| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discriminant and conditions for roots |
| Type | Find range for no real roots |
| Difficulty | Moderate -0.3 This is a standard C1 completing-the-square question with routine discriminant application. Part (a) is mechanical algebra, part (b) requires setting b²-4ac < 0 and solving a quadratic inequality (standard technique), and part (c) is basic graph sketching. Slightly easier than average due to straightforward algebraic manipulation and well-practiced techniques, though the inequality solving in (b) requires some care. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((x+2k)^2\) or \(\left(x+\frac{4k}{2}\right)^2\) | M1 | Attempt at completing the square |
| \((x \pm F)^2 \pm G \pm 3 \pm 11k\) where \(F\) and \(G\) are any functions of \(k\), not involving \(x\) | M1 | |
| \((x+2k)^2 - 4k^2 + (3+11k)\) | A1 | Accept unsimplified equivalents such as \(\left(x+\frac{4k}{2}\right)^2 - \left(\frac{4k}{2}\right)^2 + 3 + 11k\), and i.s.w. if necessary |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(4k^2 - 11k - 3 = 0 \Rightarrow (4k+1)(k-3)=0 \Rightarrow k = \ldots\) | M1 | Forming and solving a 3-term quadratic in \(k\). Or 'starting again', \(b^2-4ac=(4k)^2-4(3+11k)\) and proceed to \(k=\ldots\) |
| \(-\frac{1}{4}\) and \(3\) | A1 | Ignore any inequalities for the first 2 marks in (b) |
| Using \(b^2-4ac < 0\) for no real roots, i.e. \(4k^2-11k-3<0\), to establish inequalities involving their two critical values \(m\) and \(n\) (even if the inequalities are wrong, e.g. \(k < m,\ k < n\)) | M1 | |
| \(-\frac{1}{4} < k < 3\) | A1ft | Follow through on their critical values. Final A1ft still scored if \(m < k < n\) follows \(k < m,\ k < n\). Using \(x\) instead of \(k\) in the final answer loses only the 2nd A mark (condone use of \(x\) in earlier working) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Shape \(\cup\) (seen in (c)) | B1 | |
| Minimum in correct quadrant, not touching the \(x\)-axis, not on the \(y\)-axis, and there must be no other minimum or maximum | B1 | |
| \((0, 14)\) or \(14\) on \(y\)-axis. Allow \((14, 0)\) marked on \(y\)-axis | B1 | Final B1 dependent upon a sketch having been attempted in part (c). Note: minimum is at \((-2, 10)\) but no mark for this |
## Question 10:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(x+2k)^2$ or $\left(x+\frac{4k}{2}\right)^2$ | M1 | Attempt at completing the square |
| $(x \pm F)^2 \pm G \pm 3 \pm 11k$ where $F$ and $G$ are any functions of $k$, not involving $x$ | M1 | |
| $(x+2k)^2 - 4k^2 + (3+11k)$ | A1 | Accept unsimplified equivalents such as $\left(x+\frac{4k}{2}\right)^2 - \left(\frac{4k}{2}\right)^2 + 3 + 11k$, and i.s.w. if necessary |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $4k^2 - 11k - 3 = 0 \Rightarrow (4k+1)(k-3)=0 \Rightarrow k = \ldots$ | M1 | Forming and solving a 3-term quadratic in $k$. Or 'starting again', $b^2-4ac=(4k)^2-4(3+11k)$ and proceed to $k=\ldots$ |
| $-\frac{1}{4}$ and $3$ | A1 | Ignore any inequalities for the first 2 marks in (b) |
| Using $b^2-4ac < 0$ for no real roots, i.e. $4k^2-11k-3<0$, to establish inequalities involving their two critical values $m$ and $n$ (even if the inequalities are wrong, e.g. $k < m,\ k < n$) | M1 | |
| $-\frac{1}{4} < k < 3$ | A1ft | Follow through on their critical values. Final A1ft still scored if $m < k < n$ follows $k < m,\ k < n$. Using $x$ instead of $k$ in the final answer loses only the 2nd A mark (condone use of $x$ in earlier working) |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Shape $\cup$ (seen in (c)) | B1 | |
| Minimum in correct quadrant, not touching the $x$-axis, not on the $y$-axis, and there must be no other minimum or maximum | B1 | |
| $(0, 14)$ or $14$ on $y$-axis. Allow $(14, 0)$ marked on $y$-axis | B1 | Final B1 dependent upon a sketch having been attempted in part (c). Note: minimum is at $(-2, 10)$ but no mark for this |10.
$$\mathrm { f } ( x ) = x ^ { 2 } + 4 k x + ( 3 + 11 k ) , \quad \text { where } k \text { is a constant. }$$
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in the form $( x + p ) ^ { 2 } + q$, where $p$ and $q$ are constants to be found in terms of $k$.
Given that the equation $\mathrm { f } ( x ) = 0$ has no real roots,
\item find the set of possible values of $k$.
Given that $k = 1$,
\item sketch the graph of $y = \mathrm { f } ( x )$, showing the coordinates of any point at which the graph crosses a coordinate axis.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2010 Q10 [10]}}