(Total for Question 10 is 4 marks) ___________________________________________________________________________ 11. (a) Prove that for all positive values of and x y, ≤ (2) (b) Prove by counterexample that this is not true when and are both negative. x y (1) (Total for Question 11 is 3 marks) ___________________________________________________________________________ 12. A student was asked to give the exact solution to the equation 22x + 4 9(2x) = 0 – The student’s attempt is shown below: 22x + 4 9(2x) = 0 – 22x + 24 9(2x) = 0 – Let 2x = y 9y + 8 = 0 y2 – (y 8)(y 1) = 0 – – = 8 or = 1 y y So = 3 or = 0 x x (a) Identify the two errors made by the student. (2) (b) Find the exact solution to the equation. (2) (Total for Question 12 is 4 marks) ___________________________________________________________________________ 13. (a) Factorise completely + 10x2 + 25x x3 (2) (b) Sketch the curve with equation = + 10x2 + 25x y x3 showing the coordinates of the points at which the curve cuts or touches the x-axis. (2) The point with coordinates (3, 0) lies on the curve with equation
(Total for Question 10 is 4 marks) ___________________________________________________________________________ 11. (a) Prove that for all positive values of x and y, ≤ (2) (b) Prove by counterexample that this is not true when x and y are both negative. (1) (Total for Question 11 is 3 marks) ___________________________________________________________________________ 12. A student was asked to give the exact solution to the equation 22x + 4 – 9(2x) = 0 The student’s attempt is shown below: 22x + 4 – 9(2x) = 0 22x + 24 – 9(2x) = 0 Let 2x = y y2 – 9y + 8 = 0 (y – 8)(y – 1) = 0 y = 8 or y = 1 So x = 3 or x = 0 (a) Identify the two errors made by the student. (2) (b) Find the exact solution to the equation. (2) (Total for Question 12 is 4 marks) ___________________________________________________________________________ 13. (a) Factorise completely x3 + 10x2 + 25x (2) (b) Sketch the curve with equation y = x3 + 10x2 + 25x showing the coordinates of the points at which the curve cuts or touches the x-axis. (2) The point with coordinates (3, 0) lies on the curve with equation
01a_as_mathematics_sams_-_paper_1_pure_mathematics_lb.docx