| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Topic | 3x3 Matrices |
| Type | Geometric interpretation of systems |
| Difficulty | Standard +0.8 This question requires computing a 3×3 determinant to show singularity, then solving an underdetermined system using row reduction or substitution, and finally interpreting the solution set geometrically as a line in 3D space. It combines computational matrix skills with geometric understanding, going beyond routine exercises but using standard Further Maths techniques. |
| Spec | 4.03s Consistent/inconsistent: systems of equations4.03t Plane intersection: geometric interpretation |
**Question 4** [2+6 marks]
**(i)**
Attempt at det(**M**) M1
Det $= 0$ *Shown* A1 (Or via full alternative algebraic method)
**[2]**
**(ii)**
$-x+3y+z=1$
$5x-y+2z=16$ B1 (for all three)
$-x+y\phantom{+2z}=-2$
Parametrisation attempt (or equivalent) started: e.g. set $x=\lambda$, then $y=\lambda-2$ M1
Complete attempt: $z = 1+\lambda-3\lambda+6 = 7-2\lambda$ M1
All correct (p.v. and d.v.) … may be in vector line eqn. form: $\mathbf{r} = \begin{pmatrix}0\\-2\\7\end{pmatrix} + \lambda\begin{pmatrix}1\\1\\-2\end{pmatrix}$ A1A1
**[6]**
Alternative method 1: **B1** as above, followed by finding two distinct points on the solution line; e.g. $(2,0,3)$, $(0,-2,7)$ **M1 A1**. Then eqn. of line containing these 2 points **M1 A1** possibly **ft** for line (of intersection) of 3 planes (given by the 3 eqns.) **B1**
Alternative method 2: **B1** as above, followed by: Vector product of any two plane normals **M1A1**. Finding coords. or p.v. of any pt. on line **B1**. Eqn. of line using these results appropriately **B1** for line (of intersection) of 3 planes (given by the 3 eqns.) **B1**4 A $3 \times 3$ system of equations is given by the matrix equation $\left( \begin{array} { r r r } - 1 & 3 & 1 \\ 5 & - 1 & 2 \\ - 1 & 1 & 0 \end{array} \right) \left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \left( \begin{array} { r } 1 \\ 16 \\ - 2 \end{array} \right)$.\\
(i) Show that this system of equations does not have a unique solution.\\
(ii) Solve this system of equations and describe the geometrical significance of the solution.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2016 Q4 [6]}}