CBSE Questions for Class 11 Commerce Applied Mathematics Tangents And Its Equations Quiz 1 - MCQExams.com

The equation of normal to the curve $$ 3x^2-y^2 =8 $$ which is parallel to the line $$ x+ 3y=8 $$ is
  • $$ 3x-y=8 $$
  • $$ 3x+y+8=0 $$
  • $$ x+3y \pm 8 = 0 $$
  • $$ x+ 3y=0 $$
Let N be the set of positive integers. For all $$n \in N$$, let
$$f_n = (n + 1)^{1/3} - n^{1/3}$$ and $$A = \left\{n \in N : f_{n +1} < \dfrac{1}{3(n + 1)^{2/3}} < f_n \right\}$$
Then
  • A = N
  • A is a finite set
  • the complement of A in N is nonempty, but finite
  • A and its complement in N are both infinite
for $$f(x)=\displaystyle \int _{ 0 }^{ x }2  \left| t \right| dt$$, the tangent lines which are parallel to the bisector of the first co-ordinate angle is 
  • $$y=x-\frac{1}{4}$$
  • $$y=x+\frac{1}{4}$$
  • $$y=x-\frac{3}{2}$$
  • $$y=x+\frac{3}{2}$$
At any point on the curve $$2x^{2}y^{2}-x^{4}=c$$, the mean proportional between the abscissa and the difference between the abscissa and the sub-normal drawn to the curve at the same point is equal to 
  • ordinate
  • radius vector
  • x-intercept of tangent
  • sub-tangent
Given $$g(x)= \dfrac{x+2}{x-1}$$ and the line 3x + y -10 =0, then the line is 
  • tangent to g(x)
  • normal to g(x)
  • chord of g(x)
  • none of these
The angle formed bt the positive y-axis and the tangent to $$y = x^{2}+4x-17$$ at $$(5/2, -3/4)$$ is
  • $$\tan ^{-1}(9)$$
  • $$\dfrac{\pi }{2}\tan ^{-1}(9)$$
  • $$\dfrac{\pi }{2}\tan ^{-1}(9)$$
  • None of these
The abscissa of a point on the curve $$xy = (a+y)^{2}$$, the normal which cuts off numerically equal intercept from the coordinate axes, is 
  • $$-\dfrac{a}{\sqrt{2}}$$
  • $$\sqrt{2a}$$
  • $$\dfrac{a}{\sqrt{2}}$$
  • $$-\sqrt{2a}$$
The co-ordinates of the point (s) on the graph of the function $$f(x)= \dfrac{x^{3}}{3} - \dfrac{5x^{2}}{2} + 7x - 4$$, where the tangent drawn cuts off intercept from the co-ordinate axes which
  • (2, 8/3)
  • (3, 7/2)
  • (1, 5/6)
  • None of these
The equation of the curve $$y = be^{-x/a}$$ at the point where it crosses the y-axis is
  • $$\dfrac{x}{a}-\dfrac{y}{b}=1$$
  • $$ax = by = 1$$
  • $$ax-by=1$$
  • $$\dfrac{x}{a}+\dfrac{y}{b}=1$$
A curve is represented by the equations $$x=sec^{2}t$$ and $$y=\cot t,$$ where t is a parameter. If the tangent at the point P on the curve, where $$t=\pi /4$$, meets the curve again at the point Q, then $$\left | PQ \right |$$ is equal to
  • $$\dfrac{5\sqrt{3}}{2}$$
  • $$\dfrac{5\sqrt{5}}{2}$$
  • $$\dfrac{2\sqrt{5}}{3}$$
  • $$\dfrac{3\sqrt{5}}{2}$$
The angle between the tangents at ant point P and the line joining P to the original, where P is a point on the curve in $$(x^{2}+y^{2})=c \tan ^{-1}\dfrac{y}{x},c$$ is a constnt, is 
  • independent of x
  • independent of y
  • independent of x but dependent on y
  • independent of y but dependent on x
Let f be a continuous, differetiable and bijective function. If the tangent to y= f (x) at x = a is also the normal to y = f (x) at x = b then there  exists at least one $$c \epsilon  (a, b)$$ such that 
  • f'(c) = 0
  • $$f (c)> 0$$
  • $$f (c)< 0$$
  • None of these
The slope of the tangent to the curve $$y=\int _{ 0 }^{ x }{ \frac { dt }{ 1+{ t }^{ 3 } }  } $$ at the point where $$x=1$$ is
  • $$\frac { 1 }{ 4 } $$
  • $$\frac { 1 }{ 3 } $$
  • $$\frac { 1 }{ 2 } $$
  • $$1$$
If $$\left | f(x_{1})-f(x_{2}) \right |< (x_{1}-x_{2})^{2}$$ for all $$x_{1}$$ $$x_{2}$$ $$\in $$ R. Find the equation of tangent to the curve y = f(x) at the point (1, 2). 
  • $$x=2$$
  • $$y=2$$
  • $$y=1$$
  • $$x=1$$
The distance, from the origin, of the normal to the curve, $$x = 2\cos t + 2t\sin t, y = 2\sin t - 2t\cos t$$ at $$t = \dfrac {\pi}{4}$$, is
  • $$2$$
  • $$4$$
  • $$\sqrt {2}$$
  • $$2\sqrt {2}$$
The equation of a normal to the curve, $$\sin  y = x  \sin \displaystyle \left ( \frac{\pi}{3} + y \right ) $$ at $$x = 0$$, is
  • $$2x - \sqrt 3 y = 0$$
  • $$2y - \sqrt 3 x = 0$$
  • $$2y + \sqrt 3 x = 0$$
  • $$2x + \sqrt 3 y = 0$$
The tangent at the point $$(2, -2)$$ to the curve, $$x^2y^2-2x=4(1-y)$$ does not pass through the point.
  • $$(8, 5)$$
  • $$\left(4, \displaystyle\frac{1}{3}\right)$$
  • $$(-2, -7)$$
  • $$(-4, -9)$$
If tangent to the curve $$\displaystyle x={ at }^{ 2 },y=2at$$ is perpendicular to $$x$$-axis, then its point of contact is:
  • $$(a, a)$$
  • $$(0, a)$$
  • $$(0, 0)$$
  • $$(a, 0)$$
The normal to the curve $$x = a (\cos\theta +\theta \sin \theta ), y = a (\sin \theta -\theta \cos\theta )$$  at any point $$\theta $$ is such that 
  • it passes through the origin
  • it makes angle $$\frac{\pi }{2}+\theta $$ with the x-axis
  • it passes through $$\left ( a\frac{\pi }{2} ,-a\right )$$
  • it is at a constant distance from the origin
The normal to the curve, $$x^2+2xy-3y^2=0$$, at (1, 1)
  • does not meet the curve again
  • meets the curve again in the second quadrant
  • meets the curve again in the third quadrant
  • meets the curve again in the fourth quadrant
The equation of the tangent to the curve $$\displaystyle \mathrm{y}=\mathrm{x}+\frac{4}{\mathrm{x}^{2}}$$ , that is parallel to the $$x-axis$$, is
  • $$\mathrm{y}=1$$
  • $$\mathrm{y}=2$$
  • $$\mathrm{y}=3$$
  • $$\mathrm{y}=0$$
The tangent to the curve $$\mathrm{y}=\mathrm{e}^{\mathrm{x}}$$drawn at the point $$(\mathrm{c}, \mathrm{e}^{\mathrm{c}})$$ intersects the line joining the points $$(\mathrm{c}- \mathrm{l}, \mathrm{e}^{c-1})$$ and $$(\mathrm{c}+1, \mathrm{e}^{c-1})$$ 
  • on the left of $$x = c$$
  • on the right of $$x = c$$
  • at no point
  • at all points
If the tangent to the conic, $$y - 6 = x^2$$ at (2, 10) touches the circle, $$x^2 + y^2 + 8x - 2y = k$$ (for some fixed k) at a point $$(\alpha, \beta)$$; then $$(\alpha, \beta)$$ is;
  • $$\displaystyle \left( -\frac{4}{17}, \frac{1}{17} \right)$$
  • $$\displaystyle \left( -\frac{7}{17}, \frac{6}{17} \right)$$
  • $$\displaystyle \left( -\frac{6}{17}, \frac{10}{17} \right)$$
  • $$\displaystyle \left( -\frac{8}{17}, \frac{2}{17} \right)$$
Let $$C$$ be a curve given by $$y(x) = 1 + \sqrt {4x - 3}, x > \dfrac {3}{4}$$. If $$P$$ is a point on $$C$$, such that the tangent at $$P$$ has slope $$\dfrac {2}{3}$$, then a point through which the normal at $$P$$ passes, is:
  • $$(1, 7)$$
  • $$(3, -4)$$
  • $$(4, -3)$$
  • $$(2, 3)$$
The equation of the normal to the circle $$\displaystyle { x }^{ 2 }+{ y }^{ 2 }={ a }^{ 2 }$$ at point $$(x', y')$$ will be:
  • $$\displaystyle x'y-xy'=0$$
  • $$\displaystyle xx'-yy'=0$$
  • $$\displaystyle x'y+xy=0$$
  • $$\displaystyle xx'+yy'=0$$
What is the $$x$$-coordinate of the point on the curve $$f(x) = \sqrt {x}(7x - 6)$$, where the tangent is parallel to $$x$$-axis?
  • $$-\dfrac {1}{3}$$
  • $$\dfrac {2}{7}$$
  • $$\dfrac {6}{7}$$
  • $$\dfrac {1}{2}$$
Determine the equation of tangent at vertex of the parabola $$\displaystyle (x+4)^{2}=-4(y-2)$$.
  • $$y=0$$
  • $$y=2$$
  • $$x=0$$
  • $$x+4=0$$
What is/are the tangents to $$\displaystyle y=(x^{3}-1)(x-2)$$ at the points where the curve cuts the x-axis
  • $$y+3x=3$$
  • $$ y+2x=3$$
  • $$ y-7x+14=0$$
  • $$y -5x-14=0$$
Normal to the parabola $$\displaystyle y^{2}=4ax$$ where $$m$$ is the slope of the normal is
  • $$\displaystyle y=mx+2am-am^{3}$$
  • $$\displaystyle y=mx-2am-am^{3}$$
  • $$\displaystyle y=mx-2am+am^{3}$$
  • none of these
Tangent to parabola $$\displaystyle y^{2}=4x+5$$ which is parallel to $$y=2x+7$$
  • $$y-2x-3=0$$
  • $$ y=x+3$$
  • $$y-2x+1=0$$
  • $$y=x+1$$
The slope of tangent to the curve $$y=\int_{0}^{x}\displaystyle \frac{dx}{1+x^{3}}$$ at the point where $$x=1$$ is
  • $$\displaystyle \frac{1}{2}$$
  • $$1$$
  • $$\displaystyle \frac{1}{4}$$
  • none of these
The values of '$$a$$' for which $$y=x^{2}+ ax+25$$ touches $$x$$-axis are
  • $$\pm 10$$
  • $$\pm 2$$
  • $$\pm 1$$
  • 0
The equation of the straight line which is tangent at one point and normal at another point to the curve $$y=8{ t }^{ 3 }-1,x=4{ t }^{ 2 }+3$$ is
  • $$\displaystyle \sqrt { 2 } x-y=\frac { 89\sqrt { 2 }  }{ 27 } -1$$
  • $$\displaystyle \sqrt { 2 } x-y=\frac { 89\sqrt { 2 }  }{ 27 } +1$$
  • $$\displaystyle \sqrt { 2 } x+y=\frac { 89\sqrt { 2 }  }{ 27 } -1$$
  • $$\displaystyle \sqrt { 2 } x+y=\frac { 89\sqrt { 2 }  }{ 27 } +1$$
The equation of the normal to the curve $$\displaystyle 2y=3-x^{2}$$ at the point $$(1,1)$$
  • $$x-y=0$$
  • $$2x+y=3$$
  • $$x+2y=3$$
  • $$x-y=1$$
The equation of the normal to the curve $$\displaystyle x^{3}+y^{3}=6xy$$  at the point  $$(3,3).$$
  • $$ x+y-6=0$$
  • $$ -x-y+6=0$$
  • $$ x-y=0$$
  • $$ x+y=0$$
Normal to the curve $$\displaystyle x^{2}=4y$$ which passes through the point $$(1,2)$$
  • $$x+y=3$$
  • $$x-y=3$$
  • $$2x+y=4$$
  • $$x+2y=5$$
Find the equation of a line passing through $$(-2,3)$$ and parallel to tangent at origin for the circle $$\displaystyle x^{2}+y^{2}+x-y=0$$
  • $$x -2 y + 5 = 0$$
  • $$x -4 y + 3 = 0$$
  • $$x - y + 5 = 0$$
  • $$2x - y + 6 = 0$$
Find the equations of tangents to parabola $$\displaystyle y^{2}= 4ax$$ which are drawn from the point (2a,3a).
  • $$\displaystyle x-y+a= 0, x-2y+4a= 0$$
  • $$\displaystyle x-y-a= 0, x-2y-4a= 0$$
  • $$\displaystyle x+y+2a= 0, x-2y+a= 0$$
  • $$\displaystyle x+y-2a= 0, x+2y-4a= 0$$
If $$y = f(x)$$ be the equation of a parabola which is touched by the line $$y = x$$ at the point where $$x  = 1$$ Then
  • $$f'(1) = 1$$
  • $$f'(0) = f'(1)$$
  • $$2f(0) = 1 - f'(0)$$
  • $$f(0) + f'(0) + f"(0) = 1$$
The slope of the normal to the curve $$y = 2x^2+ 3 \sin x$$ at $$x = 0$$ is. 
  • $$3$$
  • $$\dfrac{1}{3}$$
  • $$-3$$
  • $$-\dfrac{1}{3}$$
The normal drawn at the point $$\displaystyle P\left ( at_{1}^{2},2at_{1} \right )$$ on the parabola meets the curve again at$$\displaystyle Q\left ( at_{2}^{2},2at_{2} \right ).$$ then $$\displaystyle t_{2} =?$$
  • $$\displaystyle t_{2}= -t_{1}-\dfrac{2}{t_{1}}$$
  • $$\displaystyle t_{2}= -t_{1}+\dfrac{2}{t_{1}}$$
  • $$\displaystyle t_{2}= +t_{1}-\dfrac{2}{t_{1}}$$
  • $$\displaystyle t_{2}= +t_{1}+\dfrac{2}{t_{1}}$$
The slope of the tangent to the curve $$\displaystyle y=-x^{3}+3x^{2}+9x-27$$ is maximum when x equals.
  • $$1$$
  • $$3$$
  • $$\dfrac 12$$
  • $$-\dfrac 12$$
Find the tangents and normal to the curve $$y(x-2)(x-3)-x+7=0,$$ at point (7,0) are 
  • $$x-20y-7=0,$$ $$20x+y-140=0$$.
  • $$x+20y-7=0,$$ $$20x-y-140=0$$.
  • $$ 7x-20y-1=0,$$ $$20x+7y-100=0$$.
  • $$7x+20y-1=0,$$ $$20x-7y-100=0$$.
Find the distance between the point $$(1,1)$$ and the tangent to the curve $$\displaystyle y=e^{2x}+x^{2}$$ drawn from the point where the curve cuts $$y$$-axis
  • $$\displaystyle \frac{\sqrt3}{\sqrt{5}}$$
  • $$\displaystyle \frac{3}{\sqrt{5}}$$
  • $$\displaystyle \frac{2}{\sqrt{5}}$$
  • $$\displaystyle \frac{\sqrt2}{\sqrt{5}}$$
If $$\displaystyle \frac{x}{a}+\frac{y}{b}=1$$ is a tangent to the curve $$\displaystyle x=Kt,y=\frac{K}{t},K> 0$$ than
  • $$a>0, b>0$$
  • $$a>0, b<0$$
  • $$a<0, b>0$$
  • $$a<0, b<0$$
The curve $$\displaystyle y-e^{xy}+x=0$$ has a vertical tangent at
  • $$(1, 1)$$
  • $$(0, 1)$$
  • $$(1, 0)$$
  • $$(0,0)$$
Find the equation of the tangent to the curve $$\displaystyle y=x^{2}+1$$ at the point $$(1,2).$$
  • $$2y=x$$
  • $$y=2x$$
  • $$y+x=2$$
  • $$y+2x=0$$
The equation of normal to the curve $$\displaystyle y=e^{x}$$ at the point $$(0,1)$$ is -
  • $$x + y = 1$$
  • $$x - y = 1$$
  • $$ey - x = e$$
  • $$e(y - 1) + x = 0$$
If tangent to curve at a point is perpendicular to $$x$$ - axis then at that point -
  • $$\displaystyle \frac{dy}{dx}=0 $$
  • $$\displaystyle \frac{dx}{dy}=0 $$
  • $$\displaystyle \frac{dy}{dx}=1 $$
  • $$\displaystyle \frac{dy}{dx}=-1 $$
The equation of normal to the curve $$\displaystyle y^{2}=16x$$ at the point $$(1, 4)$$ is
  • $$2x + y = 6$$
  • $$2x - y + 2 = 0$$
  • $$x + 2y = 9$$
  • None of these
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