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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 : 45
    5 x 1 = 5
  1. The value of \(\int _{ 0 }^{ \frac { \pi }{ 6 } }{ { cos }^{ 3 }3xdx } \)

    (a)

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

    (b)

    \(\frac{2}{9}\)

    (c)

    \(\frac{1}{9}\)

    (d)

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

  2. If f(x)\(f(x)=\int _{ 1 }^{ x }{ \frac { { e }^{ { sin }^{ u } } }{ u } } du,x>1\quad and\quad \int _{ 1 }^{ 3 }{ \frac { { e }^{ { sinx }^{ 2 } } }{ x } } dx=\frac { 1 }{ 2 } [f(a)-f(1)]\), then one of the possible value of a is

    (a)

    3

    (b)

    6

    (c)

    9

    (d)

    5

  3. The value of \(\int _{ -\frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ { sin }^{ 2 }xcosxdx } \) is

    (a)

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

    (b)

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

    (c)

    0

    (d)

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

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

  5. The area enclosed by the curve y2 = 4x, the x-axis and its latus rectum is ............ sq.units.

    (a)

    \(\frac23\)

    (b)

    \(\frac43\)

    (c)

    \(\frac83\)

    (d)

    \(\frac{16}{3}\)

  6. 2 x 2 = 4
  7. The area of the region bounded by the graph of y = sin x and y = cos x between x = 0 and x = \(\frac { \pi }{ 4 } \)
    (1) \(\int _{ 0 }^{ \frac { \pi }{ 4 } }{ (cos \ x-sin \ x) } dx\)
    (2) \({ \left[ sinx+cosx \right] }_{ 0 }^{ \frac { \pi }{ 4 } }\)
    (3) \(\int _{ 0 }^{ \frac { \pi }{ 4 } }{ (sinx-cosx) } dx\)
    (4) \(\sqrt { 2 } -1\)

  8. \(\int _{ a }^{ b }{ f(x) } dx=\)
    (1) \(\int _{ a }^{ b }{ f(y) } dy\)
    (2) \(-\int _{ a }^{ b }{ f(x) } dx\)
    (3) \(\int _{ a }^{ b }{ f(a+b-x) } dx\)
    (4) \(\int _{ 0 }^{ a }{ f(a-x) } dx\)​​

  9. 3 x 2 = 6
  10. Evaluate:  \(\int ^{\frac{\pi}{2}}_{\frac{\pi}{2}}\)x cox x dx.

  11. Evaluate the following definite integrals:
    \(\int _{ 3 }^{ 4 }{ \frac { dx }{ { x }^{ 2 }-4 } } \)

  12. 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.

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

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

  16. Evaluate \(\int _{ 0 }^{ \infty }{ \frac { { x }^{ n } }{ { x }^{ x } } } dx\), where n is positive integer \(\ge\)2

  17. Find, by integration, the volume of the solid generated by revolving about y-axis the region bounded between the curve y=\(\frac{3}{4} \sqrt {x^2 -16}, x\ge4\)the y-axis, and the lines y =1 and y = 6.

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

  19. 3 x 5 = 15
  20. Show that \(\int ^\frac{\pi}{2}_0\) \(\frac {dx}{4+5 sin x}\) = \(\frac {1}{3}\) log2

  21. Show that \(\int ^{1}_{0} (tan ^{-1} x + tan ^{-1}(1-x))\) dx = \(\frac {\pi}{2}\) - loge

  22. Show that the area under the curve y = sin x and y = sin 2x between x = 0 and x = \(\frac { \pi }{ 3 } \) and x axis are as 2:3

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