Q1) A coin is tossed 12 times. What is the Probability of getting exactly 8 tails? (Marks 2)

Q2) Three coins are tossed simultaneously. List the sample space for the event. (Marks 2)
              

Q3) Two cards are drawn without replacement from a well shuffled pack of 52 cards. What is the probability that one is red queen and the other is a king of black colour? (Marks 2)

Q4) Find a unit vector perpendicular to both = 3 + - 2 and = 2 + 3 - (Marks 2)

Q5) Find if = 4 - + and = - 2 + 2 are perpendicular to each other. (Marks 2)

Q6) Find the regression coefficients b_xy and b_yx given that :-
n = 7, x = 24, y = 12, x² = 374, y² = 97 and xy = 157. (Marks 2)

Q7) Solve: (1 + x)(1 + y²)dx + (1 + y)(1 + x²)dy = 0. (Marks 2)

Q8) Evaluate:- 

(x2 - 4x + 2) dx   (Marks 2)

Q9) Evaluate:- 

(2x + 4) (x2 + 4x + 3) dx   (Marks 2)

Q10) Evaluate:- 

dx     50 + 2x2    (Marks 2)

Q11) If y = tan^-1 x, Show that (1 + x²)d²y/dx² + 2x dy/dx = 0  (Marks 2)

Q12) Discuss the applicability of Rolle's Theorem for the function f(x) = x^2/3 on (-1, 1). (Marks 2)

Q13) Evaluate: lim Sin 3x + 7x    (Marks 2)
                     x->0 4x + Sin 2x

Q14) If

A =

1  -3   2 2   0   2

B =

2  -1  -1 1   0  -1

find the matrix C such that A + B + C is a zero matrix. (Marks 2)

Q15) Construct a 2 x 3 matrix whose elements in the i^th row and the j^th coloumn are given by:-
a_ij = (3i - j)/2 (Marks 2)

Q16) Show that the points with position vectors 6 - 7, 16 - 19 - 4, 3 - 6 and 2 - 5 + 10 are coplanar. (Marks 4)

Q17) Find the foot of the perpendicular from (0, 2, 7) on the line (x + 2)/-1 = (y - 1)/3 = (z - 3)/-2  (Marks 4)

Q18) Three balls are drawn without replacement from a bag containing 5 white and 4 red balls. Find the probability distribution of the number of red balls drawn. (Marks 4)

Q19) Evaluate

x2       dx   (Marks 4) (x2 - 4x + 3)

Q20) Evaluate

x cos-1 x dx  (Marks 4)

Q21) Sketch the region common to the circle x² + y² = 16 and the parabola x² = 6y. Also, find the area of the region using Integration. (Marks 4)