IGCSEFurther MathsCambridge (CIE)Additional MathsExam QuestionsVectorsVectors in Two Dimensions

Vectors in Two Dimensions (Cambridge (CIE) IGCSE Additional Maths): Exam Questions

Exam code: 606

1 hour12 questions
Medium
Hard
Very Hard
1a
Sme Calculator1 mark

Find the unit vector in the direction of open parentheses table row 5 row cell negative 12 end cell end table close parentheses.

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1b
Sme Calculator3 marks

Given that open parentheses table row 4 row 1 end table close parentheses plus k open parentheses table row cell negative 2 end cell row 3 end table close parentheses equals r open parentheses table row cell negative 10 end cell row 5 end table close parentheses, find the value of each of the constants k and r.

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1c
Sme Calculator4 marks

Relative to an origin O, the points A comma space B and C have position vectors bold p comma space 3 bold q minus bold p space and space 9 bold q minus 5 bold p respectively.

(i) Find stack A B with rightwards arrow on top in terms of bold p and bold q.

[1]

(ii) Find stack A C with rightwards arrow on top in terms of bold p and bold q.

[1]

(iii) Explain why A comma space B space and space C all lie in a straight line.

[1]

(iv) Find the ratio A B space colon space B C.

[1]

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2a1 mark

The unit vectors bold i and bold j represent due east and due north respectively.

Person A starts at a position of open parentheses negative 4 bold i plus bold j close parentheses metres and walks with a constant velocity of open parentheses 3 bold i minus bold j close parentheses metres per second.

Find the position vector of person A after 5 seconds.

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2b5 marks

Person B walks with a constant velocity of open parentheses 2 bold i plus 2 square root of 3 bold j close parentheses metres per second.

Find

(i) the speed of person B,

(ii) the bearing of the direction in which person B is walking.

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1
Sme Calculator5 marks

The parallelogram O A B C is such that stack O A with rightwards arrow on top space equals space bold a and stack space O C with rightwards arrow on top space equals space bold c bold space. The point D lies on O C such that O D space colon space D C = 1 colon space 2. The point E lies on A C such that A E space colon space E C =space 2 space colon space 1 space.

Show that stack O B with rightwards arrow on top space equals space k stack D E with rightwards arrow on top , where k is an integer to be found.

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2a
Sme Calculator1 mark

In the diagram stack O P with rightwards arrow on top space equals space 2 bold b, stack O S with rightwards arrow on top space equals space 3 bold a, stack S R with rightwards arrow on top space equals space bold b and stack P Q with rightwards arrow on top space equals space bold a. The lines O R and Q S intersect at X.

Find stack O Q with rightwards arrow on top in terms of bold a and bold b.

0606-w20-qp-22-q9a

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2b
Sme Calculator1 mark

Find stack Q S with rightwards arrow on top in terms of bold a and bold b.

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2c
Sme Calculator1 mark

Given that stack Q X with rightwards arrow on top space equals mu stack Q S with rightwards arrow on top , find stack O X with rightwards arrow on top in terms of bold a comma space bold b space and space mu.

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2d
Sme Calculator1 mark

Given that stack O X with rightwards arrow on top space equals space lambda stack O R with rightwards arrow on top, find stack O X with rightwards arrow on top in terms of bold a comma space bold b space and space lambda.

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2e
Sme Calculator3 marks

Find the value of lambda and of mu.

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2f
Sme Calculator1 mark

Find the value of fraction numerator Q X over denominator X S end fraction.

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2g
Sme Calculator1 mark

Find the value of fraction numerator O R over denominator O X end fraction.

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3a
Sme Calculator3 marks
q9-0606-w20-qp-13-additional-maths

The diagram shows the triangle O A C. The point B is the midpoint of O C. The point Y lies on A C such that O Y intersects A B at the point X where A X space colon X B space equals space 3 colon 1. It is given that stack O A with rightwards arrow on top space equals space a and stack O B with rightwards arrow on top space equals space b.

Find stack space O X with rightwards arrow on top spacein terms of a and b, giving your answer in its simplest form.

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3b
Sme Calculator1 mark

Find AC in terms of a and b.

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3c
Sme Calculator1 mark

Given that stack O Y with rightwards arrow on top space equals space h stack O X with rightwards arrow on top , find stack A Y with rightwards arrow on top in terms of a, b and h.

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3d
Sme Calculator4 marks

Given that stack A Y with rightwards arrow on top space equals space m stack A C with rightwards arrow on top, find the value of h and of m.

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4a
Sme Calculator3 marks

The vectors bold a and bold b are such that bold a space equals space alpha bold i plus bold space bold j and bold b space equals space 12 bold i plus bold space beta bold j.

Find the value of each of the constants alpha and beta such that 4 bold a minus bold b space equals space left parenthesis alpha plus 3 right parenthesis bold i minus 2 bold j.

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4b
Sme Calculator2 marks

Hence find the unit vector in the direction of bold b minus 4 bold a.

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5a3 marks

The kite A B C D has diagonals C A and B D which intersect at O, where stack O A with rightwards arrow on top equals bold a, stack O B with rightwards arrow on top equals bold b, stack O C with rightwards arrow on top equals k bold a and stack O D with rightwards arrow on top equals negative bold b.

The point X is the midpoint of A B, as shown.

Geometric diagram of a kite-shaped quadrilateral ABCD with labelled vertices A, B, C, D. The diagonals cross at O. OA is vector a, OB is vector b, OC is vector ka and OD is vector -b. The point X is the midpoint of AB.

Find

(i) stack A B with rightwards arrow on top in terms of bold a and bold b

(ii) stack O X with rightwards arrow on top in terms of bold a and bold b

(iii) stack C D with rightwards arrow on top in terms of k, bold a and bold b

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5b4 marks

The point Y lies on C D such that C Y colon Y D equals 1 colon 2.

If stack Y X with rightwards arrow on top is parallel to bold a plus bold b, find k.

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1a
Sme Calculator3 marks

A particle P is initially at the point with position vector open parentheses table row 30 row 10 end table close parentheses and moves with a constant speed of 10 space ms to the power of negative 1 end exponent in the same direction as open parentheses table row cell negative 4 end cell row 3 end table close parentheses.

Find the position vector of P after t space straight s.

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1b
Sme Calculator1 mark

As P starts moving, a particle Q starts to move such that its position vector afterspace t space straight s space is given by  open parentheses table row cell negative 80 end cell row 90 end table close parentheses plus t open parentheses table row 5 row 12 end table close parentheses.

Write down the speed of Q.

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1c
Sme Calculator3 marks

Find the exact distance between P and Q when t space equals space 10, giving your answer in its simplest surd form.

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2a
Sme Calculator3 marks

Relative to an origin O, the position vectors of the points A comma space B comma space C and D are

stack O A with rightwards arrow on top space equals open parentheses table row 6 row cell negative 5 end cell end table close parentheses comma space stack O B space with rightwards arrow on top equals open parentheses table row 10 row 3 end table close parentheses comma space stack O C with rightwards arrow on top space equals space open parentheses table row x row y end table close parentheses space a n d space stack O D with rightwards arrow on top space equals space open parentheses table row 12 row 7 end table close parentheses

Find the unit vector in the direction of stack A B with rightwards arrow on top.

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2b
Sme Calculator2 marks

The point A is the mid‑point of B C. Find the value of x and of y.

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2c
Sme Calculator3 marks

The point E lies on O D such that O E space colon space O D is 1 space colon space 1 plus lambda. Find the value of lambda such that stack B E with rightwards arrow on top is parallel to the x‑axis.

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3a
Sme Calculator1 mark

In this question all distances are in km.
A ship P sails from a point A, which has position vector open parentheses table row 0 row 0 end table close parentheses, with a speed of 52 space kmh to the power of negative 1 end exponent in the direction of open parentheses table row cell negative 5 end cell row 12 end table close parentheses

Find the velocity vector of the ship.

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3b
Sme Calculator1 mark

Write down the position vector of P at a time t hours after leaving A.

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3c
Sme Calculator1 mark

At the same time that ship P sails from A, a ship Q sails from a point B, which has position vector open parentheses table row 12 row 8 end table close parentheses, with velocity vector open parentheses table row cell negative 25 end cell row 45 end table close parentheses space kmh to the power of negative 1 end exponent.

Write down the position vector of Q at a time t hours after leaving B.

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3d
Sme Calculator1 mark

Using your answers to parts (b) and (c), find the displacement vector stack P Q with rightwards arrow on top at timespace t hours.

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3e
Sme Calculator2 marks

Hence show that P Q space equals space square root of 34 t squared space minus 168 t plus 208 end root. 

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3f
Sme Calculator2 marks

Find the value of t when P and Q are first 2 space km apart.

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4
Sme Calculator5 marks

The position vectors of three points, A comma space B space and space C, relative to an origin O, are open parentheses table row cell negative 5 end cell row cell negative 7 end cell end table close parentheses comma space open parentheses table row 10 row cell negative 4 end cell end table close parentheses space and space open parentheses table row x row y end table close parentheses respectively. Given that stack A C with rightwards arrow on top = 4 stack B C with rightwards arrow on top, find the unit vector in the direction of stack O C with rightwards arrow on top.

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5a
Sme Calculator1 mark
q8-0606-s20-qp-12-additional-maths

The diagram shows a triangle O A B such that stack O A with rightwards arrow on top space equals space bold a and stack O B with rightwards arrow on top space equals space bold b. The point P lies on O A such that O P equals 3 over 4 O A. The point Q is the mid-point of A B. The lines O B and P Q are extended to meet at the point R. Find, in terms of bold a and bold b,

Find stack A B with rightwards arrow on top

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5b
Sme Calculator3 marks

Find stack P Q with rightwards arrow on top. Give your answer in its simplest form.

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5c
Sme Calculator1 mark

It is given that n stack P Q with rightwards arrow on top space equals space stack Q R with rightwards arrow on top space and space stack B R with rightwards arrow on top space equals space k bold b, where n space and space k are positive constants.

Find stack Q R with rightwards arrow on top in terms of n comma space bold a space and space bold b.

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5d
Sme Calculator2 marks

Find stack Q R with rightwards arrow on top in terms of k comma space bold a space and space bold b.

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5e
Sme Calculator3 marks

Hence find the value of n and of k.

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