Pure: Vectors: Fluency and Exam Drill - Worksheets, Questions and Revision

20 original exam-style questions - 3 pages of questions with a full mark scheme - free printable PDF.

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A-Level · Pure Mathematics

P10D Pure: Vectors: Fluency and Exam Drill

EDEXCEL 9MA0 · Calculator allowed · about 50 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

a = 5i - 12j. Find |a|.

(Total for Question 1 is 1 mark)

b = 8i + 6j. Find |b|.

(Total for Question 2 is 1 mark)

c = 3i - 4j + 12k. Find |c|.

(Total for Question 3 is 1 mark)

Write down -2(3i - 5j) as a single vector.

(Total for Question 4 is 1 mark)

p = 4i + j, q = -i + 5j. Find p + q.

(Total for Question 5 is 1 mark)

p = 4i + j, q = -i + 5j. Find p - q.

(Total for Question 6 is 1 mark)

Find the unit vector in the direction of 6i - 8j.

(Total for Question 7 is 1 mark)

Relative to a fixed origin O, OA = 2i + 3j and OB = 7i - 9j. Find the vector AB.

(Total for Question 8 is 1 mark)

State whether the vectors 4i - 6j and -2i + 3j are parallel, giving a brief reason.

(Total for Question 9 is 1 mark)

Find the value of k for which ki + 6j is parallel to 2i + 3j.

(Total for Question 10 is 1 mark)

a = 3i - 2j + 6k. Find |a|, and hence find the unit vector in the direction of a.

(Total for Question 11 is 2 marks)

Relative to a fixed origin O, OA = 5i + 2j and OB = -i + 10j. Find the vector AB, and its magnitude.

(Total for Question 12 is 2 marks)

p = 2i - 5j. q = ti + 15j, where t is a constant. Given that p and q are parallel, find the value of t.

(Total for Question 13 is 2 marks)

Relative to a fixed origin O, OA = 2i - 3j and OB = 8i + 7j. M is the midpoint of AB. Find the position vector of M.

(Total for Question 14 is 2 marks)

Relative to a fixed origin O, the points A and B have position vectors OA = 3i + j - 2k and OB = 7i - 3j + 6k. Find the vector AB, and hence find |AB|, giving your answer as an exact simplified surd.

(Total for Question 15 is 3 marks)

Two forces acting on a particle are F1 = (3i - 2j) N and F2 = (-7i + 9j) N. Find the resultant force, and its magnitude, giving your answer in newtons to 3 significant figures.

(Total for Question 16 is 3 marks)

Relative to a fixed origin O, points P and Q have position vectors OP = i + 4j - 3k and OQ = 9i - 4j + 5k. The point R lies on PQ such that PR : RQ = 3 : 1. Find the position vector of R.

(Total for Question 17 is 4 marks)

Three forces F1 = (4i - 3j) N, F2 = (-2i + 5j) N and F3 = (π + qj) N act on a particle which is in equilibrium.

(a)Find the values of p and q.(2)

(b)Find the magnitude of F3, giving your answer in newtons to 3 significant figures.(2)

(Total for Question 18 is 4 marks)

Relative to a fixed origin O, points A and B have position vectors OA = 2i + j and OB = 6i + 3j. Show that O, A and B are collinear, and find the ratio OA : AB.

(Total for Question 19 is 3 marks)

Two survey points D and E have position vectors, relative to a fixed origin O and measured in metres, OD = 3i - 2j + 5k and OE = 9i + 4j - 7k.

(a)Find the vector DE, and hence find |DE|, giving your answer in metres to 3 significant figures.(3)

(b)The point F lies on DE such that DF : FE = 2 : 1. Find the position vector of F.(2)

(Total for Question 20 is 5 marks)

Mark scheme · P10D Pure: Vectors: Fluency and Exam Drill

Question 1

B1 |a| = 13 cao
Answer: 13

Question 2

B1 |b| = 10 cao
Answer: 10

Question 3

B1 |c| = 13 cao
Answer: 13

Question 4

B1 -6i + 10j cao
Answer: -6i + 10j

Question 5

B1 3i + 6j cao
Answer: 3i + 6j

Question 6

B1 5i - 4j cao
Answer: 5i - 4j

Question 7

B1 0.6i - 0.8j oe, using |6i-8j| = 10
Answer: 0.6i - 0.8j

Question 8

B1 AB = 5i - 12j cao
Answer: 5i - 12j

Question 9

B1 parallel, since 4i-6j = -2(-2i+3j), a scalar multiple
Answer: Parallel

Question 10

B1 k = 4 cao
Answer: k = 4

Question 11

M1 |a| = √9+4+36 = 7
A1 unit vector = (3/7)i - (2/7)j + (6/7)k oe
Answer: |a| = 7; unit vector = (3/7)i - (2/7)j + (6/7)k

Question 12

M1 AB = OB - OA = -6i + 8j
A1 |AB| = √36+64 = 10 cao
Answer: AB = -6i + 8j, |AB| = 10

Question 13

M1 sets up t/2 = 15/(-5) oe from the parallel condition
A1 t = -6 cao
Answer: t = -6

Question 14

M1 M = (OA + OB)/2
A1 OM = 5i + 2j cao
Answer: OM = 5i + 2j

Question 15

M1 OB - OA
A1 AB = 4i - 4j + 8k cao
A1 |AB| = √16+16+64 = √96 = 4sqrt(6) cao (exact)
Answer: AB = 4i - 4j + 8k, |AB| = 4sqrt(6)

Question 16

M1 add corresponding components
A1 resultant = -4i + 7j (N) cao
A1 magnitude = √16+49 = awrt 8.06 N
Answer: Resultant = -4i + 7j (N), magnitude awrt 8.06 N

Question 17

M1 find PQ = OQ - OP
A1 PQ = 8i - 8j + 8k
M1 r = OP + (3/4)PQ
A1 OR = 7i - 2j + 3k cao
Answer: OR = 7i - 2j + 3k

Question 18

(a) M1 sets the sum of i- and j-components equal to zero (equilibrium)
(a) A1 p = -2 and q = -2
(a) Answer: p = -2, q = -2
(b) M1 √p²+q² using values from (a)
(b) A1 awrt 2.83 N
(b) Answer: awrt 2.83 N

Question 19

M1 find AB = OB - OA
A1 AB = 4i + 2j = 2 x OA (scalar multiple, common point O), so O, A, B are collinear
A1 ratio OA : AB = 1 : 2 cao
Answer: O, A, B collinear; OA : AB = 1 : 2

Question 20

(a) M1 OE - OD
(a) A1 DE = 6i + 6j - 12k
(a) A1 |DE| = √36+36+144 = awrt 14.7 m
(a) Answer: DE = 6i + 6j - 12k, |DE| = awrt 14.7 m
(b) M1 F = OD + (2/3)DE (ft from (a))
(b) A1 OF = 7i + 2j - 3k oe
(b) Answer: OF = 7i + 2j - 3k

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Pure: Vectors: Fluency and Exam Drill - Worksheets, Questions and Revision

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