CBSE Questions for Class 12 Commerce Maths Application Of Derivatives Quiz 6 - MCQExams.com

The slope of the tangent to the curve $$x=3t^2+1, y=t^3-1$$ at $$x=1$$ is 
  • $$\dfrac{1}{2}$$
  • $$0$$
  • $$-2$$
  • $$\infty$$
The function $$f(x)=\frac{x}{3}+\frac{3}{x}$$ decreaes in the interval.
  • (-3, 3)
  • $$(-\infty , 3)$$
  • $$(3, \infty )$$
  • $$(-9, 9)$$
  • $$(-3,3) - \{0\}$$
Find the equation of the quadratic function $$f$$ whose graph increases over the interval $$(-\infty, -2)$$ and decreases over the interval $$(-2,+\infty)$$, $$f(0)=23$$ and $$f(1)=8$$
  • $$f(x)=3({x+2)}^{2}+35$$
  • $$f(x)=-3({x+2)}^{2}-35$$
  • $$f(x)=-3({x-2)}^{2}+35$$
  • $$f(x)=-3({x+2)}^{2}+35$$
For which region is $$f(x)=3x^2-2x+1$$ strictly increasing?
  • $$(2,5)$$
  • $$(\dfrac 13,\infty)$$
  • $$(-1,\dfrac 13]$$
  • $$(-\infty,\dfrac 13)$$
The focal length of a mirror is given by $$\displaystyle \frac{2}{f}\, =\, \displaystyle \frac{1}{v}\, -\, \displaystyle \frac{1}{u}$$. In finding the values of u and v, the errors are equal and equal to 'p'. The the relative error in f is
  • $$\displaystyle \frac{p}{2}\, \left ( \displaystyle \frac{1}{u}\, +\, \displaystyle \frac{1}{v} \right )$$
  • $$p \left ( \displaystyle \frac{1}{u}\, +\, \displaystyle \frac{1}{v} \right )$$
  • $$\displaystyle \frac{p}{2}\, \left ( \displaystyle \frac{1}{u}\, -\, \displaystyle \frac{1}{v} \right )$$
  • $$p \left ( \displaystyle \frac{1}{u}\, -\, \displaystyle \frac{1}{v} \right )$$
Suppose that $$f$$ is a polynomial of degree $$3$$ and that $$f''(x)\neq 0$$ at any of the stationary point. Then
  • $$f$$ has exactly one stationary point
  • $$f$$ must have no stationary point
  • $$f$$ must have exactly $$2$$ stationary point
  • $$f$$ has exactly $$0$$ or $$2$$ stationary point.
Let $$f:R\rightarrow R$$ be a differentiable function for all values of $$x$$ and has the peoperty that $$f(x)$$ and $$f'(x)$$ have opposite signs for all values of $$x$$. Then,
  • $$f(x)$$ is an increasing function
  • $$f(x)$$ is a decreasing function
  • $$f ^2 (x)$$ is a decreasing function
  • $$|f(x)|$$ is an increasing function
The function $$f(x) = x^2$$ is decreasing in
  • $$(- \infty, \infty)$$
  • $$(- \infty, 0)$$
  • $$(0, \infty)$$
  • $$(-2, \infty)$$
Identify the correct statements
(a) Every constant function is an increasing function.
(b) Every constant function is a decreasing function.
(c) Every identify function is an increasing function.
(d) Every identify function is a decreasing function.
  • $$(a), (b)$$ and $$(c)$$
  • $$(a)$$ and $$(c)$$
  • $$(c)$$ and $$(d)$$
  • $$(a), (c)$$ and $$(d)$$
Identify the false statement:
  • All the stationary numbers are critical numbers
  • At the stationary point the first derivative is zero
  • At critical numbers the first derivative need not exist
  • All the critical numbers are stationary numbers
The percentage error in the $$11^{th}$$ root of the number $$28$$ is approximately ____________ times the percentage error in $$28$$
  • $$\dfrac { 1 }{ 28 } $$
  • $$\dfrac { 1 }{ 11 } $$
  • $$11$$
  • $$28$$
The slope of the normal to the curve $$y = 3x^2$$ at the point whose $$x$$-coordinate $$2$$ is
  • $$\dfrac{1}{13}$$
  • $$\dfrac{1}{14}$$
  • $$\dfrac{-1}{12}$$
  • $$\dfrac{1}{12}$$
The slope of the tangent to the curve $$y=3{ x }^{ 2 }+3\sin { x } $$ at $$x=0$$ is
  • $$3$$
  • $$2$$
  • $$1$$
  • $$-1$$
Consider the following in respect of the function $$f(x) = \left\{\begin{matrix}2+ x, & x \geq 0\\ 2 - x, & x < 0\end{matrix}\right.$$
$$\displaystyle \lim_{x \rightarrow 1} f(x)$$ does not exist.
f(x) is differentiable at x = 0.
f(x) is continuous at x = 0.
Which of the above statements is/are correct?
  • $$1$$ only
  • $$3$$ only
  • $$2$$ and $$3$$ only
  • $$1$$ and $$3$$ only
Consider the following statements:
$$y = \dfrac {e^{x} + e^{-x}}{2}$$ is an increasing function on $$[0, \infty)$$.
$$y = \dfrac {e^{x} - e^{-x}}{2}$$ is an increasing function on $$(-\infty, \infty)$$.
Which of the above statements is/are correct?
  • $$1$$ only
  • $$2$$ only
  • Both $$1$$ and $$2$$
  • Neither $$1$$ nor $$2$$
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.
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
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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  • $$\dfrac{13}{8}$$
  • $$\dfrac{7}{4}$$
  • $$\dfrac{15}{8}$$
  • $$2$$
The value of $$K$$ in order that $$f(x) = \sin x - \cos x - Kx + 5$$ decreases for all positive real values of $$x$$ is given by
  • $$K < 1$$
  • $$K\geq 1$$
  • $$K > \sqrt {2}$$
  • $$K < \sqrt {2}$$
If the tangent to $$y^{2} = 4ax$$ at the point $$(at^{2}, 2at)$$ where $$|t| > 1$$ is a normal to $$x^{2} - y^{2} = a^{2}$$ at the point $$(a \sec \theta, a\tan \theta)$$, then
  • $$t = -cosec \theta$$
  • $$t = -\sec \theta$$
  • $$t = 2\tan \theta$$
  • $$t = 2\cot \theta$$
The function $$f(x)=\cfrac { \sin { x }  }{ x } $$ is decreasing in the interval
  • $$\left( -\cfrac { \pi }{ 2 } ,0 \right) $$
  • $$\left(0, \cfrac { \pi }{ 2 } \right) $$
  • $$\left( -\cfrac { \pi }{ 4 } ,0 \right) $$
  • None of these
The point on the curve $$y = \sqrt {x - 1}$$ where the tangent is perpendicular to the line $$2x + y - 5 = 0$$ is
  • $$(2, -1)$$
  • $$(10, 3)$$
  • $$(2, 1)$$
  • $$(5, -2)$$
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 approximate value of $$f(x)={ x }^{ 3 }+5{ x }^{ 2 }-7x+9=0$$ at $$x=1.1$$ is
  • $$8.6$$
  • $$8.5$$
  • $$8.4$$
  • $$8.3$$
The equation to the normal to the hyperbola $$\dfrac {x^{2}}{16} - \dfrac {y^{2}}{9} = 1$$ at $$(-4, 0)$$ is.
  • $$2x - 3y = 1$$
  • $$x = 0$$
  • $$x = 1$$
  • $$y = 0$$
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
  • $$(1,-4)$$
  • $$(2,-9)$$
  • $$(2,-4)$$
  • $$(2,-1)$$
  • $$(2,-5)$$
The equation of the tangent to the curve $$y={ x }^{ 3 }-6x+5$$ at $$(2,1)$$ is
  • $$6x-y-11=0$$
  • $$6x-y-13=0$$
  • $$6x+y+11=0$$
  • $$6x-y+11=0$$
The slope of the normal to the curve $$x=1-a\sin { \theta  } $$, $$y=b\cos ^{ 2 }{ \theta  }$$ at $$ \theta =\dfrac { \pi  }{ 2 } $$ is
  • $$\dfrac { a }{ 2b } $$
  • $$\dfrac { 2a }{ b } $$
  • $$\dfrac { a }{ b } $$
  • $$\dfrac { -a }{ 2b } $$
If an edge of a cube measure $$2$$ m with a possible error of $$0.5$$ cm. Find the corresponding error in the calculated volume of the cube.
  • $$0.6\ m^{3}$$
  • $$0.06\ m^{3}$$
  • $$0.006\ m^{3}$$
  • $$0.0006\ m^{3}$$
The tangents to curve $$y={ x }^{ 3 }-2{ x }^{ 2 }+x-2$$ which are parallel to straight line $$y=x$$, are
  • $$x+y=2$$ and $$x-y=\dfrac { 86 }{ 27 } $$
  • $$x-y=2$$ and $$x-y=\dfrac { 86 }{ 27 } $$
  • $$x-y=2$$ and $$x+y=\dfrac { 86 }{ 27 } $$
  • $$x+y=2$$ and $$x+y=\dfrac { 86 }{ 27 } $$
The slope of tangent to the curve $$ x=t^2 + 3t - 8, y = 2t^2 - 2t - 5 $$ at the point $$(2, -1)$$ is :
  • $$ \dfrac {22}{7} $$
  • $$ \dfrac {6}{7} $$
  • $$-6$$
  • None of these
The points at which the tangent to the curve $$y = x^3 - 3x^2 - 9x + 7$$ is parallel to the x-axis are 
  • $$(3, - 20)$$ and $$(- 1, 12)$$
  • $$(3, 20)$$ and $$(1, 12)$$
  • $$(1, -10)$$ and $$(2, 6)$$
  • None of these
The slope of the tangent to the curve $$y=3{ x }^{ 2 }-5x+6$$ at $$\left( 1,4 \right) $$ is
  • $$-2$$
  • $$1$$
  • $$0$$
  • $$-1$$
  • $$2$$
If $$y=8{ x }^{ 3 }-60{ x }^{ 2 }+144x+27$$ is a strictly decreasing function in the interval
  • $$(-5,6)$$
  • $$\left( -\infty ,2 \right) $$
  • $$(5,6)$$
  • $$\left( 3,\infty \right) $$
  • $$(2,3)$$
If the tangent at $$(1,1)$$ on $${ y }^{ 2 }=x{ (2-x) }^{ 2 }$$ meets the curve again at $$P$$, then $$P$$ is
  • $$(4,4)$$
  • $$(-1,2)$$
  • $$\left( \cfrac { 9 }{ 4 } ,\cfrac { 3 }{ 8 } \right) $$
  • $$(1,2)$$
The tangent to the curve $$y=a{ x }^{ 2 }+bx$$ at $$\left( 2,-8 \right) $$ is parallel to $$X$$-axis. Then,
  • $$a=2, b=-2$$
  • $$a=2, b=-4$$
  • $$a=2, b=-8$$
  • $$a=4, b=-4$$
Find the critical points of the function $$f (x)= (x - 2)^{2/3} (2x + 1)$$ 
  • $$-1$$ and $$2$$
  • $$1$$
  • $$1$$ and $$- 2$$
  • $$1$$ and $$2$$
If the straight line $$ y -2x +1=0$$ is the tangent to the curve $$xy+ax+by=0$$ at $$x=1, $$ then the values of $$a$$ and $$b$$ are respectively :
  • 1 and 2
  • 1 and -1
  • -1 and 2
  • -1 and -2
  • 1 and -2
If the angle between the curves $$ y = 2^x $$ and $$ y=3^x $$ is $$ \alpha, $$ then the value of $$ \tan \alpha $$ is equal to :
  • $$ \dfrac { \log \left( \dfrac {3}{2} \right) } { 1 + ( \log 2)( \log 3 ) } $$
  • $$ \dfrac {6}{7} $$
  • $$ \dfrac {1}{7} $$
  • $$ \dfrac { \log \left( 6 \right) } { 1 + ( \log 2)( \log 3 ) } $$
  • $$ 0^o $$
If the tangent at each point of the curve $$y=\cfrac { 2 }{ 3 } { x }^{ 3 }-2a{ x }^{ 2 }+2x+5$$ makes an acute angle with positive direction of X-axis then
  • $$a\ge 1$$
  • $$-1\le a\le 1$$
  • $$a\le -1$$
  • None of these
The equation of the tangent to the curve $$\sqrt {\dfrac {x}{a}} + \sqrt {\dfrac {y}{b}} = 1$$ at the point $$(x_{1}, y_{1})$$ is $$\dfrac {x}{\sqrt {ax_{1}}} + \dfrac {y}{\sqrt {by_{1}}} = k$$. Then, the value of $$k$$ is
  • $$2$$
  • $$1$$
  • $$3$$
  • $$7$$
  • $$\sqrt {2}$$
The slope of the normal to the curve $$y = x^2 - \dfrac{1}{x^2}$$ at $$(-1, 0) $$ is 
  • $$\dfrac{1}{4}$$
  • $$ - \dfrac{1}{4}$$
  • $$4$$
  • $$-4$$
  • $$0$$
The point on the curve $$y = 5 + x - x^{2}$$ at which the normal makes equal intercepts is
  • $$(1, 5)$$
  • $$(0, -1)$$
  • $$(-1, 3)$$
  • $$(0, 3)$$
  • $$(0, 5)$$
The function $$f(x) = 2x^3 - 15 x^2 + 36 x + 6$$ is strictly decreasing in the interval
  • $$(2, 3)$$
  • $$( - \infty, 2)$$
  • $$(3, 4)$$
  • $$(- \infty, 3) \cup (4, \infty)$$
  • $$(- \infty, 2) \cup (3, \infty)$$
A normal to parabola, whose inclination is $$30^o$$, cuts it again at an angle of.
  • $$\tan^{-1}\left(\displaystyle\frac{\sqrt{3}}{2}\right)$$
  • $$\tan^{-1}\left(\displaystyle\frac{2}{\sqrt{3}}\right)$$
  • $$\displaystyle\tan^{-1}\cdot(2\sqrt{3})$$
  • $$\displaystyle\tan^{-1}\left(\displaystyle\frac{1}{2\sqrt{3}}\right)$$
If the slope of the tangent to the curve $$y=a{ x }^{ 3 }+bx+4$$ at $$(2,14) = 21$$, then the values of $$a$$ and $$b$$ are respectively
  • $$2,-3$$
  • $$3,-2$$
  • $$-3,-2$$
  • $$2,3$$
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