Hi there looking for help to understand these questions

Algebra In this question, we consider linear transformations mapping from R2 + a2. (a) Find the standard matrix corresponding to a reection in the line 1; = :c. (Hint: it may help to draw a diagram.) (b) By appropriately multiplying together the standard matrices for each individual transformation, verify that the transformation in part (a) is equavalent to performing the following transformations in the given order: (i) A rotation through E anticlockwise. (ii) A reection in the :c-axis. (iii) A rotation through E clockwise. You must explain your calculations and show your working. Calculus Suppose we wish to solve the nonlinear equation a: = cosg(:c). (1} (a) Write down the Newton-Raphson iteration formula for solving (1)- (b) Using the initial guess 3:0 = 0.6, nd a solution of (1) correct to two decimal places. You may do this using a suitably modied version of Algorithm 2.2 from the Calculus course notes, or man- ually using a calculator, etc. Whichever method you use, you should include all the iterates (values of an) that you calculate. Algebra Let F : R* - R be a linear transformation such that x- y +z+t F a + 2z -t [x ty+ 3z - 3t Find the standard matrix of F, and a basis for each of R(F) and ker(F). Explain clearly how you arrive at your answers. (You may use technology to calculate the reduced row echelon form of any matrix.) Calculus Find the general solution to the second-order non-homogeneous differ- ential equation dax2 d'y _ 16y = 6 sinh 4x + x. You must explain each stage of your calculation clearly to receive full marks. (Hint: when looking for a particular solution, it may be helpful to recall the definition sinh x = 2