11 (a) 3² 4 × × 4 0 2 M1 for use of discriminant, eg 3² 4 × × 4 a a a − ≥ ≤ − A1 for oe a ≤ (b) = 3 M1 for at one correct equation −11, −10 = A1 for = b − −11 A1 for = c −10 12 (a) (x + )2 + 1 (x + )2 2 M1 for (x + )2 oe − − A1 for (x + )2 oe − (b) When = 0, = 1 Sketch graph 3 B1 for general shape (parabola) x y B1 for minimum point in third quadrant B1 for intercept labelled at (0, 1) y (c) (−,−) 1 B1 cao 13 Sketch graph 4 M1 for correct shape = 4 and = 0 A1 for fully correct graph with one intercept shown and not crossing x y y (0, the axis x −) B1 for equations of asymptotes = 4 and = 0 x y B1 for (0, marked on the −)
11 (a) 3² − 4 × a × 4 ≥ 0 a ≤ 2 M1 for use of discriminant, eg 3² − 4 × a × 4 A1 for a ≤ oe (b) = = − −11, −10 3 M1 for at one correct equation A1 for b = −11 A1 for c = −10 12 (a) (x + )2 − + 1 (x + )2 − 2 M1 for (x + )2 oe A1 for (x + )2 − oe (b) When x = 0, y = 1 Sketch graph 3 B1 for general shape (parabola) B1 for minimum point in third quadrant B1 for y intercept labelled at (0, 1) (c) (−,−) 1 B1 cao 13 Sketch graph x = 4 and y = 0 (0, −) 4 M1 for correct shape A1 for fully correct graph with one y intercept shown and not crossing the x axis B1 for equations of asymptotes x = 4 and y = 0 B1 for (0, −) marked on the