| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simultaneous equations |
| Type | Two curves intersecting |
| Difficulty | Moderate -0.3 This is a multi-part question involving standard C1 techniques: graphical estimation, algebraic manipulation to form a cubic, factorizing/solving the cubic (likely with one rational root), and curve translation. While it has multiple parts (9 marks total), each step uses routine methods without requiring novel insight or particularly challenging problem-solving. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02q Use intersection points: of graphs to solve equations1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| \(-5.7\) to \(-5.8\), \(-2.2\) to \(-2.3\), \(-1\) isw | B1 for 2 correct or for all 3 only stated in coordinate form, ignoring \(y\) coordinates | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1=(x+2)(x^2+7x+7)\) | M1 | Condone missing brackets if expanded correctly; or M1 for correct expansion of \((x+2)(x^2+7x+7)\) |
| Correct completion with at least one interim stage of working to given answer: \(x^3+9x^2+21x+13=0\) | A1 | |
| \([x=-1\) is root so] \((x+1)\) is factor | M1 | Implied by division of cubic by \(x+1\); condone some confusion of root/factor for this mark if division of cubic by \(x+1\) seen |
| Correctly finding other factor as \(x^2+8x+13\) | M2 | M1 for correct division of cubic by \((x+1)\) as far as obtaining \(x^2+8x\) (may be in grid) or for two correct terms of \(x^2+8x+13\) obtained by inspection; allow seen in grid without \(+\) signs |
| \(\frac{-8\pm\sqrt{8^2-4\times13}}{2}\) oe | M1 | For use of formula, condoning one error, for \(x^2+8x+13=0\); or M1 for \((x+4)^2=4^2-13\) oe or further stage, condoning one error |
| \(\frac{-8\pm\sqrt{12}}{2}\) isw or \(-4\pm\sqrt{3}\) isw and \(x=-1\) | A1 | \(x=-1\) may be stated earlier; isw wrong simplification or giving as coordinates |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Drawing the translated quadratic or showing that the horizontal gap between the relevant parts of the curve is always less than 3 | B1 | Must be a reasonable translation of given quadratic, only intersecting given curve once; intersections with \(x\) axis \(-3\) to \(-2.5\) and \(1.5\) to \(2\); ignore above \(y=1\) |
| Estimated coordinates of the point of intersection (\(1.8\) to \(2\), \(0.2\) to \(0.3\)) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y=x^2+x-5\) or \(y=\left(x+\frac{1}{2}\right)^2-\frac{21}{4}\) | 2 | M1 for \([y=](x-3)^2+7(x-3)+7\) oe or for simplified equation with '\(y=\)' omitted or for \(y=(x-a)(x-b)\) where \(a\) and \(b\) are the values \(3+\frac{-7\pm\sqrt{21}}{2}\) oe (may have been wrongly simplified); M0 for use of estimated roots in (A) |
# Question 9:
## Part (i)
$-5.7$ to $-5.8$, $-2.2$ to $-2.3$, $-1$ isw | B1 for 2 correct or for all 3 only stated in coordinate form, ignoring $y$ coordinates | [2]
## Question 9(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1=(x+2)(x^2+7x+7)$ | M1 | Condone missing brackets if expanded correctly; or M1 for correct expansion of $(x+2)(x^2+7x+7)$ |
| Correct completion with at least one interim stage of working to given answer: $x^3+9x^2+21x+13=0$ | A1 | |
| $[x=-1$ is root so] $(x+1)$ is factor | M1 | Implied by division of cubic by $x+1$; condone some confusion of root/factor for this mark if division of cubic by $x+1$ seen |
| Correctly finding other factor as $x^2+8x+13$ | M2 | M1 for correct division of cubic by $(x+1)$ as far as obtaining $x^2+8x$ (may be in grid) or for two correct terms of $x^2+8x+13$ obtained by inspection; allow seen in grid without $+$ signs |
| $\frac{-8\pm\sqrt{8^2-4\times13}}{2}$ oe | M1 | For use of formula, condoning one error, for $x^2+8x+13=0$; or M1 for $(x+4)^2=4^2-13$ oe or further stage, condoning one error |
| $\frac{-8\pm\sqrt{12}}{2}$ isw or $-4\pm\sqrt{3}$ isw and $x=-1$ | A1 | $x=-1$ may be stated earlier; isw wrong simplification or giving as coordinates |
---
## Question 9(iii)A:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Drawing the translated quadratic or showing that the horizontal gap between the relevant parts of the curve is always less than 3 | B1 | Must be a reasonable translation of given quadratic, only intersecting given curve once; intersections with $x$ axis $-3$ to $-2.5$ and $1.5$ to $2$; ignore above $y=1$ |
| Estimated coordinates of the point of intersection ($1.8$ to $2$, $0.2$ to $0.3$) | B1 | |
---
## Question 9(iii)B:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y=x^2+x-5$ or $y=\left(x+\frac{1}{2}\right)^2-\frac{21}{4}$ | 2 | M1 for $[y=](x-3)^2+7(x-3)+7$ oe or for simplified equation with '$y=$' omitted or for $y=(x-a)(x-b)$ where $a$ and $b$ are the values $3+\frac{-7\pm\sqrt{21}}{2}$ oe (may have been wrongly simplified); M0 for use of estimated roots in (A) |
---9 Fig. 9 shows the curves $y = \frac { 1 } { x + 2 }$ and $y = x ^ { 2 } + 7 x + 7$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{2ebf7ad2-638f-4378-b98d-aadd0de4c766-3_1255_1470_434_299}
\captionsetup{labelformat=empty}
\caption{Fig. 9}
\end{center}
\end{figure}
\begin{enumerate}[label=(\roman*)]
\item Use Fig. 9 to estimate graphically the roots of the equation $\frac { 1 } { x + 2 } = x ^ { 2 } + 7 x + 7$.
\item Show that the equation in part (i) may be simplified to $x ^ { 3 } + 9 x ^ { 2 } + 21 x + 13 = 0$. Find algebraically the exact roots of this equation.
\item The curve $y = x ^ { 2 } + 7 x + 7$ is translated by $\binom { 3 } { 0 }$.\\
(A) Show graphically that the translated curve intersects the curve $y = \frac { 1 } { x + 2 }$ at only one point. Estimate the coordinates of this point.\\
(B) Find the equation of the translated curve, simplifying your answer.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C1 2016 Q9 [13]}}