(a) Make m the subject of the relations \(h = \frac{mt}{d(m + p)}\).
(b) 
In the diagram, WY and WZ are straight lines, O is the centre of circle WXM and < XWM = 48°. Calculate the value of < WYZ.
(c) An operation \(\star\) is defind on the set X = {1, 3, 5, 6} by \(m \star n = m + n + 2 (mod 7)\) where \(m, n \in X\).
(i) Draw a table for the operation.
(ii) Using the table, find the truth set of : (I) \(3 \star n = 3\) ; (II) \(n \star n = 3\).
Explanation
(a) \(h = \frac{mt}{d(m + p)}\)
\(dh(m + p) = mt\)
\(dhm + dhp = mt \implies dhp = mt - dhm\)
\(dhp = m(t - dh) \implies m = \frac{dhp}{t - dh}\)
(b) 
In the diagram above, < WXM = 90° (angle in a semicircle)
< WMX = 180° - (90° + 48°)
= 42°
< XMZ = 180° - 42° (angles on a straight line)
= 138°
< WYZ = 180° - 138° (opp. angles of a cyclic quadrilateral)
= 42°
(c)
| \(\star\) | 1 | 3 | 5 | 6 |
| 1 | 4 | 6 | 1 | 2 |
| 3 | 6 | 1 | 3 | 4 |
| 5 | 1 | 3 | 5 | 6 |
| 6 | 2 | 4 | 6 | 0 |
(ii) From the table, the truth set of :
(I) \(3 \star n = 3; n = {5}\)
(II) \(n \star n = 3; n = { }\)