**Updated on April 12th, 2012: Added some video explanations**

This question tests your knowledge of numbers, sets and geometric constructions.

In **part (a)** we are given a universal set \(U\) defined as

$$

U = \{ 51, 52, 53, 54, 55, 56, 57, 58, 59 \}.

$$

Now, if \(A\) and \(B\) are subsets of \(U\), such that

$$

\begin{eqnarray}

A &=& \{ \text{ odd numbers } \} \\

B &=& \{ \text{ prime numbers } \}.

\end{eqnarray}

$$

Then, the members of the sets \(A\) and \(B\) are

$$

\begin{eqnarray}

A &=& \{ 51, 53, 55, 57, 59 \} \\

B &=& \{ 53, 59 \}.

\end{eqnarray}

$$

Here is a Venn diagram that represents the sets \(A\), \(B\) and \(U\).

In **part (b)**, we need to do some constructions using a pair of compasses, a ruler and a pencil.

Firstly, we are asked to construct a triangle \(CDE\) in which \(DE = 10 \text{ cm}\), \(DC = 8 \text{ cm}\) and \(\angle CDE = 45^\circ\).

Here is an image of the final construction along with the tools used:

And here are the steps to reproduce it.

- Draw line \(DE\) of length 10 cm.
- Construct angle \(CDE\) of \(45^\circ\), such that \(CD\) has length 8 cm.
- Using \(D\) as center and any suitable radius, draw an arc to cut at \(a\).
- Then with \(a\) as center and radius \(Da\), draw an arc to cut at \(b\).
- Next using \(Da\) again, with \(b\) as center, draw another arc to cut at \(c\).
- Now with \(b\) and \(c\) as centers and any convenient radius, draw arcs to intersect at \(d\).
- Draw a line from \(D\) through \(d\) intersecting the arc \(a\) at \(e\).
- Using \(a\) and \(e\) as centers, draw equal arcs to intersect at \(f\).
- Draw a line from \(D\) through \(f\).
- Draw line \(DC\) of length 8 cm on line \(Df\). Angle \(CDE\) is equal to \(45^\circ\).
- Draw line \(CE\) to complete the construction of triangle \(CDE\).

Secondly, we are asked to construct a line, \(CF\), perpendicular to \(DE\) such that \(F\) lies on \(DE\).

Here is an image of the final construction:

And here are the steps to reproduce it.

- Using \(C\) as center, draw an arc to cut \(DE\) at \(g\) and \(h\).
- Now with \(g\) and \(h\) as centers, draw arcs of equal radii to intersect at \(i\).
- Draw a line from \(C\) through \(i\) cutting \(DE\) at \(F\). \(CF\) is perpendicular to \(DE\) as required.

Finally, using a protractor I have measured that \(\angle DCE\) has a size of approximately \(83^\circ\).