Figure 1 shows a sketch of the curve with equation y = 2/x, x ≠ 0. The curve C has equation y = 2/x − 5, x ≠ 0, and the line l has equation y = 4x + 2. (a) Sketch and clearly label the graphs of C and l on a single diagram, showing clearly the coordinates of the points where C and l cross the coordinate axes (5 marks). (b) Write down the equations of the asymptotes of the curve C (2 marks). (c) Find the coordinates of the points of intersection of y = 2/x − 5 and y = 4x + 2 (5 marks).

(i) Sketch the curve y = 12 − x − x², giving the coordinates of all intercepts with the axes (5 marks). (ii) Solve the inequality 12 − x − x² > 0 (2 marks). (iii) Find the coordinates of the points of intersection of the curve y = 12 − x − x² and the line 3x + y = 4 (5 marks).

Solve the simultaneous equations x + y = 2 and 4y² − x² = 11 (7 marks).

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The height, h metres, of a tree, t years after being planted, is modelled by the equation h² = at + b for 0 ≤ t < 25, where a and b are constants. Given that the height of the tree was 2.60 m exactly 2 years after being planted and 5.10 m exactly 10 years after being planted: (a) find a complete equation for the model, giving the values of a and b to 3 significant figures (4 marks). Given that the height of the tree was 7 m exactly 20 years after being planted: (b) evaluate the model, giving reasons for your answer (2 marks).

Figure 4 shows a sketch of part of the curve C1 with equation y = 2x³ + 10, x > 0, and part of the curve C2 with equation y = 42x − 15x² − 7, x > 0. (a) Verify that the curves intersect at x = 1/2 (2 marks). The curves intersect again at the point P. (b) Using algebra and showing all stages of working, find the exact x coordinate of P (5 marks).

A small factory makes bars of soap. On any day, the total cost to the factory, £y, of making x bars of soap is modelled to be the sum of two separate elements: a fixed cost, and a cost that is proportional to the number of bars of soap that are made that day. (a) Write down a general equation linking y with x, for this model (1 mark). The bars of soap are sold for £2 each. On a day when 800 bars of soap are made and sold, the factory makes a profit of £500. On a day when 300 bars of soap are made and sold, the factory makes a loss of £80. Using the above information, (b) show that y = 0.84x + 428 (3 marks). (c) With reference to the model, interpret the significance of the value 0.84 in the equation (1 mark). Assuming that each bar of soap is sold on the day it is made, (d) find the least number of bars of soap that must be made on any given day for the factory to make a profit that day (2 marks).

The line l1 has equation y = (3/5)x + 6. The line l2 is perpendicular to l1 and passes through the point B(8, 0), as shown in the sketch. (a) Show that an equation for line l2 is 5x + 3y = 40 (3 marks). Given that lines l1 and l2 intersect at the point C and line l1 crosses the x-axis at the point A, (b) find the exact area of triangle ABC, giving your answer as a fully simplified fraction in the form p/q (5 marks).