A101 ASSESSMENT Quadratics, Discriminant, Inequalities 1

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Muriel Horton
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1 Do the questions as a test circle questions you cannot answer Red (1) Solve a) 7x = x 2-30 b) 4x 2-29x + 7 = 0 (2) Solve the equation x 2 6x 2 = 0, giving your answers in simplified surd form [3]

2 (3) a) Express 4x x 3 in the form p(x + q) 2 + r b) Solve the equation 4x x 3 = 0, giving your answers in simplified surd form c) The quadratic equation 4x x k = 0 has equal roots. Find the value of k [3] (4) The quadratic equation kx 2 + (3k 1)x 4 = 0 has no real roots. Find the set of possible values of k. [7]

3 (5) Write in the form: 2 f ( x) p( x q) r a) f(x) = x 2 + 4x 7 b) f(x) = x 2 8x 2 c) f(x) = 2x 2 + 4x 8 d) f(x) = 5x 2 10x + 20 [8] Amber (6) Solve: a) 6x 7 < 2x + 3 b) x > 6 [4] c) 3x 2 23x + 30 < 0 d) 2x 2 11x [6]

4 (7) Find the set of values for which: a) 5x 3 > 8x + 9 b) 3x x < 10 c) both 5x 3 > 8x + 9 and 3x x < 10 [1] [4] (8) a) Write x x 1.75 in the form: f ( x) ( x a) b b) Hence, or otherwise, solve x 2 + 5x 1.75 = 0 giving your answer in the form a ± b 2

5 (9) Solve the equation x 8 x + 13 = 0, giving your answers in the form p ± q r, where p, q and r are integers [7] (10) By using the substitution u = (3x 2) 2, find the roots of the equation: (3x 2) 4 5(3x 2) = 0. [6]

6 (11) f(x) = x 2 2x 8 a) Find the value of the discriminant of f(x) [1] b) Sketch the graph of y = x 2 2x 8 showing the intersections with the axes and the minimum point [5] (12) The width of a sports pitch is x metres, x > 0. The length of the pitch is 40 m more than its width Given that the perimeter of the pitch must be less than 400 m a) Form and solve linear inequality in x [3] b) Given that the area of the pitch must be greater than 4500 m 2, [4] form and solve a quadratic inequality in x c) By solving your inequalities, find the set of possible values of x [1]

7 Green (13) Find the real roots of the equation = 0 y 4 y 2 [5] (14) f(x) = x 2 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 b) Given that k = 4, Express f(x) in the form (x p) 2 + q, where p and q are constants to be found c) Write down the minimum value of f(x) and the value of x for which this occurs [4]

8 (15) The quadratic equation x 2 + kx + k = 0 has no real roots for x a) Write down the discriminant of x 2 + kx + k in terms of k b) Hence find the set of values that k can take [4] (16) a) Sketch the curve y = 2x 2 x 3, giving the coordinates of all points [4] of intersection with the axes b) Hence, or otherwise, solve the inequality: 2x 2 x 3 > 0 c) Given that the equation 2x 2 x 3 = k has no real roots, find the set of possible values of the constant k. [3]

9 (17) Solve the equation x 2 3 x = 0 [5] (18) Complete the table below. The first one has been done for you. For each statement you must state if it is always true, sometimes true or never true, giving a reason in each case. Statement The quadratic equation ax 2 + bx + c = 0, (a 0) has 2 real roots. (i) When a real value of x is substituted into x 2 6x + 10 = 0 the result is positive. [2 marks] Always True Sometimes True Never True Reason It only has 2 real roots when: b 2 4ac > 0. When b 2 4ac = 0, it has 1 real root and when b 2 4ac < 0, it has 0 real roots. (ii) If ax > b then x > b a. [2 marks] (iii) The difference between consecutive square numbers is odd. [2 marks] Total Marks: 116

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Core Mathematics 1 Quadratics

Regent College Maths Department Core Mathematics 1 Quadratics Quadratics September 011 C1 Note Quadratic functions and their graphs. The graph of y ax bx c. (i) a 0 (ii) a 0 The turning point can be determined