Loading [MathJax]/jax/element/mml/optable/Latin1Supplement.js

User interface language: English | Español

Date November 2018 Marks available 5 Reference code 18N.1.AHL.TZ0.H_11
Level Additional Higher Level Paper Paper 1 (without calculator) Time zone Time zone 0
Command term Express and Find Question number H_11 Adapted from N/A

Question

Let S be the sum of the roots found in part (a).

Find the roots of z24=1 which satisfy the condition 0<arg(z)<π2, expressing your answers in the form reiθ, where r, θR+.

[5]
a.

Show that Re S = Im S.

[4]
b.i.

By writing π12 as (π4π6), find the value of cos π12 in the form a+bc, where ab and c are integers to be determined.

[3]
b.ii.

Hence, or otherwise, show that S = 12(1+2)(1+3)(1+i).

[4]
b.iii.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

(r(cosθ+isinθ))24=1(cos0+isin0)

use of De Moivre’s theorem       (M1)

r24=1r=1      (A1)

24θ=2πnθ=πn12(nZ)      (A1)

0<arg(z)<π2n= 1, 2, 3, 4, 5

z=eπi12 or e2πi12 or e3πi12 or e4πi12 or e5πi12      A2

Note: Award A1 if additional roots are given or if three correct roots are given with no incorrect (or additional) roots.

 

[5 marks]

a.

Re S = cosπ12+cos2π12+cos3π12+cos4π12+cos5π12

Im S = sinπ12+sin2π12+sin3π12+sin4π12+sin5π12      A1

Note: Award A1 for both parts correct.

but sin5π12=cosπ12,  sin4π12=cos2π12,  sin3π12=cos3π12,  sin2π12=cos4π12 and sinπ12=cos5π12      M1A1

⇒ Re S = Im S       AG

Note: Accept a geometrical method.

 

[4 marks]

b.i.

cosπ12=cos(π4π6)=cosπ4cosπ6+sinπ4sinπ6      M1A1

=2232+2212

=6+24       A1

 

[3 marks]

b.ii.

 

cos5π12=cos(π6+π4)=cosπ6cosπ4sinπ6sinπ4      (M1)

Note: Allow alternative methods eg cos5π12=sinπ12=sin(π4π6).

=32221222=624      (A1)

Re S = cosπ12+cos2π12+cos3π12+cos4π12+cos5π12

Re S = 2+64+32+22+12+624      A1

=12(6+1+2+3)      A1

=12(1+2)(1+3)

S = Re(S)(1 + i) since Re S = Im S,      R1

S = 12(1+2)(1+3)(1+i)      AG

 

[4 marks]

b.iii.

Examiners report

[N/A]
a.
[N/A]
b.i.
[N/A]
b.ii.
[N/A]
b.iii.

Syllabus sections

Topic 3— Geometry and trigonometry » AHL 3.10—Compound angle identities
Show 39 related questions
Topic 1—Number and algebra » AHL 1.13—Polar and Euler form
Topic 1—Number and algebra
Topic 3— Geometry and trigonometry

View options