CBSE Questions for Class 11 Commerce Applied Mathematics Tangents And Its Equations Quiz 7 - MCQExams.com

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 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 -
  • $$(a, a)$$
  • $$\displaystyle \left ( a\left ( \frac{\pi }{2}-1 \right ),a \right )$$
  • $$\displaystyle \left ( a\left ( \frac{\pi }{2}+1 \right ),a \right )$$
  • $$\displaystyle \left ( a,a\left ( \frac{\pi }{2}+1 \right ) \right )$$
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 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
If the tangent at $$(1,\,1)$$ on $$y^2=x(2-x)^2$$ meets the curve again at $$P(a,\,b)$$ then $$a/b$$ is equal to
  • $$2$$
  • $$4$$
  • $$6$$
  • $$8$$
The abcissa of the point on the curve $$\displaystyle ay^{2}=x^{3}$$ the normal at which cuts off equal intercepts from the axes is -
  • $$1$$
  • $$\dfrac {4a } 3$$
  • $$3$$
  • $$\dfrac {4a } 9$$
The area of triangle formed by tangent to the hyperbola $$\displaystyle 2xy=a^{2}$$ and coordinates axes is -
  • $$\displaystyle a^{2}$$
  • $$\displaystyle 2a^{2}$$
  • $$\displaystyle\dfrac{ a^{2} } 2$$
  • $$\displaystyle\dfrac{ 3a^{2} } 2$$
If the tangent at any point on the curve $$\displaystyle x^{4}+y^{4}=a^{4}$$ cuts off intercept $$p$$ and $$q$$ on the axes, the value of $$\displaystyle p^{-\frac43}+q^{-\frac43}$$ is
  • $$\displaystyle a^{-\frac43}$$
  • $$\displaystyle a^{-\frac13}$$
  • $$\displaystyle a^{-\frac12}$$
  • None of these
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 slope of the tangent to the curve $$x=3t^2+1, y=t^3-1$$ at $$x=1$$ is 
  • $$\dfrac{1}{2}$$
  • $$0$$
  • $$-2$$
  • $$\infty$$
If the tangent to the curve $$\displaystyle 2y^{3}=ax^{2}+x^{3}$$ at a point $$(a, a)$$ cuts off intercepts p and q on the coordinates axes where $$\displaystyle p^{2}+q^{2}=61$$ then $$a$$ equals to
  • $$30$$
  • $$-30$$
  • $$0$$
  • $$\displaystyle \pm 30$$
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
  • $$(1, 3)$$
  • $$(0, 1)$$
  • $$(4, 6)$$
  • $$(6, 8)$$
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 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
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
  • $$\displaystyle\frac{\pi}{6}$$
  • $$\displaystyle\frac{\pi}{4}$$
  • $$\displaystyle\frac{\pi}{3}$$
  • $$\displaystyle\frac{\pi}{2}$$
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
  • $$4$$
  • $$\displaystyle \frac { 1 }{ 4 } $$
  • $$\displaystyle 16\sqrt { 2 } $$
  • $$\displaystyle \frac { 1 }{ 16\sqrt { 2 } } $$
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
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 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 equation of normal of $$x^2+y^2-2x+4y-5=0$$ at $$(2,\,1)$$ is
  • $$y=3x-5$$
  • $$2y=3x-4$$
  • $$y=3x+4$$
  • $$y=x+1$$
If the tangent at a point P, with parameter t, on the curve $$x = 4t^{2} + 3, y = 8t^{3} - 1, t\epsilon R$$ meets the curve again at a point Q, then the coordinates of Q are:
  • $$(t^{2} + 3, -t^{3} - 1)$$
  • $$(4t^{2} + 3, -8t^{3} - 1)$$
  • $$(16t^{3} + 3, -64 t^{3} - 1)$$
  • $$(t^{2} + 3, t^{3} - 1)$$
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 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
The normal to the curve $${x}^{2}=4y$$ passing $$(1,2)$$ is
  • $$x+y=3$$
  • $$x-y=3$$
  • $$x+y=1$$
  • $$x-y=1$$
  • answer required
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
  • $$\left (a\left (\dfrac {\pi}{2} - 1\right ), a\right )$$
  • $$\left (a\left (\dfrac {\pi}{2} + 1\right ), a\right )$$
  • $$\left (a \dfrac {\pi}{2}, a\right )$$
  • $$(a, a)$$
Angle between $${ y }^{ 2 }=x$$ and $${ x }^{ 2 }=y$$ at the origin is
  • $$2\tan ^{ -1 }{ \left( \dfrac { 3 }{ 4 } \right) } $$
  • $$\tan ^{ -1 }{ \left( \dfrac { 4 }{ 3 } \right) } $$
  • $$\dfrac { \pi }{ 2 } $$
  • $$\dfrac { \pi }{ 4 } $$
Equation of normal to the circle $$\displaystyle x=6\cos { \theta  } ,y=6\sin { \theta  } $$ at $$\displaystyle p\left( \frac { 2\pi  }{ 3 }  \right) $$ is
  • $$\displaystyle \sqrt { 3x } -y=0$$
  • $$\displaystyle \sqrt { 3x } +y=0$$
  • $$\displaystyle x+\sqrt { 3y } =0$$
  • $$\displaystyle x-\sqrt { 3y } =0$$
If the line $$\alpha\,x+by+c=0$$ is a tangent to the curve $$xy=4$$, then
  • $$a < 0,\,b > 0$$
  • $$a \le o,\,b > 0$$
  • $$a < 0,\,b < 0$$
  • $$a \le 0,\,b < 0$$
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
  • $$y={ x }^{ 3 }+2$$
  • $$y=-{ x }^{ 3 }-2$$
  • $$y=3{ x }^{ 3 }+4$$
  • $$y=-{ x }^{ 3 }+2$$
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 slope of the normal to the curve $$y = 3x^2$$ at the point whose $$x$$-coordinate $$2$$ is
  • $$\dfrac{1}{13}$$
  • $$\dfrac{1}{14}$$
  • $$\dfrac{-1}{12}$$
  • $$\dfrac{1}{12}$$
If $$\Delta$$ is the area of the triangle formed by the positive x-axis and the normal and tangent to the circle $$x^{2} + y^{2} = 4$$ at $$(1, \sqrt {3})$$, then $$\Delta =$$
  • $$\dfrac {\sqrt {3}}{2}$$
  • $$\sqrt {3}$$
  • $$2\sqrt {3}$$
  • $$6$$
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 where $$\alpha^{2} + \beta^{2} = 61$$ then the value of '$$a$$' is equal to
  • $$25$$
  • $$36$$
  • $$\pm 30$$
  • $$\pm 40$$
The tangent to the curve $$y=x^3+1$$ at (1, 2) makes an agnle $$\theta$$ with y axis, then the value of tan $$\theta$$ is.
  • $$3$$
  • $$\frac{1}{3}$$
  • $$-\frac{1}{3}$$
  • $$-3$$
The slope of the tangent to the curve $$y=3{ x }^{ 2 }+3\sin { x } $$ at $$x=0$$ is
  • $$3$$
  • $$2$$
  • $$1$$
  • $$-1$$
What is the slope of the tangent to the curve $$y=sin^{-1}(sin^2x)$$ at $$x=0$$ ?
  • 0
  • 1
  • 2
  • None of the above
Find the slope of the normal to the curve $$4x^3+6x^2-5xy-8y^2+9x+14=0$$T the point $$-2, 3$$.
  • $$\infty$$
  • $$11$$
  • $$\displaystyle\frac{9}{19}$$
  • $$\displaystyle-\frac{19}{9}$$
A mirror in the first quadrant is in the shape of a hyperbola whose equation is xy =A light source in the second quadrant emits a beam of light that hits the mirror at the point (2,1/2). If the reflected ray is parallel to the y-axis the slope of the incident beam is 
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  • $$\dfrac{13}{8}$$
  • $$\dfrac{7}{4}$$
  • $$\dfrac{15}{8}$$
  • $$2$$
What is the slope of the tangent to the curve $$ x = t^2 + 3t - 8, y = 2t^2 - 2t - 5$$ at t = 2 ?
  • $$\dfrac{7}{6}$$
  • $$\dfrac{6}{7}$$
  • 1
  • $$\dfrac{5}{6}$$
If the tangent to the function $$y = f(x)$$ at $$(3, 4)$$ makes an angle of $$\dfrac {3\pi}{4}$$ with the positive direction of x-axis in anticlockwise direction then $$f'(3)$$ is
  • $$-1$$
  • $$1$$
  • $$\dfrac {1}{\sqrt {3}}$$
  • $$\sqrt {3}$$
How many tangents are parallel to x-axis for the curve $$ y = x^2 - 4x + 3$$ ?
  • 1
  • 2
  • 3
  • No tangent is parallel to x-axis.
Consider the curve $$y = e^{2x}$$.Where does the tangent to the curve at (0, 1) meet the x-axis ? 
  • $$(1, 0)$$
  • $$(2, 0)$$
  • $$\left(-\dfrac{1}{2}, 0\right)$$
  • $$\left(\dfrac{1}{2}, 0\right)$$
The slope of the tangent to the curve given by $$x = 1 - \cos { \theta  }$$, $$y = \theta -\sin { \theta  } $$ at $$\theta = \dfrac { \pi  }{ 2 } $$ is
  • $$0$$
  • $$-1$$
  • $$1$$
  • Not defined
Let $$f(x)=2{ x }^{ 3 }-5{ x }^{ 2 }-4x+3,\cfrac { 1 }{ 2 } \le x\le 3$$. The point at which the tangent to the curve is parallel to the X-axis is
  • $$(1,-4)$$
  • $$(2,-9)$$
  • $$(2,-4)$$
  • $$(2,-1)$$
  • $$(2,-5)$$
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