OCR MEI Further Pure Core 2019 June — Question 8 8 marks

Exam BoardOCR MEI
ModuleFurther Pure Core (Further Pure Core)
Year2019
SessionJune
Marks8
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Mark schemeDownload PDF ↗
TopicRoots of polynomials
TypeRoots with special relationships
DifficultyStandard +0.3 This is a straightforward Further Maths question on roots of polynomials using Vieta's formulas. The special relationship α and 1/α immediately gives α·(1/α)·β = 2, so β = 2. Then substituting back yields α values and k. It requires systematic application of standard techniques rather than insight, making it slightly easier than average even for Further Maths.
Spec1.02j Manipulate polynomials: expanding, factorising, division, factor theorem4.05a Roots and coefficients: symmetric functions
8 In this question you must show detailed reasoning. The roots of the equation \(x ^ { 3 } - x ^ { 2 } + k x - 2 = 0\) are \(\alpha , \frac { 1 } { \alpha }\) and \(\beta\).
  1. Evaluate, in exact form, the roots of the equation.
  2. Find \(k\).
Question 8:
AnswerMarks Guidance
8(a) DR
1
. .2   = 2
1
 1
 2 +  + 1 = 0
1 3 1 3
   i
2 2 2
1 3 1 3
roots are [2],   i,  i
AnswerMarks
2 2 2 2M1
A1
M1
A1
M1
A1
AnswerMarks
[6]3.1a
1.1b
1.1b
1.1b
1.1b
AnswerMarks
1.1bproduct of roots used
 = 2
sum of roots used
or equivalent quadratic
(with  = 2)
AnswerMarks
solving their quadraticor (x2)(x2+x+1) = 0
M1A1
1 3 1 3
x   i
2 2 2
AnswerMarks Guidance
8(b) 1 1
k .  
 
1
12( )121
AnswerMarks
M1
A1
AnswerMarks
[2]1.1a
1.1bk = product of root pairs
or by direct substitutionor (x2)(x2+x+1)  k = 1
or by factor theorem
Question 8:
8 | (a) | DR
1
. .2   = 2

1
 1

 2 +  + 1 = 0
1 3 1 3
   i
2 2 2
1 3 1 3
roots are [2],   i,  i
2 2 2 2 | M1
A1
M1
A1
M1
A1
[6] | 3.1a
1.1b
1.1b
1.1b
1.1b
1.1b | product of roots used
 = 2
sum of roots used
or equivalent quadratic
(with  = 2)
solving their quadratic | or (x2)(x2+x+1) = 0
M1A1
1 3 1 3
x   i
2 2 2
8 | (b) | 1 1
k .  
 
1
12( )121
 | M1
A1
[2] | 1.1a
1.1b | k = product of root pairs
or by direct substitution | or (x2)(x2+x+1)  k = 1
or by factor theorem
8 In this question you must show detailed reasoning.
The roots of the equation $x ^ { 3 } - x ^ { 2 } + k x - 2 = 0$ are $\alpha , \frac { 1 } { \alpha }$ and $\beta$.
\begin{enumerate}[label=(\alph*)]
\item Evaluate, in exact form, the roots of the equation.
\item Find $k$.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Pure Core 2019 Q8 [8]}}
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