Date | November 2021 | Marks available | 1 | Reference code | 21N.2.AHL.TZ0.5 |
Level | Additional Higher Level | Paper | Paper 2 | Time zone | Time zone 0 |
Command term | Find | Question number | 5 | Adapted from | N/A |
Question
Let z=1−i.
Let w1=eix and w2=ei(x−π2), where x∈ℝ.
The current, I, in an AC circuit can be modelled by the equation I=a cos(bt−c) where b is the frequency and c is the phase shift.
Two AC voltage sources of the same frequency are independently connected to the same circuit. If connected to the circuit alone they generate currents IA and IB. The maximum value and the phase shift of each current is shown in the following table.
When the two voltage sources are connected to the circuit at the same time, the total current IT can be expressed as IA+IB.
Plot the position of z on an Argand Diagram.
Express z in the form z=aeib, where a, b∈ℝ, giving the exact value of a and the exact value of b.
Find w1+w2 in the form eix(c+id).
Hence find Re(w1+w2) in the form A cos(x-a), where A>0 and 0<a≤π2.
Find the maximum value of IT.
Find the phase shift of IT.
Markscheme
A1
[1 mark]
z=√2eiπ4 A1A1
Note: Accept an argument of 7π4. Do NOT accept answers that are not exact.
[2 marks]
w1+w2=eix+ei(x-π2)
=eix(1+e-iπ2) (M1)
=eix(1-i) A1
[2 marks]
w1+w2=eix×√2e-iπ4 M1
=√2ei(x-π4) (A1)
attempt extract real part using cis form (M1)
Re(w1+w2)=√2cos(x-π4) OR 1.4142… cos(x-0.785398…) A1
[4 marks]
It=12 cos(bt)+12 cos(bt-π2) (M1)
It=12 Re(eibt+ei(bt-π2)) (M1)
It=12√2 cos(bt-π4)
max =12√2 (=17.0) A1
[3 marks]
phase shift =π4 (=0.785) A1
[1 mark]
Examiners report
This question produced the weakest set of responses on the paper. There seemed a general lack of confidence when tackling a problem involving complex numbers. Whilst most candidates could represent a complex number on the complex plane, far fewer had the ability to move between the different forms of complex numbers. This is clearly an area of the course that needs more attention when being taught. Part (c) is challenging but it should be noted that a candidate who has answered parts (a) and (b) with confidence should find this both straightforward, and also an example of a type of problem that is mentioned in the syllabus guidance.
This question produced the weakest set of responses on the paper. There seemed a general lack of confidence when tackling a problem involving complex numbers. Whilst most candidates could represent a complex number on the complex plane, far fewer had the ability to move between the different forms of complex numbers. This is clearly an area of the course that needs more attention when being taught. Part (c) is challenging but it should be noted that a candidate who has answered parts (a) and (b) with confidence should find this both straightforward, and also an example of a type of problem that is mentioned in the syllabus guidance.
This question produced the weakest set of responses on the paper. There seemed a general lack of confidence when tackling a problem involving complex numbers. Whilst most candidates could represent a complex number on the complex plane, far fewer had the ability to move between the different forms of complex numbers. This is clearly an area of the course that needs more attention when being taught. Part (c) is challenging but it should be noted that a candidate who has answered parts (a) and (b) with confidence should find this both straightforward, and also an example of a type of problem that is mentioned in the syllabus guidance.
This question produced the weakest set of responses on the paper. There seemed a general lack of confidence when tackling a problem involving complex numbers. Whilst most candidates could represent a complex number on the complex plane, far fewer had the ability to move between the different forms of complex numbers. This is clearly an area of the course that needs more attention when being taught. Part (c) is challenging but it should be noted that a candidate who has answered parts (a) and (b) with confidence should find this both straightforward, and also an example of a type of problem that is mentioned in the syllabus guidance.
This question produced the weakest set of responses on the paper. There seemed a general lack of confidence when tackling a problem involving complex numbers. Whilst most candidates could represent a complex number on the complex plane, far fewer had the ability to move between the different forms of complex numbers. This is clearly an area of the course that needs more attention when being taught. Part (c) is challenging but it should be noted that a candidate who has answered parts (a) and (b) with confidence should find this both straightforward, and also an example of a type of problem that is mentioned in the syllabus guidance.
This question produced the weakest set of responses on the paper. There seemed a general lack of confidence when tackling a problem involving complex numbers. Whilst most candidates could represent a complex number on the complex plane, far fewer had the ability to move between the different forms of complex numbers. This is clearly an area of the course that needs more attention when being taught. Part (c) is challenging but it should be noted that a candidate who has answered parts (a) and (b) with confidence should find this both straightforward, and also an example of a type of problem that is mentioned in the syllabus guidance.