Vectors.

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1 Vectors

2 8 5 (a) y B x (i) Write down the position vector of. (ii) Find ì ì, the magnitude of. nswer(a)(i) f p [1] nswer(a)(ii)... [2] (b) S q Q p P R is the origin, = p and = q. P is extended to R so that P = PR. Q is extended to S so that Q = QS. (i) Write down in terms of p and q. nswer(b)(i) =... [1] (ii) PS and RQ intersect at M and RM = 2MQ. Use vectors to find the ratio PM : PS, showing all your working. nswer(b)(ii) PM : PS =... :... [4] UCLES /43/M/J/14

3 y B For Examiner s Use is the point ( 1, 1) and B is the point (8, 7). x (a) Write as a column vector. nswer(a) = f p [1] (b) Find. nswer(b) =... [2] (c) = 2. Write down the co-ordinates of C. nswer(c) (...,...) [1] 17 Factorise completely. (a) a + b + at + bt (b) x 2 2x 24 nswer(a)... [2] nswer(b)... [2] UCLES /21//N/13 [Turn over

4 (a) Calculate the magnitude of the vector. 5 For Examiner's Use nswer(a) [2] (b) y R P x (i) The points P and R are marked on the grid above. 3 =. Draw the vector on the grid above. [1] 5 (ii) Draw the image of vector after rotation by 90 anticlockwise about R. [2] (c) = 2a + b and = 3b a. Find in terms of a and b. Write your answer in its simplest form. nswer(c) = [2] UCLES /42//N/12

5 9 20 D For Examiner's Use d E In the diagram, is the origin. = c and = d. E is on CD so that CE = 2ED. Find, in terms of c and d, in their simplest forms, (a), c C nswer(a) = [2] (b) the position vector of E. nswer(b) [2] UCLES /23//N/12 [Turn over

6 9 (b) P Q For Examiner s Use p R s S In the pentagon PQRS, P is parallel to RQ and S is parallel to PQ. PQ = 2S and P = 2RQ. is the origin, = p and = s. Find, in terms of p and s, in their simplest form, (i) the position vector of Q, (ii). nswer(b)(i)... [2] nswer(b)(ii) =... [2] (c) Explain what your answers in part (b) tell you about the lines Q and SR. nswer(c)... [1] UCLES /41//N/13 [Turn over

7 11 24 a b B c C In the diagram, is the origin, = a, C = c and B = b. P is on the line B so that P : PB = 2 : 1. Q is the midpoint of BC. Find, in terms of a, b and c, in its simplest form (a) CB, CB =... [1] (b) the position vector of Q,... [2] (c) PQ. PQ =... [2] UCLES /22/M/J/16

8 11 2 (d) = 5 5 and =. 1 For Examiner's Use Write as a column vector. nswer(d) = [2] (e) M B C X = b and = c. (i) Find in terms of b and c. nswer(e)(i) = [1] (ii) X divides CB in the ratio 1 : 3. M is the midpoint of B. Find in terms of b and c. Show all your working and write your answer in its simplest form. nswer(e)(ii) = [4] UCLES /42//N/12 [Turn over

9 6 14 Q b R a P M S PQRS is a quadrilateral and M is the midpoint of PS. PQ = a, QR = b and SQ = a 2b. (a) Show that PS = 2b. nswer(a) [1] (b) Write down the mathematical name for the quadrilateral PQRM, giving reasons for your answer. nswer(b)... because [2] UCLES /21/M/J/15

10 14 7 R M Q T r p P PQR is a rectangle and is the origin. M is the midpoint of RQ and PT : TQ = 2 : 1. P = p and R = r. (a) Find, in terms of p and/or r, in its simplest form (i) MQ, MQ =... [1] (ii) MT, MT =... [1] (iii) T. T =... [1] (b) RQ and T are extended to meet at U. Find the position vector of U in terms of p and r. Give your answer in its simplest form.... [2] UCLES /41/M/J/16

11 15 (c) 2k MT = c m and MT = k Find the positive value of k. k =... [3] UCLES /41/M/J/16 [Turn over

12 16 9 (a) 3 m = c m 2 2 n = c - m 3 (i) Work out 2m 3n. f p [2] (ii) Calculate 2m- 3n.... [2] (b) (i) a M b B In the diagram, is the origin, = a and B = b. The point M lies on B such that M : MB = 3 : 2. Find, in terms of a and b, in its simplest form (a) B, B =... [1] (b) M, M =... [1] UCLES /43//N/16

13 17 (c) the position vector of M.... [2] (ii) M is extended to the point C. The position vector of C is a + kb. Find the value of k. k =... [1] UCLES /43//N/16 [Turn over

14 11 19 B M P b X a PB is a parallelogram. is the origin, = a and B = b. M is the midpoint of BP. (a) Find, in terms of a and b, giving your answer in its simplest form, (i) B, nswer(a)(i) B =... [1] (ii) the position vector of M. nswer(a)(ii)... [1] (b) X is on B so that BX : X = 1 : 2. Show that X lies on M. nswer(b) [4] Question 20 is printed on the next page. UCLES /23/M/J/15 [Turn over

15 (a) = c m -8 (i) Find the value of. nswer(a)(i) =... [2] (ii) Q is the point (2, 3). Find the co-ordinates of the point P. nswer(a)(ii) (...,...) [1] (b) a M L N b B In the diagram, M is the midpoint of B and L is the midpoint of M. The lines M and N intersect at L and N = 3 1 B. = a and = b. (i) Find, in terms of a and b, in its simplest form, (a), (b), nswer(b)(i)(a) =... [2] (c). nswer(b)(i)(b) =... [1] nswer(b)(i)(c) =... [2] UCLES /42/M/J/15

16 16 10 C b M a X B BC = a and C = b. (a) Find B in terms of a and b. nswer(a) B =... [1] (b) M is the midpoint of BC. X divides B in the ratio 1 : 4. Find XM in terms of a and b. Show all your working and write your answer in its simplest form. nswer(b) XM =... [4] Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright cknowledgements Booklet. This is produced for each series of examinations and is freely available to download at after the live examination series. Cambridge International Examinations is part of the Cambridge ssessment Group. Cambridge ssessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. UCLES /41//N/15

18 19 (b) C Y N a B b CB is a parallelogram. = a and = b. N : NB = 2 : 3 and Y = 5 2 C. (i) Write each of the following in terms of a and/or b. Give your answers in their simplest form. (a) (b) nswer(b)(i)(a) =... [2] nswer(b)(i)(b) =... [2] (ii) Write down two conclusions you can make about the line segments NY and BC. nswer(b)(ii) [2] UCLES /41/M/J/14 [Turn over

19 10 19 C B c X a The diagram shows a quadrilateral BC. = a, = c and = 2a. X is a point on B such that X : XB = 1 : 2. (a) Find, in terms of a and c, in its simplest form (i), nswer(a)(i) =... [1] (ii). nswer(a)(ii) =... [3] (b) Explain why the vectors and show that C, X and lie on a straight line. nswer(b) [2] UCLES /22//N/14

20 8 19 t varies inversely as the square root of u. t = 3 when u = 4. For Examiner s Use Find t when u = 49. nswer t =... [3] 20 R Q S r M p P PQR is a parallelogram, with the origin. M is the midpoint of PQ. M and RQ are extended to meet at S. = p and = r. (a) Find, in terms of p and r, in its simplest form, (i), (ii) the position vector of S. (b) When = nswer(a)(i) =... [1] nswer(a)(ii)... [1] 1 p + r, what can you write down about the position of T? 2 nswer(b)... [1] UCLES /21/M/J/13

21 9 18 Q R For Examiner's Use q M X p P is the origin and PRQ is a parallelogram. The position vectors of P and Q are p and q. X is on PR so that PX = 2XR. Find, in terms of p and q, in their simplest forms (a), nswer(a) = [2] (b) the position vector of M, the midpoint of QX. nswer(b) [2] UCLES /23/M/J/12 [Turn over

22 10 18 S For Examiner's Use Q X P M R In the diagram, PQS, PMR, MXS and QXR are straight lines. PQ = 2 QS. M is the midpoint of PR. QX : XR = 1 : 3. = q and = r. (a) Find, in terms of q and r, (i), (ii). nswer(a)(i) = [1] nswer(a)(ii) = [1] (b) By finding, show that X is the midpoint of MS. nswer (b) [3] UCLES /21/M/J/11

23 11 19 C For Examiner s Use D B c b E BCDE is a regular polygon. (a) Write down the geometrical name for this polygon. nswer(a)... [1] (b) is the origin. = b and = c. Find, in terms of b and c, in their simplest form, (i), (ii), nswer(b)(i) =... [1] (iii) the position vector of E. nswer(b)(ii) =... [2] nswer(b)(iii)... [1] Question 20 is printed on the next page. UCLES /23/M/J/13 [Turn over

24 10 19 C For Examiner s Use D c B a E is the origin. BCDEF is a regular hexagon and is the midpoint of D. = a and = c. Find, in terms of a and c, in their simplest form (a), F (b), nswer(a) =... [2] (c) the position vector of E. nswer(b) =... [2] nswer(c)... [2] UCLES /22//N/13

25 10 19 S R For Examiner's Use T Q t p P is the origin and PQRST is a regular hexagon. = p and = t. Find, in terms of p and t, in their simplest forms, (a), nswer(a) = [1] (b), nswer(b) = [2] (c) the position vector of R. nswer(c) [2] UCLES /21/M/J/12

26 16 8 P B 9b Q 6a C 3c In the diagram, is the origin and = 6a, = 9b and = 3c. The point P lies on B such that = 3b 2a. The point Q lies on BC such that = 2c 6b. (a) Find, in terms of b and c, the position vector of Q. Give your answer in its simplest form. nswer(a)... [2] UCLES /41//N/14

27 17 (b) Find, in terms of a and c, in its simplest form (i), nswer(b)(i) =... [1] (ii). nswer(b)(ii) =... [2] (c) Explain what your answers in part (b) tell you about PQ and C. nswer(c) [2] UCLES /41//N/14 [Turn over

28 15 (b) T For Examiner s Use R 4a 3b B In the diagram, = 4a and = 3b. 1 R lies on B such that = 5 (12a + 6b). 3 T is the point such that = 2. (i) Find the following in terms of a and b, giving each answer in its simplest form. (a) (b) nswer(b)(i)(a) =... [1] (c) nswer(b)(i)(b) =... [2] nswer(b)(i)(c) =... [1] (ii) Complete the following statement. The points, R and T are in a straight line because [1] (iii) Triangle R and triangle TBR are similar. area of triangle TBR Find the value of. area of triangle R nswer(b)(iii)... [2] UCLES /43//N/13 [Turn over

29 10 7 (a) P is the point (2, 5) and = 3. 2 For Examiner's Use Write down the co-ordinates of Q. nswer(a) (, ) [1] (b) D C E B c M 3a is the origin and BC is a parallelogram. M is the midpoint of B. = c, = 3a and CE = 3 1 CB. ED is a straight line with E : ED = 2 : 1. Find in terms of a and c, in their simplest forms (i), (ii) the position vector of M, nswer(b)(i) = [1] nswer(b)(ii) [2] (iii), nswer(b)(iii) = [1] (iv). (c) Write down two facts about the lines CD and B. nswer(b)(iv) = [2] nswer (c) [2] UCLES /42/M/J/12

Equation Of A Line.

Equation Of A Line. Equation Of A Line www.q8maths.com 8 17 11 y l NOT TO SCALE 3 0 4 x The diagram shows the straight line, l, which passes through the points (0, 3) and (4, 11). (a) Find the equation of line l in the form