CBSE Questions for Class 12 Commerce Maths Application Of Derivatives Quiz 3 - MCQExams.com

If there is an error of $$k%$$ in measuring the edge of a cube, then the percent error in estimating its volume is
  • $$k$$
  • $$3k$$
  • $$\displaystyle \frac{k}{3}$$
  • none of these
At what points of curve $$y= \displaystyle \frac {2}{3} x^3 + \displaystyle \frac{1}{2} x^2 $$, the tangent makes equal angle with the axis
  • $$\left( \displaystyle \frac { 1 }{ 2 } ,\displaystyle \frac { 5 }{ 24 } \right) $$ and $$\left( -1,-\displaystyle \frac { 1 }{ 6 } \right) $$
  • $$\left(\displaystyle \frac { 1 }{ 2 } ,\displaystyle \frac { 4 }{ 9 } \right)$$ and $$\left( -1,0 \right) $$
  • $$\left(\displaystyle \frac { 1 }{ 3 } ,\displaystyle \frac { 1 }{ 7 } \right) $$and $$\left( -3,\displaystyle \frac { 1 }{ 2 } \right) $$
  • $$\left(\displaystyle \frac { 1 }{ 3 } ,\displaystyle \frac { 4 }{ 47 } \right)$$ and $$\left( -1,-\displaystyle \frac { 1 }{ 3 } \right) $$
If at each point of the curve $$y=x^3-ax^2 +x+1$$, the tangent is inclined at an acute angle with the positive direction of the $$x$$-axis, then
  • $$a>0$$
  • $$a\leq \sqrt 3$$
  • $$-\sqrt 3 \leq a \leq \sqrt 3 $$
  • none of these
If $$x+ 4y=14$$ is a normal to the curve $$y^2=\alpha x ^3-\beta $$ at $$(2,3)$$, then the value of $$\alpha+\beta$$ is
  • $$9$$
  • $$-5$$
  • $$7$$
  • $$-7$$
If there is an error of $$a \%$$ in measuring the edge of a cube, then the percentage error in its surface area is
  • $$2a$$
  • $$\dfrac {a}{2}$$
  • $$3a$$
  • None of the above
The angle formed by the positive $$y-axis$$ and the tangent to $$y=x^2+4x-17$$ at $$(5/2,-3/4)$$ is
  • $${\tan^{-1}(9)}$$
  • $$\displaystyle \frac{\pi}{2}-{\tan^{-1}(9)}$$
  • $$\displaystyle \frac{\pi}{2}+{\tan^{-1}(9)}$$
  • none of these
The slope of the tangent to the curve $$y= \sqrt {4-x^2}$$ at the point where the ordinate and the abscissa are equal is
  • -1
  • 1
  • 0
  • none of these
The curve given by $$x+y=e^{xy}$$ has a tangent parallel to the y-axis at the point
  • (0,1)
  • (1,0)
  • (1,1)
  • none of these
The pressure P and volume V of a gas are connected by the relation $$PV^{1/4}=constant$$. The percentage increase in the pressure corresponding to a deminition of $$\dfrac12 \%$$ in the volume is
  • $$\dfrac {1}{2}$$ %
  • $$\dfrac {1}{4}$$ %
  • $$\dfrac {1}{8}$$ %
  • none of these
If the percentage error in the edge of a cube is 1, then error in its volume is
  • $$1 \%$$
  • $$2 \%$$
  • $$3 \%$$
  • none of these
While measuring the side of an equilateral triangle an error of $$k \%$$ is marked, the percentage error in its area is
  • $$k \%$$
  • $$2k \%$$
  • $$\dfrac {k}{2}\%$$ 
  • $$3k \%$$
If $$y=x^n$$, then the ratio of relative errors in $$y$$ and $$x$$ is
  • $$1:1$$
  • $$2:1$$
  • $$1:n$$
  • $$n:1$$
If the percentage error in measuring the surface area of a sphere is $$\alpha$$ %, then the error in its volume is
  • $$\dfrac {3}{2}\alpha\%$$ 
  • $$\dfrac {2}{3}\alpha\%$$ 
  • $$3\alpha\%$$ 
  • none of these
If there is an error of $$0.01 cm$$ in the diameter of a sphere then percentage error in surface area when the radius $$= 5 cm$$, is
  • $$0.005\%$$
  • $$0.05\%$$
  • $$0.1\%$$
  • $$0.2\%$$
The circumference of a circle is measured as $$28 cm$$ with an error of $$0.01 cm$$. The percentage error in the area is
  • $$\dfrac {1}{14}$$
  • $$0.01$$
  • $$\dfrac {1}{7}$$
  • none of these
The height of a cylinder is equal to the radius. If an error of $$\alpha$$ % is made in the height, then percentage error in its volume is
  • $$\alpha$$ %
  • $$2\alpha$$ %
  • $$3\alpha$$ %
  • none of these
If an error of $$k\%$$ is made in measuring the radius of a sphere, then percentage error in its volume is
  • $$k\%$$
  • $$3k\%$$
  • $$2k\%$$
  • $$\dfrac23k\%$$
If the ratio of base radius and height of a cone is 1:2 and percentage error in radius is $$\lambda$$ %, then the error in its volume is
  • $$\lambda$$ %
  • $$2\lambda$$%
  • $$3\lambda$$%
  • none of these
If $$T=2\pi \sqrt {\dfrac {l}{g}}$$, then relative errors in T and l are in the ratio
  • $$1/2$$
  • $$2$$
  • $$1/2\pi$$
  • none of these
In a $$\Delta ABC$$ if sides a and b remain constant such that $$\alpha$$ is the error in C, then relative error in its area is
  • $$\alpha \cot C$$
  • $$\alpha \sin C$$
  • $$\alpha\tan C$$
  • $$\alpha\cos C$$
The circumference of a circle is measured as $$56$$ cm with an error $$0.02$$ cm. The percentage error in its area is
  • $$\dfrac {1}{7}$$
  • $$\dfrac {1}{28}$$
  • $$\dfrac {1}{14}$$
  • $$\dfrac {1}{56}$$
The point(s) at each of which the tangents to the curve $$\displaystyle y = x^3 - 3x^2 - 7x + 6$$ cut off on the positive semi axis $$OX$$ a line segment half that on the negative semi axis $$OY$$, then the co-ordinates of the point(s) is/are give by:
  • $$(-1, 9)$$
  • $$(3, -15)$$
  • $$(1, -3)$$
  • none
If an error of $$1^o$$ is made in measuring the angle of a sector of radius $$30 \ cm$$, then the approximate error in its area is
  • $$450 cm^2$$
  • $$25\pi cm^2$$
  • $$2.5\pi cm^2$$
  • none of these
A line L is perpendicular to the curve $$\displaystyle  y = \dfrac {x^2}{4} - 2$$ at its point P and passes through (10, -1). The coordinates of the point P are
  • (2, -1)
  • (6, 7)
  • (0, -2)
  • (4, 2)
If the tangent at each point of the curve $$\displaystyle y=\frac { 2 }{ 3 } { x }^{ 3 }-2a{ x }^{ 2 }+2x+5$$ makes an acute angle with the positive direction of x-axis, then 
  • $$a\ge 1$$
  • $$-1\le a\le 1$$
  • $$a\le -1$$
  • none of these
Let $$f\left( x \right) =\left\{ \begin{matrix} { x }^{ { 3 }/{ 5 } }\quad \quad \quad  x\le 1 \\ -{ \left( x-2 \right)  }^{ 3 }\quad x>1 \end{matrix} \right. $$
then the number of critical points on the graph of the function is
  • $$1$$
  • $$2$$
  • $$3$$
  • $$4$$
In a $$\Delta ABC$$ the sides b and c are given. If there is an error $$\Delta A$$ in measuring angle A, then the error $$\Delta a$$ in side a is given by
  • $$\dfrac {S}{2a}\Delta A$$
  • $$\dfrac {2S}{a}\Delta A$$
  • bc sin A $$\Delta A$$
  • none of these
If errors of $$1\%$$ each are made in the base radius and height of a cylinder, then the percentage error in its volume is
  • $$1\%$$
  • $$2\%$$
  • $$3\%$$
  • none of these
Let $$\displaystyle g'(x) > 0$$ and $$\displaystyle f'(x) < 0 \forall x \epsilon R$$, then
  • $$\displaystyle f(f(x + 1)) > f(f(x - 1))$$
  • $$\displaystyle f(g(x - 1)) > f(g(x + 1))$$
  • $$\displaystyle g(f(x + 1)) > g(f(x - 1))$$
  • $$\displaystyle g(g(x + 1)) > g(g(x - 1))$$
If $$y = 4x - 5$$ is a tangent to the curve $$\displaystyle y^2 = px^3 + q$$ at $$(2, 3)$$, then
  • $$p = 2, q = -7$$
  • $$p = -2, q = 7$$
  • $$p = -2, q = -7$$
  • $$p = 2, q = 7$$
If the tangent to the curve $$xy + ax + by = 0$$ at (1, 1) makes an angle $$\displaystyle \tan ^{-1}(2)$$ with x-axis, then $$\displaystyle a + 2b$$ is equal to
  • $$\displaystyle \frac {1}{2}$$
  • $$\displaystyle - \frac {1}{2}$$
  • $$3$$
  • $$-3$$
The set of values of $$a$$ for which the function $$\displaystyle f(x) = (4a - 3) (x + \ln5) + 2(a - 7) \cot \frac {x}{2} \sin^2 \frac {x}{2}$$ does not posses critical points in its domain is
  • $$\displaystyle (- \infty, - \frac {4}{3}) \cup (2, \infty)$$
  • $$\displaystyle (- \infty, 2)$$
  • $$\displaystyle [1, \infty)$$
  • $$\displaystyle (1, \infty)$$
Let $$f$$ be a decreasing function in $$(a,b]$$, then which of the following must be true?
  • $$f$$ is continuous at $$b$$
  • $$\displaystyle f'(b)<0$$
  • $$\displaystyle \lim_{x\to b}f(x)\leq f(b)$$
  • $$\displaystyle \lim_{x\to b}f(x)\geq f(b)$$
$$\displaystyle f:(0, \infty) \rightarrow (-\frac {\pi}{2}, \frac {\pi}{2})$$ be defined as, $$\displaystyle f(x) = arc \: \tan( \: x)$$
The above function can be classified as
  • injective but not surjective
  • surjective but not injective
  • neither injective nor surjective
  • both injective as well as surjective
If at each point of the curve $$y=x^{3}-ax^{2}+x+1$$ the tangent is inclined at an acute angle with the positive direction of the x-axis then
  • $$4a>0$$
  • $$a\leq \sqrt{3}$$
  • $$-\sqrt{3}\leq a\leq \sqrt{3}$$
  • none of these
The line $$ax + by = 1$$ is tangent to the curve $$\displaystyle ax^2 + by^2 = 1$$, if $$(a, b)$$ can be equal to
  • $$\displaystyle (\frac {1}{2}, \frac {1}{2})$$
  • $$\displaystyle (\frac {1}{4}, \frac {3}{4})$$
  • $$\displaystyle (\frac {1}{2}, \frac {3}{4})$$
  • $$\displaystyle (\frac {1}{4}, \frac {1}{2})$$
If $$m$$ be the slope of a tangent to the curve $${ e }^{ 2y }=1+4{ x }^{ 2 }$$, then 
  • $$m<1$$
  • $$\left| m \right| \le 1$$
  • $$\left| m \right| >1$$ 
  • None of these
If $$m$$ be the slope of tangent to the curve $$e^{y}=1+x^{2}$$ then 
  • $$|m|>1$$
  • $$m<1$$
  • $$|m|<1$$
  • $$|m|\leq 1$$
The slope of the tangent to the curve $$y=x^{2}-x$$ at the point where the line $$y=2$$ cuts the curve in the first quadrant is
  • $$2$$
  • $$3$$
  • $$-3$$
  • none of these
The slope of the tangent to the locus $$y=\cos^{-1}\left ( \cos x \right )$$ at $$x=\displaystyle \frac{\pi }{4}$$ is
  • $$1$$
  • $$0$$
  • $$2$$
  • $$-1$$
$$P(2, 2)$$ and $$Q\left ( \displaystyle \frac{1}{2}, -1 \right )$$ are two points on the parabola $$y^{2}=2x$$. The coordinates of the point $$R$$ on the parabola, where tangent to the curve is parallel to the chord $$PQ$$ is
  • $$\left ( \displaystyle \frac{5}{4}, \sqrt{\frac{5}{2}} \right )$$
  • $$(2, -1)$$
  • $$\left ( \displaystyle \frac{1}{8}, \frac{1}{2} \right )$$
  • none of these
The function $$\: f\left( x \right) =x^{ 3 }+\lambda x^{ 2 }+5x+\sin  2x$$ will be an invertible function if $$\: \lambda $$ belongs to
  • $$\displaystyle \:\left ( -\infty, -3 \right )$$
  • $$\displaystyle \:\left ( -3, 3 \right )$$
  • $$\displaystyle \:\left ( 3, +\infty \right )$$
  • none of these
The cricital points(s) of f(x)=$$\displaystyle \frac{\left | 2-x \right |}{x^{2}}$$ is (are)
  • $$x=0$$
  • $$x=2$$
  • $$x=4$$
  • none of these
The curve given by $$x+y=e^{xy}$$ has a tangent as the $$y$$-axis at the point
  • $$(0, 1)$$
  • $$(1, 0)$$
  • $$(1, 1)$$
  • none of these
Let $$\displaystyle f(x)=e\:^{x}\sin x$$ be the equation of a curve. If at $$\displaystyle x=a,0\leq a\leq 2\pi$$, the slope of the tangent is the maximum then the value of $$a$$ is 
  • $$\displaystyle \pi /2$$
  • $$\displaystyle 3\pi /2$$
  • $$\displaystyle \pi $$
  • $$\displaystyle \pi /4$$
The slope of the tangent to the curve $$y=\sqrt{4-x^{2}}$$ at the point where the ordinate and the abscissa are equal, is
  • $$-1$$
  • $$1$$
  • $$0$$
  • none of these
The equation of the curve is given by $$x=e^{t}\sin t$$, $$y=e^{t}\cos t$$. The inclination of the tangent to the curve at the point $$t=\displaystyle \frac{\pi }{4}$$ is
  • $$\displaystyle \frac{\pi }{4}$$
  • $$\displaystyle \frac{\pi }{3}$$
  • $$\displaystyle \frac{\pi }{2}$$
  • $$0$$
The point on the curve $$\sqrt{x}+\sqrt{y}=2a^{2}$$, where the tangent is equally inclined to the axes, is
  • $$\left ( a^{4}, a^{4} \right )$$
  • $$\left ( 0, 4a^{4} \right )$$
  • $$\left ( 4a^{4}, 0 \right )$$
  • none of these
Let $$y=f(x)$$ be the equation of a parabola which is touches by the line $$y=x$$ at the point where $$x=1$$. Then
  • $$f^{'}\left ( 0 \right )=f^{'}\left ( 1 \right )$$
  • $$f^{'}\left ( 1 \right )=1$$
  • $$f\left ( 0 \right )+f^{'}\left ( 0 \right )+f^{''}\left ( 0 \right )=1$$
  • $$2f\left ( 0 \right )=1-f^{'}\left ( 0 \right )$$
The number of tangents to the curve $$y^{2}-2x^{3}-4y+8=0$$ that pass through $$(1, 0)$$ is
  • $$3$$
  • $$1$$
  • $$2$$
  • $$6$$
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