Applications Of Integrals Test

Applications Of Integrals

This is Applications of Integrals Test-04 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
For which of the following values of m , is the area of the region bounded by the curve y = x - \({x^2}\) and the line y = mx equal to \(\frac{9}{2}\) ?
  • A
    -4
  • B
    none of these
  • C
    -2
    Correct
  • D
    2
2
The area bounded by the curve y =x\(\left[ x \right]\), the x – axis and the ordinates x = 1 and x = -1 is given by
  • A
    \(\frac{2}{3}\)
    Correct
  • B
    \(\frac{1}{2}\)
  • C
    none of these
  • D
    0
3
The area bounded by the curve y = x (x – 1 ) ( x – 2 ) and the x – axis is equal to
  • A
    none of these
  • B
    1
  • C
    $y = f(x) = x(x - 1)(x - 2)$ is +ve for x > 2, is – ve for 1 < x < 2 ; +ve for 0 < x < 1 , is – ve for x< 0. Required area : $ \int\limits_0^1 {ydx + \left| {\int\limits_1^2 {ydx} } \right|} \\ \int\limits_0^1 {({x^3} - 3{x^2} + 2{x^2}} )dx \\ + \left| {\int\limits_1^2 {({x^3} - 3{x^2} + 2{x^2}} )dx} \right| \\ = \left[ {\frac{{{x^4}}}{4} - {x^3} + {x^2}} \right]_0^1 \\ + \left| {\left[ {\frac{{{x^4}}}{4} - {x^3} + {x^2}} \right]_1^2} \right| \\ = \frac{1}{2}sq.units \\ $ $\frac{1}{2}$
    Correct
  • D
    \(\frac{1}{4}\)
4
The area bounded by the curve y = 2x - \({x^2}\) and the line x + y = 0 is
  • A
    \(\frac{{35}}{6}\) sq. units
  • B
    \(\frac{{19}}{6}\) sq. units
  • C
    none of these
  • D
    \(\frac{9}{2}\) sq. units
    Correct
5
The area bounded by the curves \(y = \sqrt x ,2y + 3 = x\) and the x- axis in the first quadrant is
  • A
    36
  • B
    25
  • C
    9
    Correct
  • D
    none of these
6
The area bounded by the curves\({y^2} = 20x\) and \({x^2} = 16y\) is equal to
  • A
    none of these
  • B
    \(80\pi {\text{ }}sq.{\text{ }}units\)
  • C
    \(100\pi {\text{ }}sq.{\text{ }}units\)
  • D
    \(\frac{{320}}{3}{\text{ }}sq.{\text{ }}units\)
    Correct
7
The area bounded by the parabolas y= \({(x + 1)^2}\;and\;y = \left( {x - 1} \right){\;^2}\;and\;the\;line\;y = \frac{1}{4}\) is equal to
  • A
    \(\frac{4}{3}\)
  • B
    4
  • C
    \(\frac{1}{3}\)
    Correct
  • D
    \(\frac{1}{6}\)
8
The area of the region { ( x , y ) : \({x^2} + {y^2} \leqslant 1 \leqslant x + y\;\} \) is equal to
  • A
    \(\frac{{\pi - 2}}{4}\) sq. units
    Correct
  • B
    \(\frac{1}{2}\) sq. units
  • C
    \(\frac{{3\pi - 2}}{4}\) sq. units
  • D
    none of these
9
The area bounded by the parabolas y = \(5{x^2}and\;y - 9 = 2{x^2}\;is\)
  • A
    6\(\;\sqrt {2\;} \;sq.units\)
  • B
    \(4\;\sqrt {3\;} \;sq.units\)
  • C
    1\(2\;\sqrt 2 \;sq.units\)
  • D
    \(12\;\sqrt {3\;} \;sq.units\)
    Correct
10
The area enclosed by the parabola \({y^2} = 2x\) and its tangents through the point ( -2 , 0 ) is
  • A
    3
  • B
    \(\frac{8}{3}\)
    Correct
  • C
    none of these
  • D
    4
11
The area bounded by the angle bisectors of the lines \({x^2} - {y^2} + 2y = 1\;and\;x + y = 3\;is\)
  • A
    6 sq. units
  • B
    2 sq. units
    Correct
  • C
    4 sq. units
  • D
    3 sq.units
12
The area bounded by the curves y = \({x^2}\;and\;y = \frac{2}{{\left( {1 + {x^2}} \right)}}\;is\) equal to
  • A
    \(2\pi + \frac{1}{3}\)
  • B
    none of these
  • C
    \(\pi \) \( - \frac{2}{3}\)
    Correct
  • D
    \(2\pi - \frac{1}{3}\)
13
The area bounded by x = sin t and y = cos t +3 , where -2008 < t < 2008 , is equal to
  • A
    2\(\pi \)
  • B
    3\(\pi \)
  • C
    \(\pi \)
    Correct
  • D
    none of these
14
The area enclosed between the curves \({y^2} = x\;and\;y = \left| x \right|\) is
  • A
    none of these
  • B
    \(\frac{1}{6}sq.{\text{ }}units\)
    Correct
  • C
    \(\frac{2}{3}sq.{\text{ }}units\)
  • D
    1 sq. units
15
The area of the plane region bounded by the curves \(x + \;{y^2} = 0\;and\;x + 3{y^2} = 1\;is\;equal\;to\;\)
  • A
    \(\frac{1}{3}sq.{\text{ }}units\)
  • B
    \(\frac{4}{3}sq.{\text{ }}units\)
    Correct
  • C
    \(\frac{5}{3}sq.{\text{ }}units\)
  • D
    none of these