page 05 MARKS 6. (a) Consider the statement: For all odd numbers n, 2 4 n  is prime. Find a counterexample to show that the statement is false. (b) Prove directly that the difference between the cubes of any two consecutive integers is not divisible by 3. 7. (a) Use the substitution 2 1 u y  , or otherwise, to find the exact value of 5 2 0 4 1 y dy y    . Student engineers are using a 3D printer to make a model. Relative to a suitable set of axes, the cross-section of the model is symmetrical about the y‑axis and is represented in the first quadrant by the curve with equation 2 4 1 y x y   , y   0 5, as shown in the diagram. y 5 0 2 4 1 y x y   x (b) State the area of the cross‑section. (c) Express 2 2 1 y y  in the form 2 1 b a y   where a and b are real numbers. The curve 2 4 1 y x y   , 0 5 y  , will be rotated through π 2 radians about the y‑axis to make the model. (d) Find the volume of the model. [END OF QUESTION PAPER] 1 3 4 1 1 4 page 06 [BLANK PAGE] DO NOT WRITE ON THIS PAGE page 07 [BLANK PAGE] DO NOT WRITE ON THIS PAGE page 08 [BLANK PAGE] DO NOT WRITE ON THIS PAGE