(a) Simplify : \(\frac{\frac{1}{3}c^{2} – \frac{2}{3}cd}{\frac{1}{2}d^{2} – \frac{1}{4}cd}\)
(b) 
In the diagram, YPF is a straight line. < XPY = 44°, < MPF = 46°, < XYP = < MFP = 90°, /XY/ = 7cm and /MP/ = 9 cm.
(i) Calculate, correct to 3 significant figures, /XM/ and /YF/ ; (ii) Find < XMP.
Explanation
(a) \(\frac{\frac{1}{3}c^{2} - \frac{2}{3}cd}{\frac{1}{2}d^{2} - \frac{1}{4}cd}\)
= \(\frac{\frac{1}{3}c(c - 2d)}{\frac{1}{4}d(2d - c)}\)
= \(\frac{\frac{1}{3}c (c - 2d)}{-\frac{1}{4}d (c - 2d)}\)
= \(-\frac{4c}{3d}\)
(b)(i) In \(\Delta XYP\),
\(\frac{7}{XP} = \sin 44°\)
\(XP = \frac{7}{\sin 44}\)
= \(10.08 cm\)
< XPM + 44° + 46° = 180°
< XPM = 90°.
In \(\Delta XPM\),
\(XM^{2} = XP^{2} + PM^{2}\)
\(XM^{2} = (10.08)^{2} + 9^{2}\)
= \(101.6064 + 81 = 182.6064\)
\(XM = \sqrt{182.6064}\)
= \(15.513 cm \approxeq 13.5 cm\)
In \(\Delta XYP\),
\(\frac{7}{YP} = \tan 44\)
\(YP = \frac{7}{\tan 44}\)
\(YP = 7.249 cm \approxeq 7.2 cm\)
In \(\Delta MPF\),
\(\frac{PF}{9} = \cos 46\)
\(PF = 9 \cos 46 = 6.2519 cm\)
\(YF = YP + PF = 7.249 + 6.2519\)
= \(13.5009 cm \approxeq 13.5 cm\)
(ii) 
\(\frac{10.1}{9} = \tan \theta\)
\(\tan \theta = 1.122\)
\(\theta = \tan^{-1} (1.122)\)
= 48.296°