Given U = Odd numbers < 19, A = factors of 15, B = prime numbers in U, find A intersect B, A union B, and illustrate on a Venn diagram.

$U = \{\text{Odd numbers less than 19}\}$
$U = \{1, 3, 5, 7, 9, 11, 13, 15, 17\}$

$A = \{\text{Factors of 15}\}$
The factors of 15 are the numbers that divide 15 exactly.
$A = \{1, 3, 5, 15\}$

$B = \{\text{Prime numbers}\}$ (within $U$)
Prime numbers in $U$ are numbers greater than 1 that are only divisible by 1 and themselves.
$B = \{3, 5, 7, 11, 13, 17\}$
The elements in set A are $\boxed{\{1, 3, 5, 15\}}$ and in set B are $\boxed{\{3, 5, 7, 11, 13, 17\}}$.

ii) Find the set of $A \cap B$ and $A \cup B$.

$A \cap B$ contains elements common to both $A$ and $B$.
$A = \{1, 3, 5, 15\}$
$B = \{3, 5, 7, 11, 13, 17\}$
$A \cap B = \{3, 5\}$

$A \cup B$ contains all elements in $A$ or $B$ (or both).
$A \cup B = \{1, 3, 5, 7, 11, 13, 15, 17\}$
The set $A \cap B$ is $\boxed{\{3, 5\}}$ and $A \cup B$ is $\boxed{\{1, 3, 5, 7, 11, 13, 15, 17\}}$.

iii) Illustrate U, A and B on a Venn diagram.

$A \cap B = \{3, 5\}$. $A \text{ only} = \{1, 15\}$. $B \text{ only} = \{7, 11, 13, 17\}$. Elements in $U$ but not in $A \cup B = \{9\}$.

      U
+---------------------------------+
|         +-------+   +-------+   |
|         | A     |   | B     |   |
|         | {1,15}|   | {7,11,|   |
|         |    +--+---+--+    |   |
|         |    |3,5|   |    |   |
|         |    +--+---+--+    |   |
|         |       |   |13,17}|   |
|         +-------+   +-------+   |
|                                 |
|             {9}                 |
+---------------------------------+

iv) Shade the region for prime factors of 15 on the Venn diagram.
The prime factors of 15 are 3 and 5. This corresponds to the region $A \cap B$. Shade the overlapping region of circles A and B.

1. b) Find the value of $x$ if $3^{x-1} = 27$.

The value of $x$ is $\boxed{4}$.

2. a) Determine the combination of GH¢20, GH¢50, GH¢100, GH¢200 and GH¢500 notes to make GH¢1,000,000.
Here is one possible combination:

Number of GH¢500 notes = $\frac{\text{GH}\text{¢}500,000}{\text{GH}\text{¢}500} = 1,000$ notes.
Remaining amount = $\text{GH}\text{¢}1,000,000 - \text{GH}\text{¢}500,000 = \text{GH}\text{¢}500,000$.

Number of GH¢200 notes = $\frac{\text{GH}\text{¢}200,000}{\text{GH}\text{¢}200} = 1,000$ notes.
Remaining amount = $\text{GH}\text{¢}500,000 - \text{GH}\text{¢}200,000 = \text{GH}\text{¢}300,000$.

Number of GH¢100 notes = $\frac{\text{GH}\text{¢}200,000}{\text{GH}\text{¢}100} = 2,000$ notes.
Remaining amount = $\text{GH}\text{¢}300,000 - \text{GH}\text{¢}200,000 = \text{GH}\text{¢}100,000$.

Number of GH¢50 notes = $\frac{\text{GH}\text{¢}50,000}{\text{GH}\text{¢}50} = 1,000$ notes.
Remaining amount = $\text{GH}\text{¢}100,000 - \text{GH}\text{¢}50,000 = \text{GH}\text{¢}50,000$.

Number of GH¢20 notes = $\frac{\text{GH}\text{¢}50,000}{\text{GH}\text{¢}20} = 2,500$ notes.
Remaining amount = $\text{GH}\text{¢}50,000 - \text{GH}\text{¢}50,000 = \text{GH}\text{¢}0$.

The combination is:

• 1,000 notes of GH¢500
• 1,000 notes of GH¢200
• 2,000 notes of GH¢100
• 1,000 notes of GH¢50
• 2,500 notes of GH¢20

2. b) A rectangular sheet of metal has length 44cm and breadth 20cm. It is folded to form a cylinder with its breadth becoming the height. Calculate

i) the radius of the cylinder formed.

The length of the rectangular sheet becomes the circumference of the base of the cylinder.
Circumference $(C) = 44\text{ cm}$
The breadth of the rectangular sheet becomes the height of the cylinder.
Height $(h) = 20\text{ cm}$

The radius of the cylinder formed is $\boxed{7\text{ cm}}$.

ii) the volume of the cylinder [Take $\pi = \frac{22}{7}$]

The volume of the cylinder is $\boxed{3080\text{ cm}^3}$.

2. c) Given that $m = \begin{pmatrix} 7 \\ 2x \end{pmatrix}$ and $n = \begin{pmatrix} 5y - 3 \\ 8 \end{pmatrix}$. If $m = n$, find the value of $x$ and $y$.

For the first component:

For the second component:

The value of $x$ is $\boxed{4}$ and the value of $y$ is $\boxed{2}$.

3. a) Given that $x = -3$ and $y = \frac{3}{4}$. Find the value of

i) $x^2(y - 1)$

The value of $x^2(y - 1)$ is $\boxed{-\frac{9}{4}}$.

ii) $y^2 - \frac{3}{x}$

The value of $y^