# CSEC Math Jan 2012 P2 Q3

CSEC Math Jan 2012 Paper 2 Question 3

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 $$U$$.

A Venn diagram to represent 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:

A construction of triangle CDE

And here are the steps to reproduce it.

1. Draw line $$DE$$ of length 10 cm.
2. Construct angle $$CDE$$ of $$45^\circ$$, such that $$CD$$ has length 8 cm.
1. Using $$D$$ as center and any suitable radius, draw an arc to cut at $$a$$.
2. Then with $$a$$ as center and radius $$Da$$, draw an arc to cut at $$b$$.
3. Next using $$Da$$ again, with $$b$$ as center, draw another arc to cut at $$c$$.
4. Now with $$b$$ and $$c$$ as centers and any convenient radius, draw arcs to intersect at $$d$$.
5. Draw a line from $$D$$ through $$d$$ intersecting the arc $$a$$ at $$e$$.
6. Using $$a$$ and $$e$$ as centers, draw equal arcs to intersect at $$f$$.
7. Draw a line from $$D$$ through $$f$$.
8. Draw line $$DC$$ of length 8 cm on line $$Df$$. Angle $$CDE$$ is equal to $$45^\circ$$.
3. 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:

A construction of the line CF perpendicular to DE

And here are the steps to reproduce it.

1. Using $$C$$ as center, draw an arc to cut $$DE$$ at $$g$$ and $$h$$.
2. Now with $$g$$ and $$h$$ as centers, draw arcs of equal radii to intersect at $$i$$.
3. 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 $$83^\circ$$.

A protractor being used to measure the size of angle DCE