General Instructions :

(i) All questions are compulsory.
(ii) Question number 1 to 15 are of 2 marks each.
(iii) Question number 16 to 25 are of 4 marks each.
(iv) Question number 26 to 30 are of 6 marks each.

Q1) Find from first principle the derivative of (x² + 1)/x w.r.t. x

Q2) Evaluate : 

1 - cos 5x 1 - cos 6x

Q3) Using differentials, find the approximate value of 29

Q4) Solve the differential equation:
x (dy/dx) = x + y

Q5)  Solve the differential equation dy/dx = ysin 2x given that y(0) = 1

Q6) Using determinants, find that the area of the triangle with vertices (-3, 5), (3, -6), (7, 2).

Q7) By using elementary row transformations, find the inverse of the matrix

A =

5  2 2  1

Q8) If = - 2 + 3 and = 2 + 3 - 5 then find x . Verify that and x are perpendicular to each other. 

Q9) Using vector show that the line segment joining the mid-points of two side of a triangle is parallel to the third side.

Q10) Evaluate

1     ex     dx 0 1 + e2x

Q11) Evaluate

1         dx  (2 + 2x - x2)

Q12) In a group there are 3 men and 2 women 3 person are selected at random from this group. Find the probability that 1 man and 2 women or 2 men and 1 woman are selected.

Q13) One card is drawn from a well shuffled pack of 52 cards. If E is the event the card drawn is a king or queen and F is the event the card drawn is a queen or an ace, then find the probability of the conditional event E/F.

Q14) A die is thrown 7 times, if getting an "even number" is success, find the probability of getting at least 6 successes.

Q15) The two lines of regression for a bivariate distribution (x, y) are 3x + 2y = 7 and x + 4y = 9. Find the regression coefficient b_yx and b_xy

Q16) Find the radius of the circular section of the sphere x² + y² + z² = 49 cut by the plane 2x + 3y - z - 514 = 0

Q17) Find dy/dx when y = x^{sin x - cos x} + (x² - 1)/(x² + 1)

Q18) For the function f(x) = 2x³ - 24x + 5, find
(a) the interval (s) where it is increasing;
(b) the interval (s) where it is decreasing.

Q19) A box containing 16 bulbs out of which 4 bulbs are defective, 3 bulbs are drawn one by one from the box without replacement. Find the probability distribution of the number of defective bulbs drawn.

Q20) Prove that

1  x  x3 1  y  y3 1  z  z3

Q21) Prove{( + ) x ( + )} . ( + ) = 2[ ]