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12th Maths Sectionwise paper - 03 (Integrals and DE)

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MATHEMATICS Class II Pre-Uni Plus 2016-17 Topics Board Section-wise Test - 03 Integral Calculus, Limit of a sum, Area and Differential equations Max. Marks 100 Duration Date 3 hrs 15 min 31-12-2016 Instructions 1. Do not overwrite. 2. Draw a line using pen at the end of each answer. 3. Steps adopted in solving problems should be clearly shown. 4. In case of violation of the above conditions, the answer scripts will not be considered for valuation. PART - A I. Answer the following questions [10 1 = 10] 1. Evaluate e 2loge x dx 2. Evaluate sec 2 (7 4x)dx 3. Evaluate e x (tan x sec2 x)dx 4. Evaluate (2x 3cos x e x ) dx 1 5. Find the antiderivative of x 2 1 2 w.r.t. x. x a 6. If 3x 2 dx 8 find the value of a. 0 3 7. Evaluate 1 1 x 2 dx 1 e 8. Evaluate 1 x dx 1 4 d 2s ds 9. Find the order of 3s 2 0 dt dt 2 d2 y dy 10. Find the degree of 2 cos 0 dx dx PART - B II. Answer any TEN of the following questions 11. Evaluate sin 2 x 1 cos x dx . 12. Evaluate x 2P1617M(B)SWT3 1 x dx . 1 [10 2 = 20] 13. Evaluate sin 3x cos 4xdx 14. Evaluate x x 2 dx 15. Evaluate e5log x e 4log x e3log x e2 log x dx 2 16. Evaluate cos 2xdx 0 3 17. Evaluate x 2 1 18. Evaluate 0 x dx 1 2 (sin 1 x) 2 1 x2 dx 19. Find the area bounded by the curve y2 = 4x and the line x = 3. 20. Form the differential equation of the family of curves x y 1 by eliminating the constants a and b. a b 21. Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis. 22. Verify that the function y e 3x is a solution of the differential equation 23. Solve x 5 d 2 y dy 6y 0 . dx 2 dx dy y5 dx 24. Evaluate (x 2 1) log x dx PART - C III. Answer any TEN of the following questions 2 25. Evaluate e x dx as a limit of a sum. 0 26. Evaluate tan 1 xdx 27. Evaluate x(x 1 4 1) dx sin 8 x cos8 x 1 2sin 2 x cos2 x dx x sin x 29. Evaluate dx 1 cos x 28. Evaluate 5 30. Evaluate | x 2 | dx 5 2P1617M(B)SWT3 2 [10 3 = 30] 8 31. Evaluate 2 x 10 x x dx 2 32. Evaluate a cos x bsin x dx cos x sin x 0 33. Find the area bounded by y = sin x, x-axis and x = 0, x = 2 . 34. Find the area enclosed by the circle x2 + y2 = a2 by integration. 35. Prove that x 2 dy x 2 2y 2 xy is a homogeneous differential equation. dx 36. Find the equation of the curve through the point ( 2, 3) given that slope of the tangent at any point (x, y) 2x is 2 . y 37. In a bank principal increases continuously at the rate of 5% per year. In how many years Rs. 1,000 double itself. 38. Find the general solution of dy y cos x dx PART - D IV. Answer any SIX of the following questions 39. Find the integral of 1 w.r.t. x and hence evaluate x a2 40. Find the integral of 41. Find the integral of 2 1 x2 a2 [6 5 = 30] x w.r.t. x and hence evaluate a 2 x 2 w.r.t. x and hence evaluate 4 x dx 16 1 x 2 2x 2 dx . 1 4x x 2 dx 42. Find the area of the region enclosed by the parabola x2 = 4y and the line x = 4y 2 and x-axis. 43. Using integration, find the area bounded by the triangle whose vertices are ( 1, 0), (1, 3) and (3, 2). 44. Find the area of the region enclosed between the two circles x2 + y2 = 4 and (x 2)2 + y2 = 4 using integration. 45. Solve (x log x) dy 2 y (log x) dx x 46. Find the particular solution of the differential equation 47. Solve x 3y 2 dy y (y 0) . dx /2 1 2 48. Evaluate (a) xe x dx 0 2P1617M(B)SWT3 (b) log(tan x)dx 0 3 dy 2y tan x sin x given y 0 dx 3 PART - E IV. Answer any ONE of the following questions 2a 49. (a) Prove that [1 10 = 10] a f (x)dx 2 f (x)dx if f (2a x) f (x) and hence evaluate 0 0 a 2 0 xdx cos x b 2 sin 2 x 2 (b) Solve (x2 y2) dx + 2xy dy = 0 a 2 f (x)dx if 50. (a) Prove that f (x)dx 0 a 0 if a [4] f ( x) f (x) /2 and hence evaluate f ( x) f (x) (b) Find the area of the region bounded by the parabola y = x2 and y = | x |. *** 2P1617M(B)SWT3 [6] 4 sin 7 xdx [6] /2 [4]

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