If a*b \(=\frac { a }{ 2 } +\frac { b }{ 3 } \)then value of 2*3 is.......

3

(a)

Let P be set of all subsets of given set X. Show that \(\cup :P\times P\rightarrow P\) given by \((A,B)\rightarrow A\cup B\) and \(\cap :P\times P\rightarrow P\) given by \((A,B)\rightarrow A\cap B\) are binary operations on the set P.

3

(a)

If the mappings f and g are given by:
f = {(1, 2), (3, 5) (4, 1) and g = {(2, 3), (5, 1), (1, 3)}, write fog.

3

(a)

Find te principal values of the following:
\({ cos }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2 } \right) \)

3

(a)

If \(A=\begin{bmatrix} cos\alpha & -sin\alpha \\ sin\alpha & cos\alpha \end{bmatrix}\), then \(A+A\prime =I\), if the value of \(\alpha \)is:

(A)  \(\frac { \pi }{ 6 } \)   
(B) \(\frac { \pi }{ 3 } \)   
(C) \(\pi \quad \)   
(D) \(\frac { 3\pi }{ 2 } \).

3

(a)

If the area of the triangle is 35sq.units (2, -6), (5, 4) and (k, 4). Find the value of 'k'.

3

(a)

If \(y={ e }^{ ax }cos\quad bx\quad \)then \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -2a\frac { dy }{ dx } +\left( { a }^{ 2 }+{ b }^{ 2 } \right) y=0\)

3

(a)

Show that the differential equation \(2y{ e }^{ \frac { x }{ y } }dx+\left( y-2x\quad { e }^{ \frac { x }{ y } } \right) dy=0\): is homogeneous and find the particular solution, given that x = 0 when y = 1.

3

(a)

Solve the following Linear Programming Problems graphically:
Minimise Z = – 3x + 4 y
subject to x + 2y ≤ 8, 3x + 2y ≤ 12, x ≥ 0, y ≥ 0.

3

(a)

Two godowns A and B have grain capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops, D, E and F whose requirements are 60, 50 and 40 quintals respectively. The cost of  transportation per quintal from the godowns to the shops are given in the following table:

Transportation Cost per Quintal (in RS)
From	A	B
To
D E F	6 3 2.50	4 2 3

How should the supplies be transported in order that the transportation cost is minimum? What is the minimum cost?

3

(a)

A and B throw a die alternatively till one of them gets a ‘6’ and wins the game. Find their respective probabilities of winning, if A starts first.

3

(a)

Probability that A speaks truth is \(\frac { 4 }{ 5 } \) . A coin is tossed. A reports that a head appears.The probability that actually there was head is:
(A) \(\frac { 4 }{ 5 } \) 
(B) \(\frac { 1 }{ 2 } \)  
(C) \(\frac { 1 }{ 5 } \)  
(D) \(\frac { 2 }{ 5 } \)

3

(a)

Suppose that 90% of people are right-handed. What is the probability that at most 6 of a random sample of 10 people are right-handed.

3

(a)

A and B throw a pair of dice alternately, till one of them gets a total 10 and wins tha game.Find their respective probability of winning if A starts first.

3

(a)

Two numbers are selected at random (without replacement)from first six positive integers 2, 3, 4, 5, 6 and 7. Let X denote the larger of the two numbers obtained. Find the probability distribution of X. Find the mean and variance of this distribution

3

(a)