Edexcel C1 2007 June — Question 6 7 marks

Exam BoardEdexcel
ModuleC1 (Core Mathematics 1)
Year2007
SessionJune
Marks7
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Mark schemeDownload PDF ↗
TopicSimultaneous equations
TypeLine intersecting quadratic curve
DifficultyModerate -0.8 This is a routine C1 simultaneous equations question with clear scaffolding. Part (a) guides students through the substitution step, and part (b) requires completing the square or using the quadratic formula—both standard techniques. The surd form answer adds minor complexity but this is still easier than a typical A-level question due to the explicit guidance provided.
Spec1.02c Simultaneous equations: two variables by elimination and substitution1.02f Solve quadratic equations: including in a function of unknown
6. (a) By eliminating \(y\) from the equations $$\begin{gathered} y = x - 4 \\ 2 x ^ { 2 } - x y = 8 \end{gathered}$$ show that $$x ^ { 2 } + 4 x - 8 = 0$$ (b) Hence, or otherwise, solve the simultaneous equations $$\begin{gathered} y = x - 4 \\ 2 x ^ { 2 } - x y = 8 \end{gathered}$$ giving your answers in the form \(a \pm b \sqrt { } 3\), where \(a\) and \(b\) are integers.
(a) M1 for correct attempt to form an equation in x only. Condone sign errors/slips but attempt at this line must be seen. E.g. \(2x^2 - x^2 \pm 4x = 8\) is OK for M1.
\(2x^2 - x(x-4) = 8\)
\(x^2 + 4x - 8 = 0\) (*)
A1cso for correctly simplifying to printed form. No incorrect working seen. The \(= 0\) is required. These two marks can be scored in part (b). For multiple attempts pick best.
(b) M1 for use of correct formula. If formula is not quoted then a fully correct substitution is required. Condone missing \(x =\) or just \(+\) or \(-\) instead of \(\pm\) for M1.
\(x = \frac{-4 \pm \sqrt{4^2 - 4 \times 1 \times (-8)}}{2}\)
For completing the square must have as printed or better. If they have \(x^2 - 4x - 8 = 0\) then M1 can be given for \((x-2)^2 \pm 4 - 8 = 0\).
\(x = -2 \pm\) (any correct expression)
A1 for \(-2 +\) any correct expression. (The \(+\) is required but \(x =\) is not)
B1 for simplifying the surd e.g. \(\sqrt{48} = 4\sqrt{3}\). Must reduce to \(b\sqrt{3}\) so \(16\sqrt{3}\) or \(4\sqrt{3}\) are OK.
\(\sqrt{48} = 4\sqrt{3}\) or \(\sqrt{12} = 2\sqrt{3}\)
\(y = -2 \pm 2\sqrt{3}\)
M1 for attempting to find at least one y value. Substitution into one of the given equations and an attempt to solve for y.
\(x = -2 + 2\sqrt{3}, y = -6 + 2\sqrt{3}\) and \(x = -2 - 2\sqrt{3}, y = -6 - 2\sqrt{3}\)
A1 for correct y answers. Pairings need not be explicit but they must say which is x and which y. Mis-labelling x and y loses final A1 only.
(a) M1 for correct attempt to form an equation in x only. Condone sign errors/slips but attempt at this line must be seen. E.g. $2x^2 - x^2 \pm 4x = 8$ is OK for M1.

$2x^2 - x(x-4) = 8$

$x^2 + 4x - 8 = 0$ (*)

A1cso for correctly simplifying to printed form. No incorrect working seen. The $= 0$ is required. These two marks can be scored in part (b). For multiple attempts pick best.

(b) M1 for use of correct formula. If formula is not quoted then a fully correct substitution is required. Condone missing $x =$ or just $+$ or $-$ instead of $\pm$ for M1.

$x = \frac{-4 \pm \sqrt{4^2 - 4 \times 1 \times (-8)}}{2}$

For completing the square must have as printed or better. If they have $x^2 - 4x - 8 = 0$ then M1 can be given for $(x-2)^2 \pm 4 - 8 = 0$.

$x = -2 \pm$ (any correct expression)

A1 for $-2 +$ any correct expression. (The $+$ is required but $x =$ is not)

B1 for simplifying the surd e.g. $\sqrt{48} = 4\sqrt{3}$. Must reduce to $b\sqrt{3}$ so $16\sqrt{3}$ or $4\sqrt{3}$ are OK.

$\sqrt{48} = 4\sqrt{3}$ or $\sqrt{12} = 2\sqrt{3}$

$y = -2 \pm 2\sqrt{3}$

M1 for attempting to find at least one y value. Substitution into one of the given equations and an attempt to solve for y.

$x = -2 + 2\sqrt{3}, y = -6 + 2\sqrt{3}$ and $x = -2 - 2\sqrt{3}, y = -6 - 2\sqrt{3}$

A1 for correct y answers. Pairings need not be explicit but they must say which is x and which y. Mis-labelling x and y loses final A1 only.

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6. (a) By eliminating $y$ from the equations

$$\begin{gathered}
y = x - 4 \\
2 x ^ { 2 } - x y = 8
\end{gathered}$$

show that

$$x ^ { 2 } + 4 x - 8 = 0$$

(b) Hence, or otherwise, solve the simultaneous equations

$$\begin{gathered}
y = x - 4 \\
2 x ^ { 2 } - x y = 8
\end{gathered}$$

giving your answers in the form $a \pm b \sqrt { } 3$, where $a$ and $b$ are integers.\\

\hfill \mbox{\textit{Edexcel C1 2007 Q6 [7]}}
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