OCR MEI FP1 2010 January — Question 4 6 marks

Exam BoardOCR MEI
ModuleFP1 (Further Pure Mathematics 1)
Year2010
SessionJanuary
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
Topic3x3 Matrices
TypeInverse given/derived then solve system
DifficultyModerate -0.3 This is a straightforward Further Maths question requiring verification that MM^(-1) = I to find k, then applying the inverse to solve a system. While it involves 3×3 matrices (a Further Maths topic), the question provides the inverse formula and requires only routine matrix multiplication and arithmetic—no derivation or problem-solving insight needed. Slightly easier than average due to the scaffolding provided.
Spec4.03o Inverse 3x3 matrix4.03r Solve simultaneous equations: using inverse matrix
4 You are given that if \(\mathbf { M } = \left( \begin{array} { r r r } 4 & 0 & 1 \\ - 6 & 1 & 1 \\ 5 & 2 & 5 \end{array} \right)\) then \(\mathbf { M } ^ { - 1 } = \frac { 1 } { k } \left( \begin{array} { r r r } - 3 & - 2 & 1 \\ - 35 & - 15 & 10 \\ 17 & 8 & - 4 \end{array} \right)\).
Find the value of \(k\). Hence solve the following simultaneous equations. $$\begin{aligned} 4 x + z & = 9 \\ - 6 x + y + z & = 32 \\ 5 x + 2 y + 5 z & = 81 \end{aligned}$$
Question 4:
Part 1 - Finding k:
AnswerMarks Guidance
AnswerMark Guidance
Attempt to consider \(\mathbf{MM^{-1}}\) or \(\mathbf{M^{-1}M}\)M1 May be implied
\(\frac{1}{k}\begin{pmatrix}5&0&0\\0&5&0\\0&0&5\end{pmatrix} \Rightarrow k=5\)A1 [2] c.a.o.
Part 2 - Finding x, y, z:
AnswerMarks Guidance
AnswerMark Guidance
Attempt to pre-multiply by \(\mathbf{M^{-1}}\)M1
Attempt to multiply matricesM1
\(\frac{1}{5}\begin{pmatrix}-3&-2&1\\-35&-15&10\\17&8&-4\end{pmatrix}\begin{pmatrix}9\\32\\81\end{pmatrix} = \frac{1}{5}\begin{pmatrix}-10\\15\\85\end{pmatrix}\)A1 Correct
\(\Rightarrow x=-2,\ y=3,\ z=17\)A1 [4] All 3 correct; s.c. B1 if matrices not used 
# Question 4:

**Part 1 - Finding k:**

| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt to consider $\mathbf{MM^{-1}}$ or $\mathbf{M^{-1}M}$ | M1 | May be implied |
| $\frac{1}{k}\begin{pmatrix}5&0&0\\0&5&0\\0&0&5\end{pmatrix} \Rightarrow k=5$ | A1 **[2]** | c.a.o. |

**Part 2 - Finding x, y, z:**

| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt to pre-multiply by $\mathbf{M^{-1}}$ | M1 | |
| Attempt to multiply matrices | M1 | |
| $\frac{1}{5}\begin{pmatrix}-3&-2&1\\-35&-15&10\\17&8&-4\end{pmatrix}\begin{pmatrix}9\\32\\81\end{pmatrix} = \frac{1}{5}\begin{pmatrix}-10\\15\\85\end{pmatrix}$ | A1 | Correct |
| $\Rightarrow x=-2,\ y=3,\ z=17$ | A1 **[4]** | All 3 correct; s.c. B1 if matrices not used |

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4 You are given that if $\mathbf { M } = \left( \begin{array} { r r r } 4 & 0 & 1 \\ - 6 & 1 & 1 \\ 5 & 2 & 5 \end{array} \right)$ then $\mathbf { M } ^ { - 1 } = \frac { 1 } { k } \left( \begin{array} { r r r } - 3 & - 2 & 1 \\ - 35 & - 15 & 10 \\ 17 & 8 & - 4 \end{array} \right)$.\\
Find the value of $k$. Hence solve the following simultaneous equations.

$$\begin{aligned}
4 x + z & = 9 \\
- 6 x + y + z & = 32 \\
5 x + 2 y + 5 z & = 81
\end{aligned}$$

\hfill \mbox{\textit{OCR MEI FP1 2010 Q4 [6]}}
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