Two functions, and are defined by () = 2 ⁵ 54 ³ 76 50; () = 11 ⁴ 244 ² + 55.

a) Using a graphing software, generate the graphs of and on the same plane. Set the visible range of the x-axis to 10 and that of the y-axis to 3000. Use red and blue for the graph of and respectively. Use either labelling or legend to identify the graphs. Present the image, with a description that features number of roots and points of intersection.

For the complex numbers ₁=2+5 and ₂=1+2, a) Represent ₁ and ₂ on the same Argand diagram. For each of the following involving ₁ and ₂ as described above, a pair of expression is given. Demonstrate algebraically what single complex number each expression evaluates to. Then, for each pair, add them to the Argand diagram as in parts a) above. Use a new diagram for each pair. For each pair comment on the commutative nature of complex numbers under the operation.

The fourth and tenth terms of an arithmetic progression are defined by ₄= 7+8 ₁₀=19+14 a) Determine expressions for the first term and the common difference. b) If the sum of the first fifteen terms of this progression is 630, find the value of the thirtieth term.