Throughout this question, you should use algebra to work out your answers, showing your working clearly. You may use a graph to check that your answers are correct, but it is not sufficient to read your results from a graph.

(a) A straight line passes through the points 1 2 , 6 and 3 2 , 2

. (i) Calculate the gradient of the line. [1] (ii) Find the equation of the line. [2]

(iii) Find the x-intercept of the line. [2]

(b) Does the line y = 1 3 x + 3 intersect with the line that you found in part (a)? Explain your answer. [1]

(c) Find the coordinates of the point where the lines with the following equations intersect: 9x 1 2 y = 4, 3x + 3 2 y = 12.

[3]

(d) Using a throwing stick, Dominic can throw his dog's ball across the park. Assume that the park is flat.

The path of the ball can be modelled by the equation y = 0.02x 2 + x + 2.6, where x is the horizontal distance of the ball from where Dominic throws it, and y is the vertical distance of the ball above the ground (both measured in metres).

(i) Find the y-intercept of the parabola y = 0.02x 2 + x + 2.6 (the point at which the ball leaves the throwing stick). [1]

(ii) (1) By substituting x = 15 into the equation of the parabola, find the coordinates of the point where the line x = 15 meets the parabola. [2]

(2) Using your answer to part (d)

(ii)(1), explain whether the ball goes higher than a tree of height 4 m that stands 15 m from Dominic and lies in the path of the ball. [1]

(iii) (1) Find the x-intercepts of the parabola. Give your answers in decimal form, correct to two decimal places. [3] (2) Assume that the ball lands on the ground. Use your answer from part (d)

(iii)(1) to find the horizontal distance between where Dominic throws the ball, and where the ball first lands. [1]

(iv) Find the maximum height reached by the ball. [3

(a) A hardware engineer is looking at the temperature of Central Processing Units (CPUs) of different computers. In one experiment, the temperature of the CPU of her own computer can be modelled by the equation y = 0.05t + 45 (0 t 60), where y is the temperature of the CPU in degrees Celsius and t is the number of seconds into the experiment.

(i) Find the temperature of the CPU after 17 seconds according to this model. [1]

(ii) Explain what is meant by the inequality '(0 t 60)' that follows the equation. [1]

(iii) Using algebra, calculate the time at which the CPU is 42.6 C. [2]

(iv) Write down the gradient of the straight line represented by the equation y = 0.05t + 45. What does this measure in the practical situation being modelled? [2]

(v) What is the y-intercept of the equation y = 0.05t + 45? Explain what it means in the practical situation being modelled. [1]

(b) The same engineer decides to look into rates of cooling for liquids to experiment with different cooling solutions for servers. She finds that the rate of cooling for one liquid can be modelled by the equation: y = 97 0.95t (0 t 80) where y is the temperature of the liquid in degrees Celsius and t is the time in minutes.

(i) State whether the type of reduction for this model is linear or exponential. Describe how reduction rate differs between linear and exponential functions. [2]

(ii) Calculate the temperature when t = 15.