Section A Finding Vectors Grade A / A*

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1 Name: Teacher ssessment Section Finding Grade / * 1. PQRSTU is a regular hexagon and is the centre of the hexagon. P = p and Q = q U P p T q Q S R Express each of the following vectors in terms of p and q PQ nswer... SP nswer... (c) SQ nswer... (Total 4 marks) St Paul s Catholic School 1

2 2. The diagram shows two sets of parallel lines. Vector PQ = a and vector PS = PR = 3 PQ and PU = 2 PS U V W S P a Q R T Not to scale Write the vector PV in terms of a and nswer... Write the vector RU in terms of a and nswer... (c) Find two vectors that can e written as 3a nswer... and... (Total 4 marks) St Paul s Catholic School 2

3 3. CDEF is a regular hexagon with centre. = a and = Diagram drawn accurately C D a E F Find expressions, in terms of a and, for nswer... C nswer... (iii) EC nswer... The positions of points P and Q are given y the vectors P = a Q = a + 2 Draw and lael the positions of points P and Q on the diagram. Hence, or otherwise, deduce an expression for PQ nswer... (Total 6 marks) St Paul s Catholic School 3

4 4. In the diagram CD, D and DE are parallelograms. E a and D C Express, in terms of a and, the following vectors. Give your answers in their simplest form. a D nswer... C nswer... (iii) nswer... The point F is such that CFE is a parallelogram. Write the vector CF in terms of a and. nswer... (c) What geometrical relationship is there etween the points, D and F? Justify your answer. (Total 7 marks) St Paul s Catholic School 4

5 5. In the diagram P = 4a, P = a, = 5, R = 3 and Q = 5 2 P a Q 4a 5 3 R Not drawn accurately Find, in terms of a and, simplifying your answers, nswer PQ nswer Show clearly that points P, Q and R lie on a straight line (3) (Total 6 marks) St Paul s Catholic School 5

6 6. is a triangle. X is the midpoint of. Y is the midpoint of. Z is the point on X such that Z : ZX = 2 : 1 = 3a, = 3 3a Z X 3 Y Find, in terms of a and, the vectors Y nswer... X nswer... (iii) Z nswer..., Z and Y are on a straight line. Find the ratio Z : ZY nswer... (Total 7 marks) St Paul s Catholic School 6

7 7. P = 4a + 5 and Q = 5a. P 4 a a Q R is a point on PQ such that PR : RQ = 1 : 2. Express R in terms of a and. nswer... (3) PS = a + 4 Express S in terms of a and. nswer... (c) What two facts do R and S indicate aout the points, R and S? Give a reason for each of your answers. (Total 7 marks) St Paul s Catholic School 7

8 8. Q Not drawn accurately M P a is a triangle where M is the mid-point of. P and Q are points on such that P = PQ = Q. = a and = 2 Find, in terms of a and, expressions for nswer... MQ nswer... (iii) P nswer... What can you deduce aout quadrilateral MQP? Give a reason for your answer.... (Total 7 marks) St Paul s Catholic School 8

9 9. The diagram shows a square P. M is the mid-point of P. N is the mid-point of M. 1 P is extended to Q where Q = 1 P 2 = a and = a M N P Q Not drawn accurately Write these vectors in terms of a and. Give your answers in their simplest form. Q M nswer (iii) N nswer nswer (iv) N nswer What can you deduce aout points, N and Q? Give a reason for your answer.... (Total 7 marks) St Paul s Catholic School 9

10 10. PQR is a parallelogram. M is the mid-point of the diagonal Q. P = 2p and R = 2r R Q 2r M 2p P Express M in terms of p and r. nswer M =.. Use vectors to prove that M is also the mid-point of PR. (3) (Total 4 marks) St Paul s Catholic School 10

11 11. C is a parallelogram and M is the mid-point of C. = a and = C a Not drawn accurately M N Express the following vectors in terms of a and nswer... M nswer... M is extended to N, where N 2M. Show that N =.... (c) What does this tell you aout the position of N?... (Total 5 marks) St Paul s Catholic School 11

12 12. In triangle C, M is the mid-point of C. = s and C = t s M Find M in terms of s and t. Give your answer in its simplest form. t C nswer... (3) D = s + t The length of is not equal to the length of C. Write down the name of the shape DC. nswer... Write down one fact aout the points, M and D. Explain your answer. Fact... Explanation... (Total 6 marks) Success: Target: St Paul s Catholic School 12

VECTORS Contents Page 7.0 Conceptual Map Introduction to Vector Practice Multiplication of Vector y Scalar Practice Practice 7.2

DDITIONL MTHEMTICS FORM 5 MODULE 7 VECTORS VECTORS Contents Page 7.0 Conceptual Map 2 7.1 Introduction to Vector Practice 7.1 3 7.2 Multiplication of Vector y Scalar Practice 7.2.1 Practice 7.2.2 4 5 7.3