| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Graphical equation solving with auxiliary line |
| Difficulty | Moderate -0.3 This is a multi-part C1 question involving standard curve sketching, graphical equation solving, and algebraic manipulation. While it has many parts (9 marks total), each individual step is routine: drawing a line, reading intersections, algebraic rearrangement, factor theorem, and basic transformations. The techniques are all standard C1 content with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials1.02o Sketch reciprocal curves: y=a/x and y=a/x^21.02q Use intersection points: of graphs to solve equations1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(y = 2x + 5\) drawn | M1 | condone unruled and some doubling; tolerance: must pass within/touch at least two circles on overlay; line must be drawn long enough to intersect curve at least twice |
| \(-2, -1.4\) to \(-1.2, 0.7\) to \(0.85\) | A2 | A1 for two of these correct; condone coordinates or factors |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x^3 + 5x^2\) or \(2x + 5 - \frac{4}{x^2} = 0\) and completion to given answer | B1 | condone omission of final \(= 0\) |
| \(f(-2) = -16 + 20 - 4 = 0\) | B1 | or correct division/inspection showing that \(x+2\) is factor |
| Use of \(x+2\) as factor in long division of given cubic as far as \(2x^3 + 4x^2\) in working | M1 | or inspection or equating coefficients, with at least two terms correct; may be set out in grid format |
| \(2x^2 + x - 2\) obtained | A1 | condone omission of \(+\) sign (e.g. in grid format) |
| \([x =] \dfrac{-1 \pm \sqrt{1^2 - 4 \times 2 \times -2}}{2 \times 2}\) oe | M1 | dep on previous M1 earned; for attempt at formula or full attempt at completing square, using their other factor; not more than two errors in formula/substitution/completing square; allow even if their 'factor' has a remainder shown in working; M0 for just an attempt to factorise |
| \(\dfrac{-1 \pm \sqrt{17}}{4}\) oe isw | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{4}{x^2} = x + 2\) or \(y = x + 2\) soi | M1 | e.g. is earned by correct line drawn; condone intent for line; allow slightly out of tolerance |
| \(y = x + 2\) drawn | A1 | condone unruled; need drawn for \(-1.5 \leq x \leq 1.2\); to pass through/touch relevant circle(s) on overlay |
| 1 real root | A1 |
## Question 3(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $y = 2x + 5$ drawn | M1 | condone unruled and some doubling; tolerance: must pass within/touch at least two circles on overlay; line must be drawn long enough to intersect curve at least twice |
| $-2, -1.4$ to $-1.2, 0.7$ to $0.85$ | A2 | A1 for two of these correct; condone coordinates or factors |
---
## Question 3(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $x^3 + 5x^2$ or $2x + 5 - \frac{4}{x^2} = 0$ and completion to given answer | B1 | condone omission of final $= 0$ |
| $f(-2) = -16 + 20 - 4 = 0$ | B1 | or correct division/inspection showing that $x+2$ is factor |
| Use of $x+2$ as factor in long division of given cubic as far as $2x^3 + 4x^2$ in working | M1 | or inspection or equating coefficients, with at least two terms correct; may be set out in grid format |
| $2x^2 + x - 2$ obtained | A1 | condone omission of $+$ sign (e.g. in grid format) |
| $[x =] \dfrac{-1 \pm \sqrt{1^2 - 4 \times 2 \times -2}}{2 \times 2}$ oe | M1 | dep on previous M1 earned; for attempt at formula or full attempt at completing square, using their other factor; not more than two errors in formula/substitution/completing square; allow even if their 'factor' has a remainder shown in working; M0 for just an attempt to factorise |
| $\dfrac{-1 \pm \sqrt{17}}{4}$ oe isw | A1 | |
---
## Question 3(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{4}{x^2} = x + 2$ or $y = x + 2$ soi | M1 | e.g. is earned by correct line drawn; condone intent for line; allow slightly out of tolerance |
| $y = x + 2$ drawn | A1 | condone unruled; need drawn for $-1.5 \leq x \leq 1.2$; to pass through/touch relevant circle(s) on overlay |
| 1 real root | A1 | |
---3
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{6be6c0b0-76b7-49c0-bf1b-dc6f8f79981b-2_836_906_361_675}
\captionsetup{labelformat=empty}
\caption{Fig. 12}
\end{center}
\end{figure}
Fig. 12 shows the graph of $y = \frac { 4 } { x ^ { 2 } }$.
\begin{enumerate}[label=(\roman*)]
\item On the copy of Fig. 12, draw accurately the line $y = 2 x + 5$ and hence find graphically the three roots of the equation $\frac { 4 } { x ^ { 2 } } = 2 x + 5$.\\[0pt]
[3]
\item Show that the equation you have solved in part (i) may be written as $2 x ^ { 3 } + 5 x ^ { 2 } - 4 = 0$. Verify that $x = - 2$ is a root of this equation and hence find, in exact form, the other two roots.\\[0pt]
[6]
\item By drawing a suitable line on the copy of Fig. 12, find the number of real roots of the equation $x ^ { 3 } + 2 x ^ { 2 } - 4 = 0$.
\item You are given that $\mathrm { f } ( x ) = ( 2 x - 5 ) ( x - 1 ) ( x - 4 )$.\\
(A) Sketch the graph of $y = \mathrm { f } ( x )$.\\
(B) Show that $\mathrm { f } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 20$.
\item You are given that $\mathrm { g } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 40$.\\
(A) Show that $\mathrm { g } ( 5 ) = 0$.\\
(B) Express $\mathrm { g } ( x )$ as the product of a linear and quadratic factor.\\
(C) Hence show that the equation $\mathrm { g } ( x ) = 0$ has only one real root.
\item Describe fully the transformation that maps $y = \mathrm { f } ( x )$ onto $y = \mathrm { g } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C1 Q3 [12]}}