Further Maths Core Pure (AS/Year 1) Unit Test : Matrices Q Scheme Marks AOs Pearson Finds det M 3 p p 4 p 4 p 6 1 Completes the square to show p 4 p 6 p M1.a Concludes that (p + ) + > 0 for all values of p. Therefore det M 0 and M is non-singular. B1 3.a (3 marks) 1 1 1 1 168 Finds a a a b a b 8 b 8 ab 8b 1b 64 Deduces that 1a 168 = 0 and solves to find a = 8 A1*.a Deduces that a + 1b = 1 and solves to find b = 3 A1*.a (3 marks) Can use any of the following equations to find a and b. Award 1 mark for finding a and 1 mark for finding b. a + 1b = 1 1a 168 = 0 ab 8b = 0 1b + 64 = 1 Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 1
Further Maths Core Pure (AS/Year 1) Unit Test : Matrices 3a 3 1 1 States either cos or sin or tan 3 Finds θ = 150 and concludes this is a rotation of 150 anticlockwise about the origin. B1 3.a () 3b 3 1 1 1 3 5 Sets up a matrix equation of the form: or 3 1 1 1 3 two separate equations of the form and 3 1 1 3 5 M1 1.1a Finds 3 5 1 3 1 5 3 3 1 or 3 1 1 3 and 5 3 5 3 1 3 1 3 3 5 5 3,, States P and Q A1 3.a (3) Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.
Further Maths Core Pure (AS/Year 1) Unit Test : Matrices 3c Finds M 1 3 3 1 1 3 States either cos or sin or tan 3 Finds θ = 300 and concludes this is a rotation of 300 anticlockwise about the origin. or Finds θ = 60 and concludes that it is a rotation of 60 clockwise about the orgin. B1 3.a (3) (8 marks) Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 3
Further Maths Core Pure (AS/Year 1) Unit Test : Matrices 4a Reflection in the line y = x B1 3.a (1) 4b Calculates 0 1 a b 1 0 b a States or implies b = 4 + a and b = 4 + a = 5 + b M1.a Finds a = 1 and b = 6 A1 1.1b (3) (4 marks) Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 4
Further Maths Core Pure (AS/Year 1) Unit Test : Matrices 5 Finds 0 q 5 q 5 0 P States that this is an enlargement. A1 3.a States scale factor is q + 5 and centre is (0, 0). A1 3.a (3 marks) Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 5
Further Maths Core Pure (AS/Year 1) Unit Test : Matrices 6a Writes the matrix representing a reflection in the plane y = 0: 1 0 0 0 1 0 0 0 1 B1 3.1a 6b Finds the midpoint of the line segment = (5, 5, 9) (1) 1 0 05 a 0 1 0 5 b 0 0 19 c Makes an attempt to calculate Minimum required is setting up the calculation. M1.a Correctly finds the coordinates (5, 5, 9) A1 3.1b 6c States or implies that N is the inverse of M. M1.a (3) 1 0 0 1 Finds N = M 0 1 0 0 0 1 A1 1.1b () (6 marks) Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 6
Further Maths Core Pure (AS/Year 1) Unit Test : Matrices 7a det 1 1 M M1 1.1a M is non-singular because det (M) = 3 and so det (M) 0 A1.4 () 7b Area (S) = 3 0 = 60 B1 ft 1. (1) 7c Shows k det 1 1 M 7d States k 3 A1 ft 1.1b 1 States cos or sin or tan 3 3 θ = 15.3 Accept answers which round to 15.3 A1 1.1b () () (7 marks) Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 7
Further Maths Core Pure (AS/Year 1) Unit Test : Matrices 8a 1 3 States either cos or sin or tan 3 Finds θ = 40 and concludes this is a rotation of 40 anticlockwise about the z-axis. B1 3.a () 8b Makes an attempt to calculate 1 3 0 p 3 1 0 0 p 0 0 1 Minimum required is setting up the calculation. Correctly finds 1 3 1 0 p p 3 1 3 0 0 p p p 0 0 1 A1 1.1b () (4 marks) Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 8
Further Maths Core Pure (AS/Year 1) Unit Test : Matrices 9 Attempts to set up three equations with three unknowns. M1 3.1b At least two equations are correct, with variables defined. x = area of residential land y = area of commercial land z = area of recreational land x y z 00 A1 1.1b y z 0 1.x 0.9y 1.8z 40 Sets up a matrix equation of the form, where are numerical values.......... x............ y............ z..., M1 3.1a States the correct matrix equation: 1 1 1 x 00 0 1 1 y 0 1. 0.9 1.8 z 40 A1 1.1b Attempts to use an inverse matrix to find the values of x, y and z. 1 x 1 1 1 00 y 0 1 1 0 z 1. 0.9 1.8 40 Finds the correct answers for x, y and z: x 140 y 0 z 40 A1 1.1b Puts their answer into context. In 001, there were 140 square kilometres assigned to residential, 0 square kilometres assigned to commerical and 40 square kilometres assigned to recreation. A1 ft 3.a (7 marks) 9 Note the inverse matrix of 1 1 1 0 1 1 1. 0.9 1.8 is.18 0.38 50 1. 0.06 1 13 1. 0.3 1 Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 9
Further Maths Core Pure (AS/Year 1) Unit Test : Matrices Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 10
Further Maths Core Pure (AS/Year 1) Unit Test : Matrices 10a cos States either 1 sin or 1 or tan 1 Finds θ = 135 and concludes this is a rotation of 135 anticlockwise about the origin. or Finds θ = 45 and concludes this is a rotation of 45 clockwise about the origin. B1 3.a () 10b Finds M 1 1 1 1 1 States or implies that if a a b 3 b 3 1 M then M M1 3.1a Correctly solves to find a = 4 and b = A1 1.1b (3) (5 marks) Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 11