Miscellaneous Formulae:
· Gradient (m) of a straight line
o
o where and are points on the line
o , where is the angle the line makes with the positive direction of the x-axis.
· The distance d between two points having coordinates and is given by
· The midpoint M of a segment joining and is given by
· The quadratic formula: If , then
· Cubic formulas
o
o
o
o
· Probability formula
o
o if A and B are independent
o
Technology - free – (60 marks)
Question 1
Consider the points (-5, 4) and (7, -2)
a) Show that the midpoint of the line segment joining these two points has co-ordinates .
b) Find the gradient of the line joining these two points.
c) What would be the gradient of a line that is perpendicular to the line joining these two points?
d) Find the equation of the line that is perpendicular to the line joining these two points and that passes through the midpoint.
(1 +2 + 1 + 3 = 7 marks)
Question 2
A wildlife sanctuary has a combined total of 58 ostriches (2 legs) and giraffes (4 legs). There is a combined total of 156 legs. Define the variables and use simultaneous equations to determine how many of each animal there are.
(5 marks)
Question 3
a) Fully factorise the following polynomial expressions.
i)
ii)
b) Solve for x:
i)
ii)
[(1+1) + (3+3) = 8 marks]
Question 4
For what value(s) of does the equation have no solutions?
(3 marks)
Question 5
The point (3, -5) is reflected in the y axis then dilated by a factor of 2 from the x axis. Use matrices to find the co-ordinates of the final transformed point.
(3 marks)
Question 6
The graph of the function is shown below.
a) Find the values of , and .
The graph of a cubic function is shown below.
b) Find the equation of this function.
(4 + 3 = 7 marks)
Question 7
a) Show that is a factor of the polynomial
b) Show, by completing the square, that can be factorised to:
Important: show all your working
c) Given that , use the information from parts a) and b) to state all solutions to
(2 + 3 + 2 = 7 marks)
Question 8
Question 9
The graph has been dilated by a factor of 3 from the x axis and translated 4 units in the positive direction of x and 3 units in the negative direction of y.
a) What is the equation of the transformed graph and what are the coordinates of the point of inflection?
b) The transformed graph is now translated so that the point of inflection is at (1, 2).
Describe these translations.
[3 + 2 = 5]
Question 10
a) If , and , find .
b) State, with clear reasons, whether A and B are independent events
[2 + 2 = 4]
END OF PART I
Unit 1 Mathematical Methods (CAS) Exam – Part I Page 1