7. (i) In each of the following cases, find a 2 × 2 matrix that represents (a) a reflection in the line = y –x, (b) a rotation of 135° anticlockwise about (0, 0), (c) a reflection in the line = followed by a rotation of 135° anticlockwise about y –x (0, 0). (4) (ii) The triangle has vertices at the points (1, (3, 0) and (11, 0), where is a constant. T k), k Triangle is transformed onto the triangle by the matrix T T ʹ Given that the area of triangle is 364 square units, find the value of T k. ʹ (6) 8. The points 8k) and 4k), where is a constant, lie on the parabola with P(4k2, Q(k2, k C equation = 16x. y2 The straight line passes through the points and l1 P Q. (a) Show that an equation of the line is given by l1 3ky 4x = 8k2 – (4) The line is perpendicular to the line and passes through the focus of the parabola l2 l1 C. The line meets the directrix of at the point l2 C R. (b) Find, in terms of the coordinate of the point k, y R. (7) Prove by induction that, for , 9. f(n) = 8n 2n – is divisible by 6. (6) TOTAL FOR PAPER: 75 MARKS
7. (i) In each of the following cases, find a 2 × 2 matrix that represents (a) a reflection in the line y = –x, (b) a rotation of 135° anticlockwise about (0, 0), (c) a reflection in the line y = –x followed by a rotation of 135° anticlockwise about (0, 0). (4) (ii) The triangle T has vertices at the points (1, k), (3, 0) and (11, 0), where k is a constant. Triangle T is transformed onto the triangle T ʹ by the matrix Given that the area of triangle T ʹ is 364 square units, find the value of k. (6) 8. The points P(4k2, 8k) and Q(k2, 4k), where k is a constant, lie on the parabola C with equation y2 = 16x. The straight line l1 passes through the points P and Q. (a) Show that an equation of the line l1 is given by 3ky – 4x = 8k2 (4) The line l2 is perpendicular to the line l1 and passes through the focus of the parabola C. The line l2 meets the directrix of C at the point R. (b) Find, in terms of k, the y coordinate of the point R. (7) 9. Prove by induction that, for , f(n) = 8n – 2n is divisible by 6. (6) TOTAL FOR PAPER: 75 MARKS
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