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1.Prove that: ʃ sec x dx = log (sec x + tan x) +c

2. Evaluate: dx

3. Evaluate:

a. dx

b. dx

4. Find the value of :

a. ʃ x e x dx

b. ʃ sec 2 x tan 3 x dx

c. ʃ x sin x dx

d. ʃ x tan 2 x dx

e. ʃ sec 3 x dx

5. Find the value of :

6. If y = then prove that (1-2y)dy/dx = sin x

7. If y = then prove that (1-2y)dy/dx = cos x

8. Prove that cos -1 4/5 +tan -1 3/5 = tan -1 27/11.

9. Find the differential coefficient of

10. Differentiate (cos x) cos x with respect to x.

11. If y = then prove that

dy/ dx = y 2

x (2-ylogx)

12. Calculate correlation coefficient between x and y for the following data:

X 65 66 67 67 68 69 70 72

Y 67 68 65 68 72 72 69 71

13. Prove that correlation coefficient is the geometric mean of the regression coefficient.

14. Prove that the correlation coefficient lies between -1 to +1.

15. The following data are given:

X Y

Arithmetic mean 36 85

Standard deviation 11 8

Correlation coefficient = 0.66

Find the two regression equations for X and Y on the basis of the above data.

16. Calculate regression coefficient b yx and b xy for the variable x and y for the following data:

∑x=24, ∑y=44, xy=306, ∑x 2 =164, ∑y 2 =574, n=4.

17. Find the covariance for the pair of values:

(1,6), (2,9), (3,6), (4,7), (5,8), (6,5), (7,12), (8,3), (9,17), (10,1).

18. Given two lines of regression x+3y =11 and 2x+y =7. Find the coefficient of correlation between x

and y. Also estimate the value of x when y=4.

19. Find the equation of the plane passing through the point (4, 5, 1), (0,-1,- 1) and (-4, 4, 4).

20. Find the value of: ʃ dx

5+ 4sinx

21. Solve the differential equations: (x-1) dy/dx = 2x 3 y.

22. Solve the differential equations: dy/dx = x 2 +5xy+4y 2

x 2

23. Solve the differential equation:

a. (e x + e -x ) dy/dx = e x – e -x

b. dy/dx =

24. A card is drawn at random from well shuffled pack of 52 cards. Find the probability that it is

neither an ace nor a king.

25. A fair coin tossed six times. What is the probability of getting at least three heads.

26. Find the probability distribution of the number 6 in the three throw of a dice.

27. Tickets are marked from 1 to 12 and mixed up. One ticket is taken out at random. Find the

probability of its being a multiple of 2 or 3.

28. In single throw of two dice, what is probability of not obtaining a total of 9 or 11?

29. Differentiate √sin x from first principle.

30. Differentiate √ x from first principle.

31. Find the differential coefficients of e x (using first principle)

32. Express into partial fraction: x and 2x-3 .

1-x 3 (x-1)(x 2 +1) 2

33. Express into partial fraction: 1 and x 3 .

(x+3)(x+1)(x-2) (1-x) 4

34. Prove by vector method:

Cos (a+b) =cos a. cos b – sin a. sin b.

35. Prove by vector method:

Sin (a+b) =sin a. cos b + cos a. sin b.

36. Find the angle between those lines whose direction cosines are recognized by the following

equations:

2l + 2m – n = 0 and mn + nl + lm = 0.

37. Evaluate:

38. Evaluate the limit:

39.

Find domain and range of the function

If f(x) =; x = 0

Find whether function f(x) is continuous at x=0.

40. Find the area bound by the curve x 2 = 4y and the line x= 4y-2.

41. Find the equation of a plane through the points (-1, 3, 2) and perpendicular to the plane

x+2y+2z = 5 and 3x+3y+2z = 8.

42. Find the equation of the sphere passing through the points (1, -3, 4), (1, -5, 2), (1, -3, 0) and

its center lying on the plane x + y + z = 0.

43. If the profit is given by p(x) = 41+ 24x -18x 2 , find maximum profit.

44. A ball is thrown

45. Find the shortest distance between the lines:

X+3 = Y-6 = Z and X+2 = Y = Z-7

-4 3 2 -4 1 1

46. If = +4-3 and =2-2- , then find the modulus of .

47. a. Prove that vectors 2-3+5 and -2+2+2 are mutually perpendicular.

c. If = 2-3+ and = 3+2, then find x .

48. Find the shortest distance between two lines, whose vector equations are:

= (3-t)+(4+2t)+(t- 2)

= (1+s)+(3s-7)+(2s- 2)

Where, s and t are scalar.

49. Find the equation of the plane which passes through the intersecting line of the planes

x+ 2y+3z=5 and

2x-4y+z- 3=0

And also passes through the point (0, 1, 0).

50. By vector method show that the points A(2,3,6), B(-1,- 1,2) and C(5,7,10) are collinear.

51. Show that the angle between any two diagonals of the cube is.

52. A variable plane is at a constant distance p from the origin and meets the axis at A,B and C.

show that the locus of the centroid of tetrahedron OABC is x -2 +y -2 +z -2 =16p -2.

Best of luck for your exams. Do leave a comment below if you have any questions or suggestions.