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Applications of Integration Model Question Paper 1

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 50
    5 x 1 = 5
  1. If \(f(x)=\int _{ 0 }^{ x }{ t\ cos\ t\ dt,\ then\ \frac { dx }{ dx } } \)

    (a)

    cos x-x sin x

    (b)

    sin x+x cos x

    (c)

    x cos x

    (d)

    x sin x

  2. The value of \(\int _{ 0 }^{ 1 }{ x{ (1-x) }^{ 99 }dx } \) is

    (a)

    \(\frac{1}{11000}\)

    (b)

    \(\frac{1}{10100}\)

    (c)

    \(\frac{1}{10010}\)

    (d)

    \(\frac{1}{10001}\)

  3. The value of  \(\int _{ 0 }^{ \pi }{ { sin }^{ 4 }xdx } \) is

    (a)

    \(\frac{3\pi}{10}\)

    (b)

    \(\frac{3\pi}{8}\)

    (c)

    \(\frac{3\pi}{4}\)

    (d)

    \(\frac{3\pi}{2}\)

  4. The value of \(\int _{ \frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ \sqrt { \frac { 1-cos2x }{ 2x } } } \) dx is

    (a)

    \(\frac { 1 }{ 2 } \)

    (b)

    2

    (c)

    0

    (d)

    1

  5. The area enclosed by the curve y = \(\frac { { x }^{ 2 } }{ 2 } \) , the x - axis and the lines x = 1, x = 3 is

    (a)

    4

    (b)

    8\(\frac23\)

    (c)

    13

    (d)

    4\(\frac{1}{3}\)

  6. 5 x 2 = 10
  7. Evaluate: \(\int ^{log 2}_{-log 2} e ^{-|x|}\) dx.

  8. Evaluate the following integrals using properties of integration:
    \(\int _{ 0 }^{ 1 }{ |5x-3|dx } \)

  9. Evaluate: \(\int _{ 0 }^{ 2\pi }{ { x }^{ 2 }sin\ nx\ dx } \) where n is a positive integer.

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

  11. Find the volume of the solid obtained by revolving the area of the triangle whose sides are x = 4, y = 0 and 3x - 4y = 0 about x - axis

  12. 5 x 3 = 15
  13. Evaluate: \(\int _{ 0 }^{ 3 }{ (3{ x }^{ 2 }-4x+5) } dx\)

  14. Evaluate :\(\int _{ 0 }^{ 1 }{ \frac { 2x+7 }{ { 5x }^{ 2 }+9 } } dx\)

  15. Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { \sqrt { cot \ x } }{ \sqrt { cot \ x } +\sqrt { tan \ x } } dx } \)

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

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

  18. 4 x 5 = 20
  19. Prove that \(\int^{\frac{\pi}{4}}_{0} \frac {sin 2x dx}{ sin ^4x +cos ^4 x}\) = \(\frac{\pi}{4}\) 

  20. Evaluate : \(\int ^\frac{\pi}{4}_{0} \frac{1}{sin x+cos x}\) dx

  21. Find the area bounded by x = at2, y = at between the ordinates corresponding to t = 1 and t = 2

  22. Using integration, find the area of the triangle with sides y = 2x+1, y = 3x + 1 and x = 4.

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