| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2019 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Parameter values for unique solution |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring determinant calculation with parameters, finding singular values, then analyzing solution existence for each case. While the determinant calculation is routine, part (c) requires systematic investigation of consistency/inconsistency for singular matrices, which goes beyond standard A-level and requires careful reasoning about linear systems. |
| Spec | 4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation4.03r Solve simultaneous equations: using inverse matrix4.03s Consistent/inconsistent: systems of equations |
| Answer | Marks | Guidance |
|---|---|---|
| 10 | (a) | detA=a2 −10a+16 |
| Answer | Marks |
|---|---|
| [2] | 1.1a |
| 1.1 | Attempt to work out the |
| Answer | Marks | Guidance |
|---|---|---|
| (b) | a2 −10a+16=0⇒( a−2 )( a−8 )=0⇒a =2, 8 | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1a |
| 1.1 | Solving their quadratic |
| Answer | Marks |
|---|---|
| (c) | For both values there is no unique solution as detA = 0 |
| Answer | Marks |
|---|---|
| and 16 ≠7 so no solution | B1 |
| Answer | Marks |
|---|---|
| [7] | 2.4 |
| Answer | Marks |
|---|---|
| 2.2a | Soi by correct answers |
| Answer | Marks |
|---|---|
| of their values and solve | “correct answers” |
Question 10:
10 | (a) | detA=a2 −10a+16 | M1
A1
[2] | 1.1a
1.1 | Attempt to work out the
determinant
(b) | a2 −10a+16=0⇒( a−2 )( a−8 )=0⇒a =2, 8 | M1
A1
[2] | 1.1a
1.1 | Solving their quadratic
soi
(c) | For both values there is no unique solution as detA = 0
For a=2, equations are:
p : 2x+2y =6
1
p : 2y+2z =8
2
p : 4x+5y+z =16
3
1
2p + p = p
1 2 2 3
So there is an infinite set of solutions.
For a =8, equations are:
p :8x+2y =6
1
p :8y+2z =8
2
p : 4x+5y+z =16
3
1 1
p + p ≠ p as it gives 4x+5y+z =7
2 1 2 2 3
and 16 ≠7 so no solution | B1
M1
A1
A1
M1
A1
A1
[7] | 2.4
2.1
1.1
2.2a
2.1
1.1
2.2a | Soi by correct answers
Substitute one of their
values and solve
Substitute the other one
of their values and solve | “correct answers”
means solns are either
infinite or non-
existent.10 You are given the matrix $\mathbf { A }$ where $\mathbf { A } = \left( \begin{array} { l l l } a & 2 & 0 \\ 0 & a & 2 \\ 4 & 5 & 1 \end{array} \right)$.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $a$, the determinant of $\mathbf { A }$, simplifying your answer.
\item Hence find the values of $a$ for which $\mathbf { A }$ is singular.
You are given the following equations which are to be solved simultaneously.
$$\begin{aligned}
a x + 2 y & = 6 \\
a y + 2 z & = 8 \\
4 x + 5 y + z & = 16
\end{aligned}$$
\item For each of the values of $a$ found in part (b) determine whether the equations have
\begin{itemize}
\item a unique solution, which should be found, or
\item an infinite set of solutions or
\item no solution.
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2019 Q10 [11]}}