A LevelMathsEdexcelPureExam QuestionsVectorsVectors in 2D

Vectors in 2D (Edexcel A Level Maths: Pure): Exam Questions

Exam code: 9MA0

4 hours42 questions
Easy
Medium
Hard
Very Hard
1a
1 mark

Given that

\mathbf{a} = 3\mathbf{i} - 5\mathbf{j}

\mathbf{b} = -\mathbf{i} + 3\mathbf{j}

Find \mathbf{a} + \mathbf{b}.

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1b
1 mark

Find 5\mathbf{a}.

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1c
2 marks

Find 3\mathbf{a} - 2\mathbf{b}.

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1d
2 marks

Find an expression for \mathbf{a} - t\mathbf{b} in terms of t, where t is a constant. Give your answer in the form (p + qt)\mathbf{i} + (r + st)\mathbf{j} where p, q, r and s are integers to be found.

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2a
2 marks

Figure 1 shows a sketch of triangle ABC.

q2-11-1-vectors-in-2-dimensions-easy-a-level-maths-pure
Figure 1

Given that

\overset{\rightarrow}{AB} = 5\mathbf{i} + \mathbf{j}

\overset{\rightarrow}{AC} = 3\mathbf{i} - 2\mathbf{j}

Find \overset{\rightarrow}{BC} in terms of \mathbf{i} and \mathbf{j}.

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2b
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2 marks

Find the exact value of |\overset{\rightarrow}{BC}|.

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3a
3 marks

Given that

\mathbf{a} = 2\mathbf{i} + 4\mathbf{j}

\mathbf{b} = 3\mathbf{i} + p\mathbf{j}

\mathbf{c} = q\mathbf{i} - 2\mathbf{j}

\mathbf{d} = 6\mathbf{i} - 2\mathbf{j}

where p and q are constants.

Given that \mathbf{a} - 2\mathbf{b} = 3\mathbf{c}, find the value of p and the value of q.

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3b
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2 marks

Find the exact value of |\mathbf{d}|.

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4
3 marks

Relative to a fixed origin O, the points A, B and C have position vectors

\overset{\rightarrow}{OA} = 3\mathbf{i} + 4\mathbf{j}

\overset{\rightarrow}{OB} = -5\mathbf{i}

\overset{\rightarrow}{OC} = -8\mathbf{i} - 6\mathbf{j}

On a single set of coordinate axes, sketch the position vectors \overset{\rightarrow}{OA}, \overset{\rightarrow}{OB} and \overset{\rightarrow}{OC}.

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5
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3 marks

The vectors \mathbf{a}, \mathbf{b} and \mathbf{c} are given by

\mathbf{a} = \begin{pmatrix} 3 \\ -p \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} p \\ 4 \end{pmatrix}, \quad \mathbf{c} = \begin{pmatrix} 9 \\ 3 \end{pmatrix}

where p is a constant.

Given that \mathbf{a} + \mathbf{b} is parallel to \mathbf{c}, find the value of p.

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6a
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2 marks

Given that

\overset{\rightarrow}{AB} = 6\mathbf{i} + 3\mathbf{j}

Find |\overset{\rightarrow}{AB}|, giving your answer in the form p\sqrt{q}, where p and q are integers to be found.

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6b
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2 marks

Find the angle between \overset{\rightarrow}{AB} and the positive x-axis, giving your answer in degrees to one decimal place.

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7
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2 marks

Relative to a fixed origin O, the point A has position vector

\overset{\rightarrow}{OA} = \begin{pmatrix} -4 \\ 3 \end{pmatrix}

Find a unit vector in the direction of \overset{\rightarrow}{OA}.

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8
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4 marks

[In this question, \mathbf{i} is a unit vector due east and \mathbf{j} is a unit vector due north. Position vectors are given relative to a fixed origin O.]

A ship sails from the origin O on a bearing of 060^{\circ} for a distance of 400 km to the point P.

Find the position vector of P relative to O, giving your answer in the form (x\mathbf{i} + y\mathbf{j}) km, where x and y are exact values.

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9
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2 marks

Relative to a fixed origin O, the point A has position vector \mathbf{a} = 3\mathbf{i} - 7\mathbf{j} and the point B has position vector \mathbf{b} = -3\mathbf{i} + \mathbf{j}.

Find the distance AB.

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10
2 marks

A force \mathbf{F} acts on a particle, where

\mathbf{F} = (p\mathbf{i} + 2p\mathbf{j})\text{ N}

where p is a positive constant.

Find the magnitude of \mathbf{F}, giving your answer in exact form in terms of p.

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1a
2 marks

The triangle P Q R is such that P Q equals 3 bold i plus 5 bold j and P R equals 13 bold i minus 15 bold j

Find stack Q R with rightwards arrow on top

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1b
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2 marks

Hence find open vertical bar stack Q R with rightwards arrow on top close vertical bar giving your answer as a simplified surd.

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1c
2 marks

The point S lies on the line segment Q R so that Q S colon S R equals 3 colon 2

Find stack P S with rightwards arrow on top

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2a
3 marks

Figure 1 shows a sketch of triangle ABC.

q2-11-1-vectors-in-2-dimensions-easy-a-level-maths-pure
Figure 1

Given that

\overset{\rightarrow}{AB} = 7\mathbf{i} + \mathbf{j}

\overset{\rightarrow}{AC} = 4\mathbf{i} - 3\mathbf{j}

(i) Write down \overset{\rightarrow}{CA} in terms of \mathbf{i} and \mathbf{j}.

(ii) Find \overset{\rightarrow}{BC} in terms of \mathbf{i} and \mathbf{j}.

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2b
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2 marks

Find the exact value of |\overset{\rightarrow}{BC}|.

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3a
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3 marks

The vectors \mathbf{a}, \mathbf{b} and \mathbf{c} are given by

\mathbf{a} = \begin{pmatrix} 7 \\ 2 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} m \\ -3 \end{pmatrix}, \quad \mathbf{c} = \begin{pmatrix} 5 \\ n \end{pmatrix}

where m and n are constants.

Given that \mathbf{a} + 2\mathbf{b} = \mathbf{c}, find the value of m and the value of n.

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3b
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2 marks

The vector \mathbf{d} is given by

\mathbf{d} = \begin{pmatrix} -5 \\ k \end{pmatrix}

where k is a constant.

Given that |\mathbf{d}| = 15, find the two possible values of k, giving your answers as simplified surds.

How did you do?

4
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3 marks

Relative to a fixed origin O, the point A has position vector 2k\mathbf{i} + 7k\mathbf{j}, where k is a constant.

The point A lies on the straight line l with equation

y = 3x + 5

Find the value of k, and hence determine the coordinates of A.

How did you do?

5
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4 marks

The vectors \mathbf{a}, \mathbf{b} and \mathbf{c} are given by

\mathbf{a} = \begin{pmatrix} -5 \\ 17 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} k \\ 5 \end{pmatrix}, \quad \mathbf{c} = \begin{pmatrix} 9 \\ -29 \end{pmatrix}

where k is a constant.

Given that \mathbf{a} - \mathbf{b} is parallel to \mathbf{b} + \mathbf{c}, find the value of k.

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6a
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3 marks

The vector \overset{\rightarrow}{AB} is given by

\overset{\rightarrow}{AB} = 11\mathbf{i} - 2\mathbf{j}

Find

(i) the exact value of |\overset{\rightarrow}{AB}|,

(ii) the angle between \overset{\rightarrow}{AB} and the positive x-axis, giving your answer in degrees to two decimal places.

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6b
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2 marks

Find a unit vector in the direction of \overset{\rightarrow}{AB}, giving your answer in its simplest exact form.

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7
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4 marks

[In this question, \mathbf{i} is a unit vector due east and \mathbf{j} is a unit vector due north.]

A ship leaves a port at the fixed origin O and travels 300 km on a bearing of 120°. It then travels 500 km due south before dropping anchor at the point A.

The position vector of A relative to O is (x\mathbf{i} + y\mathbf{j}) km.

Find the exact value of x and the exact value of y.

How did you do?

8a
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3 marks

Two forces, \mathbf{F}_1 and \mathbf{F}_2, act on a particle, where

\mathbf{F}_1 = (7\mathbf{i} - 2\mathbf{j})\text{ N}

\mathbf{F}_2 = (-12\mathbf{i} - 10\mathbf{j})\text{ N}

The resultant of these two forces is \mathbf{R}.

Find the magnitude of \mathbf{R}.

How did you do?

8b
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3 marks

A third force, \mathbf{F}_3 = k\mathbf{j} N, where k is a constant, is now applied to the particle.

Given that the new resultant of the three forces acts at an angle of 45° to the positive \mathbf{j} direction, measured anticlockwise, find the value of k.

How did you do?

9a
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3 marks

Relative to a fixed origin O, the points A, B and C have position vectors

\overset{\rightarrow}{OA} = -4\mathbf{i} - 7\mathbf{j}

\overset{\rightarrow}{OB} = 3\mathbf{j}

\overset{\rightarrow}{OC} = 6\mathbf{i} + 18\mathbf{j}

Find \overset{\rightarrow}{AB} and \overset{\rightarrow}{AC}.

How did you do?

9b
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2 marks

Show that \overset{\rightarrow}{AB} is parallel to \overset{\rightarrow}{AC}, and hence state what this tells you about the points A, B and C.

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10a
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3 marks

Figure 2 shows a sketch of triangle ABC.

q9-11-1-vectors-in-2-dimensions-easy-a-level-maths-pure
Figure 2

Given that

\overset{\rightarrow}{AB} = \mathbf{a}

\overset{\rightarrow}{AC} = \mathbf{b}

and that the point P lies on BC such that BP : PC = 3 : 2,

(i) find \overset{\rightarrow}{BC} in terms of \mathbf{a} and \mathbf{b},

(ii) hence find \overset{\rightarrow}{BP} in terms of \mathbf{a} and \mathbf{b}.

How did you do?

10b
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2 marks

Given that \mathbf{a} = 7\mathbf{i} + 8\mathbf{j} and \mathbf{b} = 12\mathbf{i} + 3\mathbf{j}, find \overset{\rightarrow}{BP} in terms of \mathbf{i} and \mathbf{j}.

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1
3 marks

Relative to a fixed origin, points P, Q and R have position vectors bold p, bold q and bold r respectively.

Given that

  • P, Q and R lie on a straight line

  • Q lies one third of the way from P to R

show that

bold q equals 1 third open parentheses bold r plus 2 bold p close parentheses

How did you do?

2a
2 marks

[In this question the unit vectors bold i and bold j are due east and due north respectively.]

A stone slides horizontally across ice.

Initially the stone is at the point A open parentheses – 24 bold i – 10 bold j close parentheses m relative to the fixed point O.

After 4 seconds the stone is at the point B open parentheses 12 bold i plus 5 bold j close parentheses m relative to the fixed point O.

The motion of the stone is modelled as that of a particle moving in a straight line at constant speed.

Using the model, prove that the stone passes through O,

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2b
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3 marks

Using the model, calculate the speed of the stone.

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3a
2 marks

Relative to a fixed origin O,

  • A is the point with position vector 12 bold i

  • B is the point with position vector 16 bold j

  • C is the point with position vector open parentheses 50 bold i plus 136 bold j close parentheses

  • D is the point with position vector open parentheses 22 bold i plus 24 bold j close parentheses

Show that A D is parallel to B C.

How did you do?

3b
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4 marks

Points A, B, C and D are used to model the vertices of a running track in the shape of a quadrilateral.

Runners complete one lap by running along all four sides of the track.

The lengths of the sides are measured in metres.

Given that a particular runner takes exactly 5 minutes to complete 2 laps, calculate the average speed of this runner giving the answer in kilometres per hour.

How did you do?

4a
2 marks

Relative to a fixed origin O

  • point A has position vector 10 bold i minus 3 bold j

  • point B has position vector negative 8 bold i plus 9 bold j

  • point C has position vector negative 2 bold i plus p bold j where p is a constant

Find stack A B with rightwards arrow on top

How did you do?

4b
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2 marks

Find open vertical bar stack A B with rightwards arrow on top close vertical bar giving your answer as a fully simplified surd.

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4c
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3 marks

Given that points A, B and C lie on a straight line,

(i) find the value of p,

(ii) state the ratio of the area of triangle A O C to the area of triangle A O B.

How did you do?

5a
1 mark

Figure 1 shows a sketch of triangle ABC.

q1-11-1-vectors-in-2-dimensions-hard-a-level-maths-pure
Figure 1

Given that

\overset{\rightarrow}{AB} = 5\mathbf{i} + 8\mathbf{j}

\overset{\rightarrow}{BC} = \mathbf{i} - 5\mathbf{j}

Explain geometrically why \overset{\rightarrow}{AB} + \overset{\rightarrow}{BC} + \overset{\rightarrow}{CA} = \mathbf{0}.

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5b
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3 marks

Find \overset{\rightarrow}{CA} and hence find the exact value of |\overset{\rightarrow}{CA}|.

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6a
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2 marks

The vectors \mathbf{a}, \mathbf{b} and \mathbf{c} are given by

\mathbf{a} = \begin{pmatrix} -1 \\ n \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 5 \\ -4 \end{pmatrix}, \quad \mathbf{c} = \begin{pmatrix} m \\ 6 \end{pmatrix}

where m and n are constants.

Given that the resultant of \mathbf{a}, \mathbf{b} and \mathbf{c} is the zero vector, find the value of m and the value of n.

How did you do?

6b
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2 marks

The vector \mathbf{d} is given by

\mathbf{d} = \begin{pmatrix} -3k \\ k \end{pmatrix}

where k is a constant.

Given that |\mathbf{d}| = 2\sqrt{15}, find the two possible values of k, giving your answers as simplified surds.

How did you do?

7
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4 marks

Relative to a fixed origin O, the point A has position vector 3k\mathbf{i} - 17k\mathbf{j}, where k is a positive constant.

The point A lies on the curve C with equation

y = x^2 - 2

Find the value of k, and hence determine the coordinates of A.

How did you do?

8
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4 marks

The vectors \mathbf{a}, \mathbf{b} and \mathbf{c} are given by

\mathbf{a} = \begin{pmatrix} 3 \\ 5 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} -3k \\ k \end{pmatrix}, \quad \mathbf{c} = \begin{pmatrix} 0 \\ -4 \end{pmatrix}

where k is a constant.

Given that \mathbf{a} - \mathbf{b} is parallel to \mathbf{a} + \mathbf{c}, find the value of k.

How did you do?

9a
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3 marks

The vector \overset{\rightarrow}{AB} has magnitude 6\sqrt{3} and makes an angle of 150° with the positive x-axis, measured anticlockwise.

Find \overset{\rightarrow}{AB} in the form x\mathbf{i} + y\mathbf{j}, where x and y are exact constants to be found.

How did you do?

9b
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2 marks

Find a unit vector in the direction of \overset{\rightarrow}{AB}, giving your answer in its simplest exact form.

How did you do?

10
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6 marks

[In this question, \mathbf{i} is a unit vector due east and \mathbf{j} is a unit vector due north. Position vectors are given relative to a fixed origin O.]

In the enchanted kingdom of Vectoria, a magical flying unicorn takes off from the wizard's palace at the point O and travels 30 km on a bearing of 300°.

Chased by an evil dragon, it then travels an unknown distance of k km due north before reaching the enchanted grove at the point P, where k is a positive constant.

The position vector of P relative to O is (x\mathbf{i} + y\mathbf{j}) km.

Given that the straight-line distance between the grove and the palace is 30\sqrt{3} km, find the exact value of x and the exact value of y.

How did you do?

11a
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2 marks

Two forces \mathbf{F}_1 and \mathbf{F}_2 act on a particle, where

\mathbf{F}_1 = (5\mathbf{i} - 3\mathbf{j})\text{ N}

\mathbf{F}_2 = (x\mathbf{i} + y\mathbf{j})\text{ N}

where x and y are scalar constants.

The resultant force \mathbf{R} acting on the particle is given by \mathbf{R} = \mathbf{F}_1 + \mathbf{F}_2.

Given that \mathbf{R} acts in a direction parallel to the vector (-\mathbf{i} - 3\mathbf{j}), find the angle between \mathbf{R} and the vector \mathbf{j}, giving your answer in degrees to two decimal places.

How did you do?

11b
3 marks

Show that 3x - y = -18.

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11c
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3 marks

Given that y = -3, find the magnitude of \mathbf{R}.

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12
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5 marks

Relative to a fixed origin O, the points A, B and C have position vectors

\overset{\rightarrow}{OA} = -9\mathbf{i} + 4\mathbf{j}

\overset{\rightarrow}{OB} = -6\mathbf{i}

\overset{\rightarrow}{OC} = 3\mathbf{i} - 12\mathbf{j}

Show, using a vector method, that the points A, B and C lie on a straight line.

How did you do?

13a
1 mark

Figure 2 shows a sketch of triangle ABC.

Ydv-1zHW_q9-11-1-vectors-in-2-dimensions-easy-a-level-maths-pure
Figure 2

The point F lies on AB such that AF : FB = m : n, where m and n are positive constants.

The point G lies on the line segment BC.

The line segment FG is parallel to AC.

Explain why \overset{\rightarrow}{BG} = \lambda \overset{\rightarrow}{BC} for some scalar constant \lambda, where 0 < \lambda < 1.

How did you do?

13b
4 marks

Given that

\overset{\rightarrow}{AB} = \mathbf{a}, \quad \overset{\rightarrow}{AC} = \mathbf{b}

show that

\overset{\rightarrow}{FG} = \left(\dfrac{n}{m + n} - \lambda\right)\mathbf{a} + \lambda \mathbf{b}

How did you do?

13c
3 marks

Hence prove that BG : GC = n : m.

How did you do?

1a
2 marks
Triangle OAB with vertices labelled O, A, and B; angle A is at the top, angle O at the left, and angle B at the right.
Figure 7

Figure 7 shows a sketch of triangle O A B.

The point C is such that stack O C with rightwards arrow on top equals 2 stack O A with rightwards arrow on top.

The point M is the midpoint of A B.

The straight line through C and M cuts O B at the point N.

Given stack O A with rightwards arrow on top equals bold a and stack O B with rightwards arrow on top equals bold b, find stack C M with rightwards arrow on top in terms of bold a and bold b.

How did you do?

1b
2 marks

Show that stack O N with rightwards arrow on top equals open parentheses 2 minus 3 over 2 lambda close parentheses bold a plus 1 half lambda bold b, where lambda is a scalar constant.

How did you do?

1c
2 marks

Hence prove that space O N colon N B space equals space 2 colon 1.

How did you do?

2a
2 marks

The points A, B and C are the vertices of a triangle.

Given that

\overset{\rightarrow}{AC} = 5\mathbf{i} - 2\mathbf{j}

\overset{\rightarrow}{BC} = -3\mathbf{i} + k\mathbf{j}

where k is a constant,

Find \overset{\rightarrow}{AB} in terms of \mathbf{i}, \mathbf{j} and k.

How did you do?

2b
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3 marks

Given that |\overset{\rightarrow}{AB}| = \sqrt{89}, find the two possible values of k.

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3a
3 marks

The vectors \mathbf{a}, \mathbf{b} and \mathbf{c} are given by

\mathbf{a} = \begin{pmatrix} 8 \\ m \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} n \\ -2 \end{pmatrix}, \quad \mathbf{c} = \begin{pmatrix} m \\ n \end{pmatrix}

where m and n are constants.

Given that \mathbf{a} + \mathbf{b} = \mathbf{c} - 2\mathbf{b}, find the value of m and the value of n.

How did you do?

3b
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2 marks

The vector \mathbf{d} is given by

\mathbf{d} = \begin{pmatrix} 2k+1 \\ 2k-1 \end{pmatrix}

where k is a constant.

Given that |\mathbf{d}| = 3k\sqrt{2}, find the two possible values of k, giving your answers in simplest exact form.

How did you do?

4a
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4 marks

Relative to a fixed origin O, the point A has position vector 3k\mathbf{i} + 5k\mathbf{j}, where k is a constant.

The point A lies on the circle C with equation

(x - 11)^2 + (y - 7)^2 = 34

Find the value of k, and hence determine the coordinates of A.

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4b
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2 marks

The straight line l passes through O and A.

Explain algebraically why l must be a tangent to C.

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5
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5 marks

Relative to a fixed origin O, the points A, B and C have position vectors

\overset{\rightarrow}{OA} = -6\mathbf{i} - 2\mathbf{j}

\overset{\rightarrow}{OB} = \mathbf{i} + m\mathbf{j}

\overset{\rightarrow}{OC} = 3\mathbf{i} - 8\mathbf{j}

where m is a constant.

Given that A, B and C lie on the same straight line, use a vector method to find the exact value of m.

How did you do?

6a
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3 marks

The vector \overset{\rightarrow}{AB} has magnitude 2\sqrt{6} and makes an angle of 165° with the positive y-axis, measured anticlockwise.

Find \overset{\rightarrow}{AB} in the form a\mathbf{i} + b\mathbf{j}, where a and b are exact constants to be found.

How did you do?

6b
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2 marks

Find a unit vector in the direction of \overset{\rightarrow}{AB}, giving your answer in its simplest exact form.

How did you do?

7a
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7 marks

[In this question, \mathbf{i} is a unit vector due east and \mathbf{j} is a unit vector due north. Position vectors are given relative to a fixed origin O.]

A ship searches for a radio buoy. The ship sets out from O and moves with a constant speed of 40 km/h in a direction parallel to the vector \mathbf{i} + 3\mathbf{j}.

After 90 minutes, the ship reaches the point P. At P, the ship receives a transmission indicating that the buoy is on a bearing of 210° from the ship.

The ship immediately changes course and travels on a bearing of 210° at a constant speed of 40 km/h for a further 45 minutes, reaching the buoy at the point Q.

Given that the position vector of Q relative to O is (x\mathbf{i} + y\mathbf{j}) km, find the exact value of x and the exact value of y.

How did you do?

7b
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3 marks

Find the distance of the buoy from the ship, and its bearing from the ship, at the time the ship initially left O.

Give your distance in km to one decimal place, and your bearing to one decimal place.

How did you do?

8a
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4 marks

Three forces, \mathbf{F}_1, \mathbf{F}_2 and \mathbf{F}_3, act on a particle, where

\mathbf{F}_1 = (7\mathbf{i} - \mathbf{j})\text{ N}

\mathbf{F}_2 = (x\mathbf{i} + y\mathbf{j})\text{ N}

\mathbf{F}_3 = (k\mathbf{i} + k\sqrt{3}\,\mathbf{j})\text{ N}

where x and y are scalar constants, and k is a positive constant.

The resultant force acting on the particle is \mathbf{R}.

Given that \mathbf{R} = \mathbf{0} when |\mathbf{F}_3| = 10 N, find the exact value of x and the exact value of y.

How did you do?

8b
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3 marks

Find the magnitude of \mathbf{F}_2 and the angle it makes with the positive \mathbf{i} direction. Give both answers correct to one decimal place.

How did you do?

9a
3 marks

Figure 3 shows a sketch of triangle ABC.

~1JxTmR5_q9-11-1-vectors-in-2-dimensions-easy-a-level-maths-pure
Figure 3

The point D is the midpoint of AB. The point E is the midpoint of AC. The line segments BE and CD intersect at the point F.

Given that \overset{\rightarrow}{AB} = 2\mathbf{a} and \overset{\rightarrow}{AC} = 2\mathbf{b}, find \overset{\rightarrow}{BC}, \overset{\rightarrow}{BE} and \overset{\rightarrow}{CD} in terms of \mathbf{a} and \mathbf{b}.

How did you do?

9b
6 marks

By setting up and solving suitable vector equations, prove that each of the line segments BE and CD divides the other in the ratio 1 : 2.

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