(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 < X...
Question 1 Report
(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) \(n \star n = 3\).
Answer Details
(a)
\[ h=\frac{mt}{d(m+p)} \]
\[ dh(m+p)=mt \]
\[ dhm+dhp=mt \]
\[ dhp=mt-dhm=m(t-dh) \]
Hence,
\[ \boxed{m=\frac{dhp}{t-dh}} \]
(b)
Since \(WM\) is a diameter of the circle,
\[ \angle WXM=90^\circ \]
In \(\triangle WXM\),
\[ \angle WMX=180^\circ-(90^\circ+48^\circ)=42^\circ. \]
Since \(W, M\) and \(Z\) lie on a straight line,
\[ \angle XMZ=180^\circ-42^\circ=138^\circ. \]
Opposite angles of the cyclic quadrilateral are supplementary. Therefore,
\[ \angle WYZ=180^\circ-138^\circ=\boxed{42^\circ}. \]
(c)(i) Using \(m\star n=m+n+2\pmod 7\), the operation table is:
| \(\star\) | \(1\) | \(3\) | \(5\) | \(6\) |
|---|---|---|---|---|
| \(1\) | \(4\) | \(6\) | \(1\) | \(2\) |
| \(3\) | \(6\) | \(1\) | \(3\) | \(4\) |
| \(5\) | \(1\) | \(3\) | \(5\) | \(6\) |
| \(6\) | \(2\) | \(4\) | \(6\) | \(0\) |
(c)(ii)
(I) From the row headed \(3\), \(3\star n=3\) when \(n=5\).
\[ \boxed{\{5\}} \]
(II) The diagonal entries are \(4,1,5,0\). None is equal to \(3\). Thus,
\[ \boxed{\varnothing} \]
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