1.3: Exact values involving surds and trigonometry

4PM1/1R/June/2022 — Question 6 · 8 marks

Given that
a+55−2=11+55,\frac{a+\sqrt{5}}{\sqrt{5}-2}=11+5\sqrt{5},
(a) without using a calculator, find the value of a\displaystyle a.
Show your working clearly.
(2)
Triangle PQR\displaystyle PQR is such that
PR=(x+3) cm,QR=x cm,PR=(x+3)\,\mathrm{cm}, \qquad QR=x\,\mathrm{cm},
∠QPR=30∘,∠PQR=45∘.\angle QPR=30^\circ, \qquad \angle PQR=45^\circ.
(b) Show that
x=3+32.x=3+3\sqrt{2}.
(3)
Given that
sin 105∘=6+24\mathrm{sin}\,105^\circ=\frac{\sqrt{6}+\sqrt{2}}{4}
and that the area of triangle PQR\displaystyle PQR is A cm2\displaystyle A\,\mathrm{cm}^2,
(c) find the exact value of A\displaystyle A in the form
98(p6+q2+r3+s),\frac{9}{8}\left(p\sqrt{6}+q\sqrt{2}+r\sqrt{3}+s\right),
where p\displaystyle p, q\displaystyle q, r\displaystyle r and s\displaystyle s are integers.
(3)