CAIE FP1 2012 November — Question 2 4 marks

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2012
SessionNovember
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
Topic3x3 Matrices
TypeParameter values for unique solution
DifficultyStandard +0.3 This is a straightforward Further Maths question requiring students to recognize that a unique solution exists when the determinant of the coefficient matrix is non-zero. The calculation involves a 3×3 determinant with a parameter, followed by solving a simple inequality—standard technique practice with minimal conceptual challenge.
Spec4.03s Consistent/inconsistent: systems of equations
2 Find the set of values of \(a\) for which the system of equations $$\begin{aligned} a x + y + 2 z & = 0 \\ 3 x - 2 y & = 4 \\ 3 x - 4 y - 6 a z & = 14 \end{aligned}$$ has a unique solution.
Question 2:
AnswerMarks Guidance
Working/AnswerMarks Guidance
\(\begin{vmatrix} a & 1 & 2 \\ 3 & -2 & 0 \\ 3 & -4 & -6a \end{vmatrix} \neq 0 \Rightarrow 12a^2 + 18a - 12 \neq 0\)M1A1 Sets determinant \(\neq 0\)
\(\Rightarrow 6(2a-1)(a+2) \neq 0\)M1 Factorises or completes square
\(a \neq \frac{1}{2}\) or \(-2\)A1 Or by row operations 
## Question 2:

| Working/Answer | Marks | Guidance |
|---|---|---|
| $\begin{vmatrix} a & 1 & 2 \\ 3 & -2 & 0 \\ 3 & -4 & -6a \end{vmatrix} \neq 0 \Rightarrow 12a^2 + 18a - 12 \neq 0$ | M1A1 | Sets determinant $\neq 0$ |
| $\Rightarrow 6(2a-1)(a+2) \neq 0$ | M1 | Factorises or completes square |
| $a \neq \frac{1}{2}$ or $-2$ | A1 | Or by row operations |

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2 Find the set of values of $a$ for which the system of equations

$$\begin{aligned}
a x + y + 2 z & = 0 \\
3 x - 2 y & = 4 \\
3 x - 4 y - 6 a z & = 14
\end{aligned}$$

has a unique solution.

\hfill \mbox{\textit{CAIE FP1 2012 Q2 [4]}}
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Q1 4 Q2 4 Q3 5 Q4 6 Q5 6 Q6 7 Q7 8 Q8 9 Q9 12 Q10 12 Q11 13 Q12 EITHER Q12 OR