QUESTION 1

Syntax advice . To enter the empty set O, type { }. To enter the set {1, 2} , type (1 , 2). To enter the interval (1, 2] , type (1 , 2]. To enter the value co or -co type infinity or -infinity (a) We consider the integral I1 = / x (x - 7) dx. (i) Find a substitution u = f(x) such that we can write I1 = / (u + 7)7 ua du. Give your answer in the box below. u = f(x) = (ii) Your friend Mitchell asks you to find all values of a such that the integral J1 = x7 (x - 7)a dx diverges. Provide your answer to Mitchell as a set or an interval. Answer: a E (b) Mitchell also wants to find 12 = (In(r +9) dr. x + 9 (i) Help Mitchell by providing a substitution t = g(x) so that we can write I2 = gat1 / to at. Give your answer in the following box. t = g(2) = (ii) Mitchell also wishes to find all values of p such that the following integral J2 = (In(a + 9) -8 da 4x + 11 converges. Give your answer to Mitchell, as a set or an interval, in the box below. Answer: p E (c) Another friend Ali is challenging you to find all the values n such that the following integral arctan 13 = Jo - dx (6x + 1)n-3 converges. However, Ali has realised that you have not learnt this type of improper integral. Therefore, he/she suggests using the substitution u = 1/x to write 13 = F(u)du. Enter your answer, as a set or an interval, to Ali in the box below for all values of n such that 13 converges. Answer: n E{a} Andrew wishes to solve the equation 22 + (5)2 = l] for z E C. Solve Andrew's equation for z and answer the following questions about the solutions. yntax advice: For each part of this question, you must enter either a set or a continuous interval. To enter the empty set 525 , type {}. To enter the set {1,2}, type {1,2}. To enter the interval (1, 2] , type (1 ,. 2]. To enter the value no, type infinity. To enter the value 1r , type Pi. Please use the "Preview" button to check your syntax. Write the set or real interval of all possible values for Arg[z) {whenever it is defined}. lush) E @ Write the set or real interval of all possible values of lzl. lzl E b {b} Hence, or otherwise, consider all solutions to 4' = |z4| for z E (C and i E 3+, to answer the following questions. Only consider solutions with "11- <: arg . for a fixed non-zero value how many solutions exist your answer will depend on number of e andrew decides to fix and plot all the an argand diagram. imagines tiny version themself standing diagram at one solutions. notices circle radius centre origin passes through what is shortest distance would have travel along circumference this reach another solution yntax advice: please use button check syntax. ie consider roots el z c convert ell into cartesian form i.e. ib. syntax maple complex number. if response then enter note capital letter i. ib="(b)" let us label as z1 z2 ... explain in essay box below given say efficient method by hand not using finding other roots. b q>Equation A - A- Ix BIUS X X Styles Font . . . Words: 0 (c) Hence, or otherwise, determine the product of the 58 roots, i.e., 2122 . . . 258, for 0 = - Z. Syntax advice: If you wish to enter your answer in polar form please use Maple syntax. If your response is 9el 47/7then enter 9*exp([*4*Pi/7). Please use the "Preview" button to check your syntax. 2122 . . . 258=