Theory question 111
Algebraic Expressions(a) Make m the subject of the relation h = 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 ⊗ is defined on the set X = {1, 3, 5, 6} by m ⊗ n = m + n + 2 (mod 7), where m, n ∈ X. (i) Draw a table for the operation. (ii) Using the table, find the truth set of: I. 3 ⊗ n = 3; II. n ⊗ n = 3.
Show the model answer
(a) hd(m + p) = mt → hdm + hdp = mt → m(t − hd) = hdp → m = hdp/(t − hd)
(b) WM is a diameter, so ∠WXM = 90°; ∠XMW = 180° − 90° − 48° = 42°
∠XMZ = 180° − 42° = 138° (WMZ straight)
XYZM is cyclic, so ∠XYZ = 180° − 138° = 42°; ∠WYZ = 42°
(c)(i) Rows m = 1, 3, 5, 6; columns n = 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) I. 3 ⊗ n = 3 → n = 5: {5}
II. n ⊗ n = 3: the diagonal is 4, 1, 5, 0, so the truth set is { } (empty)
In (b), use the angle in a semicircle and then opposite angles of a cyclic quadrilateral.