2$$ and that $$f(2)=5$$. Then $$f'(3)$$ is equal to
  • 2
  • 24
  • 15
  • 19
The curve that passes through the point $$(2,3)$$ and has the property that the segment of any tangent to it lying between the coordinate axes is bisected by the point of contact, is given by
  • $${ \left( \cfrac { x }{ 2 } \right) }^{ 2 }+{ \left( \cfrac { y }{ 3 } \right) }^{ 2 }=2\quad $$
  • $$2y-3x=0$$
  • $$y=\cfrac { 6 }{ x } $$
  • $${ x }^{ 2 }+{ y }^{ 2 }=13$$
If the line $$y=4x-5$$ touches to the curve $${ y }^{ 2 }=a{ x }^{ 3 }+b$$ at the point $$(2,3)$$ then $$7a+2b=$$
  • $$0$$
  • $$1$$
  • $$-1$$
  • $$2$$
If $$f(x)$$ is an even function, where $$f(x)\ne 0$$, then which one of the following is correct?
  • $$f'(x)$$ is an even function
  • $$f'(x)$$ is an odd function
  • $$f'(x)$$ may be an even or odd function depending on the type of function
  • $$f'(x)$$ is a constant function
For the curve $$x=t^2-1$$, $$y=t^2-t$$, the tangent is perpendicular to $$x$$-axis then
  • $$t=0$$
  • $$t=\dfrac{1}{2}$$
  • $$t=1$$
  • $$t=\dfrac{1}{\sqrt{3}}$$
Let $$f\left( x \right) = {\tan ^{ - 1}}x - \frac{{In\left| x \right|}}{2},x \ne 0.$$. Then $$f\left( x \right)$$ is increasing in
  • $$\left( {0,\infty } \right)$$
  • $$\left( { - \infty ,0} \right)$$
  • $$\left( {1,\infty } \right)$$
  • none of these
If the tangent at $$({x_1},{y_1})$$ to the curve $${x^3} + {y^3} = {a^3}$$ meets the curve again at $$({x_2},{y_2})$$, then
  • $${{{x_2}} \over {{x_1}}} + {{{y_2}} \over {{y_1}}} = - 1$$
  • $${{{x_2}} \over {{y_1}}} + {{{x_1}} \over {{y_2}}} = - 1$$
  • $${{{x_1}} \over {{x_2}}} + {{{y_1}} \over {{y_2}}} = - 1$$
  • $${{{x_2}} \over {{x_1}}} + {{{y_2}} \over {{y_1}}} = 1$$
If the error committed in measuring the radius of the circle is $$0.05\%$$, then the corresponding error in calculating the area is:
  • $$0.05\%$$
  • $$0.025\%$$
  • $$0.25\%$$
  • $$0.1\%$$
A point on the curve $$y = 2{x^3} + 13{x^2} + 5x + 9$$, the tangent at which passes through the origin is 
  • $$(1, 15)$$
  • $$(1, -15)$$
  • $$(15, 1)$$
  • $$(-1, 15)$$
The value of n for which the length of the sub-normal at any point of the curve $$y^3= a^{1-n}x^{2n}$$ must be constant, is
  • $$-1$$
  • $$-\frac{1}{2}$$
  • $$\frac{3}{4}$$
  • $$1$$
The value of 'a' for which the function $$f\left( x \right) = \left( {a + 2} \right){x^3} - 3a{x^2} + 9ax - 1$$ decreases for all real values of x is
  • $$( - \infty , - 3]$$
  • $$\left( { - \infty , - 3} \right)$$
  • $$\left( { - \infty , - 2} \right)$$
  • $$( - \infty , - 3] \cup [0,\infty )$$
If the tangent at any point on the curve $$x^{4} +y^{4}=a^{4}$$ cuts off intercepts $$p$$ and $$q$$ on the coordinate axes the value of $$p^{-4/3}+q^{-4/3}$$ is
  • $$a^{-4/3}$$
  • $$a^{-1/3}$$
  • $$a^{1/2}$$
  • $$None\ of\ these$$
The slope of the tangent to the curve at a point $$(x,y) $$ on it is proportional to $$(x-2).$$ If the slope of the tangent to the curve at $$(10,-9)$$  on it is $$-3$$. The equation of the curves is .
  • $$y=k(x-2)^2$$
  • $$y=\dfrac{-3}{16}(x-2)^2+1$$
  • $$y=\dfrac{-3}{16}(x-2)^2+3$$
  • $$y=K(x+2)^2$$
At any two points of the curve represented parametrically by $$x = a\left( {2\cos t - \cos 2t} \right);y = a\left( {2\sin t - \sin 2t} \right)$$ the tangent are parallel to the axis of $$x$$ corresponding to the values of the parameter $$1$$ differing from each other by
  • $$\frac{{2\pi }}{3}$$
  • $$\frac{{3\pi }}{4}$$
  • $$\frac{\pi }{2}$$
  • $$\frac{\pi }{3}$$
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