#### 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.