Question 12:
--- 12(a) ---
12(a) | States appropriate extra piece
of information.
Condone “ and are
integers” | AO2.3 | R1 | and are real numbers.
𝑚𝑚 𝑝𝑝
(b) | Finds one 𝑚𝑚possible𝑝𝑝 pair of
values for and
or
finds the ro𝑚𝑚ot ±7/2𝑝𝑝 in Abel’s
case
or
substitutes in the
cubic and expands
𝑧𝑧=2−3i
or
uses the sum of roots or the
sum of pairwise products of
roots (not using ) to
form an equation (condone
𝑧𝑧=2+3i
sign errors) | AO2.1 | R1 | Product of roots = - 91/2
One possible solution to the cubic is
given by (for example)
, ,
7
since
𝑧𝑧=2−3i 𝑧𝑧=2+3i 𝑧𝑧=−2
7 91
This gives and
(2−3i)(2+3i)�−2�=− 2
Another solution to the cubic is (for
example) 𝑚𝑚 =−1 𝑝𝑝=−2
, ,
7
since
𝑧𝑧=2−3i 𝑧𝑧=4+6i 𝑧𝑧=−4
7 91
This gives different values of and .
(2−3i)(4+6i)�− �=−
4 2
Because there is more than o𝑚𝑚ne 𝑝𝑝
possible set of values of and ,
there is not sufficient information to
solve the problem, and B𝑚𝑚onnie is𝑝𝑝 right.
Uses product of roots
= ±91/2 to find another
possible pair of complex roots
of the cubic
or
expresses and as
complex numbers in their
expansion 𝑚𝑚 𝑝𝑝
or
uses product of roots
= ±91/2 to form another
equation (not using ) | AO3.1a | M1
Finds another possible correct
𝑧𝑧=2+3i
pair of complex roots of the
cubic (condone sign errors)
or
forms simultaneous equations
for real and imaginary parts
or
reduces their system of
equations to a system of linear
equations | AO2.1 | M1
Completes a rigorous
argument to prove that Bonnie
is correct;
for example, explains that they
have shown that there is more
than one possible set of
values for and
or
explains w𝑚𝑚hy their𝑝𝑝
simultaneous linear equations
have no unique solution | AO2.4 | R1
Total | 5
Q | Marking Instructions | AO | Marks | Typical Solution