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 $$y=4x-{ x }^{ 2 }$$ with $$\overset { - }{ OX } $$ is 
  • $${ tan }^{ -1 }(2)$$
  • $${ tan }^{ -1 }(1/3)$$
  • $${ tan }^{ -1 }(3)$$
  • $$\pi /4$$
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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