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

The tangent at the point (2,2) to the curve, x2y22x=4(1y) does not pass through the point.
  • (8,5)
  • (4,13)
  • (2,7)
  • (4,9)
If the tangent to the conic, y6=x2 at (2, 10) touches the circle, x2+y2+8x2y=k (for some fixed k) at a point (α,β); then (α,β) is;
  • (417,117)
  • (717,617)
  • (617,1017)
  • (817,217)
Let b be a nonzero real number. Suppose f:RR is a differentiable function such that f(0)=1.
If the derivative f' of f satisfies the equation f(x)=f(x)b2+x2 for all xR, 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(x)f(x)=1 for all xR
  • f(x)f(x)=0 for all xR
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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Practice Class 12 Commerce Maths Quiz Questions and Answers