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G25Vectors

Edexcel GCSE Maths (1MA1), Higher tier · Geometry and measures

Practise Vectors. 1 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.

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Sample questions

Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.

Question 1Easy3 marks
\(\mathbf{a} = \begin{pmatrix} -3 \\ -5 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}\)
(a) Work out \(\mathbf{a} - \mathbf{b}\) as a column vector.[1]
(b) Work out \(2\mathbf{a} - 3\mathbf{b}\) as a column vector.[2]
Show the answer and mark scheme
(a) Answer: \(\begin{pmatrix} -4 \\ -6 \end{pmatrix}\)
  • B1 for \(\begin{pmatrix} -4 \\ -6 \end{pmatrix}\)

Worked solution: \(\begin{pmatrix} -3 \\ -5 \end{pmatrix} - \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} -4 \\ -6 \end{pmatrix}\)

(b) Answer: \(\begin{pmatrix} -9 \\ -13 \end{pmatrix}\)
  • M1 for \(2\mathbf{a} = \begin{pmatrix} -6 \\ -10 \end{pmatrix}\) or \(-3\mathbf{b} = \begin{pmatrix} -3 \\ -3 \end{pmatrix}\)
  • A1 for \(\begin{pmatrix} -9 \\ -13 \end{pmatrix}\)

Worked solution: \(2\mathbf{a} - 3\mathbf{b} = \begin{pmatrix} -6 \\ -10 \end{pmatrix} + \begin{pmatrix} -3 \\ -3 \end{pmatrix} = \begin{pmatrix} -9 \\ -13 \end{pmatrix}\)

Question 2Medium5 marks
\(\mathbf{a} = \begin{pmatrix} 3 \\ 6 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\)
(a) The vector \(\mathbf{c}\) is such that \(4\mathbf{a} + \mathbf{c} = 2\mathbf{b}\)
Work out \(\mathbf{c}\) as a column vector.[2]
(b) \(m\mathbf{a} + n\mathbf{b} = \begin{pmatrix} -12 \\ -20 \end{pmatrix}\)
Find the value of \(m\) and the value of \(n\).[3]
Show the answer and mark scheme
(a) Answer: \(\mathbf{c} = \begin{pmatrix} -6 \\ -16 \end{pmatrix}\)
  • M1 for \(\mathbf{c} = 2\mathbf{b} - 4\mathbf{a}\) or for \(4\mathbf{a} = \begin{pmatrix} 12 \\ 24 \end{pmatrix}\) and \(2\mathbf{b} = \begin{pmatrix} 6 \\ 8 \end{pmatrix}\)
  • A1 for \(\begin{pmatrix} -6 \\ -16 \end{pmatrix}\)

Worked solution: \(\mathbf{c} = 2\mathbf{b} - 4\mathbf{a} = \begin{pmatrix} 6 \\ 8 \end{pmatrix} - \begin{pmatrix} 12 \\ 24 \end{pmatrix} = \begin{pmatrix} -6 \\ -16 \end{pmatrix}\)

(b) Answer: \(m = -2, \ n = -2\)
  • M1 for two correct equations, e.g. \(3m + 3n = -12\) and \(6m + 4n = -20\)
  • M1 for a correct method to eliminate one variable
  • A1 for m = −2 and n = −2

Worked solution: Top: \(3m + 3n = -12\)
Bottom: \(6m + 4n = -20\)
Solving simultaneously: \(m = -2\), \(n = -2\)
Check: \(-2\mathbf{a} - 2\mathbf{b} = \begin{pmatrix} -12 \\ -20 \end{pmatrix}\)

Question 3Hard4 marks
OPR and OQS are straight lines.
\(\overrightarrow{OP} = \mathbf{s}\) and \(\overrightarrow{OQ} = \mathbf{t}\).
\(\overrightarrow{OR} = 3\mathbf{s}\) and \(\overrightarrow{OS} = 3\mathbf{t}\).
Diagram NOT accurately drawn
[object Object]
(a) Prove that PQSR is a trapezium.[3]
(b) Write down the ratio of the length of PQ to the length of RS.[1]
Show the answer and mark scheme
(a) Answer: \(\overrightarrow{PQ} = \mathbf{t} - \mathbf{s}\) and \(\overrightarrow{RS} = 3\mathbf{t} - 3\mathbf{s} = 3(\mathbf{t} - \mathbf{s})\), so \(\overrightarrow{RS} = 3\overrightarrow{PQ}\): PQ is parallel to RS and 3 times as long, so PQSR is a trapezium.
  • M1 for \(\overrightarrow{PQ} = \mathbf{t} - \mathbf{s}\) or \(\overrightarrow{RS} = 3\mathbf{t} - 3\mathbf{s}\)
  • M1 for \(\overrightarrow{RS} = 3(\mathbf{t} - \mathbf{s})\) or \(3\overrightarrow{PQ}\)
  • C1 for a complete proof: \(\overrightarrow{RS}\) is a multiple of \(\overrightarrow{PQ}\), so PQ is parallel to RS, and they are not equal in length, so PQSR is a trapezium

Worked solution: \(\overrightarrow{PQ} = \overrightarrow{PO} + \overrightarrow{OQ} = -\mathbf{s} + \mathbf{t} = \mathbf{t} - \mathbf{s}\)
\(\overrightarrow{RS} = \overrightarrow{RO} + \overrightarrow{OS} = -3\mathbf{s} + 3\mathbf{t} = 3(\mathbf{t} - \mathbf{s}) = 3\overrightarrow{PQ}\)
So RS is parallel to PQ and 3 times as long: PQSR has one pair of parallel sides of different lengths, so it is a trapezium.

(b) Answer: 1 : 3
  • B1 for 1 : 3 oe

Worked solution: \(\overrightarrow{RS} = 3\overrightarrow{PQ}\), so RS is 3 times as long as PQ: PQ : RS = 1 : 3.

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