1.47: In triangle , cm, cm, cm and angle (a) Show that where is a prime number.

4PM1/2/June/2025 — Question 5 · 8 marks

In triangle ABC\displaystyle ABC, AC=12\displaystyle AC = 12 cm, BC=14\displaystyle BC = 14 cm, AB=x\displaystyle AB = x cm and angle ABC=30∘\displaystyle ABC = 30^\circ
(a) Show that x=P3±95\displaystyle x = P\sqrt{3} \pm \sqrt{95} where P\displaystyle P is a prime number.
(5)
(b) Hence or otherwise, find in cm2\displaystyle \mathrm{cm}^2, the difference between the two possible areas of triangle ABC\displaystyle ABC
Give your answer in the form mn\displaystyle m\sqrt{n} where m\displaystyle m is a prime number and n\displaystyle n is an integer.
(3)