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CBSE Questions for Class 12 Commerce Maths Application Of Derivatives Quiz 2 - MCQExams.com

The angle made by the tangent line at (1, 3) on the curve y=4xx2 with OX is 
  • tan1(2)
  • tan1(1/3)
  • tan1(3)
  • π/4
equation of tangent at (0,0) for the equation y^2=16x
  • y=0
  • x=0
  • x+y=0
  • x-y=0
The intercept on x-axis made by tangent to the curve, \displaystyle y=\int _{ 0 }^{ x }{ \left| t \right|  } dt,x\in R, which are parallel to the line y=2x, are equal to
  • \pm 1
  • \pm 2
  • \pm 3
  • \pm 4
The area of triangle formed by tangent and normal at point (\sqrt{3}, 1) of the curve x^2+y^2=4 and x-axis is?
  • \dfrac{4}{\sqrt{3}}
  • \dfrac{2}{\sqrt{3}}
  • \dfrac{8}{\sqrt{3}}
  • \dfrac{5}{\sqrt{3}}
The function f(x)=x^3-6x^2+9x+3 is decreasing for
  • 1 < x < 3
  • x > 1
  • x < 1
  • x < 1 or x > 3
A curve y=me^{mx} where m > 0 intersects y-axis at a point P.
What is the equation of tangent to the curve at P ?
  • y=mx+m
  • y=-mx+2m
  • y=m^2x+2m
  • y=m^2x+m
If f(x)=kx^3 -9x^2 +9x+3 is increasing for every real number x, then
  • k > 3
  • k \ge 3
  • k < 3
  • k \le 3
The tangent to the curve y=e^{kx} at a point (0, 1) meets the x-axis at (q, 0) where a \epsilon  \left [ -2,-1 \right ] then k \epsilon  
  • [-1/2,0]
  • [-1,-1/2]
  • [0,1]
  • [1/2, 1]
The function f(x)=x^3-27x+8 is increasing when
  • |x| < 3
  • |x| > 3
  • -3 < x < 3
  • none\ of\ these
A stationary point of \mathrm{f}(\mathrm{x})=\sqrt{16 -\mathrm{x}^{2}} is 
  • (4, 0)
  • (-4, 0)
  • (0, 4)
  • (-4, 4)
\mathrm{f}(\mathrm{x})=\mathrm{x}+2 cosx is increasing in
  • (0,\displaystyle \frac{\pi}{2})
  • (\displaystyle \frac{-\pi}{2},\frac{\pi}{6})
  • (\displaystyle \frac{\pi}{2},{\pi})
  • (\displaystyle \frac{-\pi}{2}\frac{\pi}{2})

 \displaystyle \mathrm{f}(\mathrm{x})=\frac{\mathrm{x}}{\log \mathrm{x}}-\frac{\log 5}{5} is decreasing in
  • (\mathrm{e}, \infty)
  • (0, 1)\mathrm{U}(1, \mathrm{e})
  • (0, 1)
  • (1, e)
The condition that \mathrm{f}(\mathrm{x}) =\mathrm{x}^{3}+\mathrm{a}\mathrm{x}^{2}+\mathrm{b}\mathrm{x}+\mathrm{c}\mathrm \ {i}\mathrm{s}\mathrm \ {a}\mathrm{n} increasing function for all real values of \mathrm{x} is
  • \mathrm{a}^{2}<12\mathrm{b}
  • \mathrm{a}^{2}<3\mathrm{b}
  • \mathrm{a}^{2}<4\mathrm{b}
  • \mathrm{a}^{2}<16\mathrm{b}
I. lf f'(\mathrm{a})>0 then \mathrm{f} is increasing at \mathrm{x}=\mathrm{a}
II:  If f is increasing at \mathrm{x}=\mathrm{a} then f'(\mathrm{a}) need not to be positive
  • only I
  • only II
  • both I and II
  • neither I nor II
\mathrm{f}(\mathrm{x})= sinx-ax is decreasing in R if
  • \mathrm{a}>1
  • \mathrm{a}<13
  • \displaystyle \mathrm{a}>\frac{1}{2}
  • \displaystyle \mathrm{a}<\frac{1}{2}
\displaystyle x \epsilon \ (0,\frac{\pi}{2}) , \displaystyle \mathrm{f}(\mathrm{x})=\mathrm{x}\sin \mathrm{x}+\cos \mathrm{x}+\frac{1}{2}\cos^{2}\mathrm{x} is
  • Increasing
  • Decreasing
  • Constant
  • Nothing can be determi ned
A stationary value of \mathrm{f}(\mathrm{x})=\mathrm{x}(\ln \mathrm{x})^{2} is
  • 2\mathrm{e}^{-2}
  • 4\mathrm{e}^{-2}
  • 2\mathrm{e}^{2}
  • 4\mathrm{e}^{2}

\displaystyle \mathrm{f}(\mathrm{x})=\frac{\mathrm{a}\sin \mathrm{x}+\mathrm{b}\cos \mathrm{x}}{\mathrm{a}\cos \mathrm{x}-\mathrm{b}\sin \mathrm{x}} \ \ \ \ \ (\tan \mathrm{x}\neq\frac{\mathrm{a}}{\mathrm{b}}) is
  • increasing in domain \mathrm{f}
  • Decreasing in domain \mathrm{f}
  • Constant
  • Nothing can be determined

\displaystyle \mathrm{f}(\mathrm{x})=\frac{\mathrm{a}\sin \mathrm{x}+\mathrm{b}\cos \mathrm{x}}{\mathrm{c}\sin \mathrm{x}+\mathrm{d}\cos \mathrm{x}} is an increasing funtion if
  • ad-bc =0
  • ad -bc < 0
  • ad - bc > 0
  • ab + cd = 0
Assertion A: The curves x^{2}=y,\ x^{2}=-y  touch each other at (0, 0).
Reason R: The slopes of the tangents at (0, 0) for both the curves are equal.
  • Both A and R are true R is the correct explanation of A
  • Both A and R are true but R is not correct explanation of A
  • A is true but R is false
  • A is false but R is true
If the slope of the tangent to the curve xy+ ax+ by=0 at the point (1, 1) on it is 2, then values of a and b are
  • 1, 2
  • 1, -2
  • -1, 2
  • -1, -2
If the slope of the tangent to the curve y = x^{3} at a point on it is equal to the ordinate of the point then the point is
  • (27, 3)
  • (3, 27)
  • (3, 3)
  • (1, 1)
The critical point of \mathrm{f}(\mathrm{x})=|2\mathrm{x}+7| at \mathrm{x}=
  • 0
  • 7
  • \dfrac{-7}{2}
  • -7
P(1, 1) is a point on the parabola  y=x^{2} whose vertex is A. The point on the curve at which the tangent drawn is parallel to the chord  \overline{AP}   is
  • (\displaystyle \frac{1}{2}, \frac{1}{4})
  • (\displaystyle \frac{-1}{2}, \frac{1}{4})
  • (2, 4)
  • (4, 2)
The stationary point of \mathrm{f}(\mathrm{x})=\mathrm{e}^{\mathrm{x}}+\mathrm{e}^{-\mathrm{x}} is
  • (1, 2)
  • (2, 3)
  • (1, 3)
  • (0, 2)
The number of ciritical points of \displaystyle \mathrm{f}(\mathrm{x})=\frac{|x-1|}{x^{2}} is
  • 1
  • 2
  • 3
  • 0
For the parabola y^{2}=8x, tangent and normal are drawn at P(2, 4) which meet the axis of the parabola in A and B, then the length of the diameter of the circle through A, P, B is
  • 2
  • 4
  • 8
  • 6
\mathrm{f}(\mathrm{x})=(\sin^{-1}\mathrm{x})^{2}+(\cos^{-1}\mathrm{x})^{2} is stationary at
  • \displaystyle \mathrm{x}=\frac{1}{\sqrt{2}}
  • \displaystyle \mathrm{x}=\frac{\pi}{4}
  • \mathrm{x} = 1
  • \mathrm{x} = 0
The point on the hyperbola y = \dfrac {x - 1}{x + 1} at which the tangents are parallel to y = 2x + 1 are
  • (0, -1) only
  • (-2, 3) only
  • (0, -1), (-2, 3)
  • (-2, 3), (5, 4)
The arrangment of the slopes of the normals to the curve  y=e^{\log(cosx)} in the ascending order at the points given below.
A) \displaystyle x=\frac{\pi}{6},  B) \displaystyle x=\frac{7\pi}{4},  C)x=\frac{11\pi}{6},  D)x=\frac{\pi}{3}
  • C, B, D, A
  • B, C, A, D
  • A, D, C, B
  • D, A, C, B
Assertion(A): The tangent to the curve y=x^{3}-x^{2}-x+2 at (1, 1) is parallel to the x axis.
Reason(R): The slope of the tangent to the above curve at (1, 1) is zero.
  • Both A and R are true R is the correct explanation of A
  • Both A and R are true but R is not correct explanation of A
  • A is true but R is false
  • A is false but R is true
The number of tangents to the curve x^{3/2}+y^{3/2}=a^{3/2}, where the tangents are equally inclined to the axes, is
  • 2
  • 1
  • 0
  • 4
Match the points on the curve  2y^{2}=x+1 with the slope of normals at those points and choose 
the correct answer.
Point
Slope of normal
I : (7, 2)

a){-4\sqrt{2}}

II: (0, \displaystyle \frac{1}{\sqrt{2}})

b) -8
III : (1, 1)
c) -4
IV:  (3, \sqrt{2})


d){-2\sqrt{2}}



  • i-b,ii- d,iii- c,iv- a
  • i-b,ii- a,iii- d,iv- c
  • i-b,ii- c,iii- d,iv- a
  • i-b,ii- d,iii- a,iv- c
lf the tangent to the curve f(x)=x^{2} at any point (c, f(c)) is parallel to the line joining points (a, f(a)) and (b,f(b)) on the curvel then a,\ c,\ b are in
  • AP
  • GP
  • HP
  • AGP
lf the parametric equation of a curve given by x=e^{t}\cos t,\ y=e^{t}\sin t, then the tangent to the curve at the point t=\dfrac{\pi}{4} makes with axis of x the angle.
  • 0
  • \dfrac{\pi}{4}
  • \dfrac{\pi}{3}
  • \dfrac{\pi}{2}
lf the curve y=px^{2}+qx+r passes through the point (1, 2) and the line y=x touches it at the origin, then the values of p,\ q and r are
  • p=1,q=-1,r=0
  • p=1,q=1,r=0
  • p=-1,q=1,r=0
  • p=1,q=2,r=3
lf the chord joining the points where x= p,\ x =q on the curve y=ax^{2}+bx+c is parallel to the tangent drawn to the curve at (\alpha, \beta) then \alpha=
  • 2pq
  • \sqrt{pq}
  • \displaystyle \frac{p+q}{2}
  • \displaystyle \frac{p-q}{2}
The arrangement of the following curves in the ascending order of slopes of their tangents at the given points.
A) \displaystyle y=\frac{1}{1+x^{2}} at x=0

B) y=2e^{\frac{-x}{4}}, where it cuts the y-axis
C) y= cos(x) at \displaystyle x=\frac{-\pi}{4}
D) y=4x^{2} at x=-1
  • DCBA
  • ACBD
  • ABCD
  • DBAC
Observe the following lists for the curve y=6+x-x^{2} with the slopes of tangents at the given points; I, II, III, IV
Point
Tangent slope
I: (1, 6)
a) 3
II: (2, 4)
b) 5
III: (-1, 4)
c) -1
IV: (-2, 0)
d) -3
  • a ,b, c ,d
  • b, c, d ,a
  • c, d ,b ,a
  • c, d ,a ,b
If the circle x^{2}+y^{2}+2gx+2fy+c=0 is touched by y=x at P such that 
OP=6\sqrt{2}, then the value of c is
  • 36
  • 144
  • 72
  • None of these
The points on the hyperbola x^{2}-y^{2}=2 closest to the point (0, 1) are
  • (\displaystyle \pm\frac{3}{2},\frac{1}{2})
  • (\displaystyle \frac{1}{2},\pm\frac{3}{2})
  • (\displaystyle \frac{1}{2},\frac{1}{2})
  • (\displaystyle \pm\frac{3}{4}\pm\frac{3}{2})
The number of tangents to the curve x^{3/2} + y^{3/2}= 2a^{3/2}, a>0, which are equally inclined to the axes, is
  • 2
  • 1
  • 0
  • 4
If m is the slope of a tangent to the curve e^y= 1+x^2, then 
  • \left | m \right | > 1
  • m>1
  • m>-1
  • \left | m \right | \leq 1
If the circle x^2 + y^2 + 2gx + 2fy + c =0 is touched by y = x at P in the first quadrant, such that OP = 6 \sqrt2, then the value of c is
  • 36
  • 144
  • 72
  • None of these
\Delta (x)=\begin{vmatrix} \sin  x & \cos  x  &\sin  2x+\cos  2x \\ 0 &1  &1 \\ 1 &0  &-1 \end{vmatrix}
{\Delta }'(x) vanishes at least once in
  • \left(0, \dfrac{\pi}2\right)
  • \left(\dfrac{\pi}{2}, \pi \right)
  • \left(0, \dfrac{\pi}4\right)
  • \left(-\dfrac{\pi}{2}, 0\right)
lf \alpha = \cos 10^{\circ} - \sin 10^{\circ}, \beta = \cos 45^{\circ} - \sin 45^{\circ}, \gamma = \cos 70^{\circ} - \sin 70^{\circ} then the descending order of \alpha, \beta, \gamma is
  • \alpha, \beta, \gamma
  • \gamma, \beta, \alpha
  • \alpha, \gamma, \beta
  • \beta, \alpha, \gamma
For the curve y=3\sin \theta\cos\theta,  x=e^{\theta}\sin \theta,  0\leq \theta\leq\pi; the tangent is parallel to x -axis when \theta is
  • 0
  • \displaystyle \frac{\pi}{2}
  • \displaystyle \frac{\pi}{4}
  • \displaystyle \frac{\pi}{6}
The curve given by the equation y-e^{xy}+x=0 has a vertical tangent at the point
  • (0, 1)
  • (1, 1)
  • (- 1, 1)
  • (1, 0)

The sum of the intercepts made on the axes of coordinates by any tangent to the curve \sqrt{x}+\sqrt{y}=2 is equal to

  • 4
  • 2
  • 8
  • None of these
If there is an error of 2% in measuring the length of a simple pendulum, then percentage error in its period is
  • 1\%
  • 2\%
  • 3\%
  • 4\%
0:0:1


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Practice Class 12 Commerce Maths Quiz Questions and Answers