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C3 – Homework 13 – (33 marks) Q1. A cuboid has a rectangular cross-section where the length of the rectangle is equal to twice its width, x cm, as shown in Figure 2. The volume of the cuboid is 81 cubic centimetres. (a) Show that the total length, L cm, of the twelve edges of the cuboid is given by (3) (b) Use calculus to find the minimum value of L. (6) (c) Justify, by further differentiation, that the value of L that you have found is a minimum. (2) (Total 11 marks) Q2. The curve C has equation y =12√ (x) − x − 10, x>0 (a) Use calculus to find the coordinates of the turning point on C. (7) (b) Find . (2) (c) State the nature of the turning point. (1) (Total 10 marks) Q3. (a) Use the identity cos(A + B) = cos A cos B − sin Asin B, to show that cos 2A = 1 − 2 sin2A (2) The curves C1 and C2 have equations (b) Show that the x-coordinates of the points where C1 and C2 intersect satisfy the equation (3) (c) Express 4cos 2x + 3sin 2x in the form R cos (2x − α), where R > 0 and 0 < α < 90°, giving the value of α to 2 decimal places. (3) (d) Hence find, for 0 ≤ x < 180°, all the solutions of giving your answers to 1 decimal place. (4) (Total 12 marks)