CBSE Questions for Class 12 Commerce Maths Application Of Derivatives Quiz 5 - MCQExams.com

The angle at which the curve $$\displaystyle y=ke^{kx}$$ intersects the y - axis is
  • $$\displaystyle \tan ^{-1}k^{2}$$
  • $$\displaystyle \cot ^{-1}\left ( k^{2} \right )$$
  • $$\displaystyle \sin ^{-1\left ( \dfrac{1}{\sqrt{1+k^{4}}} \right )}$$
  • $$\displaystyle \sec {-1\left ( \sqrt{1+k^{4}} \right )}$$
The abscissa of the point on the curve $$\displaystyle \sqrt{xy}=a+x$$ the tangent at which cuts off equal intercepts from the co-ordinate axes is (a > 0)
  • $$\displaystyle \dfrac{a}{\sqrt{2}}$$
  • $$\displaystyle- \dfrac{a}{\sqrt{2}}$$
  • $$\displaystyle a\sqrt{2}$$
  • $$\displaystyle -a\sqrt{2}$$
The curve $$\displaystyle y=ax^{3}+bx^{2}+cx+5$$ touches the $$x$$ - axis at $$P(-2, 0)$$ and cuts the $$y$$-axis at a point $$Q$$, where its gradient is $$3$$. Find $$a, b, c$$.
  • $$\displaystyle a=-\frac{1}{5}, b=1,c=3$$
  • $$\displaystyle a=-\frac{1}{4}, b=-1,c=4$$
  • $$\displaystyle a=-\frac{1}{4}, b=0,c=3$$
  • $$\displaystyle a=-\frac{1}{3}, b=1,c=-3$$
Let h be a twice continuously differentiable positive function on an open interval $$J$$. Let $$\displaystyle g\left ( x \right )=ln\left ( h(x) \right ) $$ for each $$\displaystyle x\epsilon J $$
Suppose $$\displaystyle \left ( h'\left (  x \right )\right )^{2}> h''\left ( x \right )h\left ( x \right )$$ for each $$\displaystyle x\epsilon J$$ Then
  • $$g$$ is increasing on $$J$$
  • $$g$$ is decreasing on $$J$$
  • $$g$$ is concave up on $$J$$
  • $$g$$ is concave down on $$J$$
If the curve $$\displaystyle { \left( \frac { x }{ a }  \right)  }^{ n }+{ \left( \frac { y }{ b }  \right)  }^{ n }=2$$ touches the straight line $$\displaystyle \frac { x }{ a } +\frac { y }{ b } =2$$, then find the value of $$n$$.
  • $$2$$
  • $$3$$
  • $$4$$
  • any real number
For the curve represented parametrically by the equations $$\displaystyle x=2\ln\cot t+1$$ and $$\displaystyle y=\tan t+\cot t$$
  • tangent at $$t = \displaystyle \dfrac {\pi }4$$ is parallel to x-axis
  • normal at $$t = \displaystyle \dfrac {\pi }4$$ is parallel to y=axis
  • tangent at $$t = \displaystyle \dfrac {\pi }4$$ is parallel to the line $$y = x$$
  • normal at $$t = \displaystyle \dfrac {\pi }4$$ is parallel to the line $$y = x$$
The coordinates of the point(s) on the graph of the function $$\displaystyle f(x)=\frac{x^{3}}3{-\frac{5x^{2}}{2}}+7x-4$$ where the tangent drawn cut off intercepts from the coordinate axes which are equal in magnitude but opposite in sign is
  • $$(2,\dfrac 83)$$
  • $$(3, \dfrac 72)$$
  • $$(1,\dfrac  56)$$
  • none
If the radius of a sphere is measured as $$9 \ cm$$ with an error of $$ 0.03 \ cm$$ then, find the approximate error in calculating its volume.
  • $$\displaystyle 9.72\pi\:\: cm^{3}$$
  • $$\displaystyle 7.92\pi\:\: cm^{3}$$
  • $$\displaystyle 8.72\pi\:\: cm^{3}$$
  • None of these
At what point of the curve $$\displaystyle y=2x^{2}-x+1$$ tangent is parallel to $$y = 3x + 4$$
  • $$(0, 1)$$
  • $$(1, 2)$$
  • $$(-1, 4)$$
  • $$(2, 7)$$
The slope of the normal to the curve $$\displaystyle x=a\left ( \theta -\sin \theta  \right ),\: \: y=a\left ( 1-\cos \theta  \right )$$ at point $$\displaystyle \theta =\dfrac{\pi }2$$ is
  • $$0$$
  • $$1$$
  • $$-1$$
  • $$\dfrac 1{\displaystyle \sqrt{2}}$$
asymptotes of the graph
  • $$\displaystyle x=\frac{3\pi }{2}$$
  • $$\displaystyle x=-\frac{\pi }{2}$$
  • $$\displaystyle x=\frac{\pi }{2}$$
  • $$\displaystyle x=-\frac{3\pi }{2}$$
The slope of the tangents to the curve $$y = (x + 1) (x - 3)$$ at the points where it crosses x - axis are
  • $$\displaystyle \pm 2 $$
  • $$\displaystyle \pm 3 $$
  • $$\displaystyle \pm 4$$
  • None of these
Let $$f$$ be a continuous, differentiable and bijective function. If the tangent to $$y=f\left( x \right) $$ at $$x=b$$, then there exists at least one $$c\in \left( a,b \right) $$ such that 
  • $$f'\left( c \right) =0$$
  • $$f'\left( c \right) >0$$
  • $$f'\left( c \right) <0$$
  • none of these
The normal to the curve $$\displaystyle \sqrt{x}+\sqrt{y}=\sqrt{a}$$ is perpendicular to $$x$$ axis at the point
  • $$(0, a)$$
  • $$(a, 0)$$
  • $$(\dfrac a  4, \dfrac a  4)$$
  • No where
If equation of normal at a point $$\displaystyle \left ( m^{2},-m^{3} \right )$$ on the curve $$\displaystyle x^{3}-y^{2}=0\: \: is\: \: y=3mx-4m^{3}$$ then $$\displaystyle m^{2}$$ equals
  • $$0$$
  • $$1$$
  • $$-\dfrac 2  9$$
  • $$\dfrac 2  9$$
The slope of normal to the curve $$\displaystyle y^{2}=4ax$$ at a point $$\displaystyle \left ( at^{2},2at \right )$$ is
  • $$\dfrac 1  t$$
  • $$t$$
  • $$-t$$
  • $$-\dfrac 1  t$$
On the ellipse, $$4x^2\, +\, 9y^2\, =\, 1$$, the points at which the tangents are parallel to the line $$8x = 9y$$ are
  • $$\left ( \displaystyle \frac{2}{5},\,\frac{1}{5} \right )$$
  • $$\left ( -\displaystyle \frac{2}{5},\,\frac{1}{5} \right )$$
  • $$\left ( -\displaystyle \frac{2}{5},\,-\frac{1}{5} \right )$$
  • $$\left ( \displaystyle \frac{2}{5},\,-\frac{1}{5} \right )$$
The slope of the tangent to the curve $$xy + ax - by = 0$$ at the point $$(1, 1)$$ is $$2$$ then values of $$a$$ and $$b$$ are respectively -
  • $$1, 2$$
  • $$2, 1$$
  • $$3, 5$$
  • None of these
At what point the tangent line to the curve $$\displaystyle y=\cos \left ( x+y \right ),\left ( -2\pi \leq x\leq 2\pi  \right )$$ is parallel to $$x + 2y = 0$$
  • $$\displaystyle \left ( \dfrac {\pi }2, 0 \right )$$
  • $$\displaystyle \left ( -\dfrac {\pi }2, 0 \right )$$
  • $$\displaystyle \left (\dfrac{ 3\pi }2, 0 \right )$$
  • $$\displaystyle \left (-\dfrac{ 3\pi }2,\dfrac { \pi }2 \right )$$
The line $$\dfrac x a +\dfrac  y  b = 1$$ touches the curve $$\displaystyle y=be^{-\tfrac xa}$$ at the point -
  • $$(0, a)$$
  • $$(0. 0)$$
  • $$(0, b)$$
  • $$(b, 0)$$
At what values of $$a$$, the curve $$x^4+3ax^3+6x^2+5$$ is not situated below any of its tangent lines
  • $$|a|\,>\,\displaystyle\frac{4}{3}$$
  • $$|a|\,<\,\displaystyle\frac{4}{3}$$
  • $$|a|\,>\,1$$
  • $$|a|\,<\,\displaystyle\frac{1}{3}$$
The point at which the tangent to the curve $$\displaystyle y=x^{3}+5$$ is perpendicular to the line $$x + 3y = 2$$ are
  • $$(6, 1), (-1, 4)$$
  • $$(6, 1) (4, -1)$$
  • $$(1, 6), (1, 4)$$
  • $$(1, 6), (-1, 4)$$
The points on the curve $$\displaystyle y^{2}=4a\left ( x+a\sin \frac{x}{a} \right )$$ at which the tangent is parallel to x axis lie on -
  • a straight line
  • a parabola
  • a circle
  • an ellipse
The equation of normal to the curve $$x+y=x^{y}$$, where it cuts x-axis is
  • $$y=x+1$$
  • $$y=-x+1$$
  • $$y=x-1$$
  • $$y=-x-1$$
The lines tangent to the curve $$x^3-y^3+x^2y-yx^2+3x-2y=0$$ and $$x^5-y^4+2x+3y=0$$ at the origin intersect at an angle $$\theta$$ equal to
  • $$\displaystyle\frac{\pi}{6}$$
  • $$\displaystyle\frac{\pi}{4}$$
  • $$\displaystyle\frac{\pi}{3}$$
  • $$\displaystyle\frac{\pi}{2}$$
The points on the curve $$\displaystyle 9y^2=x^{3}$$ where the normal to the curve makes equal intercepts with coordinates axes is :
  • $$\displaystyle \left ( 4,\frac{8}{3} \right )\: \: or\: \: \left ( 4,-\frac{8}{3} \right )$$
  • $$\displaystyle \left ( -4,\frac{8}{3} \right )$$
  • $$\displaystyle \left ( -4,-\frac{8}{3} \right )$$
  • None of these
If the line $$x -y = 0$$ is tangent to $$f(x) = b \ln x - x$$, then $$b$$ lies in the interval
  • $$(1, 3)$$
  • $$(0, 1)$$
  • $$(4, 6)$$
  • $$(6, 8)$$
The coordinates of the points on the curve $$\displaystyle x=a\left ( \theta +\sin \theta  \right ),y=a\left ( 1-\cos \theta  \right )$$ where tangent is inclined an angle $$\displaystyle \dfrac{\pi }4$$ to the $$x-$$axis are -
  • $$(a, a)$$
  • $$\displaystyle \left ( a\left ( \frac{\pi }{2}-1 \right ),a \right )$$
  • $$\displaystyle \left ( a\left ( \frac{\pi }{2}+1 \right ),a \right )$$
  • $$\displaystyle \left ( a,a\left ( \frac{\pi }{2}+1 \right ) \right )$$
If the curve $$y^2=ax^3-6x^2+b$$ passes through $$(0,\,1)$$ and has its tangent parallel to y-axis at $$x=2$$, then
  • $$a=2,\,b=1$$
  • $$a=\displaystyle\frac{23}{8},\,b=1$$
  • $$a=-\displaystyle\frac{8}{23},\,b=1$$
  • $$a=-\displaystyle\frac{23}{8},\,b=1$$
Let tangent at a point P on the curve $$\displaystyle { x }^{ 2m }={ y }^{ \tfrac { n }{ 2 }  }={ a }^{ \tfrac { 4m+n }{ 2 }  }$$ meets the x-axis and y-axis at A and B respectively, If AP:PB is $$\displaystyle \frac { n }{ \lambda m } $$, where P lies between A and B, then find the value of $$\displaystyle \lambda $$
  • $$4$$
  • $$3$$
  • $$-4$$
  • $$-3$$
The minimum value of the polynomial.
$$p(x)=3{ x }^{ 2 }-5x+2$$
  • $$-\frac { 1 }{ 6 }$$
  • $$\frac { 1 }{ 6 }$$
  • $$\frac { 1 }{ 12 }$$
  • $$-\frac { 1 }{ 12 }$$
If the function $$\displaystyle f\left( x \right)=\left( { a }^{ 2 }-3a+2 \right) \cos { \frac { x }{ 2 }  } +\left( a-1 \right) x$$ possesses critical points, then $$a$$ belongs to the interval
  • $$\left( -\infty ,0 \right) \cup \left( 4,\infty  \right) $$
  • $$(-\infty ,0]\cup [4,\infty )$$
  • $$(-\infty ,0]\cup \left\{ 1 \right\} \cup [4,\infty )$$
  • None of these
If the tangent to the curve $$x = a(8 + sin \theta), y = a(1 + cos \theta )$$ at $$\theta = \displaystyle \frac{\pi}{3}$$ makes an angle $$\alpha$$ with x-axis, then $$\alpha$$ is equal to
  • $$\displaystyle \frac{\pi}{3}$$
  • $$\displaystyle \frac{2\pi}{3}$$
  • $$\displaystyle \frac{\pi}{6}$$
  • $$\displaystyle \frac{5\pi}{6}$$
The curve which passes through $$(1, 2)$$ and whose tangent at any point has a slope that is half of slope of the line joining origin to the point of contact, is -
  • A rectangle hyperbola
  • A circle
  • A parabola
  • A straight line through origin
  • Answer required
If f(x) = $$\dfrac{x}{ sin x}$$ and g(x) = $$\dfrac{x}{tanx}$$ where 0<x $$\leq$$ 1, then in this interval $$f(x)$$ is
  • 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 curve $$\displaystyle y=f\left( x \right) ;\left( y>0 \right) $$  passes thorugh $$(1,1)$$ and at point $$\displaystyle P(x,y)$$ tangents cuts x-axis and y-axis at A and B respectively. If P divides AB  internally in the ratio $$3 : 2$$, then the value of $$\displaystyle f\left( \frac { 1 }{ 8 }  \right) $$ is
  • $$4$$
  • $$\displaystyle \frac { 1 }{ 4 } $$
  • $$\displaystyle 16\sqrt { 2 } $$
  • $$\displaystyle \frac { 1 }{ 16\sqrt { 2 } } $$
Determine the intervals over which the function is decreasing, increasing, and constant.
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  • Increasing $$[3, \infty )$$; Decreasing $$(-\infty , 3]$$
  • Increasing $$(-\infty , 3]$$; Decreasing $$[3, \infty )$$
  • Increasing $$(-\infty , 3]$$; Decreasing $$(-\infty , 3]$$
  • Increasing $$[3, \infty)$$; Decreasing $$[3, \infty )$$
The slope of the tangent to the curve $$x={t}^{2}+3t-8$$, $$y=2{t}^{2}-2t-5$$ at the point $$(2,-1)$$ is
  • $$\cfrac{22}{7}$$
  • $$\cfrac{6}{7}$$
  • $$\cfrac{7}{6}$$
  • $$\cfrac{-6}{7}$$
  • answer required
The line $$y=mx+1$$ is a tangent to the curve $${y}{^2}=4x$$, if the value of $$m$$ is
  • $$1$$
  • $$2$$
  • $$3$$
  • $$\cfrac{1}{2}$$
  • answer required
The abscissa of the points, where the tangent to curve $$y={x}^{3} - 3{x}^{2} - 9x+5$$ is parallel to x-axis, are
  • $$x=0$$ and $$0$$
  • $$x=1$$ and $$-1$$
  • $$x=1$$ and $$-3$$
  • $$x=-1$$ and $$3$$
The points on the curve $$9{y}^{2}={x}^{3}$$, where the normal to the curve makes equal intercepts with the axes are
  • $$\left( 4,\pm \cfrac { 8 }{ 3 } \right) $$
  • $$\left( 4,\cfrac { -8 }{ 3 } \right) $$
  • $$\left( 4,\pm \cfrac { 3 }{ 8 } \right) $$
  • $$\left( \pm 4,\cfrac { 3 }{ 8 } \right) $$
  • answer required
Angle between $${ y }^{ 2 }=x$$ and $${ x }^{ 2 }=y$$ at the origin is
  • $$2\tan ^{ -1 }{ \left( \dfrac { 3 }{ 4 } \right) } $$
  • $$\tan ^{ -1 }{ \left( \dfrac { 4 }{ 3 } \right) } $$
  • $$\dfrac { \pi }{ 2 } $$
  • $$\dfrac { \pi }{ 4 } $$
If the line $$\alpha\,x+by+c=0$$ is a tangent to the curve $$xy=4$$, then
  • $$a < 0,\,b > 0$$
  • $$a \le o,\,b > 0$$
  • $$a < 0,\,b < 0$$
  • $$a \le 0,\,b < 0$$
Let $$y=e^{x^2}$$ and $$y=e^{x^2}\sin\, x$$ be two given curves. Then, angle between the tangents to the curves at any point their intersection is 
  • $$0$$
  • $$\pi$$
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{\pi}{4}$$
The slope at any point of a curve $$y=f\left( x \right) $$ is given by $$\dfrac { dy }{ dx } =3{ x }^{ 2 }$$ and it passes through $$\left( -1,1 \right) $$. The equation of the curve is
  • $$y={ x }^{ 3 }+2$$
  • $$y=-{ x }^{ 3 }-2$$
  • $$y=3{ x }^{ 3 }+4$$
  • $$y=-{ x }^{ 3 }+2$$
Suppose that the equation $$f\left( x \right) ={ x }^{ 2 }+bx+c=0$$ has two distinct real roots $$\alpha $$ and $$\beta $$. The angle between the tangent to the curve $$y=f\left( x \right) $$ at the point $$\left( \dfrac { \alpha +\beta  }{ 2 } ,f\left( \dfrac { \alpha +\beta  }{ 2 }  \right)  \right) $$ and the positive direction of the $$x$$-axis is
  • $${ 0 }^{ }$$
  • $${ 30 }^{ }$$
  • $${ 60 }^{ }$$
  • $${ 90 }^{ }$$
The equation of one of the curves whose slope at any point is equal to $$y+2x$$ is
  • $$y=2(e^x+x-1)$$
  • $$y=2(e^x-x-1)$$
  • $$y=2(e^x-x+1)$$
  • $$y=2(e^x+x+1)$$
The function $$f(x)=ax+b$$ is strictly increasing for all real $$x$$, if
  • $$a> 0$$
  • $$a< 0$$
  • $$a=0$$
  • $$a\le 0$$
The equation of normal of $$x^2+y^2-2x+4y-5=0$$ at $$(2,\,1)$$ is
  • $$y=3x-5$$
  • $$2y=3x-4$$
  • $$y=3x+4$$
  • $$y=x+1$$
The coordinates of the point P on the curve $$x = a(\theta + \sin \theta), y = a(1 - \cos \theta)$$ where the tangent is inclined at an angle $$\dfrac {\pi}{4}$$ to the x-axis, are
  • $$\left (a\left (\dfrac {\pi}{2} - 1\right ), a\right )$$
  • $$\left (a\left (\dfrac {\pi}{2} + 1\right ), a\right )$$
  • $$\left (a \dfrac {\pi}{2}, a\right )$$
  • $$(a, a)$$
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