Mathematics – Important Question Bank for Madhya Pradesh HSSC 12th (HSC) Board Exam 2018

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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.