Challenging +1.2 This is a standard Further Pure 1 question on solving systems with parameters. Students must apply two constraints (sum = 6, sum of squares = 26) to the general solution form to find specific values of λ and μ. While it requires careful algebraic manipulation and understanding of solution spaces, it follows a predictable template for this topic with clear steps and no novel insight required.
has the form \(\left( \begin{array} { r } 1 \\ - 2 \\ 2 \\ - 1 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 }\), where \(\lambda\) and \(\mu\) are scalars and \(\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}\) is a basis for \(K\).
Hence obtain a solution \(\mathbf { x } ^ { \prime }\) of ( \(*\) ) such that the sum of the components \(\mathbf { x } ^ { \prime }\) is 6 and the sum of the squares of the components of \(\mathbf { x } ^ { \prime }\) is 26 .
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has the form $\left( \begin{array} { r } 1 \\ - 2 \\ 2 \\ - 1 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 }$, where $\lambda$ and $\mu$ are scalars and $\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}$ is a basis for $K$.
Hence obtain a solution $\mathbf { x } ^ { \prime }$ of ( $*$ ) such that the sum of the components $\mathbf { x } ^ { \prime }$ is 6 and the sum of the squares of the components of $\mathbf { x } ^ { \prime }$ is 26 .
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\hfill \mbox{\textit{CAIE FP1 2016 Q15}}