The complex numbers and are given by 1. = 2 + 8i and = 1 i – Find, showing your working, (a) in the form + where and are real, a bi, a b (3) (b) the value of , (2) (c) the value of arg , giving your answer in radians to 2 decimal places. (2) f(x) = 3x2 . 2. – (a) Write down, to 3 decimal places, the value of f(1.3) and the value of f(1.4). (1) The equation f(x) = 0 has a root between 1.3 and 1.4 α (b) Starting with the interval [1.3, 1.4], use interval bisection to find an interval of width 0.025 which contains α. (3) (c) Taking 1.4 as a first approximation to apply the Newton-Raphson procedure once α, to f(x) to obtain a second approximation to giving your answer to 3 decimal places. α, (5) A sequence of numbers is defined by 3. = 2, = 5 4, 1. – n Prove by induction that, for ℤ, = 5n 1 + 1. n – (4) 4. Figure 1 Figure 1 shows a sketch of part of the parabola with equation = 12x . y2 The point on the parabola has . P x-coordinate The point is the focus of the parabola. S (a) Write down the coordinates of S. (1)
1. The complex numbers and are given by = 2 + 8i and = 1 – i Find, showing your working, (a) in the form a + bi, where a and b are real, (3) (b) the value of , (2) (c) the value of arg , giving your answer in radians to 2 decimal places. (2) 2. f(x) = 3x2 – . (a) Write down, to 3 decimal places, the value of f(1.3) and the value of f(1.4). (1) The equation f(x) = 0 has a root α between 1.3 and 1.4 (b) Starting with the interval [1.3, 1.4], use interval bisection to find an interval of width 0.025 which contains α. (3) (c) Taking 1.4 as a first approximation to α, apply the Newton-Raphson procedure once to f(x) to obtain a second approximation to α, giving your answer to 3 decimal places. (5) 3. A sequence of numbers is defined by = 2, = 5 – 4, n 1. Prove by induction that, for n ℤ, = 5n – 1 + 1. (4) 4. Figure 1 Figure 1 shows a sketch of part of the parabola with equation y2 = 12x . The point P on the parabola has x-coordinate . The point S is the focus of the parabola. (a) Write down the coordinates of S. (1)
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