CBSE Questions for Class 12 Commerce Maths Application Of Derivatives Quiz 4 - MCQExams.com

Let the parabolas $$y=x^{2}+ax+b$$ and $$y=x(c-x)$$ touch each other at the point $$(1, 0)$$. Then 
  • $$a=-3$$
  • $$b=1$$
  • $$c=2$$
  • $$b+c=3$$
The curve $$\displaystyle \frac{x^{n}}{a^{n}}+\frac{y^{n}}{b^{n}}=2$$ touches the line $$\displaystyle \frac{x}{a}+\frac{y}{b}=2$$ at the point
  • $$(b, a)$$
  • $$(a, b)$$
  • $$(1, 1)$$
  • $$\displaystyle \left ( \frac{1}{a}, \frac{1}{b} \right )$$
If the line joining the points $$(0, 3)$$ and $$(5, -2)$$ is the tangent to the curve $$\displaystyle y=\frac{c}{x+1}$$ then the value of $$c$$ is
  • $$1$$
  • $$-2$$
  • $$4$$
  • none of these
A point on the ellipse $$4x^{2}+9y^{2}=36$$ where the tangent is equally inclided to the axes is
  • $$\displaystyle \left ( \frac{9}{\sqrt{13}}, \frac{4}{\sqrt{13}} \right )$$
  • $$\displaystyle \left ( -\frac{9}{\sqrt{13}}, \frac{4}{\sqrt{13}} \right )$$
  • $$\displaystyle \left ( \frac{9}{\sqrt{13}}, -\frac{4}{\sqrt{13}} \right )$$
  • $$\displaystyle \left ( \frac{4}{\sqrt{13}}, -\frac{9}{\sqrt{13}} \right )$$
If error in  measuring the edge of a cube is $$k$$% then the percentage error in estimating its volume is
  • $$k$$
  • $$3k$$
  • $$\displaystyle \frac{k}{3}$$
  • none of these
The angle between two tangents to the ellipse $$\displaystyle \frac{x^{2}}{16}+\frac{y^{2}}{9}=1$$ at the points where the line $$y=1$$ cuts the curve is
  • $$\displaystyle \frac{\pi }{4}$$
  • $$\tan^{-1}\displaystyle \frac{6\sqrt{2}}{7}$$
  • $$\displaystyle \frac{\pi }{2}$$
  • none of these
A tangent to the curve $$y=\displaystyle \int_{0}^{x}\left | t \right |dt$$, which is parallel to the line y=x, cuts off an intercept from the y-axis equal to
  • $$1$$
  • $$\displaystyle -\frac{1}{2}$$
  • $$\displaystyle \frac{1}{2}$$
  • $$-1$$
Angle between the tangents to the curve $$y= x^{2}-5x+6$$ at the points $$(2,0)$$ and $$\left ( 3,0 \right )$$ is
  • $$\displaystyle \frac{\pi}{2}$$
  • $$\displaystyle \frac{\pi}{3}$$
  • $$\displaystyle \frac{\pi}{6}$$
  • $$\displaystyle \frac{\pi}{4}$$
The number of tangents to the curve $$\displaystyle y= e^{\left | x \right |}$$ at the point $$(0,1)$$ is
  • 2
  • 1
  • 4
  • 0
The curve $$y+e^{xy}+x= 0$$ has a tangent parellel to y-axis at a point
  • $$\left ( -1,\:0 \right )$$
  • $$\left ( 1,\:0 \right )$$
  • $$\left ( 1,\:1 \right )$$
  • $$\left ( 0,\:0 \right )$$
The tangent to the curve $$\displaystyle y=e^{x}$$ drawn at the point $$\displaystyle \left ( c, e^{c} \right )$$ intersects the line joining the points $$\displaystyle \left ( c-1, e^{c-1} \right )$$$$\displaystyle \left ( c+1, e^{c+1} \right )$$
  • on the left of $$\displaystyle x=c$$
  • on the right of $$\displaystyle x=c$$
  • at no point
  • at all points
For $$a\in \left[ \pi ,2\pi  \right] $$ and $$n\in Z$$, the critical points of $$\displaystyle f\left( x \right)=\frac { 1 }{ 3 } \sin { a } \tan ^{ 3 }{ x } +\left( \sin { a } -1 \right) \tan { x } +\sqrt { \frac { a-2 }{ 8-a }  } $$ are 
  • $$x=n\pi$$
  • $$x=2n\pi$$
  • $$x=(2n+1)\pi$$
  • None of these
  • Assertion is true and Reason is true; Reason is a correct explanation for Assertion.
  • Assertion is True, Reason is true; Reason is not a correct explanation for Assertion.
  • Assertion is true, Reason is false
  • Assertion is false, Reason is true
$$\displaystyle y=4x^{2}$$ and $$\displaystyle y= x^{2}.$$
The two curves
  • intersect each other
  • touch each other
  • do not meet
  • represent parabola
If the normal to the curve $$\displaystyle y= f\left ( x \right )$$ at the point $$\displaystyle \left ( 3, 4 \right )$$ makes an angle $$\displaystyle \frac{3\pi}{4}$$ with the positive x-axis then $$\displaystyle f'\left ( 3 \right )$$ is equal to
  • $$-1$$
  • $$\displaystyle -\frac{3}{4}$$
  • $$\displaystyle \frac{4}{3}$$
  • $$1$$
Find the slopes of the tangents of the curve $$y=(x+1)(x-3)$$ at the points where it cuts the X-axis.
  • $$4$$
  • $$-4$$
  • $$2$$
  • $$-2$$
Find the points on the curve $$y=x^{3}$$, the tangents at which are inclined at an angle of $$60^{\circ}$$ to x-axis.
  • $$x=\pm\dfrac{1}{\sqrt{\sqrt{3}}}, y =\dfrac{1}{\sqrt{3}}.\dfrac{1}{\sqrt{\sqrt{3}}}$$.
  • $$x=\dfrac{1}{\sqrt{\sqrt{3}}}, y =\dfrac{1}{\sqrt{3}}.\dfrac{1}{\sqrt{\sqrt{3}}}$$.
  • $$x=\pm\dfrac{1}{\sqrt{\sqrt{3}}}, y =\pm \dfrac{1}{\sqrt{3}}.\dfrac{1}{\sqrt{\sqrt{3}}}$$.
  • $$x=-\dfrac{1}{\sqrt{\sqrt{3}}}, y =\pm \dfrac{1}{\sqrt{3}}.\dfrac{1}{\sqrt{\sqrt{3}}}$$.
Find the points on the curve $$y=x/(1-x^{2})$$ where the tangents makes an angle of $$\pi /4$$ with x-axis
  • $$(\sqrt { 3 } ,-\sqrt { \dfrac { 2 }{ 3 } } ),(-\sqrt { 2 } ,\sqrt { \dfrac { 2 }{ 3 } } )$$
  • $$(\sqrt { 3 } ,-\sqrt { \dfrac { 3 }{ 4 } } ),(-\sqrt { 3 } ,\sqrt { \dfrac { 3 }{ 4 } } )$$
  • $$(\sqrt { 3 } ,-\sqrt { \dfrac { 3 }{ 2 } } ),(-\sqrt { 3 } ,\sqrt { \dfrac { 3 }{ 2 } } )$$
  • none of these
Find the condition that the line $$\displaystyle Ax+By= 1$$ may be a normal to the curve $$\displaystyle a^{n-1}y=x^{n}.$$
  • $$\displaystyle a^{n}B\left ( B^{2}+nA^{2} \right )^{n}=A^{n}n^{n}.$$
  • $$\displaystyle a^{n-1}B\left ( B^{2}+nA^{2} \right )^{n-1}=A^{n}n^{n}.$$
  • $$\displaystyle a^{n}B\left ( B^{2}+nA^{2} \right )^{n-1}=A^{n}n^{n}.$$
  • $$\displaystyle a^{n-1}B\left ( B^{2}-nA^{2} \right )^{n-1}=A^{n}n^{n}.$$
If the normal to the curve $$y=f(x)$$ at the point $$(3,4) $$ makes an angle $$3\pi /4 $$ with the positive x-axis, then $$f'(3)=$$
  • $$-1$$
  • $$0$$
  • $$1$$
  • $$\sqrt 3$$
The curve $$\displaystyle y-e^{xy}+x=0$$ has a vertical tangent at the point 
  • $$(1,\ 1)$$
  • $$no\ point$$
  • $$(0,\ 1)$$
  • $$(1,\ 0)$$
The set of all values of x for which the function $$\displaystyle f\left ( x \right )= \left ( k^{2}-3k+2 \right )\left ( \cos ^{2}\frac{x}{4}-\sin ^{2}\frac{x}{4} \right )+\left ( k-1 \right )x+\sin 1$$ does not posses critical points is 
  • $$\displaystyle \left ( -4,4 \right )$$
  • $$\displaystyle \left ( 0,4 \right)$$
  • $$\displaystyle \left ( 0,1 \right )\cup \left ( 1,4 \right )$$
  • $$\displaystyle \left ( 0,2 \right )\cup \left ( 2,4 \right )$$
 Determine the intervals of monotonicity of $$\displaystyle f \left ( x \right )= \log \left | x \right |.$$ 
  • increasing for $$x>0$$
  • increasing for $$x<0$$
  • decreasing for $$x>0$$
  • decreasing for $$x<0$$
If $$\displaystyle x\cos \alpha +y\sin \alpha =p$$ touches $$\displaystyle x^{2}+a^{2}y^{2}=a^{2},$$ then
  • $$\displaystyle p^{2}=a^{2}\sin^{2}\alpha +\cos^{2}\alpha $$
  • $$\displaystyle p^{2}=a^{2}\cos^{2}\alpha +\sin^{2}\alpha $$
  • $$\displaystyle 1/p^{2}=\sin^{2}\alpha +\alpha^{2}\cos^{2}\alpha $$
  • $$\displaystyle 1/p^{2}=\cos^{2}\alpha +\alpha^{2}\sin^{2}\alpha $$
If the line $$ax+by+c=0$$ is a normal to the curve $$xy=1$$, then
  • $$\displaystyle a> 0, b> 0$$
  • $$\displaystyle a> 0, b< 0$$
  • $$\displaystyle a< 0, b> 0$$
  • $$\displaystyle a< 0, b< 0$$
Find the co-ordinates of the points on the curve $$\displaystyle y= x/\left ( 1+x^{2} \right )$$ where the tangent to the curve has greatest slope.
  • $$\left(\displaystyle \sqrt 3, \frac {\sqrt 3}4\right)$$
  • $$(\displaystyle 0, 0)$$
  • $$\left(\displaystyle -\sqrt 3, -\frac {\sqrt 3}4\right)$$
  • $$\left(\displaystyle 1, \frac {1}2\right)$$
The line $$y=x$$ is a tangent to the parabola $$\displaystyle y= ax^{2}+bx+c$$ at the point $$x=1$$.If the parabola passes through the point $$(-1,0)$$, then determine $$a, b, c.$$
  • $$\displaystyle a= \frac{1}{2}, b= \frac{1}{4}, c= \frac{1}{3}.$$
  • $$\displaystyle a= \frac{1}{4}, b= \frac{1}{2}, c= \frac{1}{4}.$$
  • $$\displaystyle a= 2, b= 1, c= 4.$$
  • $$\displaystyle a= 4, b= 2, c= 4.$$
Find $$\displaystyle \frac{dy}{dx}$$ if$$ \:y= \left [ x+\sqrt{x+} \sqrt{x}\right ]^{1/2}$$, at $$x=1$$
  • $$\dfrac{3+4\sqrt{2}}{8\sqrt{2}(\sqrt{1+\sqrt{2}})}$$
  • Not defined
  • $$0$$
  • $$e$$
A and B are points $$(-2,0)$$ and $$(1,3)$$ on the curve $$\displaystyle y=4-x^{2}$$. If the tangent at P on the curve be parallel to chord AB, then co-ordinates of point P are 
  • $$\displaystyle \left ( -\frac{1}{3}, \frac{5}{3} \right )$$
  • $$\displaystyle \left ( \frac{1}{2}, -\frac{15}{4} \right )$$
  • $$\displaystyle \left ( -\frac{1}{2}, \frac{15}{4} \right )$$
  • $$\displaystyle \left ( -\frac{1}{3}, \frac{1}{5} \right )$$
The line $$\dfrac xa+\dfrac yb=1$$ touches the curve $$\displaystyle y=be^{-x/a}$$ at the point
  • $$(a,b/a)$$
  • $$(-a,b/a)$$
  • $$(a,a/b)$$
  • None of these
If the line, $$\displaystyle ax+by+c= 0$$ is a normal to the curve $$xy=2,$$ then
  • $$a < 0, b > 0$$
  • $$a > 0, b < 0$$
  • $$a > 0, b > 0$$
  • $$a < 0, b < 0$$
The function $$\displaystyle f\left ( x \right )=2\log \left ( x-2 \right )-x^{2}+4x+1$$  increases in the interval
  • $$\displaystyle \left ( 1, 2 \right )$$
  • $$\displaystyle \left (2, 3 \right )$$
  • $$\displaystyle \left ( 5/2, 3 \right )$$
  • $$\displaystyle \left ( 2, 4 \right )$$
The critical points of the function $$\displaystyle f\left ( x \right )= \frac{\left | x-1 \right |}{x^{2}}$$ are
  • 0
  • 1
  • 2
  • -1
If $$\displaystyle f\left ( 0 \right )=0$$ and $$\displaystyle f''\left ( x \right )>0$$ for all $$x > 0$$, then $$\displaystyle \frac{f(x)}{x}$$
  • decreases on $$\displaystyle \left ( 0, \infty \right )$$
  • increases on $$\displaystyle \left ( 0, \infty \right )$$
  • decreases on $$\displaystyle \left ( 1, \infty \right )$$
  • neither increases nor decreases on $$\displaystyle \left ( 0, \infty \right )$$
The interval(s) of decrease of  of the function $$\displaystyle f\left ( x \right )= x^{2}\log 27-6x\log 27+\left ( 3x^{2}-18x+24 \right )\log \left ( x^{2}-6x+8 \right )$$ is
  • $$\displaystyle \left ( 3-\sqrt{1+1/3e}, 2\right )$$
  • $$\displaystyle \left ( 4, 3+\sqrt{1+1/3e}\right )$$
  • $$\displaystyle \left ( 3, 4 +\sqrt{1+1/3e}\right )$$
  • none of these
The slope of the tangent to the curve represented by $$x= t^{2}+3t-8$$ and $$y= 2t^{2}-2t-5$$ at the point $$M\left ( 2,-1 \right )$$ is

  • 7/6
  • 2/3
  • 3/2
  • 6/7
The number of critical points of the fuction $$\displaystyle f'\left ( x \right ),$$ where $$\displaystyle f'\left ( x \right )= \frac{\left | x-2 \right |}{x^{3}}$$ are
  • $$0$$
  • $$1$$
  • $$3$$
  • $$4$$
The value of a for which the function $$\displaystyle f\left ( x \right )= \left ( 4a-3 \right )\left ( x+\log 5 \right )+2\left ( a-7 \right )\cot\left ( x/2 \right )\sin ^{2}\left ( x/2 \right )$$ does not possess critical points is
  • $$\displaystyle \left ( -\infty , -4/3 \right )$$
  • $$\displaystyle \left ( -\infty , -1 \right )$$
  • $$\displaystyle \left ( 1, \infty \right )$$
  • $$\displaystyle \left ( 2, \infty \right )$$
The critical points of the function $$\displaystyle f\left ( x\right )=\left ( x-2 \right )^{2/3}\left ( 2x+1\right )$$ are
  • $$-1,2$$
  • $$1$$
  • $$\displaystyle 1,-\frac{1}{2}$$
  • $$1,2$$
The coordinates of the point on the curve $$\displaystyle \left ( x^{2}+1 \right )\left ( y-3 \right )=x$$ where a tangent to the curve has the greatest slope are given by
  • $$\displaystyle \left ( \sqrt{3}, 3+\sqrt{3}/4 \right )$$
  • $$\displaystyle \left ( -\sqrt{3}, 3-\sqrt{3}/4 \right )$$
  • $$\displaystyle \left ( 0, 3 \right )$$
  • none of these
The angle at which the curve $$y=ke^{kx}$$ intersects the $$y$$ -axis is
  • $$\tan ^{-1}(k^{2})$$
  • $$\cot ^{-1}(k^{2})$$
  • $$\sin ^{-1}\left ( 1/\sqrt{1+k^{4}} \right )$$
  • $$



    \sec ^{-1}\left ( 1/\sqrt{1+k^{4}} \right )$$
The lines tangent to the curves $$\displaystyle y^{3}-x^{2}y+5y-2x=0$$ and $$\displaystyle x^{4}-x^{3}y^{2}+5x+2y=0$$ at the origin intersect at an angle $$\displaystyle \theta $$ equal to
  • $$\displaystyle \frac{\pi }{6}$$
  • $$\displaystyle \frac{\pi }{4}$$
  • $$\displaystyle \frac{\pi }{3}$$
  • $$\displaystyle \frac{\pi }{2}$$
Let $$\displaystyle f\left ( x \right )=x^{3}+ax+b$$ with $$\displaystyle a\neq b$$ and suppose the tangent lines to the graph of $$f$$ at $$x = a$$ and $$x = b$$ have the same gradient Then the value of $$f (1)$$ is equal to
  • $$0$$
  • $$1$$
  • $$\displaystyle -\frac{1}{3}$$
  • $$\displaystyle \frac{2}{3}$$
A curve with equation of the form $$\displaystyle y=ax^{4}+bx^{3}+cx+d$$ has zero gradient at the point (0, 1) and also touches the x-axis at the point (-1, 0) then the values of x for which the curve has a negative gradient are
  • x > -1
  • x < 1
  • x < -1
  • $$\displaystyle -1\leq \times \leq 1$$
If $$f'(x) = g(x)\left ( x-a \right )^{2}$$, where $$g(a)\neq 0$$ and $$g$$ is continuous at $$x = a$$ then
  • $$f$$ is increasing near a if $$g(a) > 0$$
  • $$f$$ is increasing near a if $$g(a) < 0$$
  • $$f$$ is decreasing near a if $$g(a) > 0$$
  • $$f$$ is decreasing near a if $$g(a) < 0$$
The curve $$y= ax^{3}+bx^{2}+cx+8$$  touches $$x$$-axis at $$P\left ( -2,0 \right )$$ and cuts the $$y$$-axis at a point $$Q(0,8)$$ where its gradient isThe values of $$a$$, $$b$$, $$c$$ are respectively

  • $$-\displaystyle \frac{1}{2},-\frac{3}{4},3$$
  • $$\displaystyle 3, -\frac{1}{2},-4$$
  • $$\displaystyle -\frac{1}{2},-\frac{7}{4},2$$
  • none of these
Suppose $$f'(x)$$ exists for each $$x$$ and $$h(x) = f(x) - (f(x))^{2}+(f(x))^{3}$$ for every real number $$x$$. Then
  • $$h$$ is increasing whenever f is increasing
  • $$h$$ is increasing whenever f is decreasing
  • $$h$$ is decreasing whenever f is decreasing
  • nothing can be said in general.
The critical points of the function $$f\left( x \right)={ \left( x-2 \right)  }^{ 2/3 }\left( 2x+1 \right) $$ are
  • $$1$$ and $$2$$
  • $$1$$ and $$\displaystyle-\frac{1}{2}$$
  • $$-1$$ and $$2$$
  • $$1$$
The graph a function $$f$$ is given. On what interval is $$f$$ increasing ?
255196.png
  • $$(-1, 3]$$
  • $$(-3,1)$$
  • $$(-3,1]$$
  • none of these
The points of contact of the vertical tangents $$x= 2-3\sin \theta $$, $$y= 3+2\cos \theta $$ are
  • $$\left ( 2,5 \right ),\left ( 2,1 \right )$$
  • $$\left ( -1,3 \right ),\left ( 5,3 \right )$$
  • $$\left ( 2,5 \right ),\left ( 5,3 \right )$$
  • $$\left ( -1,3 \right ),\left ( 2,1 \right )$$
0:0:1


Answered Not Answered Not Visited Correct : 0 Incorrect : 0

Practice Class 12 Commerce Maths Quiz Questions and Answers