| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Parameter values for unique solution |
| Difficulty | Standard +0.3 This is a standard FP1 question testing routine determinant calculation and understanding of when systems have unique solutions. Part (i) is mechanical 3×3 determinant expansion, part (ii) requires recognizing that a=3 makes det=0 (from factoring the quadratic) so no unique solution exists, then checking consistency. These are textbook exercises requiring no novel insight, though slightly above average difficulty due to being Further Maths content. |
| Spec | 4.02h Square roots: of complex numbers4.03j Determinant 3x3: calculation4.03s Consistent/inconsistent: systems of equations |
| Answer | Marks | Guidance |
|---|---|---|
| \(a^2 - 8a + 15\) | M1 | Show correct expansion process for \(3 \times 3\), condone sign errors |
| M1 | Correct expansion for any \(2 \times 2\), may use Cramer's rule which is M2 | |
| A1 | Obtain given answer correctly |
| Answer | Marks | Guidance |
|---|---|---|
| \(\det \mathbf{X} = 3^2 - 3 \times 8 + 15 = 0\) | B1 | And must state not a unique solution or equivalent |
| e.g. \(3y + 5z = 5\), or \(3x - 7z = -4\), or \(5x + 7y = 5\) | M1 | Attempt to solve equations |
| A1 | Find a repeated equation and state consistency; N.B. They may solve the equations first then deduce consistent and non-unique and gets 3/3 (possibly) |
# Question 9:
## Part (i):
$a^2 - 8a + 15$ | M1 | Show correct expansion process for $3 \times 3$, condone sign errors
| M1 | Correct expansion for any $2 \times 2$, may use Cramer's rule which is M2
| A1 | Obtain **given** answer correctly
**[3]**
## Part (ii):
$\det \mathbf{X} = 3^2 - 3 \times 8 + 15 = 0$ | B1 | And must state not a unique solution or equivalent
e.g. $3y + 5z = 5$, or $3x - 7z = -4$, or $5x + 7y = 5$ | M1 | Attempt to solve equations
| A1 | Find a repeated equation and state consistency; N.B. They may solve the equations first then deduce consistent and non-unique and gets 3/3 (possibly)
**[3]**
---9 (i) The matrix $\mathbf { X }$ is given by $\mathbf { X } = \left( \begin{array} { r r r } a & 3 & - 2 \\ 0 & a & 5 \\ 1 & 2 & 1 \end{array} \right)$. Show that the determinant of $\mathbf { X }$ is $a ^ { 2 } - 8 a + 15$.\\
(ii) Explain briefly why the equations
$$\begin{array} { r }
3 x + 3 y - 2 z = 1 \\
3 y + 5 z = 5 \\
x + 2 y + z = 2
\end{array}$$
do not have a unique solution and determine whether these equations are consistent or inconsistent.\\
(i) Use an algebraic method to find the square roots of the complex number $9 + 40 \mathrm { i }$.\\
(ii) Show that $9 + 40 \mathrm { i }$ is a root of the quadratic equation $z ^ { 2 } - 18 z + 1681 = 0$.\\
(iii) By using the substitution $z = \frac { 1 } { u ^ { 2 } }$, find the roots of the equation $1681 u ^ { 4 } - 18 u ^ { 2 } + 1 = 0$. Give your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
\hfill \mbox{\textit{OCR FP1 2016 Q9 [6]}}