Moderate -0.8 This is a straightforward linear combination problem requiring students to expand the vector equation, equate coefficients of i and j, then solve two simultaneous linear equations. It's purely procedural with no conceptual challenges or problem-solving required, making it easier than average but not trivial since it involves algebraic manipulation with parameters.
4 The vectors \(\mathbf { v } _ { 1 }\) and \(\mathbf { v } _ { 2 }\) are defined by \(\mathbf { v } _ { 1 } = 2 \mathrm { a } \mathbf { i } + \mathrm { bj }\) and \(\mathbf { v } _ { 2 } = b \mathbf { i } - 3 \mathbf { j }\) where \(a\) and \(b\) are constants. Given that \(3 \mathbf { v } _ { 1 } + \mathbf { v } _ { 2 } = 22 \mathbf { i } - 9 \mathbf { j }\), find the values of \(a\) and \(b\).
Attempt to scalar multiply \(\mathbf{v}_1\) and add to \(\mathbf{v}_2\). Allow vector expression or 2 separate components
\(6a+b=22\) and \(3b-3=-9\)
M1
Equate coefficients of \(\mathbf{i}\) and \(\mathbf{j}\) to form two equations. Allow if \(\mathbf{i}\) and \(\mathbf{j}\) still seen in every term of these equations
\(a = 4\)
A1
cao
\(b = -2\)
A1
cao
[4]
## Question 4:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $3(2a\mathbf{i}+b\mathbf{j})+(b\mathbf{i}-3\mathbf{j}) [= 22\mathbf{i}-9\mathbf{j}]$ | M1 | Attempt to scalar multiply $\mathbf{v}_1$ and add to $\mathbf{v}_2$. Allow vector expression or 2 separate components |
| $6a+b=22$ and $3b-3=-9$ | M1 | Equate coefficients of $\mathbf{i}$ and $\mathbf{j}$ to form two equations. Allow if $\mathbf{i}$ and $\mathbf{j}$ still seen in every term of these equations |
| $a = 4$ | A1 | cao |
| $b = -2$ | A1 | cao |
| [4] | | |
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4 The vectors $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ are defined by $\mathbf { v } _ { 1 } = 2 \mathrm { a } \mathbf { i } + \mathrm { bj }$ and $\mathbf { v } _ { 2 } = b \mathbf { i } - 3 \mathbf { j }$ where $a$ and $b$ are constants. Given that $3 \mathbf { v } _ { 1 } + \mathbf { v } _ { 2 } = 22 \mathbf { i } - 9 \mathbf { j }$, find the values of $a$ and $b$.
\hfill \mbox{\textit{OCR MEI Paper 1 2024 Q4 [4]}}