Mark scheme Pure Mathematics Year 1 (AS) Unit Test 2: Coordinate geometry in the (x, y) plane
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1 Mark scheme Pure Mathematics Year 1 (AS) Unit Test : Coordinate in the (x, y) plane Q Scheme Marks AOs Pearson 1a Use of the gradient formula to begin attempt to find k. k 1 ( ) or 1 (k 4) ( k 1) (i.e. correct k 4 1 substitution into gradient formula and equating to k + 6 = 1 + k 1 = 7k ). k = * (must show sufficient, convincing and correct working). M1.a 1st A1* 1.1b Assumed knowledge. 1b Student identifies the coordinates of either A or B. Can be seen or implied, for example, in the subsequent step when student attempts to find the equation of the line. A(, ) or B(1, 4). Correct substitution of their coordinates into y = mx + b or y y 1 = m(x x 1) o.e. to find the equation of the line. For example, b 4 1 b y 4 x1 or y x or 11 y x or x y11 0 or () B1 1.1b nd () of a straight line given the gradient and a point on the line. Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.
2 Mark scheme Pure Mathematics Year 1 (AS) Unit Test : Coordinate in the (x, y) plane 1c Midpoint of AB is (, 1) seen or implied. B1.a rd Slope of line perpendicular to AB is, seen or implied. B1.a Attempt to find the equation of the line (i.e. substituting their midpoint and gradient into a correct equation). For example, 1 b or y1 x of a perpendicular bisector. x y 0 or y x 0. Also accept any multiple of x y 0 providing a, b and c are still integers. ( marks) Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.
3 Mark scheme Pure Mathematics Year 1 (AS) Unit Test : Coordinate in the (x, y) plane a 11 ( 7) 18 m Correct substitution of (4, 7) or ( 6, 11) and their gradient into y = mx + b or y y 1 = m(x x 1) o.e. to find the equation of the line. For example, 7 4 b 11 6 b or or y 7 x 4. or y 11 x 6 B1 1.1b nd y + x 1 = 0 or y x + 1 = 0 only () of a straight line given two points. b 1 1 y0, x so,0 A. Award mark for 1 x seen. 1 x0, y so Area = B 1 0,. Award mark for 1 y seen. B1 1.1b rd B1 1.1b B1 1.1b Solve involving length and area in the context of straight line graphs. () (6 marks) Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.
4 Mark scheme Pure Mathematics Year 1 (AS) Unit Test : Coordinate in the (x, y) plane y = mx seen or implied. 4th Substitutes their y = mx into x 6x mx 8( mx ) 4 o.e. Rearranges to a term quadratic in x (condone one arithmetic error). 1 m x (6 1 m) x 16 0 x 6x y 8y 4 M1.1a Use the discriminant to determine conditions for the intersection of circles and straight lines. Uses b 4ac 0, 6 1m 41 m 16 0 M1.1a Rearranges to 0m 6m 7 0 or any multiple of this. Attempts solution using valid method. For example, M1.a m m or 10 m o.e. (NB decimals A0). 10 (7) (7 marks) y Elimination of x follows the same scheme. x leading to m y y 6 y 8y 4 m m This leads to (1 m ) y (4 6m 8 m ) y 4 1m 4m 0 Use of b 4ac 0 gives 4 6m 8m 41 m 4 1m 4m 0 which reduces to 4m 0m 6m 7 0. m cannot equal 0, so this must be discarded as a solution for the final A mark. b 4ac 0could be used implicitly within the quadratic equation formula. Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.
5 Mark scheme Pure Mathematics Year 1 (AS) Unit Test : Coordinate in the (x, y) plane 4a Student attempts to complete the square twice for the first equation (condone sign errors). x y 6 6 x y 6 64 Centre (, 6) A1.a Radius = 8 A1.a M1.a 4th Find the centre and radius of a circle, given the equation, by completing the square. Student attempts to complete the square twice for the second equation (condone sign errors). x y q q x y q 18 q M1.a Centre (, q) A1.a Radius = 18 q A1.a (6) 4b Uses distance formula for their centres and 80. For example, 6 q 80 Student simplifies to term quadratic. For example, q 1q 0 0 M1.a th Concludes that the possible values of q are and 10 () ( marks) Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.
6 Mark scheme Pure Mathematics Year 1 (AS) Unit Test : Coordinate in the (x, y) plane a Student completes the square twice. Condone sign errors. x y x y 4 40 So centre is (4, ) and radius is 40 4th () Find the centre and radius of a circle, given the equation, by completing the square. b Substitutes x = 10 into equation (in either form). or y y 10y 1 0 Rearranges to term quadratic in y y 10y 1 0 (could be in completed square form y 4) M1.a th Obtains solutions y =, y = 7 (must give both). Rejects y = 7 giving suitable reason (e.g. 7 < ) or it would be below the centre or AQ must slope upwards o.e. B1. c ( ) 1 m AQ = 10 4 m (i.e. 1 over their m AQ ) B1ft.a l Substitutes their Q into a correct equation of a line. For example, 10 b or y x 10 B1 1.1b th y = x + 7 of the tangent to a given circle at a specified point. Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.
7 Mark scheme Pure Mathematics Year 1 (AS) Unit Test : Coordinate in the (x, y) plane d 6 AQ o.e. (could just be in coordinate form). M1.1a th AP o.e. so student concludes that point P has 6 coordinates (, 1). Substitutes their P and their gradient 1 ( m from c) into a correct equation of a line. For example, y x b 1 or y1 x AQ M1.1a M1.a e PA 40 Uses Pythagoras theorem to find 40 EP. Area of EPA = (could be in two parts). Area = 0 B1.1a th B1.a (1 marks) Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.
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