OCR MEI Further Pure Core 2024 June — Question 3 4 marks

Exam BoardOCR MEI
ModuleFurther Pure Core (Further Pure Core)
Year2024
SessionJune
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicRoots of polynomials
TypeSum of powers of roots
DifficultyStandard +0.3 This is a standard Further Maths question on sum of powers of roots using Newton's identities or the algebraic identity (α+β+γ)² = α²+β²+γ² + 2(αβ+βγ+γα). Students apply Vieta's formulas to find the elementary symmetric functions, then substitute into a well-known formula. It requires recall of technique rather than problem-solving, making it slightly easier than average even for Further Maths.
Spec4.05a Roots and coefficients: symmetric functions
3 The equation \(2 x ^ { 3 } - 2 x ^ { 2 } + 8 x - 15 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
Determine the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).
Question 3:
AnswerMarks
3𝛼+𝛽+𝛾 = 1
𝛼𝛽+𝛽𝛾+𝛾𝛼 = 4
[𝛼2+𝛽2+𝛾2 =](𝛼+𝛽+𝛾)2−2(𝛼𝛽+𝛽𝛾+𝛾𝛼)
AnswerMarks
= −7B1
B1
M1
AnswerMarks
A13.1a
1.1
3.1a
AnswerMarks
1.1May be embedded
May be embedded
oe Accept numerical values in terms of their 1 and 4, i.e.
12−2×4. No slips allowed.
AnswerMarks Guidance
Alternative methodM1 Correct substitution chosen and attempted
Let 𝑧 = 𝑥2
3 2
2(√𝑧) −2(√𝑧) +8√𝑧−15 = 0
2
AnswerMarks Guidance
(√𝑧) (2𝑧+8)2 = (15+2𝑧)22
(√𝑧) (2𝑧+8)2 = (15+2𝑧)2M1 Rearranging and squaring both sides to remove square
root
AnswerMarks Guidance
4𝑧3+28𝑧2+4𝑧−225 = 0A1 Correct equation
𝛼2+𝛽2+𝛾2 = −7A1
[4]
M1
Correct substitution chosen and attempted
Question 3:
3 | 𝛼+𝛽+𝛾 = 1
𝛼𝛽+𝛽𝛾+𝛾𝛼 = 4
[𝛼2+𝛽2+𝛾2 =](𝛼+𝛽+𝛾)2−2(𝛼𝛽+𝛽𝛾+𝛾𝛼)
= −7 | B1
B1
M1
A1 | 3.1a
1.1
3.1a
1.1 | May be embedded
May be embedded
oe Accept numerical values in terms of their 1 and 4, i.e.
12−2×4. No slips allowed.
Alternative method | M1 | Correct substitution chosen and attempted
Let 𝑧 = 𝑥2
3 2
2(√𝑧) −2(√𝑧) +8√𝑧−15 = 0
2
(√𝑧) (2𝑧+8)2 = (15+2𝑧)2 | 2
(√𝑧) (2𝑧+8)2 = (15+2𝑧)2 | M1 | Rearranging and squaring both sides to remove square
root
4𝑧3+28𝑧2+4𝑧−225 = 0 | A1 | Correct equation
𝛼2+𝛽2+𝛾2 = −7 | A1
[4]
M1
Correct substitution chosen and attempted
3 The equation $2 x ^ { 3 } - 2 x ^ { 2 } + 8 x - 15 = 0$ has roots $\alpha , \beta$ and $\gamma$.\\
Determine the value of $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$.

\hfill \mbox{\textit{OCR MEI Further Pure Core 2024 Q3 [4]}}
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Q1 4 Q2 7 Q3 4 Q4 4 Q5 6 Q6 6 Q7 5 Q8 10 Q9 8 Q10 6 Q11 14 Q12 12 Q13 10 Q14 12 Q15 10 Q16 6 Q17 20