| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 2 (Further Pure Core 2) |
| Year | 2018 |
| Session | March |
| Marks | 12 |
| Topic | 3x3 Matrices |
| Type | Geometric interpretation of systems |
| Difficulty | Challenging +1.2 This is a structured Further Maths question on systems of linear equations requiring matrix methods (likely row reduction) and geometric interpretation. While it involves multiple parts and concepts (solving systems, determining singularity conditions, linear combinations, and geometric interpretation of planes), each step is fairly guided and uses standard techniques. The linear combination in (iii)(a) is straightforward coefficient matching, and the overall problem follows a predictable path for FM students familiar with singular matrices and plane geometry. |
| Spec | 4.03r Solve simultaneous equations: using inverse matrix4.03s Consistent/inconsistent: systems of equations |
| Answer | Marks | Guidance |
|---|---|---|
| \(\begin{pmatrix}3 & 2 & -1\\2 & -4 & 7\\10 & 20 & -25\end{pmatrix}^{-1} = \frac{1}{40}\begin{pmatrix}-40 & 30 & 10\\120 & -65 & -23\\80 & -40 & -16\end{pmatrix}\) | M1 | BC |
| \(\frac{1}{40}\begin{pmatrix}-40 & 30 & 10\\120 & -65 & -23\\80 & -40 & -16\end{pmatrix}\begin{pmatrix}5\\60\\b\end{pmatrix} = \begin{pmatrix}40 + \frac{1}{4}b\\\frac{3300+23b}{40}\\-50-\frac{2b}{5}\end{pmatrix}\) | A1 | For any correct component |
| \(x = 40 + \frac{1}{4}b, y = -\frac{3300+23b}{40}, z = -50-\frac{2b}{5}\) | A1 | All three correct (oe) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\begin{vmatrix}3 & 2 & -1\\2 & -4 & 7\\a & 20 & -25\end{vmatrix}\) | M1 | Consideration of determinant of correct matrix and genuine attempt to find determinant. |
| \(= 3(100 - 140) - 2(-50 - 7a) - l(40 + 4a)\) | A1 | |
| \(10a - 60\) | A1 | |
| \(a = 6\) | A1 [3] | Set to 0 and solve |
| Answer | Marks |
|---|---|
| \(2\alpha - 4\beta = 20\) or \(-\alpha + 7\beta = -25\) | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(4 \times 5 + (-3) \times 60\) | M1 | Their \(\alpha, \beta\). |
| \(-160\) | A1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| The three planes form a sheaf with a common line of intersection oe | EI [1] | Do not ISW (eg if "or two of the planes may be parallel" then E0). |
## 7(i)
$\begin{pmatrix}3 & 2 & -1\\2 & -4 & 7\\10 & 20 & -25\end{pmatrix}^{-1} = \frac{1}{40}\begin{pmatrix}-40 & 30 & 10\\120 & -65 & -23\\80 & -40 & -16\end{pmatrix}$ | M1 | BC | eg $\begin{pmatrix}-1 & \frac{3}{4} & \frac{1}{4}\\\frac{3}{1} & -\frac{13}{8} & -\frac{23}{40}\\2 & -1 & -\frac{2}{5}\end{pmatrix}$
$\frac{1}{40}\begin{pmatrix}-40 & 30 & 10\\120 & -65 & -23\\80 & -40 & -16\end{pmatrix}\begin{pmatrix}5\\60\\b\end{pmatrix} = \begin{pmatrix}40 + \frac{1}{4}b\\\frac{3300+23b}{40}\\-50-\frac{2b}{5}\end{pmatrix}$ | A1 | For any correct component
$x = 40 + \frac{1}{4}b, y = -\frac{3300+23b}{40}, z = -50-\frac{2b}{5}$ | A1 | All three correct (oe)
[3]
## 7(ii)
$\begin{vmatrix}3 & 2 & -1\\2 & -4 & 7\\a & 20 & -25\end{vmatrix}$ | M1 | Consideration of determinant of correct matrix and genuine attempt to find determinant.
$= 3(100 - 140) - 2(-50 - 7a) - l(40 + 4a)$ | A1
$10a - 60$ | A1
$a = 6$ | A1 [3] | Set to 0 and solve
## 7(iii)(a)
$2\alpha - 4\beta = 20$ or $-\alpha + 7\beta = -25$ | M1
## 7(iii)(b)
$4 \times 5 + (-3) \times 60$ | M1 | Their $\alpha, \beta$.
$-160$ | A1 [2]
## 7(iii)(c)
The three planes form a sheaf with a common line of intersection oe | EI [1] | Do not ISW (eg if "or two of the planes may be parallel" then E0).
---In the following set of simultaneous equations, $a$ and $b$ are constants.
\begin{align}
3x + 2y - z &= 5\\
2x - 4y + 7z &= 60\\
ax + 20y - 25z &= b
\end{align}
\begin{enumerate}[label=(\roman*)]
\item In the case where $a = 10$, solve the simultaneous equations, giving your solution in terms of $b$. [3]
\item Determine the value of $a$ for which there is no unique solution for $x$, $y$ and $z$. [3]
\item \begin{enumerate}[label=(\alph*)]
\item Find the values of $\alpha$ and $\beta$ for which $\alpha(2y - z) + \beta(-4y + 7z) = 20y - 25z$ for any $y$ and $z$. [3]
\item Hence, for the case where there is no unique solution for $x$, $y$ and $z$, determine the value of $b$ for which there is an infinite number of solutions. [2]
\item When $a$ takes the value in part (ii) and $b$ takes the value in part (iii)(b) describe the geometrical arrangement of the planes represented by the three equations. [1]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 2018 Q7 [12]}}