CBSE Questions for Class 11 Commerce Applied Mathematics Tangents And Its Equations Quiz 5 - MCQExams.com

The sum of the intercepts made on the axes of coordinates by any tangent to the curve $$\sqrt{x}+\sqrt{y}=2$$ is equal to
  • $$4$$
  • $$2$$
  • $$8$$
  • 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
  • $$1$$
  • $$\displaystyle -\frac{1}{2}$$
  • $$\displaystyle \frac{1}{2}$$
  • $$-1$$
Slope of Normal to the curve $$y = x^{2} - \dfrac {1}{x^{2}}$$ at $$(-1, 0)$$ is
  • $$\dfrac {1}{4}$$
  • $$-\dfrac {1}{4}$$
  • $$4$$
  • $$-4$$
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
  • $$-1$$
  • $$\displaystyle -\frac{3}{4}$$
  • $$\displaystyle \frac{4}{3}$$
  • $$1$$
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
  • 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
The normal to the curve $$x=a\left ( 1-\cos \theta  \right )$$, $$y=a\sin \theta $$ at $$\theta $$ always passes through the fixed point
  • $$(0, 0)$$
  • $$(0, a)$$
  • $$(a, 0)$$
  • $$(a, a)$$
The normal to the curve $$x=a\left ( \cos \theta +\theta \sin \theta  \right )$$, $$y=a\left ( \sin \theta -\theta \cos \theta  \right )$$ at any point $$\theta $$ is such that
  • it makes angle $$\displaystyle \frac{\pi }{2}+\theta $$ with x-axis
  • it passes through the origin
  • it is at a constant distance from the origin
  • it passes through $$\displaystyle \left ( a\frac{\pi }{2}, -a \right )$$
The curve possessing the property text the intercept made by the tangent at any point of the curve on the $$y-$$ axis is equal to square of the abscissa of the point of tangency, is given by
  • $$y^{2}=x+C$$
  • $$y=2x^{2}+C$$
  • $$y=-x^{2}+cx$$
  • $$None\ of\ these$$
Which of this/these is/are tangent(s) to $$\displaystyle 3x^{2}+y^{2}+x+2y=0$$ and also is/are perpendicular to the line 4x-2y=1 ? 
  • $$2y+x=0$$
  • $$ 2y-x = 2$$
  • $$ 2y +x +4 = 0$$
  • $$\displaystyle 2y+x+\frac{13}{3}=0$$
If the tangent at P  on the curve $$\displaystyle x^{m}y^{n}=a^{m+n}$$ meets the co-ordinates axes at A and B, then $$AP: PB= $$
  • $$m^2:n^2$$
  • $$m^3:n^3$$
  • $$m:n$$
  • $$2m:n$$
Find the equation of the tangent to the curve at any point $$(X, Y)$$.
$$\displaystyle \frac{x^{m}}{a^{m}}+\frac{y^{m}}{b^{m}}=1.$$
  • $$\displaystyle \frac{X}{a}\left ( \frac{x}{a} \right )^{m-1}+\frac{Y}{b}\left ( \frac{y}{b} \right )^{m-1}=1$$
  • $$\displaystyle \frac{X}{b}\left ( \frac{x}{a} \right )^{m-1}+\frac{Y}{a}\left ( \frac{y}{b} \right )^{m-1}=1$$
  • $$\displaystyle \frac{X}{a}\left ( \frac{x}{b} \right )^{m-1}+\frac{Y}{b}\left ( \frac{y}{a} \right )^{m-1}=1$$
  • $$\displaystyle \frac{X}{b}\left ( \frac{x}{b} \right )^{m-1}+\frac{Y}{b}\left ( \frac{y}{b} \right )^{m-1}=1$$
For the equation $$\displaystyle x^{2/3}+y^{2/3}=a^{2/3}$$, find the equation of tangent at the point $$\displaystyle x=a\sin ^{3}\theta, y=a\cos ^{3} \theta$$.
  • $$\displaystyle y-a\cos ^{3}\theta =\frac{\cos \theta }{\sin \theta }(x-a \sin ^{3}\theta ) $$
  • $$\displaystyle y-a\cos ^{3}\theta =-\frac{\cos \theta }{\sin \theta }(x-a \sin ^{3}\theta ) $$
  • $$\displaystyle y-a\cos ^{3}\theta =-\frac{\cos \theta }{\sin \theta }(x+a \sin ^{3}\theta ) $$
  • 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 $$f'(3)=$$
  • $$-1$$
  • $$0$$
  • $$1$$
  • $$\sqrt 3$$
Find the slopes of the tangents of the curve $$y=(x+1)(x-3)$$ at the points where it cuts the X-axis.
  • $$4$$
  • $$-4$$
  • $$2$$
  • $$-2$$
If the normal to the curve y = f(x) at the point (3, 4) makes an angle $$\dfrac{3\pi }{4}$$ with the positive x-axis, then f'(3) is equal to
  • -1
  • $$-\dfrac{3 }{4}$$
  • $$\dfrac{4}{5}$$
  • 1
If the tangent at the point $$\displaystyle \left ( at^{2},at^{3} \right )$$ on the curve $$\displaystyle ay^{2}= x^{3}$$ meets the curve again at Q.then the co-ordinates of Q is/are 
  • $$\displaystyle \left ( -\frac{1}{4}at^{2},-\frac{1}{8}at^{3} \right ).$$
  • $$\displaystyle \left ( \frac{1}{4}at^{2},-\frac{1}{8}at^{3} \right ).$$
  • $$\displaystyle \left ( \frac{1}{4}at^{3},-\frac{1}{8}at^{2} \right ).$$
  • $$\displaystyle \left ( -\frac{1}{4}at^{3},-\frac{1}{8}at^{3} \right ).$$
What are the tangent and normal to the curve $$x=\displaystyle \frac{2at^{2}}{1+t^{2}}$$, $$ y= \displaystyle \frac{2at^{3}}{a+t^{2}}$$ at the point for which $$\displaystyle t=\frac{1}{2}$$
  • $$\displaystyle 16x+13y=9a$$ , $$\displaystyle 13x-16y=2a$$
  • $$\displaystyle 13x-16y=2a$$ , $$\displaystyle 16x+13y=9a$$
  • $$\displaystyle 16x-13y=9a$$ , $$\displaystyle 13x+16y=2a$$
  • $$\displaystyle 13x+16y=2a$$ , $$\displaystyle 16x-13y=9a$$
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 
  • $$\displaystyle \left ( -\frac{1}{3}, \frac{5}{3} \right )$$
  • $$\displaystyle \left ( \frac{1}{2}, -\frac{15}{4} \right )$$
  • $$\displaystyle \left ( -\frac{1}{2}, \frac{15}{4} \right )$$
  • $$\displaystyle \left ( -\frac{1}{3}, \frac{1}{5} \right )$$
Normal to the curve $$y=\displaystyle x^{3}-2x^{2}+4$$ at the point where $$x=2$$
  • $$3x+4y=18$$
  • $$x+4y=18$$
  • $$4x+3y=18$$
  • $$4x+y=18$$
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 equations of the tangents drawn to the curve $$\displaystyle y= x^{4}$$ which are drawn from the point (2,0).
  • $$\displaystyle y= 0$$ and $$\displaystyle y-\left ( \frac{4}{3} \right )^{4}= 4\left ( \frac{4}{3} \right )^{3}\left ( x-\frac{4}{3} \right )$$
  • $$\displaystyle y= 0$$ and $$\displaystyle y-\left ( \frac{8}{3} \right )^{4}= 4\left ( \frac{8}{3} \right )^{3}\left ( x-\frac{8}{3} \right )$$
  • $$\displaystyle y= 0$$ and $$\displaystyle y-\left ( \frac{4}{3} \right )^{4}= 4\left ( \frac{8}{3} \right )^{2}\left ( x-\frac{8}{3} \right )$$
  • $$\displaystyle y= 0$$ and $$\displaystyle y-\left ( \frac{8}{3} \right )^{4}= 4\left ( \frac{8}{3} \right )^{2}\left ( x-\frac{8}{3} \right )$$
The points of contact of the tangents drawn from the origin to the curve $$y=\sin{x}$$ lie on the curve
  • $${x}^{2}-{y}^{2}=xy$$
  • $${x}^{2}+{y}^{2}={x}^{2}{y}^{2}$$
  • $${x}^{2}-{y}^{2}={x}^{2}{y}^{2}$$
  • $$None\ of\ these$$
The curve $$\displaystyle y-e^{xy}+x=0$$ has a vertical tangent at the point 
  • $$(1,\ 1)$$
  • $$no\ point$$
  • $$(0,\ 1)$$
  • $$(1,\ 0)$$
Let $$f(x, y)$$ be a curve in the $$x-y$$ plane having the property that distance from the origin of any tangent to the curve is equal to distance of point of contact from the $$y-$$ axis. Of $$f(1, 2)=0$$, then all such possible curves are 
  • $$x^2+y^2=5x$$
  • $$x^2-y^2=5x$$
  • $$x^2y^2=5x$$
  • $$All\ of\ these$$
At what point p(x,y)of the curve $$\displaystyle y=e^{-\left | x \right |}$$ should a tangent  be drawn so that area of the triangle bounded by the tangent and the co-ordinate axes be greatest ?
  • $$\displaystyle \left ( \pm e, 1 \right )$$
  • $$\displaystyle \left ( \pm 1,\dfrac 1e \right )$$
  • $$\displaystyle \left ( 1,\pm \dfrac 1e \right )$$
  • $$\displaystyle \left ( \pm 1, e \right )$$
If $$\displaystyle x\cos \alpha +y\sin \alpha =p$$ touches $$\displaystyle x^{2}+a^{2}y^{2}=a^{2},$$ then
  • $$\displaystyle p^{2}=a^{2}\sin^{2}\alpha +\cos^{2}\alpha $$
  • $$\displaystyle p^{2}=a^{2}\cos^{2}\alpha +\sin^{2}\alpha $$
  • $$\displaystyle 1/p^{2}=\sin^{2}\alpha +\alpha^{2}\cos^{2}\alpha $$
  • $$\displaystyle 1/p^{2}=\cos^{2}\alpha +\alpha^{2}\sin^{2}\alpha $$
The point of intersection of the tangents drawn to the curve $$\displaystyle x^{2}y=1-y$$ at the points where it is met by the curve $$\displaystyle xy=1-y$$ is given by
  • $$\displaystyle \left ( 0, -1\right )$$
  • $$\displaystyle \left ( 1, 1\right )$$
  • $$\displaystyle \left ( 0, 1\right )$$
  • none of these
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 sum of the intercepts of a tangent to $$\displaystyle \sqrt{x}+\sqrt{y}=\sqrt{a}, a> 0$$ upon the coordinate axes is
  • $$2a$$
  • $$a$$
  • $$a/2$$
  • $$\displaystyle \sqrt{a}$$
The equation of tangent to the curve $$\displaystyle y=1-e^{\tfrac x2}$$ at the point where it meets y - axis is -
  • $$x + 2y = 0$$
  • $$2x + y = 0$$
  • $$x - y = 2$$
  • None of these
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 $$a, b, c.$$
  • $$\displaystyle a= \frac{1}{2}, b= \frac{1}{4}, c= \frac{1}{3}.$$
  • $$\displaystyle a= \frac{1}{4}, b= \frac{1}{2}, c= \frac{1}{4}.$$
  • $$\displaystyle a= 2, b= 1, c= 4.$$
  • $$\displaystyle a= 4, b= 2, c= 4.$$
The parametric equations of a curve are given by $$\displaystyle x= \sec ^{2}t, y= \cot t.$$ Tangent at $$\displaystyle P \,t=\dfrac { \pi }4$$ meets the curve again at Q; then $$\displaystyle PQ=? $$
  • $$ \left ( 2\sqrt{5} \right ).$$
  • $$\dfrac {\left ( 5\sqrt{5} \right )}2.$$
  • $$ \dfrac {\left ( 3\sqrt{5} \right )}2.$$
  • $$ \left ( 3\sqrt{5} \right ).$$
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 tangent at $$(1, 1)$$ on  $$\displaystyle y^{2}= x\left ( 2-x^{2} \right )$$ meets the curve again at $$P$$, then $$P$$ is
  • $$\displaystyle \left (4, 4 \right )$$
  • $$\displaystyle \left (-1, 2 \right )$$
  • $$\displaystyle \left (9/4, 3/8 \right )$$
  • $$None\ of\ these$$
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 coordinates of the point $$M(x, y)$$ on $$\displaystyle y= e^{-\left | x \right |}$$ so that the area formed by the coordinates axes and the tangent at $$M$$ is greatest, are
  • $$(e, 1)$$
  • $$\displaystyle \left (1, e^{-1} \right )$$
  • $$\displaystyle \left (-1, e^{-1} \right )$$
  • $$(0, 1)$$
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 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
The value of m for which the area of the triangle included between the axes and any tangent to the $$\displaystyle x^{m}y= b^{m}$$ curve is constant, is
  • $$\dfrac 12$$
  • $$1$$
  • $$\dfrac 32$$
  • $$2$$
If the tangent at any point on the curve $$\displaystyle x^{4}+y^{4}= a^{4}$$ cuts off intercepts p and q on the coordinate axes, the value of $$\displaystyle p^{-\tfrac 43}+q^{-\tfrac 43}$$ is
  • $$\displaystyle a^{-\tfrac 43}$$
  • $$\displaystyle a^{-\tfrac 12}$$
  • $$\displaystyle a^{\tfrac 12}$$
  • none of these
The equation of the tangent to the curve $$\displaystyle y= \left ( 2x-1 \right )e^{2\left ( 1-x \right )}$$ at the point of its maximum is
  • $$y=1$$
  • $$x=1$$
  • $$x+y=1$$
  • $$x-y=-1$$
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
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