CBSE Questions for Class 12 Commerce Maths Application Of Derivatives Quiz 8 - MCQExams.com

The inclination of the tangent at $$\theta = \dfrac {\pi}{3}$$ on the curve $$x = a(\theta + \sin \theta), y = a(1 + \cos \theta)$$ is
  • $$\dfrac {\pi}{3}$$
  • $$\dfrac {\pi}{6}$$
  • $$\dfrac {2\pi}{3}$$
  • $$\dfrac {5\pi}{6}$$
The greatest slope among the lines represented by the equation $$4x^2 - y^2 + 2y - 1 = 0 $$ is - 
  • $$-3$$
  • $$-2$$
  • $$2$$
  • $$3$$
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
  • $$\dfrac{x_{2}}{x_{1}}+\dfrac{y_{2}}{y_{1}}=-1$$
  • $$\dfrac{x_{2}}{y_{1}}+\dfrac{x_{1}}{y_{2}}=-1$$
  • $$\dfrac{x_{1}}{x_{2}}+\dfrac{y_{1}}{y_{2}}=-1$$
  • $$\dfrac{x_{2}}{x_{1}}+\dfrac{y_{2}}{y_{1}}=1$$
The ordinate of all points on the curve $$y=\dfrac{1}{2\sin^{2}x+3\cos^{2}x}$$  where the tangent is horizontal, is
  • Always equal to $$\dfrac{1}{2}$$
  • Always equal to $$\dfrac{1}{3}$$
  • $$\dfrac{1}{2}$$ or $$\dfrac{1}{3}$$ according as $$n$$ is an even or an odd integer.
  • $$\dfrac{1}{2}$$ or $$\dfrac{1}{3}$$ according as $$n$$ is an even integer
The curve given by $$x + y = {e^{xy}}$$ has an tangents parallel to the y-axis at the point
  • $$(0,1)$$
  • $$(1,0)$$
  • $$(1,1)$$
  • $$(0,0)$$
Let $$f$$ be continuous and differentiable function such that $$f(x)$$ and $$f^{'}(x)$$ have opposite sign everywhere. Then
  • $$f$$ is increasing
  • $$f$$ is decreasing
  • $$|f|$$ is non-monotonic
  • none of the above
If $$- 4 \leq x \leq 4$$ then critical points of $$f ( x ) = x ^ { 2 } - 6 | x | + 4$$ are 
  • $$3 , - 2$$
  • $$6 , - 6$$
  • $$3 , - 3$$
  • $$0,1$$

Number of possible tangents to the curve $$y = \cos \left( {x + y} \right), - 3\pi  \leqslant x \leqslant 3\pi $$, that are parallel to the line $$x + 2y = 0$$, is

  • 1
  • 2
  • 3
  • 4
The normal to the curve, $${x}^{2}+2xy-{3y}^{2}=0,\ at\left (1,1\right)$$:
  • Meets the curve, again in the fourth quadrant
  • Does not meet the curve again
  • Meets the curve again in the second quadrant
  • Meets the curve again in the third quadrant
Number of tangents drawn from the point $$\left (-1/2,0\right)$$ to the curve $$y={e}^{x}$$. (Here { } denotes fractional part function ). 
  • $$2$$
  • $$1$$
  • $$3$$
  • $$4$$
If the tangent at P of the curve $$y^2=x^3$$ intersects the curve again at Q and the straight lines OP, OQ ma angles $$\alpha, \beta$$ with the x-axis where 'O' is the origin then $$\tan\alpha/\tan\beta$$ has the value equal to?
  • $$-1$$
  • $$-2$$
  • $$2$$
  • $$\sqrt{2}$$
A curve C has the property that if the tangent drawn at any point 'P' on C meets the coordinate axes at A and B, and P is midpoint of AB. If the curve passes through the point $$(1, 1)$$ then the equation of the curve is?
  • $$xy=2$$
  • $$xy=3$$
  • $$xy=1$$
  • $$xy=4$$
Two lines drawn through the point $$A ( 4,0 )$$  divide the area bounded by the curve $$y = \sqrt { 2 } \sin ( \pi x / 4 )$$  and  $$x$$ - axis between the lines $$x = 2$$  and   $$x = 4$$  into three equal parts. Sum of the slopes of the drawn lines is:
  • $$- 4 \sqrt { 2 } / \pi$$
  • $$- \sqrt { 2 } / \pi$$
  • $$- 2 \sqrt { 2 } / \pi$$
  • None
The number of critical points of the function $$f(x)=|x-1||x-2|$$ is 
  • $$1$$
  • $$2$$
  • $$3$$
  • none of these
The line $$3x-4y=0$$
  • is a tangent to the circle $${x}^{2}+{y}^{2}=25$$
  • is a normal to the circle $${x}^{2}+{y}^{2}=25$$
  • does not meet the circle $${x}^{2}+{y}^{2}=25$$
  • does not pass thro' the origin
 The curves $$x = y^2   and.  xy = k$$ cut at right angles, If $$6k^2$$ = 1.
  • True
  • False
The tangent to the curve $$2a^2y=x^3-3ax^2$$ is parallel to the x-axis at the points
  • $$(0 , 0) , (2a, - 2a)$$
  • $$(0 , 0) , (-2a,  2a)$$
  • $$(0 , 0) , (-2a, - 2a)$$
  • $$(2 , 2) , (0 , 0)$$
$$f(x)=\dfrac{x}{5}+\dfrac{4}{x}(x\neq 0)$$ in increasing in
  • $$(-5,\ 0)$$
  • $$(0,\ 5)$$
  • $$(-\infty ,\ -5)\cup (5,\ \infty )$$
  • $$(-5,\ 5)$$
If $$x-2y+k=0$$ is a common tangent to $$\displaystyle{ y }^{ 2 }=4x\quad \& \frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { 3 } } =1\left( a>\sqrt { 3 }  \right)  $$, then the value of a, k and other common tangent are given by
  • $$a = 2$$
  • $$a = -2$$
  • $$x+2y+4 = 0$$
  • $$k=4$$
The tangent at any point of the curve $$x={ at }^{ 3 },y={ at }^{ 4 }$$ divides the abscissa of the point of contact in the ratio
  • $$1:4$$
  • $$3:2$$
  • $$1:3$$
  • $$3:1$$
The values of $$x$$ satisfying $$\left| sinx \right| ^{\left| cosx \right|} +log_\left| cosx \right| \left| sinx \right| =2,$$ where $$x\epsilon (0,\dfrac{\pi}{2}),$$ is  
  • $${\{\dfrac{\pi}{2}}\} $$
  • $${\{\dfrac{\pi}{3}}\} $$
  • $$(0, \pi 4) $$
  • $$( \pi 4, \pi 2) $$
Slope of tangent to the circle $$( x - r ) ^ { 2 } + y ^ { 2 } = r ^ { 2 }$$ at the point $$( x , y )$$ lying on the circle is
  • $$\frac { x } { y - r }$$
  • $$\frac { r - x } { y }$$
  • $$\frac { y ^ { 2 } - x ^ { 2 } } { 2 x y }$$
  • $$\frac { y ^ { 2 } + x ^ { 2 } } { 2 x y }$$
If the slope of one of the lines represented $${a^3}{x^2} + 2hxy + {b^3}{y^2} = 0$$ be the square of the other, then $$ab(a+b)$$ is equal to:
  • $$2h$$
  • $$-2h$$
  • $$8h$$
  • $$-8h$$
State true or false.
The curves $$y = {x^2} - 3x + 1$$ and $$x\left( {y + 3} \right) = 4$$ intersect at the right angles at their point of intersection
  • True
  • False
The slope of the straight line which is both tangent and normal to the curve $$4x^3=27y^2$$ is 
  • $$\underline { + } 1$$
  • $$\underline { + } \dfrac { 1 }{ 2 } $$
  • $$\underline { + } \dfrac { 1 }{ \sqrt { 2 } } $$
  • $$\underline { + } \sqrt { 2 } $$
If the line $$x+y=0$$ touches the curve $$2y^2=\alpha x^2+\beta $$ at $$(1,-1),$$ then $$(\alpha ,\beta )=$$
  • $$(-2,4)$$
  • $$(-1,3)$$
  • $$(4,-2)$$
  • $$(2,0)$$
The angle between the curves $$y = \sin x$$ and $$y = \cos x$$ is 
  • $$\tan^{-1}(2\sqrt{2})$$
  • $$\tan^{-1}(3\sqrt{2})$$
  • $$\tan^{-1}(3\sqrt{3})$$
  • $$\tan^{-1}(5\sqrt{2})$$
The function $$f(x)=log x$$
  • has maxima at x=e
  • has minima at x=e
  • has neither maxima nor minima
  • all of these
Three normals are drawn from the point $$\left(c,0\right)$$ to the curve $${y}^{2}=x.$$If two of the normals are perpendicular to each other,then $$c=$$
  • $$\dfrac{1}{4}$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{3}{4}$$
  • $$1$$
Let $$g(x)=\displaystyle \int _{1-x}^{1+x}t|f'(t)|dt$$, where $$f(x)$$ does not behave like a constant function in any interval $$(a,b)$$ and the graph of $$y=f'(x)$$ is symmetric about the line $$x=1$$, then
  • $$g(x)$$ is increasing $$\forall \ x\ \in\ R$$
  • $$g(x)$$ is increasing only if $$x<1$$
  • $$g(x)$$ is increasing only if $$f(x)$$ is increasing
  • $$g(x)$$ is decreasing $$\forall \ x\ \in \ R$$
Let $$f ( x ) = \frac { \csc x + \cot x - 1 } { 1 + \cot x - \csc x }$$. The primitive of $$f ( x )$$ with respect to $$x$$ is equal to (Where $$C$$ is constant of integration.)
  • $$\ln \left( \sin \frac { x } { 2 } \right) + C$$
  • $$2 \ln \left( \cos \frac { x } { 2 } \right) + C$$
  • $$\ln ( 1 - \sin x ) + C$$
  • $$\ln ( 1 - \cos x ) + C$$
The function $$f(x)=\dfrac{x}{x^2+1}$$ increasing, if 
  • $$-1 < x$$
  • $$x > 1$$
  • $$-1 < x$$ or $$x > 1$$
  • $$-1 < x < 1$$
The approximate value of $$\sqrt[10]{0.999}$$ is 
  • 0.0998
  • 0.9998
  • 0.0999
  • 0.9999
Let A = {1, 2, ......... 10} and B = {1, 2, .......... 5}
f : A $$\rightarrow$$ B is a non-decreasing into function, then number of such function is 
  • 1001
  • 1876
  • 205
  • 875
Equation of the tangent at (1, -1) to the curve
$${ x }^{ 3 }-x{ y }^{ 2 }-4{ x }^{ 2 }-xy+5x+3y+1=0$$ is 
  • $$x-4y-5=0$$
  • $$x+1=0$$
  • $$y-1=0$$
  • $$y+1=0$$
The angle made by the tangent at any point on the curve $$x=a(t+\sin { t } \cos { t } ),y=a{ (1+\sin { t } ) }^{ 2 }$$ with x-axis is
  • $$\dfrac { \pi }{ 2 } $$
  • $$\dfrac { \pi }{ 4 } $$
  • $$\pi +\dfrac { t }{ 2 } $$
  • $$\dfrac { \pi }{ 4 } +\dfrac { t }{ 2 } $$
The normal to the curve $$x=\quad a(cos\theta +\theta sin\theta ),\quad y=\quad a(sin\theta -\theta cos\theta )$$ at any point $$'\theta '$$ is such that
  • it passes through the origin
  • it makes an angle $$\dfrac { \pi }{ 2 } +\theta $$ with the x-axis
  • it is at a constant distance from the origin 
  • it passes through $$\left(a,\dfrac{\pi}{2}\right)$$
If $$-4\le x\le 4$$, then critical points of $$f\left( x \right) ={ x }^{ 2 }-6\left| x \right| +4$$ are 
  • $$3, -2$$
  • $$6, -6$$
  • $$3, -3, 0$$
  • $$0, 1, 3$$
If tangent at any point on the curve $${ y }^{ 2 }=1+{ x }^{ 2 }\ makes\ an\ angle\ \theta $$ with positive direction of the x-axis then
  • $$\left| \tan { \theta } \right| >1$$
  • $$\left| \tan { \theta } \right| <1$$
  • $$\left| \tan { \theta } \right| \ge1$$
  • $$\left| \tan { \theta } \right| \le 1$$
The tangent to the curve, $$y = xe^{x^2}$$ passing through the point $$(1, e)$$ also passes through the point:
  • $$\left(\dfrac{4}{3}, 2e\right)$$
  • $$(2, 3e)$$
  • $$\left(\dfrac{5}{3}, 2e\right)$$
  • $$(3, 6e)$$
The equation of the normal to the curve$$y=\left( 1+x \right) ^{ y }+{ sin }^{ -1 }\left( { sin }^{ 2 }x \right) at\quad x=0$$ is
  • x+y=1
  • x-y+1=0
  • 2x+y=2
  • 2x-y+1
If the function $$f ( x ) = 2 x ^ { 2 } - k x + 5$$ is increasing in $$[ 1,2 ]$$, then '$$k$$' lies in the interval
  • $$( - \infty , 4 )$$
  • $$( 4, \infty )$$
  • $$( - \infty , 8 ]$$
  • $$( 8 , \infty )$$
A particle moves along a line by $$s = \dfrac {1}{3} t^{3} - 3t^{2} + 8t + 5$$, it changes its direction when
  • $$t = 1, t = 2$$
  • $$t =2, t = 4$$
  • $$t =0, t = 4$$
  • $$t = 2, t = 3$$
Length of the normal to the curve at any point on the curve $$y=\dfrac { a\left( { e }^{ x/a }+{ e }^{ -x/a } \right)  }{ 2 } $$ varies as 
  • $$x$$
  • $${ x }^{ 2 }$$
  • $$y$$
  • $${ y }^{ 2 }$$
Function $$f(x)=\dfrac { \lambda sinx+6cosx }{ 2sinx+3cosx } $$ is monotonic increasing If 
  • $$\lambda >4$$
  • $$\lambda >1$$
  • $$\lambda <4$$
  • $$\lambda <2$$
For x greater than or equal to zero and less than or equal to  $$2\pi$$, $$\sin x$$ and $$\cos x$$ are both decreasing on the intervals 
  • $$(0 , \dfrac{\pi}{2})$$
  • $$ (\dfrac{\pi}{2}, \pi)$$
  • $$(\pi ,3\dfrac{\pi}{2})$$
  • $$(\dfrac{3\pi} 2 , 2\pi)$$
$$f(x)=\cfrac{x}{5}+\cfrac{5}{x}$$ is increaing in
  • $$(-5,0)$$
  • $$(0,5)$$
  • $$(-\infty,-5)\cup(5,\infty)$$
  • $$(-5,5)$$
The complex number $$\frac{(-\sqrt 3 + 3i)(1-i)}{(3 + \sqrt 3i) (i) (\sqrt 3 + \sqrt 3i)}$$ when represented in the Argand diagram is 
  • in the second quadrant
  • in the first quadrant
  • on the y-axis (imaginary axis)
  • on the x-axis (real axis)
If the line $$ax+y=c$$, touches both the curves $${x}^{2}+{y}^{2}=1$$ and $${y}^{2}=4\sqrt{2}x$$, then $$\left| c \right| $$ is equal to:
  • $$1/2$$
  • $$2$$
  • $$\sqrt{2}$$
  • $$\cfrac { 1 }{ \sqrt { 2 } } $$
Let $$f"(x) > 0$$ and $$\phi (x) = f(x) + f(2 - x) , x \in (0,2)$$ be a function, then the function $$\phi (x)$$ is
  • increasing in $$(0, 1)$$ and decreasing $$(1, 2)$$
  • decreasing in $$(0,1)$$ and increasing $$(1, 2)$$
  • increasing in $$(0, 2)$$
  • decreasing in $$(0, 2)$$
0:0:1


Answered Not Answered Not Visited Correct : 0 Incorrect : 0

Practice Class 12 Commerce Maths Quiz Questions and Answers