General Instructions :

(i) All questions are compulsory.
(ii) Question number 1 to 15 are of 2 marks each.
(iii) Question number 16 to 30 are of 3 marks each.
(iv) Question number 31 to 35 are of 5 marks each.

Q1) Using elementary row transformations, find the inverse of the matrix

A =

4  5 3  4

Q2) For the matrices A and B, verify that (A, B)' = A'B' where 

A =

1  2 3  4

B =

4  5 1  2

Q3) Obtain the equation of the sphere with the points (1, -1, 1) and (3, -3, 3) as the extremities of a diameter and find the co-ordinates of its centre.

Q4) Find the work done by the force = 2 + + , acting on a particle, if the particle is displaced from the point with position vector 2 + 2 + 2 to the point with position vector 3 + 4 + 5.

Q5)  Verify Rolle's theorem for the following f(x) in the interval |2, 3|:
f(x) = x² - 5x + 6

Q6) Evaluate

cosec x - cot x           x

Q7) Differentiate tan^-1 [2x/(1 - x²)] w.r.t. tan^-1 x

Q8) Evaluate

1 2 sin-1 xdx 0  (1 - x2)3/2

Q9) Evaluate

dx      x2 + 4x + 8

Q10) Evaluate

xcot -1 xdx

Q11) Evaluate

dx          (x2 + 4x + 2)

Q12) Evaluate

ex [tan x + log sec x]dx

Q13) Five dice are thrown simultaneously, if the occurrence of 3, 4, or 5 in a single die is considered as a success, find the probability of at least  3 success.

Q14) Two dice are thrown together. What is the probability that the sum of the numbers on the two faces is neither 9 or 11?

Q15) If A and B are two independent events such that P(AUB) = 0.6 and P(A) = 0.2 find P(B).

Q16) Using properties of determinants prove that

x + 4    x       x x       x + 4    x  x       x     x + 4

= 16(3x + 4)

Q17) 

If A =

3  2 2  1

verify that A2 - 4A - I = O, where

I =

1  0 0  1

and O =

0  0 0  0

Hence, find A-1

Q18) Find the equation of the Plane passing through point(1, 2, 1) and perpendicular to the line joining the points (1, 4, 2) and (2, 3, 5) find also the perpendicular distance of the origin from the plane.

Q19) Prove that
( x ) x ( x ) = [ ] - [ ]

Q20) P, Q, R and S are the mid points of the sides of a quadrilateral ABCD using vectors, show that PQRS is a parallelogram?

Q21) There are 3 urns A, B, and C, urn A contains 4 red balls and 3 black balls. Urn B contains 5 red balls and 4 black balls. Urn C contains 4 red balls and 4 black balls. One ball is drawn from each of these urns, what is the probability that the 3 balls drawn consists of 2 red balls and a black ball?

 

Next