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

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 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 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 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 equation of tangents to the curve $$ y(1+x^2 )=2-x, $$ where it crosses x-axis is:
  • $$ x+5y=2 $$
  • $$ x-5y=2 $$
  • $$ 5x-y=2 $$
  • $$ 5x+y=2 $$
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 equation of the tangent to the curve $$y=1-e^{\dfrac{x}{2}}$$ at the point of intersection with $$Y-$$ axis 
  • $$x+2y=0$$
  • $$2x+y=0$$
  • $$x-y=2$$
  • $$x+y=2$$
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 intercepts on $$x$$-axis made by tangents to the curve, $$y=\int_{0}^{x}|t|$$ dt, $$x\in R$$, which are parallel to the line $$y=2x$$, are equal to
  • $$\pm 2$$
  • $$\pm 3$$
  • $$\pm 4$$
  • $$\pm 1$$
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 slope of the normal to the curve $$ y = 2x ^{2} + 3 \sin x $$ at $$ x = 0 $$ is 
  • $$3$$
  • $$1/3$$
  • $$-3$$
  • $$-1 /3$$
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 ...........
  • $$\left ( 4, \pm \dfrac{8}{3} \right )$$
  • $$\left ( 4, \dfrac{-8}{3} \right )$$
  • $$\left ( 4, + \dfrac{8}{3} \right )$$
  • $$\left (\pm 4, \dfrac{8}{3} \right )$$
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 coordinates of a point P(x, y) lying in the first quadrant of the ellipse $$\displaystyle \frac{x^{2}}{8}+\frac{y^{2}}{18}=1$$ so that the area of the triangle formed by the tangent at P and the axes is the smallest are
  • (3,2)
  • (-2,3)
  • (2,-3)
  • (2,3)
The greatest inclination between the tangents is
  • $$\displaystyle \tan^{-1} \left ( \dfrac{\mathrm{a}+\mathrm{b}}{2\sqrt{\mathrm{a}\mathrm{b}}}\right )$$
  • $$\displaystyle \tan^{-1} \left (\dfrac{\mathrm{a}-\mathrm{b}}{2\sqrt{\mathrm{a}\mathrm{b}}}\right )$$
  • $$\tan^{-1}\sqrt{\dfrac{\mathrm{a}}{\mathrm{b}}}$$
  • $$\tan^{-1}\sqrt{\dfrac{\mathrm{b}}{\mathrm{a}}}$$
If $$OT$$ and $$ON$$ are perpendiculars dropped from the origin to the tangent and normal to the curve $$x=a\sin^{3}t, y=a\cos^{3}t$$ at an arbitrary point, then which of the following is/are correct?
  • $$4{OT}^{2}+ON{}^{2}=a^{2}$$
  • $${OT}^{2}+{ON}^{2}=a^{2}$$
  • $${OT}^{2}+{ON}^{2}=2a^{2}$$
  • $${OT}^{2}+2{ON}^{2}=4a^{2}$$
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
  • $$(x-1)^{2}$$
  • $$(x+1)^{2}$$
  • $$(x+1)^{3}$$
  • $$(x-1)^{3}$$
The portion of the tangent to the curve $$x=\displaystyle \sqrt{a^{2}-y^{2}}+\frac{a}{2}\log\dfrac{a-\sqrt{a^2-y^2}}{a+\sqrt{a^{2}-y^{2}}}$$ intercepted between the curve and $$x-$$axis, is of length
  • $$\displaystyle \frac{a}{2}$$
  • $$a$$
  • $$2 a$$
  • $$\displaystyle \frac{a}{4}$$
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 number of different points on the curve $$y^2=x(x+1)^2$$, where the tangent to the curve drawn at (1, 2) meets the curve again, is
  • 0
  • 1
  • 2
  • 3
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
  • $$7$$
  • $$8$$
  • $$9$$
  • $$10$$
The number of values of c such that the straight line $$3x + 4y = c$$ touches the curve $$\dfrac{x^{4}}{2}=x+y$$, is :
  • $$0$$
  • $$1$$
  • $$2$$
  • $$4$$
Let tangent at a point P on the curve $$x^{2m}\: Y^{\frac{n}{2}}=a^{\frac{4m+ n}{2}}(m, \: n\in \: N, \: n \: is \: even)$$, meets the x-axis and y-axis at A and B respectively, if $$AP : PB \:is \:\dfrac{n}{\lambda m}$$, where P lies between A and B, then find the value of $$\lambda$$
  • 2
  • 4
  • 6
  • 8
The equation of the normal to the curve $$y = e^{-2|x|}$$ at the point where the curve cuts the line $$x=\displaystyle \frac{1}{2}$$ is
  • $$2e(ex + 2y) = e^{2} - 4$$
  • $$2e(ex - 2y) = e^{2} - 4$$
  • $$2e(ey - 2x) = e^{2} - 4$$
  • none of these
The  perpendicular  distance between the point (1, 1) and the tangent to the curve y  $$=e^{2x}+x^2$$ drawn at the point x $$=$$ 0 is
  • $$\dfrac{1}{\sqrt{5}}$$
  • $$\dfrac{3}{\sqrt{5}}$$
  • $$\dfrac{2}{\sqrt{5}}$$
  • $$\dfrac{4}{\sqrt{5}}$$
The sum of the intercepts on the coordinate axis by any tangent to the curve $$\sqrt{x} + \sqrt{y} = 2$$ is
  • $$2$$
  • $$4$$
  • $$6$$
  • $$8$$
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 $$(2, 1)$$
The equation of the tangent to the curve $$y=\sqrt{9-2x^2}$$ at the point, where the ordinate & the abscissa are equal , is 
  • $$2x+y-\sqrt3=0$$
  • $$2x+y-3=0$$
  • $$2x-y-3\sqrt3=0$$
  • $$2x+y-3\sqrt{3}=0$$
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
  • $$\left (\displaystyle \frac{1}{2},\displaystyle \frac{1}{4} \right )$$
  • $$\left ( \displaystyle \frac{1}{2}, \displaystyle \frac{1}{2} \right )$$
  • $$(2,4)$$
  • $$\left ( \displaystyle \frac{1}{4}, \displaystyle \frac{1}{2} \right )$$
At the point $$P(a,a'')$$ on the graph of $$y=x^n$$, $$(n \epsilon N)$$, in the first quadrant , a normal is drawn. The normal intersects the $$y$$-axis at the point $$(0,b)$$. If $$\lim_{a\rightarrow 0}b=\displaystyle \frac{1}{2}$$, then n equals
  • $$1$$
  • $$3$$
  • $$2$$
  • $$4$$
Let $$f(x)$$=$$\begin{cases} -x^2, {for  \   x<0} \\x^2+8,  {for \   x\geq 0}  \end{cases}$$. Then $$x$$-intercept of the line, that is, the tangent to the graph of $$f(x)$$ in both the intervals of its domain, is
  • zero
  • $$-1$$
  • $$-2$$
  • $$-4$$
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 
  • $$1$$
  • $$-1$$
  • $$2$$
  • $$0$$
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 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 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
  • $$\displaystyle \frac {1}{4}(\pi +2t)$$
  • $$\displaystyle \frac {1-\sin t}{\cos t}$$
  • $$\displaystyle \frac {1}{4}(2t-\pi)$$
  • $$\displaystyle \frac {1+\sin t}{\cos 2t}$$
An equation for the line that passes through $$(10, -1)$$ and is perpendicular to $$y \displaystyle = \frac{x^2}{4} - 2$$ is
  • $$4x + y = 39$$
  • $$2x + y = 19$$
  • $$x + y = 9$$
  • $$x + 2y = 8$$
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 $$\ln \left (\displaystyle \frac {\sqrt 3+\sqrt 5} {2}\right )$$
  • $$\ln \left (\displaystyle \frac {\sqrt 3+\sqrt 7} {2}\right )$$
  • $$\ln \left (\displaystyle \frac {\sqrt 7-\sqrt 3} {7}\right )$$
  • $$\pm\ln \left (\displaystyle \frac {\sqrt 3+\sqrt 7} {2}\right )$$
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
A curve is represented by the equations, $$\displaystyle x = \sec^2t$$ and $$y =\cot t$$, where $$t$$ is a parameter. If the tangent at the point $$P$$ on the curve where $$\displaystyle t = \dfrac{\pi}{4}$$ meets the curve again at the point $$Q$$ then $$|PQ|$$ is equal to
  • $$\displaystyle \frac {5\sqrt 3}{2}$$
  • $$\displaystyle \frac {5\sqrt 5}{2}$$
  • $$\displaystyle \frac {2\sqrt 5}{3}$$
  • $$\displaystyle \frac {3\sqrt 5}{2}$$
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 x-intercept of the tangent at any arbitrary point of the curve $$\displaystyle \frac {a}{x^2} + \frac {b}{y^2} = 1$$ is proportional to:
  • square of the abscissa of the point of tangency
  • square root of the abscissa of the point of tangency
  • cube of the abscissa of the point of tangency
  • cube root of the abscissa of the point of tangency
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
  • $$\displaystyle \frac {1}{4} (\pi + 2t)$$
  • $$\displaystyle \frac {1 - \sin t}{ \cos t}$$
  • $$\displaystyle \frac {1}{4} (2t - \pi)$$
  • $$\displaystyle \frac {1 + \sin t}{\cos 2 t}$$
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