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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 θ p be the angle APO, θ q be the angle AQO and θ r be the angle ARO.

Consider the case when  θ p = θ q + θ r and QR = 1.

Find an expression for  θ p in terms of  p .

[3]
a.

Show that  p = q 2 + q 1 2 q + 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)

tan θ p = 1 p        (A1)

θ p = arctan ( 1 p )       A1

 

METHOD 2

AP = p 2 + 1       (A1)

use of sin, cos, sine rule or cosine rule using the correct length of AP      (M1)

θ p = arcsin ( 1 p 2 + 1 )   or   θ p = arccos ( p p 2 + 1 )       A1

 

[3 marks]

a.

QR = 1 ⇒  r = q + 1       (A1)

Note: This may be seen anywhere.

 

tan θ p = tan ( θ q + θ r )

attempt to use compound angle formula for tan       M1

tan θ p = tan θ q + tan θ r 1 tan θ q tan θ r       (A1)

1 p = 1 q + 1 r 1 ( 1 q ) ( 1 r )       (M1)

1 p = 1 q + 1 q + 1 1 ( 1 q ) ( 1 q + 1 )   or   p = 1 ( 1 q ) ( 1 q + 1 ) ( 1 q ) + ( 1 q + 1 )       A1

1 p = q + q + 1 q ( q + 1 ) 1        M1

Note: Award M1 for multiplying top and bottom by q ( q + 1 ) .

 

p = q 2 + q 1 2 q + 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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