**Question 1)**

The x-coordinate of a point on the line joining the points P(2, 2, 1) and Q(5, 1, – 2) is 4. Find its z-coordinate

**Question 2)**

Write the principle value of tan^{-1}(1) +cos^{-1}(-1/2)

**Question 3)**

A fair coin is tossed 8 times, find the probability of

(i) exactly 5 heads

(ii) at least six heads

(iii) at most six heads

**Question 4)**

Three cards are drawn successively with replacement from a well shuffled pack of

52 cards. Find the mean and variance of the number of red cards

**Question 5)**

There are three coins. One coin is two headed coin.Second coin is biased one which comes up tail 25% times and third coin is unbiased one.One of the three coin is chosen at random and tossed and head comes. what is the probability that it was the two-headed coin?

**Question 6)**

Determine the values of ‘a’ and ‘b’ such that the following function is continuous

at x = 0

**Question 7)**

Determine the value of ‘k’ for which the following function is continuous

at x = 3 :

**Question 8)**

Find the value of k for which the function

is continuous at x=?/4

**Question 9)**

if x^{y} + y^{x}= a^{b} find dy/dx

**Question 10)**

Find the intervals in which the function f given by

f(x) = sin x + cos x, 0 ? x ? 2?

is strictly increasing or strictly decreasing.

**Question 11)**

**Question 12)**

Find the intervals in which the function given by

f(x) = 2x^{3} – 3x^{2} – 36x + 7 is

(a) Strictly increasing

(b) Strictly decreasing

**Question 13)**

**Question 14)**

Find the value of the below

**Question 15)**

If **a** ,**b** and **c** are mutually perpendicular vectors of equal magnitudes, find the angles which the vector 2**a** +**b**+ 2**c** makes with the vectors **a** ,**b** and **c** .

**Question 16)**

Solve the following linear programming problem graphically:

Minimize: z = 3x + 9y

When: x + 3y ? 60

x + y ? 10

x? y

x ? 0, y ? 0

**Question 17)**

There are 4 cards numbered 1, 3, 5 and 7, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two drawn cards. Find the mean and variance of X

**Question 18)**

Using Integration, find the area of the following region

{(x, y) : y^{2} ? ax, x^{2} + y^{2} ? 2ax,a > 0}

**Question 19)**

Find the equation of the plane passing through the point (–1, 3, 2) and perpendicular to

each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0

**Question 20)**

Let f : R – (-4/3) -> R be function defined as f(x)=4x/3x + 4. Show that f is a one-one

function. Also check whether f is an onto function or not. Hence find f^{-1} in

(Range of f) -> R – (-4/3)