CBSE Questions for Class 12 Commerce Maths Application Of Derivatives Quiz 1 - MCQExams.com

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 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 b be a nonzero real number. Suppose $$f : R \rightarrow R$$ is a differentiable function such that $$f(0) = 1$$.
If the derivative f' of f satisfies the equation $$f'(x) = \dfrac{f(x)}{b^2 + x^2}$$ for all $$x \in R$$, then which of the following statements is/are TRUE?
  • If $$b > 0$$, then f is an increasing function
  • If $$b < 0$$, then f is a decreasing function
  • $$f\left( x \right) f\left( -x \right) =1$$ for all $$x\in R$$
  • $$f(x) f(x) = 0$$ for all $$x \in R$$
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?
  • $$-\dfrac {1}{3}$$
  • $$\dfrac {2}{7}$$
  • $$\dfrac {6}{7}$$
  • $$\dfrac {1}{2}$$
Consider the following statements in respect of the function $$f(x) = x^{3} - 1, \quad x\epsilon [-1, 1]$$
I. $$f(x)$$ is increasing in $$[-1, 1]$$
II. $$f'(x)$$ has no root in $$(-1, 1)$$.
Which of the statements given above is/ are correct?
  • Only I
  • Only II
  • Both I and II
  • Neither I nor II
If $$\dfrac{x^2}{f(4a)}=\dfrac{y^2}{f(a^2-5)}$$ respresents and ellipse with major axis as y-axis and $$f$$ is a decreasing function, then 
  • $$a \in (-\infty, 1)$$
  • $$a \in (5, \infty)$$
  • $$a \in (1, 4)$$
  • $$a \in (-1, 5)$$
The values of $$\mathrm{x}$$ at which $$\mathrm{f}(\mathrm{x})=\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{x}$$ is stationary are given by
  • $$\mathrm{n}\pi,\ \forall n\in Z$$
  • $$(2\displaystyle \mathrm{n}+1)\frac{\pi}{2},\ \forall n\in Z$$
  • $$\displaystyle \frac{\mathrm{n}\pi}{4},\ \forall n\in Z$$
  • $$\displaystyle \frac{\mathrm{n}\pi}{2},\ \forall n\in Z$$

The number of stationary points of $$\mathrm{f}(\mathrm{x})=\mathrm{s}\mathrm{i}\mathrm{n} \mathrm{x}$$ in $$[0, 2{\pi}]$$ are
  • 1
  • 2
  • 3
  • 4
Find the equation of a line passing through $$(-2,3)$$ and parallel to tangent at origin for the circle $$\displaystyle x^{2}+y^{2}+x-y=0$$
  • $$x -2 y + 5 = 0$$
  • $$x -4 y + 3 = 0$$
  • $$x - y + 5 = 0$$
  • $$2x - y + 6 = 0$$

 Stationary point of $$\displaystyle \mathrm{y}=\frac{\log \mathrm{x}}{\mathrm{x}}(\mathrm{x}>0)$$ is
  • $$(1, 0)$$
  • $$(\displaystyle \mathrm{e},\frac{1}{\mathrm{e}})$$
  • $$(\displaystyle \frac{1}{\mathrm{e}},-\mathrm{e})$$
  • $$(\displaystyle \frac{1}{\mathrm{e}'}\frac{1}{\mathrm{e}})$$
I: lf $$\mathrm{f}'(\mathrm{a})<0$$ then the function $$\mathrm{f}$$ is decreasing at $$\mathrm{x}=\mathrm{a}$$
II: lf $$\mathrm{f}$$ is decreasing at $$\mathrm{x}=\mathrm{a}$$ then $$\mathrm{f}'(\mathrm{a})<0$$ 
Which of the above statements are true ?
  • onlyI
  • only II
  • both I and II
  • neither I nor II
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 value of $$\mathrm{x}$$ at which $$\mathrm{f}(\mathrm{x})=$$ cosx is stationary are given by
  • $$\mathrm{n}\pi,\ \forall n\in Z$$
  • $$(2\displaystyle \mathrm{n}+1)\frac{\pi}{2},\ \forall n\in Z$$
  • $$\displaystyle \frac{\mathrm{n}\pi}{4},\ \forall n\in Z$$
  • $$\displaystyle \frac{\mathrm{n}\pi}{2},\ \forall n\in Z$$
The number of stationary points of $$\mathrm{f}(\mathrm{x})=\cos \mathrm{x}$$ in $$[0, 2{\pi}]$$ are
  • $$1$$
  • $$2$$
  • $$3$$
  • $$4$$
The curve $$\displaystyle y-e^{xy}+x=0$$ has a vertical tangent at
  • $$(1, 1)$$
  • $$(0, 1)$$
  • $$(1, 0)$$
  • $$(0,0)$$
The stationary point of $$\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}-10\mathrm{x}+43$$ is
  • (5, 18)
  • (18, 5)
  • (5, 5)
  • (5, 15)
The point on the curve $$\displaystyle y=x^{2}-3x+2$$ at which the tangent is perpendicular to the line $$y = x$$ is -
  • $$(0, 2)$$
  • $$(1, 0)$$
  • $$(-1, 6)$$
  • $$(2, -2)$$
If tangent to curve at a point is perpendicular to $$x$$ - axis then at that point -
  • $$\displaystyle \frac{dy}{dx}=0 $$
  • $$\displaystyle \frac{dx}{dy}=0 $$
  • $$\displaystyle \frac{dy}{dx}=1 $$
  • $$\displaystyle \frac{dy}{dx}=-1 $$
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 curve $$\displaystyle y=\sin x+\cos ^{2}x $$ is zero at the point where -
  • $$\displaystyle x=\frac{\pi }{4}$$
  • $$\displaystyle x=\frac{\pi }{2}$$
  • $$\displaystyle x=\pi$$
  • No where
The slope of the tangent to the curve $$\displaystyle y=\sin x$$ at point $$(0, 0)$$ is
  • $$1$$
  • $$0$$
  • $$\displaystyle \infty $$
  • None of these
If tangent at a point of the curve $$y = f(x)$$ is perpendicular to $$2x - 3y = 5$$ then at that point $$\displaystyle \dfrac{dy}{dx}$$ equals
  • $$\dfrac 2  3$$
  • $$-\dfrac 2  3$$
  • $$\dfrac 3  2$$
  • $$-\dfrac 3  2$$
The inclination of the tangent w.r.t. $$x$$ - axis to the curve $$\displaystyle x^{2}+2y=8x-7$$ at the point $$x = 5$$ is
  • $$\displaystyle\dfrac{ \pi }4$$
  • $$\displaystyle\dfrac{ \pi }3$$
  • $$\displaystyle\dfrac{3 \pi }4$$
  • $$\displaystyle\dfrac{ \pi }2$$
The slope of the tangent to the curve $$\displaystyle y=-x^{3}+3x^{2}+9x-27$$ is maximum when x equals.
  • $$1$$
  • $$3$$
  • $$\dfrac 12$$
  • $$-\dfrac 12$$
At what point the tangent to the curve $$\displaystyle \sqrt{x}+\sqrt{y}=\sqrt{a}$$ is perpendicular to the $$x$$ - axis
  • $$(0, 0)$$
  • $$(a, a)$$
  • $$(a, 0)$$
  • $$(0, a)$$
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 line $$y = x + 1$$ is a tangent to the curve $$ y^2 = 4x$$ at the point.
  • $$(1, 2)$$
  • $$(2, 1)$$
  • $$(1, 4)$$
  • $$( 2, 2)$$
If a tangent to the curve $$\displaystyle y=6x-{ x }^{ 2 }$$ is parallel to the line $$\displaystyle 4x-2y-1=0$$, then the point of tangency on the curve is:
  • (2, 8)
  • (8, 2)
  • (6, 1)
  • (4, 2)
The slope of the tangent to the curve $$y = \int_{0}^{x} \dfrac {dt}{1 + t^{3}}$$ at the point where $$x = 1$$ is
  • $$\dfrac {1}{4}$$
  • $$\dfrac {1}{3}$$
  • $$\dfrac {1}{2}$$
  • $$1$$
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 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 slope of the tangent to the curve $$y=\displaystyle\int_{0}^{x}\dfrac{dt}{1+t^3}$$ at the point where x=1 is 
  • $$\dfrac{1}{4}$$
  • $$\dfrac{1}{3}$$
  • $$\dfrac{1}{2}$$
  • 1
Consider the curve $$y = e^{2x}$$.What is the slope of the tangent to the curve at (0, 1) ?
  • 0
  • 1
  • 2
  • 4
The gradient of the tangent line at the point $$(a cos \alpha, a sin \alpha)$$ to the circle $$x^2 + y^2 = a^2$$, is
  • $$tan (\pi - \alpha)$$
  • $$ tan \alpha$$
  • $$ cot \alpha$$
  • - $$ cot \alpha$$
The function $$x^{x}$$ is increasing, when
  • $$x > \dfrac {1}{e}$$
  • $$x < \dfrac {1}{e}$$
  • $$x < 0$$
  • For all $$x$$
Find the approximate error in the volume of a cube with edge $$x$$ cm, when the edge is increased by $$2\%$$
  • $$4\%$$
  • $$2\%$$
  • $$6\%$$
  • $$8\%$$
Which one of the following be the gradient of the hyperbola $$xy=1$$ at the point $$\left(t,\dfrac{1}{t}\right)$$
  • $$-\dfrac{1}{t}$$
  • $$-\dfrac{1}{t^2}$$
  • $$\dfrac{1}{t}$$
  • $$-\dfrac{2}{t^2}$$
If the product of the slope of tangent to curve at $$(x,y)$$ and its y-co-ordinate is equal to the x-co-ordinate of the point, then it represent.
  • circle
  • parabola
  • ellipse
  • rectangular hyperbola
The slope of the tangent to the curve $$xy+ax-by=0$$ at the point $$(1,1)$$ is $$2$$, then value of $$a$$ and $$b$$ are respectively:
  • $$1,2$$
  • $$2,1$$
  • $$3,5$$
  • None of these
The graph of the function $$f(x) = 2x^3 - 7$$ goes :
  • up to the right and down to the left
  • down to the right and up to the left
  • up to the right and up to the left
  • down to the right and down to the left
  • none of these ways.
A curve with equation of the form $$y=a{x}^{4}+b{x}^{3}+cx+d$$ has zero gradient at the point $$(0,1)$$ and also touches the x-axis at the point $$(-1,0)$$ then
  • $$a=3$$
  • $$b=4$$
  • $$c+d=1$$
  • for $$x< -1$$ the curve has a negative gradient
The local maximum value of $$x{(1-x)}^{2},0\le x\le 2$$ is
  • $$2$$
  • $$\dfrac {4}{27}$$
  • $$5$$
  • $$2,\dfrac {4}{27}$$
Function $$f(x)=x-\ell nx$$ is decreasing, when
  • $$x \in (0,1)$$
  • $$x \in (-1,1)$$
  • $$x \in (1,\infty)$$
  • $$None\ of\ these$$
If the curves $${y}^{2}=6x,9{x}^{2}+b{y}^{2}=16$$ intersect each other at right angles, then the values of $$b$$ is
  • $$6$$
  • $$\cfrac{7}{2}$$
  • $$4$$
  • $$\cfrac{9}{2}$$
The interval in which the  function $$f(x) = {x^3}$$ increases less rapidly than $$\,g(x) = 6{x^2} + 15x + 5$$ is :
  • $$( - \infty , - 1)\,\,\,\,$$
  • $$( - 5,1)\,\,\,\,$$
  • $$( - 1,5)$$
  • $$(5,\infty )$$
The values of $$x$$ for which the tangents to the curves $$y=x\cos{x},y=\cfrac{\sin{x}}{x}$$ are parallel to the axis of $$x$$ are roots of  (respectively)
  • $$\sin{x}=x,\tan{x}=x$$
  • $$\cot{x}=x,\sec{x}=x$$
  • $$\cot{x}=x,\tan{x}=x$$
  • $$\tan{x}=x,\cot{x}=x$$
Among all the critical points of a function f(x)=(4-x)|2-x|. Let 'a' and 'b' be the maximum and minimum values of their abscissate respectively then match the correct option. 
  • a+2b=7
  • 2a+b=7
  • 2a+b=5
  • 2a-b=5
The slope of the tangent to the curve $$y=sinx$$ where it crosses the $$x-axis$$ is 
  • $$1$$
  • $$-1$$
  • $$ \pm 1$$
  • $$ \pm 2$$
The equation of normal to the curve $$y=\left| { x }^{ 2 }-\left| x \right|  \right| $$ at $$x=-2$$ is
  • $$3y=2x+10$$
  • $$3y=x+8$$
  • $$2y=x+6$$
  • $$2y=3x+10$$
The Point (s) on the cure $${ y }^{ 3 }+{ 3x }^{ 2 }=12y$$ where the tangent is vertical (parallel to y-axis), is/are.
  • $$\left[ \pm \dfrac { 4 }{ \sqrt { 3 } } ,-2 \right] $$
  • $$\left( \pm \dfrac { \sqrt { 11 } }{ 3 } ,1 \right) $$
  • $$(0,0)$$
  • $$\left( \pm \dfrac { 4 }{ \sqrt { 3 } } ,2 \right) $$
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