can you do my undergraduate maths assignment for me please

can you do my undergraduate maths assignment for me please

Question 1

In this question, positions are given with reference to a Cartesian

coordinate system whose x- and y-axes point due East and due North,

respectively. Distances are measured in kilometres.

An aeroplane flies in a straight line from City A, at (200,−100), to City B,

at (500,−700).

(a) (i) Find the equation of the line of flight of the aeroplane. [3]

(ii) Find the direction of travel of the aeroplane, as a bearing, with

the angle correct to one decimal place. [3]

(b) After landing at City B, the aeroplane flies in a straight line in the

direction N57◦W, to City C, before finally flying in a straight line in

the direction N16◦E back to City A.

(i) Draw a diagram, in the form of a triangle, showing the three

flights of the aeroplane. Calculate and mark the three angles of

the triangle on your diagram, in degrees to one decimal place. [4]

(ii) Find the distance flown by the aeroplane from City C to City A,

to the nearest kilometre. [5]

(c) This part of the question is about the aeroplane’s flight from City A

to City B. There is an aircraft detection tower at position (500,−400).

(i) Find parametric equations for the line of flight of the aeroplane.

Your equations should be in terms of the parameter t, and should

be such that the aeroplane is at City A when t = 0 and at City B

when t = 1. [2]

(ii) Write down an expression, in terms of t, for the square of the

distance between the aircraft detection tower and the point with

parameter t on the line of flight of the aeroplane. Simplify your

answer. [3]

(iii) Use your answer to part (c)(ii), and the method of completing the

square, to determine the distance, to the nearest kilometre,

between the aeroplane and the aircraft detection tower at the

point on the line of flight of the aeroplane where it is closest to the

aircraft detection tower. (The distance required is the ‘horizontal’

distance; that is, the distance between the aircraft detection tower

and the point on the ground immediately below the aeroplane.) [5]

Question 2

As for other questions, remember to show your working explicitly

throughout your answer to this question.

(a) (i) Use the Composite Rule to show

that the function

h(x) = [5]

has derivative

h’ (x) =
(ii) Use the Quotient Rule and your answer to part (a)(i) to show

that the function

k(x) =

has derivative

k’(x) = [4]

(iii) Find any stationary points of the function k(x) defined in

part (a)(ii), and use the First Derivative Test to classify each

stationary point as a local maximum or a local minimum of k(x). [5]

(iv) Using your answers to parts (a)(ii) and (a)(iii), find the area

below the graph of

y =

and above the x-axis. Give your answer to five significant figures. [4]

(b) (i) Using your answer to part (a)(i), find the general solution of the

differential equation

dy =

dx

giving the solution in implicit form. [4]

(ii) Find the particular solution of the differential equation in

part (b)(i) for which y = 0 when x = 1, and then give this

particular solution in explicit form. [3]