Date | November 2018 | Marks available | 4 | Reference code | 18N.2.AHL.TZ0.H_11 |
Level | Additional Higher Level | Paper | Paper 2 | Time zone | Time zone 0 |
Command term | Determine and Sketch | Question number | H_11 | Adapted from | N/A |
Question
Consider the rectangle OABC such that AB = OC = 10 and BC = OA = 1 , with the points P , Q and R placed on the line OC such that OP = p, OQ = q and OR = r, such that 0 < p < q < r < 10.
Let θp be the angle APO, θq be the angle AQO and θr be the angle ARO.
Consider the case when θp=θq+θr and QR = 1.
Find an expression for θp in terms of p.
Show that p=q2+q−12q+1.
By sketching the graph of p as a function of q, determine the range of values of p for which there are possible values of q.
Markscheme
* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.
METHOD 1
use of tan (M1)
tanθp=1p (A1)
θp=arctan(1p) A1
METHOD 2
AP =√p2+1 (A1)
use of sin, cos, sine rule or cosine rule using the correct length of AP (M1)
θp=arcsin(1√p2+1) or θp=arccos(p√p2+1) A1
[3 marks]
QR = 1 ⇒ r=q+1 (A1)
Note: This may be seen anywhere.
tanθp=tan(θq+θr)
attempt to use compound angle formula for tan M1
tanθp=tanθq+tanθr1−tanθqtanθr (A1)
1p=1q+1r1−(1q)(1r) (M1)
1p=1q+1q+11−(1q)(1q+1) or p=1−(1q)(1q+1)(1q)+(1q+1) A1
1p=q+q+1q(q+1)−1 M1
Note: Award M1 for multiplying top and bottom by q(q+1).
p=q2+q−12q+1 AG
[6 marks]
increasing function with positive q-intercept A1
Note: Accept curves which extend beyond the domain shown above.
(0.618 <) q < 9 (A1)
⇒ range is (0 <) p < 4.68 (A1)
0 < p < 4.68 A1
[4 marks]