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1. (a) Show that x2 + 6x + 11 can be written as (x + p)2 + q where p and q are integers to be found. (2) (b) In the space at the top of page 7, sketch the curve with equation y = x2 + 6x + 11, showing clearly any intersections with the coordinate axes. (2) (c) Find the value of the discriminant of x2 + 6x + 11 (2) (Total 6 marks) El Alsson British International School 1 2. (a) On the axes below sketch the graphs of (i) y = x (4 – x) (ii) y = x2 (7 – x) showing clearly the coordinates of the points where the curves cross the coordinate axes. (5) (b) Show that the x-coordinates of the points of intersection of y = x (4 – x) and y = x2 (7 – x) are given by the solutions to the equation x(x2 – 8x + 4) = 0 (3) El Alsson British International School 2 The point A lies on both of the curves and the x and y coordinates of A are both positive. (c) Find the exact coordinates of A, leaving your answer in the form (p + q√3, r + s√3), where p, q, r and s are integers. (7) (Total 15 marks) f(x) = x2 + 4kx + (3 + 11k), 3. (a) where k is a constant. Express f(x) in the form (x + p)2 + q, where p and q are constants to be found in terms of k. (3) Given that the equation f(x) = 0 has no real roots, (b) find the set of possible values of k. (4) Given that k = 1, (c) sketch the graph of y = f(x), showing the coordinates of any point at which the graph crosses a coordinate axis. (3) (Total 10 marks) 4. The equation x2 + 3px + p = 0, where p is a non-zero constant, has equal roots. Find the value of p. (Total 4 marks) 5. (a) Factorise completely x3 – 6x2 + 9x (3) El Alsson British International School 3 (b) Sketch the curve with equation y = x3 – 6x2 + 9x showing the coordinates of the points at which the curve meets the x-axis. (4) Using your answer to part (b), or otherwise, (c) sketch, on a separate diagram, the curve with equation y = (x – 2)3 – 6(x – 2)2 + 9(x – 2) showing the coordinates of the points at which the curve meets the x-axis. (2) (Total 9 marks) 6. The equation kx2 4 x (5 k ) 0 , where k is a constant, has 2 different real solutions for x. (a) Show that k satisfies k 2 5k 4 0. (3) (b) Hence find the set of possible values of k. (4) (Total 7 marks) 7. Given that the equation 2qx2 + qx – 1 = 0, where q is a constant, has no real roots, (a) show that q2 + 8q < 0. (2) (b) Hence find the set of possible values of q. (3) (Total 5 marks) El Alsson British International School 4 8. The equation x2 + kx + 8 = k has no real solutions for x. (a) Show that k satisfies k2 + 4k – 32 < 0. (3) (b) Hence find the set of possible values of k. (4) (Total 7 marks) 9. Find the set of values of x for which x 2 – 7x – 18 > 0. (Total 4 marks) 10. The equation x2 + 2px + (3p + 4) = 0, where p is a positive constant, has equal roots. (a) Find the value of p. (4) (b) For this value of p, solve the equation x2 + 2px + (3p + 4) = 0. (2) (Total 6 marks) 11. x2 + 2x + 3 (x + a)2 + b. (a) Find the values of the constants a and b. (2) (b) In the space provided below, sketch the graph of y = x2 + 2x + 3, indicating clearly the coordinates of any intersections with the coordinate axes. (3) El Alsson British International School 5 (c) Find the value of the discriminant of x2 + 2x + 3. Explain how the sign of the discriminant relates to your sketch in part (b). (2) The equation x2 + kx + 3 = 0, where k is a constant, has no real roots. (d) Find the set of possible values of k, giving your answer in surd form. (4) (Total 11 marks) 12. y C A P R O Q x The diagram above shows part of the curve C with equation y = x2 – 6x + 18. The curve meets the y-axis at the point A and has a minimum at the point P. (a) Express x2 – 6x + 18 in the form (x – a)2 + b, where a and b are integers. (3) El Alsson British International School 6 (b) Find the coordinates of P. (2) (c) Find an equation of the tangent to C at A. (4) The tangent to C at A meets the x-axis at the point Q. (d) Verify that PQ is parallel to the y-axis. (1) The shaded region R in the diagram is enclosed by C, the tangent at A and the line PQ. (e) Use calculus to find the area of R. (5) (Total 15 marks) 13. Given that the equation kx2 + 12x + k = 0, where k is a positive constant, has equal roots, find the value of k. (Total 4 marks) El Alsson British International School 7 14. Given that f(x) = x2 – 6x + 18, (a) x 0, express f(x) in the form (x – a)2 + b, where a and b are integers. (3) The curve C with equation y = f(x), x 0, meets the y-axis at P and has a minimum point at Q. (b) Sketch the graph of C, showing the coordinates of P and Q. (4) The line y = 41 meets C at the point R. (c) Find the x-coordinate of R, giving your answer in the form p + q2, where p and q are integers. (5) (Total 12 marks) 15. f(x) = x2 – kx + 9, where k is a constant. (a) Find the set of values of k for which the equation f(x) = 0 has no real solutions. (4) Given that k = 4, (b) express f(x) in the form (x – p)2 + q, where p and q are constants to be found, (3) (c) write down the minimum value of f(x) and the value of x for which this occurs. (2) (Total 9 marks) 16. (a) Solve the equation 4x2 + 12x = 0. (3) El Alsson British International School 8 f(x) = 4x2 + 12x + c, where c is a constant. (b) Given that f(x) = 0 has equal roots, find the value of c and hence solve f(x) = 0. (4) (Total 7 marks) El Alsson British International School 9