#### 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.