Date | November 2019 | Marks available | 1 | Reference code | 19N.1.AHL.TZ0.H_9 |
Level | Additional Higher Level | Paper | Paper 1 (without calculator) | Time zone | Time zone 0 |
Command term | Show that | Question number | H_9 | Adapted from | N/A |
Question
In the following diagram, the points A, B, C and D are on the circumference of a circle with centre O and radius r. [AC] is a diameter of the circle. BC=r, AD=CD and A∧BC=A∧DC=90∘.
Given that cos75∘=q, show that cos105∘=−q.
Show that B∧AD=75∘.
By considering triangle ABD, show that BD2=5r2−2r2q√6.
By considering triangle CBD, find another expression for BD2 in terms of r and q.
Use your answers to part (c) to show that cos75∘=1√6+√2.
Markscheme
cos105∘=cos(180∘−75∘)=−cos75∘ R1
=−q AG
Note: Accept arguments using the unit circle or graphical/diagrammatical considerations.
[1 mark]
AD=CD⇒C∧AD=45∘ A1
valid method to find B∧AC (M1)
for example: BC=r⇒B∧CA=60∘
⇒B∧AC=30∘ A1
hence B∧AD=45∘+30∘=75∘ AG
[3 marks]
AB=r√3, AD=(CD)=r√2 A1A1
applying cosine rule (M1)
BD2=(r√3)2+(r√2)2−2(r√3)(r√2)cos75∘ A1
=3r2+2r2−2r2√6cos75∘
=5r2−2r2q√6 AG
[4 marks]
B∧CD=105∘ (A1)
attempt to use cosine rule on ΔBCD (M1)
BD2=r2+(r√2)2−2r(r√2)cos105∘
=3r2+2r2q√2 A1
[3 marks]
5r2−2r2q√6=3r2+2r2q√2 (M1)(A1)
2r2=2r2q(√6+√2) A1
Note: Award A1 for any correct intermediate step seen using only two terms.
q=1√6+√2 AG
Note: Do not award the final A1 if follow through is being applied.
[3 marks]