CBSE Questions for Class 11 Commerce Applied Mathematics Tangents And Its Equations Quiz 4 - MCQExams.com

The curve given by the equation $$y-e^{xy}+x=0$$ has a vertical tangent at the point
  • $$(0, 1)$$
  • $$(1, 1)$$
  • $$(- 1, 1)$$
  • $$(1, 0)$$

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
If the parabola $$y = ax^2 - 6x + b$$ passes through $$(0, 2)$$ and has its tangent at $$x = \dfrac{3}{2}$$ parallel to the $$x-$$axis then
  • $$a = 2, b = -2$$
  • $$a = 2, b = 2$$
  • $$a = - 2, b = 2$$
  • $$a = - 2, b = - 2$$
At what points of curve $$y= \displaystyle \frac {2}{3} x^3 + \displaystyle \frac{1}{2} x^2 $$, the tangent makes equal angle with the axis
  • $$\left( \displaystyle \frac { 1 }{ 2 } ,\displaystyle \frac { 5 }{ 24 } \right) $$ and $$\left( -1,-\displaystyle \frac { 1 }{ 6 } \right) $$
  • $$\left(\displaystyle \frac { 1 }{ 2 } ,\displaystyle \frac { 4 }{ 9 } \right)$$ and $$\left( -1,0 \right) $$
  • $$\left(\displaystyle \frac { 1 }{ 3 } ,\displaystyle \frac { 1 }{ 7 } \right) $$and $$\left( -3,\displaystyle \frac { 1 }{ 2 } \right) $$
  • $$\left(\displaystyle \frac { 1 }{ 3 } ,\displaystyle \frac { 4 }{ 47 } \right)$$ and $$\left( -1,-\displaystyle \frac { 1 }{ 3 } \right) $$
The point on the curve  $$3y=6x-5x^3 $$, the normal at which passes through the origin is
  • $$\left( 1,\displaystyle \frac { 1 }{ 3 } \right) $$
  • $$\left( \displaystyle \frac { 1 }{ 3 } ,1 \right) $$
  • $$\left( 2,\displaystyle \frac { -28 }{ 3 } \right) $$
  • none of these
If at each point of the curve $$y=x^3-ax^2 +x+1$$, the tangent is inclined at an acute angle with the positive direction of the $$x$$-axis, then
  • $$a>0$$
  • $$a\leq \sqrt 3$$
  • $$-\sqrt 3 \leq a \leq \sqrt 3 $$
  • none of these
Consider a curve $$y=f(x)$$ in $$xy$$- plane. The curve passes through $$(0,0)$$ and has the property that a segment of tangent drawn at any point $$P(x, f(x))$$ and the line $$y=3$$ gets bisected by the line $$x+y=1$$, then the equation of the curve is 
  • $$y^2=9(x-y)$$
  • $$(y-3)^2=9(1-x-y)$$
  • $$(y+3)^2=9(1-x-y)$$
  • $$(y-3)^2-9(1+x+y)$$
The value of $$'c'$$ such that the line joining $$(0,3),(5,-2)$$ is a tangent to the curve $$y=\dfrac{c}{x+1}$$
  • $$4$$
  • $$3$$
  • $$23$$
  • $$1$$
The equation of the tangent to the curve $$y=be^{-x/a}$$ at the point where it crosses the $$y$$-axis is
  • $$\displaystyle \frac {x}{a}-\displaystyle \frac{y}{b} =1$$
  • $$ax+by=1$$
  • $$ax-by=1$$
  • $$\displaystyle \frac {x}{a}+\displaystyle \frac{y}{b} =1$$
The distance between the origin and the tangent to the curve $$y = e^{2x} + x^{2}$$ drawn at the point $$x = 0$$ is
  • $$\dfrac {1}{\sqrt 5}$$
  • $$\dfrac {2}{\sqrt 5}$$
  • $$\dfrac {-1}{\sqrt 5}$$
  • $$\dfrac {2}{\sqrt 3}$$
The slope of the tangent to the curve $$y= \sqrt {4-x^2}$$ at the point where the ordinate and the abscissa are equal is
  • -1
  • 1
  • 0
  • none of these
The curve given by $$x+y=e^{xy}$$ has a tangent parallel to the y-axis at the point
  • (0,1)
  • (1,0)
  • (1,1)
  • none of these
The normal to the curve $$2x^2 +y^2=12$$  at the point $$(2,2)$$ cuts the curve again at
  • $$\left (-\displaystyle \frac {22}{9}, -\displaystyle \frac {2}{9}\right )$$
  • $$\left (\displaystyle \frac {22}{9}, \displaystyle \frac {2}{9}\right )$$
  • $$(-2,-2)$$
  • none of these
If a variable tangent to the curve $$x^2y=c^3$$ makes intercepts $$a , b$$ on $$X$$-axes and $$Y$$- axes, respectively, then the value of $$a^2b$$ is
  • $$27c^3$$
  • $$\displaystyle \frac {4}{27}c^3$$
  • $$\displaystyle \frac {27}{4}c^3$$
  • $$\displaystyle \frac {4}{9} c^3$$
If $$y = 4x - 5$$ is a tangent to the curve $$\displaystyle y^2 = px^3 + q$$ at $$(2, 3)$$, then
  • $$p = 2, q = -7$$
  • $$p = -2, q = 7$$
  • $$p = -2, q = -7$$
  • $$p = 2, q = 7$$
The point(s) at each of which the tangents to the curve $$\displaystyle y = x^3 - 3x^2 - 7x + 6$$ cut off on the positive semi axis $$OX$$ a line segment half that on the negative semi axis $$OY$$, then the co-ordinates of the point(s) is/are give by:
  • $$(-1, 9)$$
  • $$(3, -15)$$
  • $$(1, -3)$$
  • none
If the tangent to the curve $$xy + ax + by = 0$$ at (1, 1) makes an angle $$\displaystyle \tan ^{-1}(2)$$ with x-axis, then $$\displaystyle a + 2b$$ is equal to
  • $$\displaystyle \frac {1}{2}$$
  • $$\displaystyle - \frac {1}{2}$$
  • $$3$$
  • $$-3$$
A line L is perpendicular to the curve $$\displaystyle  y = \dfrac {x^2}{4} - 2$$ at its point P and passes through (10, -1). The coordinates of the point P are
  • (2, -1)
  • (6, 7)
  • (0, -2)
  • (4, 2)
If the curve $$y=ax^3+bx^2+cx$$ is inclined at $$45^0$$ to the x-axis at $$(0,0)$$ but it touches the x-axis at $$(1,0)$$ then the value of $$a,b$$ and $$c$$ are given by
  • $$a=-1,b=2,c=1$$
  • $$a=1,b=-2,c=1$$
  • $$a=1,b=1,c=-2$$
  • $$a=-2,b=1,c=1$$
If the tangent at each point of the curve $$\displaystyle y=\frac { 2 }{ 3 } { x }^{ 3 }-2a{ x }^{ 2 }+2x+5$$ makes an acute angle with the positive direction of x-axis, then 
  • $$a\ge 1$$
  • $$-1\le a\le 1$$
  • $$a\le -1$$
  • none of these
The angle formed by the positive $$y-axis$$ and the tangent to $$y=x^2+4x-17$$ at $$(5/2,-3/4)$$ is
  • $${\tan^{-1}(9)}$$
  • $$\displaystyle \frac{\pi}{2}-{\tan^{-1}(9)}$$
  • $$\displaystyle \frac{\pi}{2}+{\tan^{-1}(9)}$$
  • none of these
The coordinates of the point(s) on the graph of the function $$f(x)=\displaystyle \frac {x^3}{3}-\displaystyle \frac{5x^2}{2}+7x-4$$, where the tangent drawn cuts off intercepts from the coordinate axes which are equal in magnitude but opposite in sign, are
  • $$(2,8/3)$$
  • $$(3,7/2)$$
  • $$(1,5/6)$$
  • none of these
If a variable tangent to the curve $$\displaystyle x^2y = c^3$$ makes intercepts a, b on x and y axis respectively, then the value of $$\displaystyle a^2b$$ is
  • $$\displaystyle 27 c^3$$
  • $$\displaystyle \frac {4}{27} c^3$$
  • $$\displaystyle \frac {27}{4} c^3$$
  • $$\displaystyle \frac {4}{9} c^3$$
The line $$ax + by = 1$$ is tangent to the curve $$\displaystyle ax^2 + by^2 = 1$$, if $$(a, b)$$ can be equal to
  • $$\displaystyle (\frac {1}{2}, \frac {1}{2})$$
  • $$\displaystyle (\frac {1}{4}, \frac {3}{4})$$
  • $$\displaystyle (\frac {1}{2}, \frac {3}{4})$$
  • $$\displaystyle (\frac {1}{4}, \frac {1}{2})$$
The slope of the tangent to the curve $$y=x^{2}-x$$ at the point where the line $$y=2$$ cuts the curve in the first quadrant is
  • $$2$$
  • $$3$$
  • $$-3$$
  • none of these
The slope of the tangent to the locus $$y=\cos^{-1}\left ( \cos x \right )$$ at $$x=\displaystyle \frac{\pi }{4}$$ is
  • $$1$$
  • $$0$$
  • $$2$$
  • $$-1$$
The point of contact of a tangent from the point $$(1, 2)$$ to the circle $$x^{2}+y^{2}=1$$ has the coordinates 
  • $$(1,0)$$
  • $$(-1,0)$$
  • $$\displaystyle (\frac{3}{5},-\frac{4}{5})$$
  • $$\displaystyle (-\frac{3}{5},\frac{4}{5})$$
Let $$\displaystyle f(x)=e\:^{x}\sin x$$ be the equation of a curve. If at $$\displaystyle x=a,0\leq a\leq 2\pi$$, the slope of the tangent is the maximum then the value of $$a$$ is 
  • $$\displaystyle \pi /2$$
  • $$\displaystyle 3\pi /2$$
  • $$\displaystyle \pi $$
  • $$\displaystyle \pi /4$$
If at each point of the curve $$y=x^{3}-ax^{2}+x+1$$ the tangent is inclined at an acute angle with the positive direction of the x-axis then
  • $$4a>0$$
  • $$a\leq \sqrt{3}$$
  • $$-\sqrt{3}\leq a\leq \sqrt{3}$$
  • none of these
The slope of the tangent to the curve $$y=\sqrt{4-x^{2}}$$ at the point where the ordinate and the abscissa are equal, is
  • $$-1$$
  • $$1$$
  • $$0$$
  • none of these
The equation of the curve is given by $$x=e^{t}\sin t$$, $$y=e^{t}\cos t$$. The inclination of the tangent to the curve at the point $$t=\displaystyle \frac{\pi }{4}$$ is
  • $$\displaystyle \frac{\pi }{4}$$
  • $$\displaystyle \frac{\pi }{3}$$
  • $$\displaystyle \frac{\pi }{2}$$
  • $$0$$
The triangle by the tangent to the curve $$f(x) = x^{2} bx -b$$ at the point (1, 1) and the co-ordinate axes lies in the first quadrant. If its area is 2, then the value of b is 
  • -1
  • 3
  • -3
  • 1
If the tangent to the curve $${ 2y }^{ 3 }={ ax }^{ 2 }+{ x }^{ 3 }$$ at the point $$(a,a)$$ cuts off intercepts $$\alpha $$ and $$\beta $$ on the coordinate axes such that $${ \alpha  }^{ 2 }+{ \beta  }^{ 2 }=61,$$ then $$a=$$
  • $$\pm 30$$
  • $$\pm 5$$
  • $$\pm 6$$
  • $$\pm 61$$
If $$m$$ be the slope of a tangent to the curve $${ e }^{ 2y }=1+4{ x }^{ 2 }$$, then 
  • $$m<1$$
  • $$\left| m \right| \le 1$$
  • $$\left| m \right| >1$$ 
  • None of these
If $$m$$ be the slope of tangent to the curve $$e^{y}=1+x^{2}$$ then 
  • $$|m|>1$$
  • $$m<1$$
  • $$|m|<1$$
  • $$|m|\leq 1$$
If the tangent to the curve $$\sqrt{x}+\sqrt{y}=\sqrt{a}$$ at any points on it cuts the axes $$OX$$ and $$OY$$ at $$P$$ and $$Q$$ respectively then $$OP+OQ$$ is
  • $$2a$$
  • $$a$$
  • $$\displaystyle \frac{1}{2}a$$
  • none of these
$$P(2, 2)$$ and $$Q\left ( \displaystyle \frac{1}{2}, -1 \right )$$ are two points on the parabola $$y^{2}=2x$$. The coordinates of the point $$R$$ on the parabola, where tangent to the curve is parallel to the chord $$PQ$$ is
  • $$\left ( \displaystyle \frac{5}{4}, \sqrt{\frac{5}{2}} \right )$$
  • $$(2, -1)$$
  • $$\left ( \displaystyle \frac{1}{8}, \frac{1}{2} \right )$$
  • none of these
The number of tangents to the curve $$y^{2}-2x^{3}-4y+8=0$$ that pass through $$(1, 0)$$ is
  • $$3$$
  • $$1$$
  • $$2$$
  • $$6$$
The curve given by $$x+y=e^{xy}$$ has a tangent as the $$y$$-axis at the point
  • $$(0, 1)$$
  • $$(1, 0)$$
  • $$(1, 1)$$
  • none of these
The equation of tangent to the curve $$y=e^{-\left | x \right |}$$ at the point where the curve cuts the line $$x=1$$ is
  • $$x+y=e$$
  • $$e(x+y)=1$$
  • $$y+ex=1$$
  • none of these
The equation of tangent to the curve $$y=be^{-x/a}$$ where it cuts the $$y$$-axis is
  • $$\displaystyle \frac{x}{a}+\frac{y}{b}=1$$
  • $$\displaystyle \frac{x}{a}+\frac{y}{b}=-1$$
  • $$\displaystyle \frac{x}{a}-\frac{y}{b}=1$$
  • none of these
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
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
The triangle formed by the tangent to the curve $$\displaystyle f(x)=x^{2}+bx-b$$ at the point $$(1, 1)$$ and the coordinates axes, lies in the first quadrant. If its area is $$2$$ then the value of b is
  • -1
  • 3
  • -3
  • 1
Let $$y=f(x)$$ be the equation of a parabola which is touches by the line $$y=x$$ at the point where $$x=1$$. Then
  • $$f^{'}\left ( 0 \right )=f^{'}\left ( 1 \right )$$
  • $$f^{'}\left ( 1 \right )=1$$
  • $$f\left ( 0 \right )+f^{'}\left ( 0 \right )+f^{''}\left ( 0 \right )=1$$
  • $$2f\left ( 0 \right )=1-f^{'}\left ( 0 \right )$$
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
  • $$(b, a)$$
  • $$(a, b)$$
  • $$(1, 1)$$
  • $$\displaystyle \left ( \frac{1}{a}, \frac{1}{b} \right )$$
The normal to the curve $$2x^{2}+y^{2}=12$$ at the point $$(2, 2)$$ cuts the curve again at
  • $$\displaystyle \left ( -\frac{22}{9}, -\frac{2}{9} \right )$$
  • $$\displaystyle \left ( \frac{22}{9}, \frac{2}{9} \right )$$
  • $$(-2, -2)$$
  • none of these
The area bounded by the axes of reference and the normal to $$y=\log_{e}x$$ at the point $$(1, 0)$$ is
  • 1 unit$$^{2}$$
  • 2 unit$$^{2}$$
  • $$\displaystyle \frac{1}{2}$$ unit$$^{2}$$
  • none of these
A point on the ellipse $$4x^{2}+9y^{2}=36$$ where the tangent is equally inclided to the axes is
  • $$\displaystyle \left ( \frac{9}{\sqrt{13}}, \frac{4}{\sqrt{13}} \right )$$
  • $$\displaystyle \left ( -\frac{9}{\sqrt{13}}, \frac{4}{\sqrt{13}} \right )$$
  • $$\displaystyle \left ( \frac{9}{\sqrt{13}}, -\frac{4}{\sqrt{13}} \right )$$
  • $$\displaystyle \left ( \frac{4}{\sqrt{13}}, -\frac{9}{\sqrt{13}} \right )$$
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Practice Class 11 Commerce Applied Mathematics Quiz Questions and Answers