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CBSE Questions for Class 12 Commerce Applied Mathematics Applications Of Integrals Quiz 12 - MCQExams.com

If the line x=α divides the area of region R={(x,y)R2:x3yx,0x1} into two equal parts, then
  • 2α44α2+1=0
  • α4+4α21=0
  • 0<α12
  • 12<α<1
The area bounded by the parabolas y2=4a(x+a) and y2=4a(xa)  is
  • 163a2 sq units
  • 83 sq units
  • 43a2 sq units
  • None of these
The area of the region bounded by the curves y=2x,y=2xx2 and x=2 is
  • 3log243
  • 3log249
  • 32log29
  • 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
The area bounded byu the curve y = cos x, the line joining  (- \pi / 4 , \cos (- \pi / 4 )) and (0, 2) and the line joining  ( \pi / 4 , \cos ( \pi / 4 )) and (0, 2) is :
  • \frac{4 + \sqrt{2}}{8} \pi - \sqrt{2}
  • \frac{4 + \sqrt{2}}{8} \pi + \sqrt{2}
  • \frac{4 + \sqrt{2}}{4} \pi - \sqrt{2}
  • \frac{4 + \sqrt{2}}{4} \pi + \sqrt{2}
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
Let function f_n be the number of way in which a positive integer n can be written as an ordered sum of several positive integers. For example, for n=3, {f_3} = 3,since, 3 = 3, 3 = 2 + 1 and 3 = 1+1+1. Then{f_5} =
  • 4
  • 5
  • 6
  • 7
Area bounded by the curves y= [\frac{x^{2}}{64}+2],y=x-1 and x=0 above x-axis is where [.] denotes greatest x.
  • 2 sq. unit
  • 3 sq. unit
  • 4 sq. unit
  • none of these
The area bounded by circles x^2+y^2=r^2, r=1, 2 and rays given by 2x^2-3xy-2y^2=0(y > 0) is?
  • \pi
  • \dfrac{3\pi}{4}
  • \dfrac{\pi}{2}
  • \dfrac{\pi}{4}
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
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.
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
Consider two curves {C_1}\,:\,y = \frac{1}{x}\,and\,{C_2}:\,y = \,\ell nx  on the xy plane. Let {D_1} denotes the region surrounded by {C_1},{C_2} and the lines x=1  and {D_2} denotes the region the region surrounded by {C_1},{D_2} and the line x=a. If a {D_1}={D_2} then the value of 'a' -
  • \frac{e}{2}
  • e
  • 1
  • 2\left( {e - 1} \right)
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 }
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
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
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
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}
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
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
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
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}
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 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
The area bounded by the curves y=xe^{x},y=xe^{-x} and the line x=1, is
  • \dfrac {2}{e}
  • 1-\dfrac {2}{e}
  • \dfrac {1}{e}
  • 1-\dfrac {1}{e}
The area (in sq.units) of the region \left\{ ( x , y ) :{ y} ^ { 2 } \ge 2 x\right. and {x} ^ { 2 } +{ y} ^ { 2 } \le 4 x , x \ge 0 , y \ge 0 is 
  • \pi - \dfrac { 8 } { 3 }
  • \pi - \dfrac { 4 \sqrt { 2 } } { 3 }
  • \dfrac { \pi } { 2 } - \dfrac { 2 \sqrt { 2 } } { 3 }
  • \pi - \dfrac { 4 } { 3 }
The area bounded by the curves \sqrt{x}+\sqrt{y}=1 and {x}+{y}=1 is ?
  • \dfrac{1}{3}
  • \dfrac{1}{6}
  • \dfrac{1}{2}
  • \dfrac{5}{6}
  • \dfrac{1}{4}
In a system of three curves C_{1}, C_{2} and C_{3}, C_{1} is a circle whose equation is x^{2}+y^{2}=4. C_{2} is the locus of orthogonal tangents drawn on C_{1}. C_{3} is the intersection of perpendicular tangents drawn on C_{2}. Area enclosed between the curve C_{2} and C_{3} is-
  • 8\pi\ sq.\ units
  • 16\pi\ sq.\ units
  • 32\pi\ sq.\ units
  • None\ of\ these
Area bounded between the curves y=\sqrt{4-x^2} and y^2=3|x| is/are?
  • \dfrac{\pi -1}{\sqrt{3}}
  • \dfrac{2\pi -1}{3\sqrt{3}}
  • \dfrac{2\pi -\sqrt{3}}{3}
  • \dfrac{2\pi -\sqrt{3}}{3\sqrt{3}}
Area bounded by the curves y=\cos^{-1}(\sin x) and y=\sin^{-1}(\sin x) in the interval [0, \pi] is 
  • \dfrac{\pi^{2}}{16}
  • \dfrac{\pi^{2}}{32}
  • \dfrac{\pi^{2}}{4}
  • \dfrac{\pi^{2}}{8}
Area bounded between asymptomes of curves f(x) and f^{-1}(x) is 
  • 4
  • 9
  • 16
  • 25
The area of the region bounded by the X-axis and the curves defined by y=tanx\left( \dfrac { -\pi  }{ 3 } \le x\le \dfrac { \pi  }{ 3 }  \right) and\quad y=cotx\left( \dfrac { \pi  }{ 6 } \le x\le \dfrac { 3\pi  }{ 2 }  \right)
  • log\dfrac { 3 }{ 2 }
  • log\sqrt { \dfrac { 3 }{ 2 } }
  • 2log\dfrac { 3 }{ 2 }
  • log\left( \dfrac { 3 }{ \sqrt { 2 } } \right)
The area bounded by the curves is \sqrt{\left|x\right|}+\sqrt{\left|y\right|}=\sqrt{a} and x^{2}+y^{2}=a^{2} (where a>0) is 
  • \left(\pi-\dfrac{2}{3}\right)a^{2}\ sq\ units
  • \left(\pi+\dfrac{2}{3}\right)a^{2}\ sq\ units
  • \left(\pi+\dfrac{2}{3}\right)a^{3}\ sq\ units
  • \left(\pi-\dfrac{2}{3}\right)a^{3}\ sq\ units
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Practice Class 12 Commerce Applied Mathematics Quiz Questions and Answers