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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.

[3]
a.

Show that p=q2+q12q+1.

[6]
b.

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.

[4]
c.

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(1p2+1)  or  θp=arccos(pp2+1)      A1

 

[3 marks]

a.

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+tanθr1tanθqtanθr      (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+q12q+1       AG

 

[6 marks]

b.

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]

 

c.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c.

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.2—2d and 3d trig, sine rule, cosine rule, area
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