All India - 2016 (Central)

If and then find the value of x.

Use elementary column operation in the following matrix equation:

Write the number of all possible matrices of order with each entry 1, 2 or 3.

Write the position vector of the point which divides the join of points with position vectors and in the ratio
2:1.

Write the number of vectors of unit length perpendicular to both the vectors and

Find the vector equation of the plane with intercepts 3, -4 and 2 on x, y and z-axis respectively.

Find the coordinates of the point where the line through the points A(3, 4, 1) and B(5, 1, 6) crosses the XZ plane. Also find the angle which this line makes with the XZ plane.

The two adjacent sides of a parallelogram are and Find the two-unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram.

In a game, a man wins ₹ 5 for getting a number greater than 4 and loses ₹ 1 otherwise, when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a number greater than 4. Find the expected value of the amount he wins/loses.

OR

A bag contains 4 balls. Two balls are drawn at random (without replacement) and are found to be white. What is the probability that all balls in the bag are white?

Differentiate with respect to x.

OR

If prove that