Question 4 - A-Level Maths

OCR FP1 2006 January — Question 4 5 marks

Exam Board	OCR
Module	FP1 (Further Pure Mathematics 1)
Year	2006
Session	January
Marks	5
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Roots of polynomials
Type	Substitution to find new equation
Difficulty	Standard +0.3 This is a straightforward substitution question where the substitution is given explicitly. Students expand (u+2)³, collect terms to get a simpler cubic in u, then solve. The arithmetic is routine and the depressed cubic likely has an obvious integer root, making this slightly easier than average despite being Further Maths content.
Spec	4.05b Transform equations: substitution for new roots

4 Use the substitution $x = u + 2$ to find the exact value of the real root of the equation $$x ^ { 3 } - 6 x ^ { 2 } + 12 x - 13 = 0$$

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Answer	Marks	Guidance
$u^2 + 4u + 4$	B1	$u + 2$ squared and cubed correctly
$u^3 + 6u^2 + 12u + 8$
$M1$	Substitute these and attempt to simplify
$u^3 - 5 = 0$ or equivalent	A1	Obtain $u^3 - 5 = 0$ or equivalent
$u = \sqrt[3]{5}$	A1ft	Correct solution to their equation
$x = 2 + \sqrt[3]{5}$	A1ft	Obtain $2 +$ their answer [Decimals score 0/2 of final A marks]

Total: 5 marks

$u^2 + 4u + 4$ | B1 | $u + 2$ squared and cubed correctly

$u^3 + 6u^2 + 12u + 8$ | 

$M1$ | Substitute these and attempt to simplify

$u^3 - 5 = 0$ or equivalent | A1 | Obtain $u^3 - 5 = 0$ or equivalent

$u = \sqrt[3]{5}$ | A1ft | Correct solution to their equation

$x = 2 + \sqrt[3]{5}$ | A1ft | Obtain $2 +$ their answer [Decimals score 0/2 of final A marks]

**Total: 5 marks**

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4 Use the substitution $x = u + 2$ to find the exact value of the real root of the equation

$$x ^ { 3 } - 6 x ^ { 2 } + 12 x - 13 = 0$$

\hfill \mbox{\textit{OCR FP1 2006 Q4 [5]}}

This paper (10 questions)

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Q1 5 Q2 5 Q3 4 Q4 5 Q5 6 Q6 7 Q7 10 Q8 9 Q9 10 Q10 11

Answer	Marks	Guidance
\(u^2 + 4u + 4\)	B1	\(u + 2\) squared and cubed correctly
\(u^3 + 6u^2 + 12u + 8\)
\(M1\)	Substitute these and attempt to simplify
\(u^3 - 5 = 0\) or equivalent	A1	Obtain \(u^3 - 5 = 0\) or equivalent
\(u = \sqrt[3]{5}\)	A1ft	Correct solution to their equation
\(x = 2 + \sqrt[3]{5}\)	A1ft	Obtain \(2 +\) their answer [Decimals score 0/2 of final A marks]