Figure 1 shows a sketch of the parabola with equation = 8x. P y2 The point lies on where > 0, and the point lies on where < 0. P C, y Q C, y The line segment is parallel to the PQ y-axis. Given that the distance is 12, PQ (a) write down the of y-coordinate P, (1) (b) find the of x-coordinate P. (2) Figure 1 shows the point which is the focus of S C. The line passes through the point and the point l P S. (c) Find an equation for in the form + + = 0, where and are integers. l ax by c a, b c (4) 6. f(x) = tan + 3x 6, < < . x – – (a) Show that the equation f(x) = 0 has a root in the interval [1, 2]. (2) (b) Use linear interpolation once on the interval [1, 2] to find an approximation to . Give your answer to 2 decimal places. (3) = 2 7. z − i√3. (a) Calculate arg giving your answer in radians to 2 decimal places. z, (2) Use algebra to express (b) + in the form + where and are integers, z z2 a bi√3, a b (3) (c) in the form + where and are integers. c di√3, c d (4) Given that 3i, w = – where is a real constant, and arg (4 5i + 3w) = – –, (d) find the value of .
Figure 1 shows a sketch of the parabola P with equation y2 = 8x. The point P lies on C, where y > 0, and the point Q lies on C, where y < 0. The line segment PQ is parallel to the y-axis. Given that the distance PQ is 12, (a) write down the y-coordinate of P, (1) (b) find the x-coordinate of P. (2) Figure 1 shows the point S which is the focus of C. The line l passes through the point P and the point S. (c) Find an equation for l in the form ax + by + c = 0, where a, b and c are integers. (4) 6. f(x) = tan + 3x – 6, – < x < . (a) Show that the equation f(x) = 0 has a root in the interval [1, 2]. (2) (b) Use linear interpolation once on the interval [1, 2] to find an approximation to . Give your answer to 2 decimal places. (3) 7. z = 2 − i√3. (a) Calculate arg z, giving your answer in radians to 2 decimal places. (2) Use algebra to express (b) z + z2 in the form a + bi√3, where a and b are integers, (3) (c) in the form c + di√3, where c and d are integers. (4) Given that w = – 3i, where is a real constant, and arg (4 – 5i + 3w) = –, (d) find the value of .
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