CBSE Questions for Class 11 Commerce Applied Mathematics Tangents And Its Equations Quiz 13 - MCQExams.com

For the curve $$x=t^2-1$$, $$y=t^2-t$$, the tangent is perpendicular to $$x$$-axis then
  • $$t=0$$
  • $$t=\dfrac{1}{2}$$
  • $$t=1$$
  • $$t=\dfrac{1}{\sqrt{3}}$$
If the tangent at any point on the curve $$x^{4} +y^{4}=a^{4}$$ cuts off intercepts $$p$$ and $$q$$ on the coordinate axes the value of $$p^{-4/3}+q^{-4/3}$$ is
  • $$a^{-4/3}$$
  • $$a^{-1/3}$$
  • $$a^{1/2}$$
  • $$None\ of\ these$$
The slope of the tangent to the curve at a point $$(x,y) $$ on it is proportional to $$(x-2).$$ If the slope of the tangent to the curve at $$(10,-9)$$  on it is $$-3$$. The equation of the curves is .
  • $$y=k(x-2)^2$$
  • $$y=\dfrac{-3}{16}(x-2)^2+1$$
  • $$y=\dfrac{-3}{16}(x-2)^2+3$$
  • $$y=K(x+2)^2$$
If the line $$y=4x-5$$ touches to the curve $${ y }^{ 2 }=a{ x }^{ 3 }+b$$ at the point $$(2,3)$$ then $$7a+2b=$$
  • $$0$$
  • $$1$$
  • $$-1$$
  • $$2$$
$$f(x) = \left\{\begin{matrix} -x^2, & \text{for} \ x < 0 \\ x^2 + 8, & \text{for} \ x \ge 0 \end{matrix}\right.$$
Let . Then x-intercept of the line, thet is , the tangent to the graph of f(x) is 
  • zero
  • -1
  • -2
  • -4
If the tangent at any point on the curve $$x+y^4=a$$ cuts off intercepts p and q on the co-ordinate axes, the value of $$p^{-4/3}+q^{-4/3}$$ is 
  • $$a^{-4/3}$$
  • $$a^{-1/2}$$
  • $$a^{1/3}$$
  • None of these
If the tangent at $$({x_1},{y_1})$$ to the curve $${x^3} + {y^3} = {a^3}$$ meets the curve again at $$({x_2},{y_2})$$, then
  • $${{{x_2}} \over {{x_1}}} + {{{y_2}} \over {{y_1}}} = - 1$$
  • $${{{x_2}} \over {{y_1}}} + {{{x_1}} \over {{y_2}}} = - 1$$
  • $${{{x_1}} \over {{x_2}}} + {{{y_1}} \over {{y_2}}} = - 1$$
  • $${{{x_2}} \over {{x_1}}} + {{{y_2}} \over {{y_1}}} = 1$$
A point on the curve $$y = 2{x^3} + 13{x^2} + 5x + 9$$, the tangent at which passes through the origin is 
  • $$(1, 15)$$
  • $$(1, -15)$$
  • $$(15, 1)$$
  • $$(-1, 15)$$
The point on the curve $$y = b e^{\dfrac {-x}{a}}$$ at which the tangent drawn is $$\dfrac {x}{a} + \dfrac {y}{b} = 1$$ is
  • $$(0, b)$$
  • $$\left (a, \dfrac {1}{e}\right )$$
  • $$(0, 1)$$
  • $$(1, 0)$$
The slope of the tangent to the curve $${r^2} = {a^2}\cos 2\theta$$, where $$x = r\cos \theta ,y = r\sin \theta $$, at the point $$\theta=\frac{\pi}{6}$$ is
  • $$\frac{1}{2}$$
  • $$-1$$
  • $$1$$
  • $$0$$
The line $$\dfrac{x}{a}+\dfrac{y}{b}=1$$ touches the curve $$y=be^{-x/a}$$ at the point.
  • $$(a, b/a)$$
  • $$(-a, b/a)$$
  • $$(0, b)$$
  • None of these

The abscissa of the point on the curve $$\sqrt {xy}  = a + x$$ , the tangent at which cuts off equal intercepts from the co-ordinate axes is (a >0)

  • $$\dfrac{a}{{\sqrt 2 }}$$
  • $$ - \dfrac{a}{{\sqrt 2 }}$$
  • $$a\sqrt 2 $$
  • - $$a\sqrt 2 $$
if $$m$$ is the slope of a tangent to the curve $$e^{y}=1+x^{2}$$, then  $$m$$ belongs to the interval
  • $$[-1, 1]$$
  • $$[-2, -1]$$
  • $$[1, 2]$$
  • $$[1, 3]$$
Find the slope of tangent of the curve$$x = a\,{\sin ^3}t,y = b\,\,{\cos ^3}t$$ at $$t = \frac{\pi }{2}$$
  • $$cott$$
  • $$-tant$$
  • $$-cott$$
  • $$\text{not defined  at}$$ $$\frac{\pi}{2}$$
The slope of the curve $$y=\sin { x } +\cos ^{ 2 }{ x }$$ is zero at a point , whose x-coordinate can be ?
  • $$\dfrac { \pi }{ 4 } $$
  • $$\dfrac { \pi }{ 2 } $$
  • $${ \pi }$$
  • $$\dfrac { \pi }{ 3 } $$
A tangent drawn to the curve $$y = f\left( x \right)$$ at $$P\left( {x,y} \right)$$
cuts the x and y axes at A and B, respectively, such that $$AP:PB = 1:3$$. If $$f\left( 1 \right) = 1$$ then the curve passes through $$\left( {k,\frac{1}{8}} \right)$$ where $$k$$ is
  • $$1$$
  • $$2$$
  • $$3$$
  • $$4$$
Equation of tangent to the circle $$x^{2}+ y^{2}-6x+4y-12=0$$ which are parallel to the line $$4x+3y+5=0$$ is ?
  • $$4x+3y+19=0$$
  • $$4x+3y+31=0$$
  • $$4x+3y-19=0$$
  • $$4x+3y+29=0$$
Let $$f ( x ) = \frac { 1 } { 1 + x ^ { 2 } }$$. Let $$m$$ be the slope, $$'a'$$ be the $$x$$-intercept and $$'b'$$ be the $$y$$-intercept of tangent to $$y = f ( x )$$.Abscissa of point of contact of the tangent for which $$'m'$$ is greatest is:
  • $$\frac { 1 } { \sqrt { 3 } }$$
  • $$1$$
  • $$0$$
  • $$\frac { - 1 } { \sqrt { 3 } }$$
If $$V$$ is the set of points on the curve $$y^{3} - 3xy +2 = 0$$ where the tangent is vertical then $$V =$$.
  • $$\phi$$
  • $$\left \{(1 , 0)\right \}$$
  • $$\left \{(1, 1)\right \}$$
  • $$\left \{(0, 0), (1, 1)\right \}$$
Paraboals $$(y-\alpha )^{2}=4a(x-\beta )and (y-\alpha )^{2}=4a'(x-\beta ')$$ will have a common normal (other than the normal passing through vertex ofparabola)if:
  • $$\frac{2(a-a')}{\beta '-\beta }< 1$$
  • $$\frac{2(a-a')}{\beta -\beta' }< 1$$
  • $$\frac{2(a'-a)}{\beta -\beta' }< 1$$
  • $$\frac{2(a'a)}{\beta -\beta' }> 1$$
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 passes through origin
  • it passes through the point $$(1,1)$$
  • it passes through $$\left( \frac { a\pi }{ 2 } ,-a \right) $$
  • it is at a constant distance from the origin
The tangent to the circle $$x^2+y^2=5$$ at the point $$(1,-2)$$  also touches the circle $$x^2+y^2-8x+6y+20=0$$ at 
  • $$(-2,1)$$
  • $$(-3,0)$$
  • $$(-1,-1)$$
  • $$(3,-1)$$
If for a curve represented parametrically by $$x={ sec }^{ 2 }t,\quad y=cot\quad t\quad $$ , the tangent  at a point $$P(t=\frac { \pi  }{ 4 } )$$ meets the curve again at the point Q, then $$\begin{vmatrix} PQ \end{vmatrix}$$is equal to 
  • $$\frac { 2\sqrt { 5 } }{ 3 } $$
  • $$\frac { 3\sqrt { 5 } }{ 2 } $$
  • $$\frac { 5\sqrt { 3 } }{ 3 } $$
  • $$\frac { 5\sqrt { 5 } }{ 4 } $$
If the subnormal to the curve $${ x }^{ 2 }.{ y }^{ n }={ a }^{ 2 }$$ is a constant then n=
  • $$-4$$
  • $$-3$$
  • $$-2$$
  • $$-1$$
If $$\theta$$ is angle of intersection between $$y=10-x^{2}$$ and $$y=4+x^{2}$$ then $$|\tan \theta|$$ is-
  • $$\dfrac {5\sqrt {3}}{11}$$
  • $$\dfrac {7\sqrt {3}}{15}$$
  • $$\dfrac {4\sqrt {3}}{11}$$
  • $$none$$
The tangent to the curve $$y  =x^2 - 5x + 5$$, parallel to the line $$2y = 4x + 1$$, also passes through the point
  • $$\lgroup \dfrac{1}{4}, \dfrac{7}{2} \rgroup$$
  • $$\lgroup \dfrac{7}{2}, \dfrac{1}{4} \rgroup$$
  • $$\lgroup -\dfrac{1}{8}, 7 \rgroup$$
  • $$\lgroup \dfrac{1}{8}, -7 \rgroup$$
$$y = 6\tan \,x\left( {{e^x} - x - 1} \right) - 3{x^3} - {x^4} - \frac{5}{4}{x^5},\,$$ if $${n^{th}}$$ derivative at x=0 is non zero then least value of n is
  • 3
  • 4
  • 5
  • 6
The slope of the tangent to the curve at a point (x, y) on it is proportional to (x-2). If the slope of the tangent to the curve at  (10,-9) on it -The equation of the curve is
  • $$y=k{ \left( x-2 \right) }^{ 2 }$$
  • $$y=\frac { -3 }{ 16 } { \left( x-2 \right) }^{ 2 }\ +1$$
  • $$y=\frac { -3 }{ 16 } { \left( x-2 \right) }^{ 2 }\ +3$$
  • $$y=k{ \left( x+2 \right) }^{ 2 }$$
If the curves $$\dfrac {x^{2}}{a^{2}} + \dfrac {y^{2}}{4} = 1$$ and $$y^{3} = 16x$$ intersect at right angles, then $$a^{2}$$ is equal to
  • $$5/3$$
  • $$4/3$$
  • $$6/11$$
  • $$3/2$$
If the slope of one of the lines given by $${a^2}{x^2} + 2hxy+by^2 = 0$$ be three times of the other , then h is equal to 
  • $$2\sqrt 3 ab$$
  • $$-2\sqrt 3 ab$$
  • $$\frac{2}{{\sqrt 3 }}ab$$
  • $$-\frac{2}{{\sqrt 3 }}ab$$
The equation of normal to the curve $$y=log^x_e$$ at the point $$P(1, 0)$$ is ___________.
  • $$2x+y=2$$
  • $$x-2y=1$$
  • $$x-y=1$$
  • $$x+y=1$$
If the tangent to the curve $$y=\cfrac { x }{ { x }^{ 2 }-3 } ,x\in R,\left( x\neq \pm \sqrt { 3 }  \right) $$, at a point $$\left( \alpha ,\beta  \right) \neq \left( 0,0 \right) $$ on it is parallel to the line $$2x+6y-11=0$$, then:
  • $$\left| 6\alpha +2\beta \right| =19\quad $$
  • $$\left| 2\alpha +6\beta \right| =11$$
  • $$\left| 6\alpha +2\beta \right| =9$$
  • $$\left| 2\alpha +6\beta \right| =19\quad $$
If the line joining the point (0, 3 ) and (5, -2) is a tangent to the curve $$y= \dfrac{c}{x+1}$$, then the value of c is 
  • 1
  • -2
  • 4
  • None of these
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