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12th Standard Maths English Medium Applications of Integration Reduced Syllabus Important Questions 2021

12th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 100

      Multiple Choice Questions


    15 x 1 = 15
  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  \(\int _{ 0 }^{ \pi }{ { sin }^{ 4 }xdx } \) is

    (a)

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

    (b)

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

    (c)

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

    (d)

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

  3. 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)

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

  4. The value of \(\int _{ 0 }^{ a }{ { (\sqrt { { a }^{ 2 }-{ x }^{ 2 } } ) }^{ 2 } } dx\)

    (a)

    \(\frac { { \pi a }^{ 2 } }{ 16 } \)

    (b)

    \(\frac { 3\pi { a }^{ 4 } }{ 16 } \)

    (c)

    \(\frac { 3\pi { a }^{2 } }{ 8} \)

    (d)

    \(\frac { 3\pi { a }^{ 4 } }{ 8} \)

  5. For any value of n∈Z, \(\int _{ 0 }^{ \pi }{ e{ cos }^{ 2x }{ cos }^{ 3 } } [(2n+1)x]\) is

    (a)

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

    (b)

    \(\pi\)

    (c)

    0

    (d)

    2

  6. \(\int _{ 1 }^{ \sqrt { 3 } }{ \frac { dx }{ 1+{ x }^{ 2 } } } \) is

    (a)

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

    (b)

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

    (c)

    \(\frac { \pi }{ 12 } \)

    (d)

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

  7. 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

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

  9. The area bounded by the parabola y = x2 and the line y = 2x is

    (a)

    \(\frac43\)

    (b)

    \(\frac23\)

    (c)

    \(\frac{51}{3}\)

    (d)

    \(\frac{30}{3}\)

  10. If \(\int _{ 0 }^{ a }{ f(x) } dx+\int _{ 0 }^{ a }{ f(2a-x) } dx=\)

    (a)

    \(\int _{ 0 }^{ a }{ f(x) } dx\)

    (b)

    \(2\int _{ 0 }^{ a }{ f(x) } dx\)

    (c)

    \(\int _{ 0 }^{ 2a }{ f(x) } dx\)

    (d)

    \(\int _{ 0 }^{ 2a }{ f(a-x) } dx\)

  11. \(\int _{ -1 }^{ 1 }{ x \ dx } \) = ...............

    (a)

    -1

    (b)

    1

    (c)

    0

    (d)

    2

  12. 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)

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

  13. The area of the ellipse \(\frac { { x }^{ 2 } }{ 9 } +\frac { { y }^{ 2 } }{ 4 } =1\)

    (a)

    (b)

    36π

    (c)

    2

    (d)

    36π2

  14. \(\int _{ -\frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ \frac { sinx }{ 2+cosx } dx= } \)

    (a)

    0

    (b)

    2

    (c)

    log 2

    (d)

    log 4

  15. The volume when  \(y=\sqrt { 3+{ x }^{ 2 } } \) from x = 0 to x = 4 is rotated about x-axis is .................

    (a)

    \(100\pi \)

    (b)

    \(\frac { 100\pi }{ 9 } \)

    (c)

    \(\frac { 100\pi }{ 3 } \)

    (d)

    \(\frac { 100 }{ 3 } \)

    1. 2 Marks


    10 x 2 = 20
  16. Evaluate: \(\int _{ 0 }^{ 3 }{ (3{ x }^{ 2 }-4x+5) } dx\)

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

  18. Evaluate the following integrals using properties of integration:
    \(\int _{ -\frac { \pi }{ 4 } }^{ \frac { \pi }{ 4 } }{ { sin }^{ 2 }xdx } \)

  19. Evaluate the following
    \(\int _{ 0 }^{ \pi /2 }{ { sin }^{ 10 }x\quad dx } \)

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

  21. Find the slope of the tangent to the curve \(y=\int _{ 0 }^{ x }{ \cfrac { dt }{ 1+{ t }^{ 3 } } stx=1 } \) 

  22. Find the area bounded by the curve y=sin2x between the ordinates x=0.x=π and x-axis.

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

  24. Find the area of the region bounded by the curve y=sin x and the ordinate x=0.\(x=\cfrac { \pi }{ 3 } \)

  25. Find the area bounded by the curve y=cosax in one arc of the curve.

    1. 3 Marks


    10 x 3 = 30
  26. Find an approximate value of \(\int _{ 1 }^{ 1.5 }{ xdx } \) by applying the left-end rule with the partition {1.1,1.2,1.3,1.4,1.5}.

  27. Evaluate\(\int _{ 0 }^{ 1 }{ x^3dx } \), as the limit of a sum.

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

  29. Evaluate: \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { cos\theta }{ (1+sin\theta )(2+sin\theta ) } } d\theta \)

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

  31. Evaluate the following integrals using properties of integration:
    \(\int _{ -5 }^{ 5 }{ xcos } \left( \frac { { e }^{ x }-1 }{ { e }^{ x }+1 } \right) dx\)

  32. Evaluate the following integrals using properties of integration:
    \(\int _{ 0 }^{ 2\pi }{ xlog\left( \frac { 3+cos\quad x }{ 3-cos\quad x } \right) } dx\)

  33. Evaluate \(\\ \int _{ 0 }^{ 1 }{ { e }^{ -2x }(1+x-{ 2x }^{ 3 })dx } \)

  34. Evaluate the following:
    \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { dx }{ 1+5{ cos }^{ 2 }x } } \)

  35. Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { dx }{ 4{ sin }^{ 2 }x+5{ cos }^{ 2 }x } } \)

    1. 5 Marks


    7 x 5 = 35
  36. Estimate the value of \(\int _{ 0 }^{ 0.5 }{ { x }^{ 2 } } dx\)using the Riemann sums corresponding to 5 subintervals of equal width and applying (i) left-end rule (ii) right-end rule (iii) the mid-point rule.

  37. Evaluate\(\int ^{\pi}_{0} \frac{x}{1+sin x}\) dx

  38. Find the volume of the solid generated by the revolution of the loop of the curve x = t2 y = t - \(\frac { { t }^{ 3 } }{ 3 } \) about x-axis.

  39. Find the area of the curve y2=(x-5)2(x-6) between
    (i) x=5 and x=6
    (ii) x=6 and x=7

  40. Find the area of the region bounded by the curves x2+2y2=0 and x+3y2=1.

  41. Find the area bounded by the curves y=|x|-1 and y=-|x|+1

  42. Find the area enclosed by the parabolas 5x2-y=0 and 2x2-y+9=0.

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