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MEMORANDUM GRADE 12 MATHEMATICS PAPER 2 JUNE 2011 This memo paper consists of 8 pages. Grade 12 Mathematics Paper 2 Memorandum June 2011 Question 1 1.1.1 A reflection about the y-axis 1 for reflection 1 for y-axis (2) 1.1.2 A rotation through 900 in an anti clockwise 1 for rotation direction 1 for size and direction (2) 1.1.3 a rotation through θ in an anti clockwise direction 1 for rotation 1 for size and direction (2) 1.1.4 An enlargement by a scale factor of 2 1 for enlargement 1 for the scale factor (2) 1.2 1.1.4 ( the enlargement). It changes the size. 1 for the correct transformation 1 for the reason (2) 1.3 Accept calculator answers as well 1 for substitution 1 for values of special angles 1 for answer 1 for substitution 1 for values of special angles 1 for answer (6) [16] 2 Grade 12 Mathematics Paper 2 Memorandum June 2011 Question 2 2.1 A reflection in the line y=-x 1 for reflection (2) 1 for line of reflection 2.2 (2) 2.3 A rotation through 1800 1 for rotation (2) 1 for angle of rotation 2.4 (2) 2.5.1 k = 1,5 (2) 2.5.2 F is ( -4,5 ; -1,5) 1 per coordinate (2) 2.6 Area ΔABC (2) [14] Question 3 3.1.1 1 for substitution (2) 1 for answer 3.1.2 the equation of AC will be: 1 for substitution of gradient 1 for substitution of point (3) 1 for answer 3.1.3 1 for tan… 1 for gradient 1 for answer (3) 3.1.4 ( product of gradients = -1) 1 for gradient Midpt. of DB is (3;-2). 1 for conclusion Substitute this to check if it is on line AC 1 for reason LHS = -2 2 for midpoint RHS = 2 for showing midpoint on AC (7) So AC bisect DB perpendicularly. 3 Grade 12 Mathematics Paper 2 Memorandum June 2011 3.1.5 =6,48 2 for AM 2 for BD 2 for area (6) 3.2.1 M is 1 per substitution for each co-ordinate. (2) 3.2.2 M is (1;4) 1 per gradient (3) 1 for conclusion Therefore the gradients are equal So the points are collinear 3.2.3 Midpoint of AC is (1;4). So diagonals bisect each 1 for midpoint other. 1 for conclusion 1 for gradient So diagonals are perpendicular 1 for conclusion (4) Hence diagonals of ABCD bisect each other perpendicularly. [30] Question 4 4.1 1 for LHS ( completing the squares) 1 for RHS 1 for final equation So M is (-2 ; 6) and r = 6 2 for centre (6) 1 for radius 4.2.1 When y=2, 1 for recognising the value of y x = -5 ( from the equation of the line) 1 for the value of x And the radius r= 5 1 for the value of r 1 for equation in Hence the circle is centre radius form Therefore: (6) 1 for expanding So: 4 Grade 12 Mathematics Paper 2 Memorandum June 2011 4.2.2 Let D be Then 1 for the calculation of x And 1 for the calculation of y 1 for the gradient of the radius Thus the equation of the tangent is given by: 1 for the gradient of the tangent 1 for substitution 1 for the answer. (6) [18] Question 5 5.1 1 for simplification 1 for each answer 1 for each answer = 0,78 or (5) = 0,37 5.2.1 2 per diagram 17 1 for expansion α -4 8 β 1 for substitution -3 1 for answer (5) 5 -15 = 5.2.2 1 for expansion 1 for substitution (3) 1 for answer. 5 Grade 12 Mathematics Paper 2 Memorandum June 2011 5.3 1 per reduction (6) [19] Question 6 6.1 1 for getting all to LHS 1 for grouping 1 for factors 1 for roots 1 for working with sinx only (7) Therefore 1 for each answer 6.2.1 (4) 6.2.2 2 for numerator 1 for denominator 1 for simplification 1 for simplification 1 for conclusion (6) [17] Question 7 7.1.1 1330 (1) 7.1.2 1 for Substitution (2) 1 for answer So AC = 183,12 7.1.3 1 for substitution (2) So 1 for answer 6 Grade 12 Mathematics Paper 2 Memorandum June 2011 7.1.4 1 substitution 1 for the value of θ 2 for the answer (4) 7.2.1 (1) 7.2.2 (2) 7.2.3 1 for substitution 1 for expanding compound angle 1 for special angle values 1 for isolating p 1 for simplification (5) [17] 7 Grade 12 Mathematics Paper 2 Memorandum June 2011 Question 8 8.1.1 (1) 8.1.2 (1) 8.1.3 (300 ; 0) (1) 8.1.4 (1) 8.2.1 cos graph: (4) sin graph: (3) 1 for turning points 1 for turning points 1 for y-intercept 1 for x-intercept 1 for endpoints 1 for endpoints (8) 1 for x-intercepts 1 for y-intercept 8.2.2 (1) 8.2.3 1 for each endpoint 1 for correct notation (3) 8.2.4 1 for each value 1 for the answer (3) [19]] 8 Grade 12 Mathematics Paper 2 Memorandum June 2011 Total: 150 marks 9