Max Marks : 100
Time allowed : 3 hrs

Q1) Evaluate

{(1 - Cos x)/x2}

Q2) A particle moves along a straight line so that S = t . Show that the acceleration is negative and is proportional to the cube of velocity.

Q3) Evaluate

Cos4xdx

Q4) Evaluate           

(1 + tan x)/(x + log sec x) dx

Q5)  

2 1

(x - 1)/x2 . ex dx

Q6) A die is rolled. If the outcome is an even number, what is the probability that it is a prime number ?

Q7) Calculate Spearman's rank correlation from the following Data:

Q8) Solve the differential equation: dy/dx = 1 + x + y + xy 

Q9) Find a matrix X such that 2A + B + X = 0,
where           

A =

-1  2  3   4

B =

3 -2 1   5

Q10) Using differentials, find the approximate value of 26.

Q11) Three bags contain 7 white, 8 red and 

Q12) If x^p y^q = (x + y)^{p +q} , prove that
 dy/dx = y/x

Q13) Evaluate as limit of a sum.

2 0

(x2 + 2)dx

Q14) Solve the differential equation (x + 2y²)dy/dx = y , given that when x = 2, y = 1

Q15) Using properties of determinants prove that:

1   a   a3 1   b   b3 1   c   c3

=(a - b)(b - c)(c - a)(a + b + c)

Q16) Find if lagranges mean value theorem is applicable to the function f(x) = x + (1/x) on [1, 3]

Q17) Evaluate

2x + 1       dx (x2 + 4x + 3)

Q18) Find the regression coefficients and hence the coefficient of correlation from the following regression lines:
 3x + 2y - 3 = 0 ; 2x + y - 4 = 0

Q19) A window is in the form of a rectangle surmounted by a semi-circular opening. If the perimeter of the window is 20m, find the dimensions of the window so that the maximum possible light is admitted through the whole opening.

Q20) Using matrices solve the following system of equation for x, y and z:
            x + 2y - 3z = 6
            3x + 2y - 2z = 3
            2x - y + z = 2

Q21) Using the properties of integral, Evaluate:

/2   -/2

f(x)dx, where f(x) = Sin |x| + Cos |x|