Essential Mathematics 1: Algebra and Trigonometry - Mathematics Assignment Help

Assignment Task:

Task:

Question 1 (3+2+3+3+4 = 15 marks)
(a) Find the equation of the line passing through the points (2, −1) and (1, 1). Give your answer in standard form Ax + By = C.
(b) Find the slope of a line that is perpendicular to the line obtained in part (a) and give a reason for your answer.
(c) Find the equation of the line that is perpendicular to the line obtained in part (a) and  that passes through the point (−1, 0). Give your answer in standard form.
(d) Graph (by hand) the two equations obtained in part (a) and part (c) on the same Cartesian axes. Label the coordinates of the point of intersection between these lines.
(e) Solve the system of equations obtained in part (a) and part (c) by using the elimination method. Marks will be awarded for checking your solution.

Question 2 (5+(4+4) = 13 marks)
(a) From the given equation of the circle, find the centre and the radius:

2x
2 + 2y
2 − 10x − 12y = 7

.
(b) (i) For what values of c does the line y = x+c never meet the parabola y = 2x 2−3x−7?
(ii) Choose one of the values of c from part (i) and sketch the graphs to show that these curves never meet.

Question 3 (3+4 = 7 marks)
Verify the following trigonometric identities. In each case you might like to start with the most complicated side, and use algebraic manipulation to show that it can be written like the other side.
(a) sin4
t − cos4
t = sin2
t − cos2
t

(b) tan x + tan y
1 − tan x tan y = sin x cos y + cos x sin y cos x cos y − sin x sin y

Question 4 (6 marks)

A car leaves an intersection travelling at an average speed of 56 kilometres per hour. Five min- utes later, a second car leaves the same intersection and travels on the road making an angle of 112? with the first, at an average speed of 48 kilometres per hour. Assuming the roads are straight, how far apart are the cars 15 minutes after the first car has left? Leave your answer in 3 decimal places. Sketch a figure of the scenario.

Question 5 (3+4+4+1 = 12 marks)
Consider the function y = 2 cos(2x + π) for −3π 4≤ x ≤ 3π4
. Follow the steps below to sketch the graph of y.
(a) State the amplitude, period and phase shift in the graph of this function.
(b) Solve y = 2 cos(2x+π) for − 3π 4 ≤ x ≤ 3π4 to find the horizontal intercepts (x-intercepts)of the function.
(c) Find the values of x for −3π4≤ x ≤3π4
for which the maximum and the minimum values of the function occur.
(d) State the range of the function.

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