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Applications of Integration Model Question Paper

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 40
    5 x 1 = 5
  1. The value of \(\int _{ -4 }^{ 4 }{ \left[ { tan }^{ -1 }\left( \frac { { x }^{ 2 } }{ { x }^{ 4 }+1 } \right) +{ tan }^{ -1 }\left( \frac { { x }^{ 4 }+1 }{ { x }^{ 2 } } \right) \right] dx } \) is

    (a)

    \(\pi\)

    (b)

    \(2\pi\)

    (c)

    \(3\pi\)

    (d)

    \(4\pi\)

  2. The value of \(\int _{ 0 }^{ \infty }{ { e }^{ -3x }{ x }^{ 2 }dx } \\ \) is

    (a)

    \(\frac{7}{27}\)

    (b)

    \(\frac{5}{27}\)

    (c)

    \(\frac{4}{27}\)

    (d)

    \(\frac{2}{27}\)

  3. The value of \(\int _{ 0 }^{ \frac { 2 }{ 3 } }{ \frac { dx }{ \sqrt { 4-9{ x }^{ 2 } } } } \) is 

    (a)

    \(\frac{\pi}{6}\)

    (b)

    \(\frac{\pi}{2}\)

    (c)

    \(\frac{\pi}{4}\)

    (d)

    \({\pi}\)

  4. If \(\int _{ 0 }^{ 2a }{ f(x) } dx=2\int _{ 0 }^{ a }{ f(x) } \) then

    (a)

    f(2a -x) = - f(x)

    (b)

    f(2a - x) = f(x)

    (c)

    f(x) is odd

    (d)

    f(x) is even

  5. The ratio of the volumes generated by revolving the ellipse \(\frac { { x }^{ 2 } }{ 9 } +\frac { { y }^{ 2 } }{ 4 } \) = 1 about major and minor axes is

    (a)

    4:9

    (b)

    9:4

    (c)

    2:3

    (d)

    3:2

  6. 5 x 2 = 10
  7. Evaluate the following definite integrals:
    \(\int _{ -1 }^{ 1 }{ \frac { dx }{ { x }^{ 2 }+2x+5 } } \)

  8. Evaluate the following integrals using properties of integration:
    \(\int _{ 0 }^{ 2\pi }{ { sin }^{ 4 }{ x\quad cos }^{ 3 }dx } \)

  9. Evaluate the following:
    \(\int _{ 0 }^{ 1 }{ { x }^{ 3 }{ e }^{ -2x }dx } \)

  10. Evaluate \(\int _{ 0 }^{ 1 }{ \frac { { e }^{ x } }{ 1+{ e }^{ 2x } } dx } \)

  11. Find the area of the region enclosed by the curve y = \(\sqrt x\) + 1, the axis of x and the lines x=0, x=4.

  12. 5 x 3 = 15
  13. Evaluate \(\int _{ 1 }^{ 2 }{ \frac { x }{ (x+1)(x+2) } dx } \)

  14. Evaluate:  \(\int ^{\frac{\pi}{2}}_{\frac{\pi}{2}}\)x cox x dx.

  15. Find, by integration, the volume of the solid generated by revolving about y-axis the region bounded between the parabola x = y2 +1, the y-axis, and the lines y =1 and y = −1.

  16. Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { sin \ x }{ 9+{ cos }^{ 2 } } dx } \)

  17. Evaluate \(\int _{ 0 }^{ 1 }{ \sqrt { 9-4{ x }^{ 2 } } dx } \)

  18. 2 x 5 = 10
  19. Evaluate \(\int ^\frac {\pi}{2}_{0} \)( sin2 x + cos4 x ) dx

  20. Find the area of the region bounded by x−axis, the curve y = cos x, the lines x = 0 and x =\(\pi\).

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