Foundation Program

Mathematics S

Term 1 Examination
Time Allowed: 2 hours
Reading Time: 5 minutes

SAMPLE A
 TERM 1 EXAMINATION

Directions to Candidates

1. Attempt ALL questions.

2. Answer each question in a separate answer book. Clearly write the question
number in the box provided on the front of each answer book

3. All necessary working must be shown in every question. Answers must be

correctly numbered and clearly arranged.

4. All questions are of equal value. Total marks available – 72.

5. Marks will not necessarily be awarded if directions to candidates have not

been followed.
Question 1 (12 marks)
Work on separate sheets of paper clearly marked Question 1. At the end of the exam,
scan and upload these as a single PDF file to the Moodle Assignment activity labelled
“Question 1”.

(i) Fully factorise 9 x 2 − 6 xy − 3 x + 2 y .

1
(ii) Find the domain of the function y = 1− x + .
x+2

(iii) If cos10° =a , express cos170° in terms of a .

4
(iv) Evaluate ∑5
k =2
k −1
.

(v) Find the value of x if log 3 (7 − x) =

2 .

10
(vi) If x = log 2 3 and y = log 2 5 , express log 2 in terms of x and y .
9

x3 + 1
(vii) Find lim .
x → −1 x +1

(viii) A and M are the points ( −3, 8) and (1, 3) respectively. M is the midpoint of the
line segment AC .

(a) Find the coordinates of the point C .

(b) B is the point ( 3, 5) and ABCD is a parallelogram. Find the length of the
shorter diagonal of the parallelogram.
Question 2 (12 marks)
Work on separate sheets of paper clearly marked Question 2. At the end of the exam,
scan and upload these as a single PDF file to the Moodle Assignment activity labelled
“Question 2”.

(i) Consider the parabola y =− x2 + 4 x + 1 .

(a) Find the coordinates of the vertex.

(b) If the parabola has the restricted domain x ≥ 3 , state the range.

(ii) A bag contains 5 red balls, 3 green balls and 4 blue balls. Two balls are selected at
random without replacement from the bag. Find the probability of obtaining:

(a) two red balls.

(b) two balls with different colours.

(iii) The quadratic equation 3 x 2 + 7 x − 2 =0 has roots α and β . Find the values of:

(a) α +β .

(b) α2 + β2 .

1 1
(iv) Simplify − .
cos θ cot 2 θ
2

(v) The table below shows the probability distribution for a discrete random variable X .

xi 2 3 4 5
pi 0.2 0.4 0.3 0.1

(a) Calculate the expected value E ( X ) .

(b) Calculate the variance Var ( X ) .

Question 3 (12 marks)
Work on separate sheets of paper clearly marked Question 3. At the end of the exam,
scan and upload these as a single PDF file to the Moodle Assignment activity labelled
“Question 3”.

x +1
(i) Determine whether the function f ( x) = is even, odd or neither even nor odd.
x2

(ii) Find the value of c for which the quadratic equation 3 x 2 + 2 x + c =0 has exactly
one solution.

(iii) The polynomial 2 x 3 + x 2 + kx + 10 is divisible by ( x + 2) . Find the value of k .

(iv) For two events A and B it is given that P ( A) = 0 ⋅ 7 , P ( B )= 0 ⋅ 5 and

P( A ∪ B ) =0 ⋅ 9 .

(a) Find P( A ∩ B ) .

(b) State whether A and B are dependent or independent events. Give a reason
for your answer.

(v) Differentiate each of the following with respect to x :

(a) (3 x − 7) 4 .

3x − 2
(b) .
2x +1

(c) x ( x + 1) 5 .

(vi) On separate diagrams, sketch graphs of the following functions showing all their
essential features:

1
(a) =y −1 .
x+2

(b) y= 3
x .
Question 4 (12 marks)
Work on separate sheets of paper clearly marked Question 4. At the end of the exam,
scan and upload these as a single PDF file to the Moodle Assignment activity labelled
“Question 4”.

(i) The first term of an arithmetic series is 5 , the last term is 296 and the common
difference between successive terms is 3 . Find the sum of the terms of the arithmetic
series.

(ii) The infinite geometric series 4 + 4k + 4k 2 + ... has a limiting sum of 16 .

Find the value of k .

 mx + 10 x≤2
(iii) Consider the function f ( x) =  2
where m and a are constants.
 a ( x − 3) x>2

(a) Given that the function is continuous at x = 2 , find an equation relating the
constants m and a .

(b) Given that the function is continuous and differentiable at x = 2 , find values
for the constants m and a .

1  1
(iv) The tangent to the curve y = at the point  a,  intersects the y-axis at 3 . Find
x  a
the value of a and the equation of the tangent.

2 1
(v) Solve the inequality ≥ .
x +1 x − 2
Question 5 (12 marks)
Work on separate sheets of paper clearly marked Question 5. At the end of the exam,
scan and upload these as a single PDF file to the Moodle Assignment activity labelled
“Question 5”.

(i) Linda scored 77% in a Maths test in which the class mean and standard deviation
were 65% and 8% respectively. In a second Maths test the class mean and standard
deviation were 72% and 6% respectively. Linda was absent for the second test.
Calculate a fair estimated mark for Linda based on her performance in the first test.

(ii) By using the principle of mathematical induction prove that

1 2
n ( n + 1) for all integers n ≥ 1 .
2
13 + 23 + 33 + ... +=
n3
4

x2
(iii) Consider the curve y = f ( x) where f ( x) = .
x +1

(a) Find the equations of all asymptotes of the curve.

(b) Find the coordinates of any stationary points on the curve and determine their
nature.

(c) Sketch the curve showing all of the information above.

Question 6 (12 marks)
Work on separate sheets of paper clearly marked Question 6. At the end of the exam,
scan and upload these as a single PDF file to the Moodle Assignment activity labelled
“Question 6”.

(i) Find the gradient of the tangent to the curve x 2 + xy − y 2 =

1 at the point ( 2, −1) .

(ii) Solve the following equations for 0° ≤ θ ≤ 360° :

θ
(a) tan = −1 .
2

(b) cot θ + tan θ =

2sec θ .

(iii) A cylindrical can, including base and top, is made from a total of 150π cm 2 of sheet
metal. The sheet metal used to make the can is shown in the diagram below.

Diagram not
to scale

(a) If the volume of the can is given by V cm3 and the radius of the base is r cm,
show that= V 75π r − π r 3 .

(b) Determine the height of the can for which the volume of the can is maximum.