NCEA Level 3 Mathematics and Statistics 91577 (3.5) — page 1 of 6
SAMPLE ASSESSMENT SCHEDULE
This assessment schedule has been updated (4 July 2013). Question parts have been rearranged so that all three questions involve complex number calculations, in line with the intent of the standard.
Student work exemplars were completed before this update and have not been amended.
Mathematics and Statistics (Mathematics) 91577 (3.5): Apply the algebra of complex
numbers in solving problems
Assessment Criteria
Achievement
Achievement with Merit
Apply the algebra of complex
numbers in solving problems
involves:
Achievement with Excellence
Relational thinking involves one or
more of:
Extended abstract thinking
involves one or more of:
 selecting and using methods
 carrying out a logical sequence of
steps
 devising a strategy to
investigate a problem
 demonstrating knowledge of
concepts and terms
 connecting different concepts or
representations
 identifying relevant concepts in
context
 communicating using appropriate
representations.
 demonstrating understanding of
concepts
 developing a chain of logical
reasoning, or proof
 forming and using a model
 forming a generalisation
and also relating findings to a
context, or communicating thinking
using appropriate mathematical
statements.
and also using correct
mathematical statements, or
communicating mathematical
insight.
Evidence Statement
One
Expected Coverage
Achievement
15 5
2
7 7
Correct solution.
(a)
(b)
–22
Correct solution.
x - m = x - 10 x + 25
Correct expres-
10 x = m + 25
sion for
(c)
100x = ( m + 25 )
2
m + 25 )
(
x=
100
2
x.
Merit
Correct solution.
Excellence
NCEA Level 3 Mathematics and Statistics 91577 (3.5) — page 2 of 6
( z - 3+ i) ( z - 3 - i)
= z 2 - 6z +10
(z
2
- 6z +10 ) ( z - a )
Correct value of
either n OR other roots.
Correct value of
n other AND
roots.
= z 3 - (6 + a)z 2 + (10 + 6a)z -10a
(d)
= z 3 - 4z 2 - 2z + n
a = -2
n = 20
Other roots are z = –2
and z = 3 – i
(e)
1
3

 2i
x  iy x  iy
x  iy  3( x  iy )
 2i
x2  y 2
4x
2 yi
 2
 2i
2
2
x y
x  y2
4x
2
2
x  y2
2y
1
2
x  y2
x y
Correct expression with real
denominator.
Correct answer
with a logical
chain of reasoning and correct
mathematical
statements.
4x
2
2 x2
x  1, y  1
z  1 i

Not Achieved
NØ
No response; no relevant evidence.
N1
Candidate provides ONE correct relevant algebraic step towards a solution.
N2
Candidate provides ONE correct solution.
A3
Candidate provides TWO of the Achievement criteria.
A4
Candidate provides THREE of the Achievement criteria.
M5
Candidate provides ONE of the Merit criteria.
M6
Candidate provides TWO of the Merit criteria.
E7
Candidate provides ALL of the Excellence criteria with minor errors ignored.
E8
Candidate provides ALL of Excellence criteria correctly.
Achievement
Merit
Excellence
NCEA Level 3 Mathematics and Statistics 91577 (3.5) — page 3 of 6
Two
Expected Coverage
Achievement
(a)
3cis
π
12
Correct solution.
(b)
x = 6 ± 5i
Correct solution.
z 4 = ncis
π
2
One correct solution.
z2 = 4 ncis
5π
8
z3 = 4 ncis
-3π
8
z4 = 4 ncis
-7π
8
Correct expression with irrational denominator.
144k 2  36k 2

3 6 2 6

12k  6k
6

6k
6
(d)
Excellence
Four correct solutions.
(Accept positive
arguments.)
π
z1 = ncis
8
4
(c)
Merit
Correct solution.
 6k
Correct equation
for
- b ± b - 4 ac
2
x=
2a
x1 = 2 x 2 Þ
(
- b - b - 4 ac = 2 - b + b - 4 ac
2
(e)
2
)
b = 3 b 2 - 4 ac
b = 9b - 36 ac
2
2
36 ac = 8 b
relationship between a, b, c.
NOTE: Can get
to line
Correct answer
with a logical
chain of reasoning and correct
mathematical
statements.
b = 3 b 2 - 4ac
by logic.
2
9 ac = 2 b 2
Not Achieved
NØ
No response; no relevant evidence.
N1
Candidate provides ONE correct relevant algebraic step towards a solution.
N2
Candidate provides ONE correct solution.
A3
Candidate provides TWO of the Achievement criteria.
A4
Candidate provides THREE of the Achievement criteria.
M5
Candidate provides ONE of the Merit criteria.
M6
Candidate provides TWO of the Merit criteria.
Achievement
Merit
NCEA Level 3 Mathematics and Statistics 91577 (3.5) — page 4 of 6
E7
Candidate provides ALL of Excellence criteria with minor errors ignored.
E8
Candidate provides ALL of Excellence criteria correctly.
Excellence
NCEA Level 3 Mathematics and Statistics 91577 (3.5) — page 5 of 6
Three
Expected Coverage
Achievement
uv = 2 + 6i
uv placed at
correct point
on Argand diagram.
u  v  8  4ki  2ki  k 2 i 2
Correct solution.
(a)


Merit
 8  k 2  6ki
(b)
k=5
x=
(b - c) y
c-a
(a - b) y
z=
c-a
\x + y+ z =
(c)
(b - c) y
Correct solution.
Correct expression for u
with a rational
denominator.
Accept
Correct solution.
(a - b)y
c-a
c-a
(b - c)y + (c - a)y + (a - b)y
=
c-a
y(b - c + c - a + a - b)
=
c-a
=0
+y+
1
+i
1+ 3i
1- 3i
=
+i
10
1 7
= + i
10 10
u=
(d)
Correct expression of x, y
or z in terms of
one other variable.
0.1+ 0.7i
2
æ 1ö æ 7 ö
u = ç ÷ +ç ÷
è 10 ø è 10 ø
=
0. 5 or
1
2
2
Excellence
NCEA Level 3 Mathematics and Statistics 91577 (3.5) — page 6 of 6
(e)
x -1 x +1 2
+
=0
t - 1 2x 2x
x -1 x -1
=0
t - 1 2x
x -1 x -1
=
t -1
2x
2x(x - 1) = (x - 1)(t - 1)
Correct equation with denominator of 1.
(eg, line 4.)
Correct answer with a
logical chain of
reasoning and
correct mathematical
statements.
2x(x - 1) - (x - 1)(t - 1) = 0
(x - 1)(2x - (t - 1)) = 0
x =1
or
x=
t -1
2
Not Achieved
NØ
No response; no relevant evidence.
N1
Candidate provdes ONE correct relevant algebraic step towards a solution.
N2
Candidate provides ONE correct solution.
A3
Candidate provides TWO of the Achievement criteria.
A4
Candidate provides THREE of the Achievement criteria.
M5
Candidate provides ONE of the Merit criteria.
M6
Candidate provides TWO of the Merit criteria.
E7
Candidate provides ALL of Excellence criteria with minor errors ignored.
E8
Candidate provides ALL of Excellence criteria correctly.
Achievement
Merit
Excellence