CBSE Questions for Class 12 Commerce Maths Application Of Integrals Quiz 11 - MCQExams.com

The smaller area enclosed by $$y=f(x)$$, where $$f(x)$$ is polynomial of least degree satisfying $$\displaystyle{ \left[ \lim _{ x\rightarrow 0 }{ 1+\frac { f\left( x \right)  }{ { x }^{ 3 } }  }  \right]  }^{ \tfrac { 1 }{ x }  }=e$$ and the circle $$x^2+y^2=2$$ above the $$x-$$axis is
  • $$\displaystyle\frac { \pi }{ 2 } +\frac { 3 }{ 5 } $$
  • $$\displaystyle\frac { \pi }{ 2 } -\frac { 3 }{ 5 } $$
  • $$\dfrac { \pi }{ 2 } -\dfrac { 6 }{ 5 } $$
  • None of these
The area of the region described by $$\left \{(x, y)/ x^{2} + y^{2} \leq 1\ and\ y^{2} \leq 1 - x\right \}$$ is
  • $$\dfrac {\pi}{2} - \dfrac {2}{3}$$
  • $$\dfrac {\pi}{2} + \dfrac {2}{3}$$
  • $$\dfrac {\pi}{2} + \dfrac {4}{3}$$
  • $$\dfrac {\pi}{2} - \dfrac {4}{3}$$
The area (in square units) of the region bounded by the curves $$x=y^2$$ and $$x=3-2y^2$$ is
  • $$\dfrac{3}{2}$$
  • 2
  • 3
  • 4
The area (in square units) bounded by the curves $$x\, =\, -2y^2$$ and $$x\, =\, 1-3y^2$$ is
  • $$\displaystyle \frac{2}{3}$$
  • $$1$$
  • $$\displaystyle \frac{4}{3}$$
  • $$\displaystyle \frac{5}{3}$$
The area of the region bounded by the curves $$x^{2} + y^{2} = 8$$ and $$y^{2} = 2x$$ is
  • $$2\pi + \dfrac {1}{3}$$
  • $$\pi + \dfrac {1}{3}$$
  • $$2\pi + \dfrac {4}{3}$$
  • $$\pi + \dfrac {4}{3}$$
Area common to the curves $$5x^2  = 0$$ and $$ 2x^2  + 9 = 0$$ is equal to
  • $$12 \sqrt 3 $$
  • $$ 6 \sqrt3$$
  • $$36$$
  • $$18$$
The area of the region, bounded by the curves $$y = \sin^{-1} x + x (1 - x)$$ and $$y = \sin^{-1} x - x (1 - x)$$ in the first quadrant, is
  • $$1$$
  • $$\dfrac {1}{2}$$
  • $$\dfrac {1}{3}$$
  • $$\dfrac {1}{4}$$
If the area bounded by the curves $$y=a{ x }^{ 2 }$$ and $$x=a{ y }^{ 2 }$$, $$(a>0)$$ is $$1$$ sq.units, then the value of $$a$$ is
  • $$\cfrac { 2 }{ 3 } $$
  • $$\cfrac { 1 }{ \sqrt 3 } $$
  • $$1 $$
  • $$4$$
Area bounded by curve $$y=x^2$$ and $$y=2-x^2$$ is ?
  • $$\dfrac{8}{3}$$ sq units
  • $$\dfrac{3}{8}$$ sq units
  • $$\dfrac{3}{2}$$ sq units
  • None of these
If the line $$x = \alpha $$ divides the area of region $$R=\left\{ \left( x,y \right) \in { R }^{ 2 }:{ x }^{ 3 }\le y\le x,0\le x\le 1 \right\} $$ into two equal parts, then
  • $$2{ \alpha }^{ 4 }-4{ \alpha }^{ 2 }+1=0$$
  • $${ \alpha }^{ 4 }+4{ \alpha }^{ 2 }-1=0$$
  • $$0 < \alpha \le \dfrac { 1 }{ 2 } $$
  • $$\dfrac { 1 }{ 2 } < \alpha < 1$$
The area bounded by the parabolas $$y^2 = 4a(x + a)$$ and $$y^2 = - 4a (x - a)$$  is
  • $$\dfrac{16}{3} a^2$$ sq units
  • $$\dfrac{8}{3} $$ sq units
  • $$\dfrac{4}{3} a^2$$ sq units
  • None of these
The area of the region bounded by the curves $$y = 2^{x}, y = 2x - x^{2}$$ and $$x = 2$$ is
  • $$\dfrac {3}{\log 2} - \dfrac {4}{3}$$
  • $$\dfrac {3}{\log 2} - \dfrac {4}{9}$$
  • $$\dfrac {3}{2} - \dfrac {\log 2}{9}$$
  • None of these
The area of the portion of the circle $${ x }^{ 2 }+{ y }^{ 2 }=64$$ which is exterior to the parabola $${ y }^{ 2 }=12x$$, is
  • $$\left( 8\pi -\sqrt { 3 } \right) $$ sq units
  • $$\dfrac { 16 }{ 3 } \left( 8-\sqrt { 3 } \right) $$ sq units
  • $$\dfrac { 16 }{ 3 } \left( 8\pi -\sqrt { 3 } \right) $$ sq units
  • None of the above
Area common to the curves $$y^{2} = ax$$ and $$x^{2} + y^{2} = 4ax$$ is equal to
  • $$(9\sqrt {3} + 4\pi) \dfrac {a^{2}}{3}$$
  • $$(9\sqrt {3} + 4\pi)a^{2}$$
  • $$(9\sqrt {3} - 4\pi) \dfrac {a^{2}}{3}$$
  • None of these
The area bounded by $$x^2+y^2-2x=0$$ & $$y=\sin\displaystyle\frac{\pi x}{2}$$ in the upper half of the circle is?
  • $$\displaystyle\frac{\pi}{2}-\frac{4}{\pi}$$
  • $$\displaystyle\frac{\pi}{4}-\frac{2}{\pi}$$
  • $$\displaystyle \pi -\frac{8}{\pi}$$
  • None
On the real line R, we define two functions f and g as follows:
$$f(x) = min [x - [x], 1 - x + [x]]$$,
$$g(x) = max [x - [x], 1 - x + [x]]$$,
where [x] denotes the largest integer not exceeding x. 
The positive integer n for which $$\displaystyle \int_{0}^{n}{(g(x) - f(x) ) dx = 100}$$ is?
  • $$100$$
  • $$193$$
  • $$200$$
  • $$202$$
The parabola $$y^2=4x+1$$ divides the disc $$x^2+y^2\leq 1$$ into two regions with areas $$A_1$$ and $$A_2$$. Then $$|A_1-A_2|$$ equals.
  • $$\displaystyle\frac{1}{3}$$
  • $$\displaystyle\frac{2}{3}$$
  • $$\displaystyle\frac{\pi}{4}$$
  • $$\displaystyle\frac{\pi}{3}$$
The area bounded by the curves $$y = \sin x, y = \cos x$$ and x-axis from $$x = 0$$ to $$x = \pi /2$$ is
  • $$2 + \sqrt {2}$$
  • $$\sqrt {2}$$
  • $$2$$
  • $$2 - \sqrt {2}$$
The area bounded by min (|x|, |y|) = 2 and max (|x|, |y|) = 4 is
  • 8 sq unit
  • 16 sq unit
  • 24 sq unit
  • 32 sq unit
The area of the region $$\left\lfloor x \right\rfloor +\left\lfloor y \right\rfloor =1,-1\le x\le 1$$ and $$xy\le 1/2$$
  • rational
  • $$\cfrac { 1 }{ 2 } \left( \cfrac { 3 }{ 2 } +\ln { 2 } \right) $$
  • $$\cfrac { 1 }{ 2 } \left( \cfrac { 5 }{ 2 } +\ln { 2 } \right) $$
  • Irrational
Area bounded by the curves $$\displaystyle y = \left[ \frac{x^2}{64} + 2 \right]$$ ([.] denotes the greatest integer function) $$y = x - 1$$ and $$x = 0$$ above the x-axis is
  • 2
  • 3
  • 4
  • none of these
The area bounded by the curves $$y = \dfrac {1}{4} |4 - x^{2}|$$ and $$y = 7 -|x|$$ is
  • $$18$$
  • $$32$$
  • $$36$$
  • $$64$$
Consider two curves $$C_1 : (y - \sqrt 3)^2 = 4 ( x - \sqrt2) $$ and $$ C_2 : x^2 + y^2 = ( 6 + 2 \sqrt2 ) x + 2 \sqrt{3y} - 6 ( 1 + \sqrt2)$$ then
  • $$C_1 and C-2$$ touch each other only at one point.
  • $$C_1 and C-2$$ touch each other exactly at two points.
  • $$C_1 and C-2$$ intersect (but do not touch) at exactly two points.
  • $$C_1 and C-2$$ neither intersect nor touch each other.
What is the area of the region bounded by the parabola $${ y }^{ 2 }=6(x-1)$$ and $${ y }^{ 2 }=3x$$
  • $$\cfrac { \sqrt { 6 } }{ 3 } $$
  • $$\cfrac { 2\sqrt { 6 } }{ 3 } $$
  • $$\cfrac { 4\sqrt { 6 } }{ 3 } $$
  • $$\cfrac { 5\sqrt { 6 } }{ 3 } $$
The area bounded by the curves $$x= a \cos^3t, y= a \sin^3 t$$ is 
  • $$\dfrac{3\pi a^2}{8}$$
  • $$\dfrac{3\pi a^2}{16}$$
  • $$\dfrac{3\pi a^2}{32}$$
  • None of the above
The area of the region lying between the line x-y+2=0 and the curve x=$$\sqrt y $$.
  • 9
  • 9/2
  • 10/3
  • none of these
$$Let\quad f(x)=2-\left| x-1 \right| and\quad g(x)={ \left( x-1 \right)  }^{ 2 },\quad then\quad $$
  • area bounded by $$f(x)$$ and $$g(x)$$ is $$\cfrac { 7 }{ 6 } $$
  • area bounded by $$f(x)$$ and $$g(x)$$ is $$\cfrac { 7 }{ 3 } $$
  • area bounded by $$f(x)$$$$g(x)$$ and $$x-$$ axis is $$\cfrac { 5 }{ 3 } $$
  • area bounded by $$f(x)$$$$g(x)$$ and $$x-$$ axis is $$\cfrac { 5 }{ 6 } $$
In the square ABCD, the "shaded" region is the intersection of two circular regions centered at B and D respectively. If AB= 10, then what is the area of the shaded region?
1035047_d234b612381142cbb1a35321b59c1619.png
  • $$25(\pi-2)$$
  • $$50(\pi-2)$$
  • $$25\pi$$
  • $$50\pi$$
  • $$ 40\pi (5-\sqrt{2})$$
The area bounded by the curves $$y={ \left( x-1 \right)  }^{ 2 },y={ \left( x+1 \right)  }^{ 2 }$$ and $$y=\dfrac { 1 }{ 4 }$$ is 
  • $$\dfrac { 1 }{ 3 } sq\ unit$$
  • $$\dfrac { 2 }{ 3 } sq\ unit$$
  • $$\dfrac { 1 }{ 4 } sq\ unit$$
  • $$\dfrac { 1 }{ 5 } sq\ unit$$
Area common to the circle $$x^{2}+y^{2}=64$$ and the parabola $$y^{2}=4x$$ is
  • $$\dfrac{16}{3}(4\pi + \sqrt{3})$$
  • $$\dfrac{16}{3}(8\pi + \sqrt{3})$$
  • $$\dfrac{16}{3}(4\pi - \sqrt{3})$$
  • $$none\ of\ these$$
If $$f\left(x\right)=$$max$$\left\{\sin{x},\cos{x},\dfrac{1}{2}\right\}$$, then the area of the region bounded by the curves $$y=f\left(x\right),x-$$axis $$y-$$axis and $$x=2\pi$$ is
  • $$\left(\dfrac{5\pi}{12}+3\right)$$.sq.unit
  • $$\left(\dfrac{5\pi}{12}+\sqrt{2}\right)$$.sq.unit
  • $$\left(\dfrac{5\pi}{12}+\sqrt{3}\right)$$.sq.unit
  • $$\left(\dfrac{5\pi}{12}+\sqrt{2}+\sqrt{3}\right)$$.sq.unit
The parabola $$y=\dfrac{x^2}{2}$$ divides the circle $$x^2+y^2=8$$ into two parts. Find the area of both parts.
  • $$6\pi+\dfrac{4}{3}$$ , $$2\pi-\dfrac{4}{3}$$
  • $$6\pi-\dfrac{4}{3}$$ , $$2\pi+\dfrac{4}{3}$$
  • $$6\pi+\dfrac{2}{3}$$ , $$2\pi-\dfrac{2}{3}$$
  • $$6\pi-\dfrac{2}{3}$$ , $$2\pi+\dfrac{2}{3}$$
The area enclosed by the curve $$y=\sqrt{(4-x^2)}, y\geq \sqrt{2}\sin\left(\dfrac{x\pi}{2\sqrt{2}}\right)$$ and x-axis is divided by y-axis in the ratio.
  • $$\dfrac{\pi^2-8}{\pi^2+8}$$
  • $$\dfrac{\pi^2-4}{\pi^2+4}$$
  • $$\dfrac{\pi -4}{\pi +4}$$
  • $$\dfrac{2\pi^2}{\pi^2+2\pi -8}$$
If $$k=2$$ then $$f\left(x\right)$$ attains point of inflection at
  • $$0$$
  • $$\sqrt{2}$$
  • $$-\sqrt{2}$$
  • None of these
The area of the region enclosed between by the $${x^2} + {y^2} = 16$$  and the parabola  $${y^2} = 6x$$.
  • $$\dfrac{2}{3} (\sqrt 3 + 4\pi)$$ sq. units
  • $$\dfrac{4}{3} (\sqrt 3 + 4\pi)$$ sq. units
  • $$\dfrac{2}{3} (\sqrt 3 + 8\pi)$$ sq. units
  • $$\dfrac{4}{3} (\sqrt 3 + 8\pi)$$ sq. units
Area bounded by $$|x-1| \le 2$$ and $$x^{2}-y^{2}=1$$, is
  • $$6 \sqrt{2}+\dfrac{1}{2}$$ In $$|3+2\sqrt{2}|$$
  • $$6 \sqrt{2}+\dfrac{1}{2}$$ In $$|3-2\sqrt{2}|$$
  • $$6 \sqrt{2}-$$ In $$|3+2\sqrt{2}|$$
  • $$none\ of\ these$$
Find area curved by three circles 
  • $$(5\pi-3\sqrt{3})units^{2}$$
  • $$(5\pi+4\sqrt{3})units^{2}$$
  • $$(5\pi+3\sqrt{3})units^{2}$$
  • $$(5\pi+3\sqrt{2})units^{2}$$
The whole area of the curves $$x=a\cos^3t, y=b\sin^3t$$ is given by?
  • $$\dfrac{3}{8}\pi ab$$
  • $$\dfrac{5}{8}\pi ab$$
  • $$\dfrac{1}{8}\pi ab$$
  • None of these
The triangle formed by the tangent to the parabola $$y^2=4x$$ at the point whose abscissa lies in the interval $$\left[a^2, 4a^2\right]$$, the ordinate and the x-axis, has the greatest area equal to?
  • $$12a^3$$
  • $$8a^3$$
  • $$16a^3$$
  • $$20a^3$$
Consider the two curves 
$${ C }_{ 1 } :{ y }^{ 2 }=4x $$
$$ { C }_{ 2 } : { x }^{ 2  }+ { y }^{ 2 } - 6x + 1 = 0$$
Then, the area of region between these curves?
  • $$\dfrac{20}{3}-2\pi$$
  • $$\dfrac{10}{3}-2\pi$$
  • $$\dfrac{20}{3}-\pi$$
  • $$\dfrac{10}{3}-\pi$$
Area bounded by the curves $$y=\log _{ e }{ x } \quad$$ and  $$y={ \left( \log _{ e }{ x }  \right)  }^{ 2 }$$ is ?
  • $$e-2$$
  • $$3-e$$
  • $$e$$
  • $$e-1$$
The area enclosed between the curve $$y=x^3$$ and $$y=\sqrt{x}$$ is 
  • $$\dfrac{5}{3}$$
  • $$\dfrac{5}{4}$$
  • $$\dfrac{5}{12}$$
  • None of these
The area between the curves y=tan x, cot x and axis in the interval $$\left[0,\pi   \right/2$$]is ?
  • log $$2$$
  • log $$3$$
  • log $$5$$
  • none of these
The area bounded by the curves $$y = \sin \left( {x - \left[ x \right]} \right),\,y = \sin 1$$ and the x-axis is
  • $$\sin 1$$
  • $$1 - \sin 1$$
  • $$1 + \sin 1$$
  • None of these
The area (in square units) bounded by the curves $$y = {\cos ^{ - 1}}\left| {\cos \,x} \right|$$ and $$y = {\left( {{{\cos }^{ - 1}}\left| {\cos \,x} \right|} \right)^2},x \in \left[ {0,\pi } \right]$$ is
  • $$\frac{4}{3} + \frac{{{\pi ^2}}}{4}\left( {\frac{\pi }{3} - 1} \right)$$
  • $$\frac{4}{3} + \frac{{{\pi ^2}}}{4}\left( {\frac{\pi }{3} + 1} \right)$$
  • $$\frac{2}{3} + \frac{{{\pi ^2}}}{4}\left( {\frac{\pi }{3} - 1} \right)$$
  • $$\frac{2}{3} + \frac{{{\pi ^2}}}{4}\left( {\frac{\pi }{3} + 1} \right)$$
The area bounded by the curves $$y = sin^{-1} |sin \, x| $$ and $$y = (sin^{-1} | sin \, x|)^2 , \, 0 \le x \le 2 \pi$$ is
  • $$\left(\dfrac{\pi^3}{3} + \dfrac{4}{3} \right)$$ sq. unit
  • $$\left(\dfrac{\pi^3}{6} - \dfrac{\pi^2}{2} + \dfrac{4}{3} \right)$$ sq. unit
  • $$\left(\dfrac{\pi^2}{2} - \dfrac{4}{3} \right)$$ sq. unit
  • $$\left(\dfrac{\pi^2}{6} - \dfrac{\pi}{4} + \dfrac{4}{3} \right)$$ sq. unit
The area bounded by the curves $${y^2} = 4x$$ and $${x^2} = 4y$$ is : 
  • $$\frac{{32}}{3}$$
  • $$\frac{{16}}{3}$$
  • $$\frac{8}{3}$$
  • $$0$$
The area bounded by $$y=2-\left| 2-x \right|$$ and $$y=\frac { 3 }{ \left| x \right|  }$$ is :
  • $$\frac { 4+3\ell n3 }{ 2 } $$
  • $$\frac { 4-3\ell n3 }{ 2 } $$
  • $$\frac { 3 }{ 2 } +\ell n3$$
  • $$\frac { 1 }{ 2 } +\ell n3$$
Area of the region defined by $$1\ \le |x|+|y|$$ and $$x^{2}-2x+1 \le 1-y^{2}$$ is $$k \pi$$ then $$k=.....sq$$ units 
  • $$\dfrac {3}{4}$$
  • $$\dfrac {7}{6}$$
  • $$\dfrac {128}{5}$$
  • $$\dfrac {10}{3}$$
The maximum area bounded by the curves $${y^2} = 4ax,\,\,\,\,y = ax\,\,a$$ and $$y = \frac{x}{a}\,\,,1 \le a \le 2$$  is 
  • $$ 44 sq. units $$
  • $$ 74 sq. units $$
  • $$84 sq.units $$
  • $$ 114 sq. units $$
0:0:1


Answered Not Answered Not Visited Correct : 0 Incorrect : 0

Practice Class 12 Commerce Maths Quiz Questions and Answers