OCR MEI Further Pure Core AS 2018 June — Question 4 5 marks

Exam BoardOCR MEI
ModuleFurther Pure Core AS (Further Pure Core AS)
Year2018
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicRoots of polynomials
TypeComplex roots with real coefficients
DifficultyModerate -0.3 This is a straightforward application of the complex conjugate root theorem and polynomial construction. Students need to recognize that 2+i must also be a root, then expand (x-3)(x-(2-i))(x-(2+i)) using standard techniques. While it requires multiple steps, it's a routine textbook exercise with no novel insight required.
Spec4.02g Conjugate pairs: real coefficient polynomials4.02j Cubic/quartic equations: conjugate pairs and factor theorem
Find a cubic equation with real coefficients, two of whose roots are \(2 - i\) and \(3\). [5]
Question 4:
AnswerMarks Guidance
4third root is 2 + i B1
2  i + 2 + i + 3 = 7
(2  i)(2 + i) + (2 + i)3 + 3(2  i) = 17
(2  i)(2 + i)3 = 15
so eqn is z3  7z2 + 17z  15 = 0
OR
(z – 2 + i)(z  2 – i) = z2 – 4z + 5
(z2  4z + 5)(z  3) = z3  7z2 + 17z  15
AnswerMarks
so eqn is z3  7z2 + 17z  15 = 0B1ft
B1ft
B1ft
B1cao
M1
A1
M1
A1
AnswerMarks
[5]3.1a
1.1
1.1
AnswerMarks
1.1ft their 2 + i
must include ‘ = 0’
(z – 2 + i)(z  2 – i)
= z2 – 4z + 5
AnswerMarks
their (z2  4z + 5)  (z  3)any variable , e.g. x 
Question 4:
4 | third root is 2 + i | B1 | 1.2 | soi
2  i + 2 + i + 3 = 7
(2  i)(2 + i) + (2 + i)3 + 3(2  i) = 17
(2  i)(2 + i)3 = 15
so eqn is z3  7z2 + 17z  15 = 0
OR
(z – 2 + i)(z  2 – i) = z2 – 4z + 5
(z2  4z + 5)(z  3) = z3  7z2 + 17z  15
so eqn is z3  7z2 + 17z  15 = 0 | B1ft
B1ft
B1ft
B1cao
M1
A1
M1
A1
[5] | 3.1a
1.1
1.1
1.1 | ft their 2 + i
must include ‘ = 0’
(z – 2 + i)(z  2 – i)
= z2 – 4z + 5
their (z2  4z + 5)  (z  3) | any variable , e.g. x
Find a cubic equation with real coefficients, two of whose roots are $2 - i$ and $3$. [5]

\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2018 Q4 [5]}}
This paper (10 questions)
View full paper
Q1 4 Q2 3 Q3 5 Q4 5 Q5 7 Q6 4 Q7 9 Q8 6 Q9 9 Q10 8