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12th Standard Maths English Medium Applications of Integration Reduced Syllabus Important Questions With Answer Key 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 \(\int _{ -\frac { \pi }{ 4 } }^{ \frac { \pi }{ 4 } }{ \left( \frac { { 2x }^{ 7 }-{ 3x }^{ 5 }+{ 7x }^{ 3 }-x+1 }{ { cos }^{ 2 }x } \right) dx } \) is 

    (a)

    4

    (b)

    3

    (c)

    2

    (d)

    0

  2. The value of \(\int _{ 0 }^{ \frac { \pi }{ 6 } }{ { cos }^{ 3 }3xdx } \)

    (a)

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

    (b)

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

    (c)

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

    (d)

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

  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 volume of solid of revolution of the region bounded by y2 = x(a − x) about x-axis is

    (a)

    \(\pi a^2\)

    (b)

    \(\frac { \pi { a }^{ 2 } }{ 4 } \)

    (c)

    \(\frac { \pi { a }^{ 2 } }{ 5 } \)

    (d)

    \(\frac { \pi { a }^{ 2 } }{ 6 } \)

  5. 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)

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

  6. 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)

    \({\pi}\)

  7. The value of \(\int _{ -1 }^{ 2 }{ |x|dx } \)

    (a)

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

    (b)

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

    (c)

    \(\frac{5}{2}\)

    (d)

    \(\frac{7}{2}\)

  8. 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

  9. 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

  10. The value of \(\int _{ -\pi }^{ \pi }{ { sin }^{ 3 }x \ { cos }^{ 3 }x \ } dx\) is

    (a)

    0

    (b)

    \(\pi \)

    (c)

    2\(\pi \)

    (d)

    4\(\pi \)

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

  12. 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

  13. 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)

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

  14. The volume generated by the curve y2 = 16x from x = 2 to x = 3 rotating about x - axis ......... cu. units

    (a)

    72π

    (b)

    \(\frac { 256\times 19 }{ 3 } \)

    (c)

    40ㅠ

    (d)

    80ㅠ

     

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

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

  19. Evaluate the following
    \(\int _{ 0 }^{ \pi /2 }{ { cos}^{ 7}x\quad dx } \)

  20. 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.

  21. Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { e }^{ 3x } } cosxdx\)

  22. Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { e }^{ -x } } \)

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

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

  25. Find the volume of the solid y=x3,x=0,y=1 is revolved about the y-axis.

    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: \(\int _{ 0 }^{ \frac { 1 }{ \sqrt { 2 } } }{ \frac { { sin }^{ -1 }x }{ { (1-{ x }^{ 2 }) }^{ \frac { 3 }{ 2 } } } dx } \)

  29. Show that \(\int ^\frac{2\pi}{0}_{0}\) g(cos x)dx = 2 \(\int ^{\pi}_{0}\) g(cosx)dx where g(cos x) is a function of cos x

  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 definite integrals:
    \(\int _{ 0 }^{ 1 }{ \frac { 1-{ x }^{ 2 } }{ { (1+{ x }^{ 2 }) }^{ 2 } } } dx\)

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

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

  34. Evaluate \(\int _{ 0 }^{ \pi }{ \sqrt { 1+4{ sin }^{ 2 }\cfrac { x }{ 2 } -4sin\cfrac { x }{ 2 } dx } } \) 

  35. Find the area bounded by the curve y=x|x|,x-axis and the ordinates x=-1,x=1

    1. 5 Marks


    7 x 5 = 35
  36. Find the area of the region bounded between the parabolas y2 x = 4 and x2 y = 4.

  37. Find, by integration, the volume of the container which is in the shape of a right circular conical frustum.

  38. Find the value of ‘c’ for which the area bounded by the curve y=8x2-x5,the lines x=1,x=c and x-axis \(\cfrac { 16 }{ 3 } \)

  39. AOB is the positive quadrant of the ellipse \(\cfrac { { x }^{ 2 } }{ { a }^{ 2 } } +\cfrac { { y }^{ 2 } }{ { b }^{ 2 } } =1\) where OA=a and OB=b.Find the area between the arc AB and chord AB of the elipse.

  40. Show that the ratio of the area under the curve y=sinx and y=sin2x between x=0 and \(x=\cfrac { \pi }{ 3 } \) and x- axis are as 2 : 3.

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

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

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