Foundation Program Mathematics C SAMPLE D
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Foundation Program
SAMPLE D
Mathematics C
Final Term 1 Examination
Question 1 (12 marks) Use a SEPARATE book clearly marked Question 1
(i) Factorise 1 − 4m2 .
(ii) Evaluate π− 1 correct to 2 decimal places.
e
(iii) Simplify x 2x
(iv) Express y in terms of x when x − 1 = 4 .
y
(v) M and F are sets defined as follows:
M = { multiples of 3 between 1 and 17 }
F = { factors of 24 between 5 and 17 }
Find n(M ∩F).
(vi) Solve 2x − 1 < 3 .
(vii) Sketch the graph of y = x + 1 − 1 showing the essential features.
(viii) If − 1 = 3 − , find the value of p.
(ix) The infinite geometric series x − x2 + x3 − x4 + ⋯ has a limiting sum of 1 . Find
the value of x.
Question 2 (12 marks) Use a SEPARATE book clearly marked Question 2
(i) Find the equation ofthe axis of symmetry of the parabola y = 1 − x − x2 .
(ii) dy for each of the following functions expressing the answers in simplified
form.
(a) y = (2x2 + 1)4 .
(b) x2 − 1
(iii) Find the gradient ofthe tangent to the curve y = 2 at the point where x = 9 .
(iv) Solve the equation 22x − 3(2x+1 )= 16 .
(v) A polynomial is given by P(x) = x3 − 6x2 + 11x − 6 .
(a) Show that (x − 1) is a factor of P(x), and hence use polynomial division to
(b) Hence solve x3 − 6x2 + 11x − 6 < 0 .
Question 3 (12 marks) Use a SEPARATE book clearly marked Question 3
(i) In a particular town, 40% of the population are aged under 30 years
and 45% are aged from 30 years to 60 years inclusive. Ifthis information is represented on a sector graph, find the size of the sector angle representing the percentage ofthe population aged over 60 years.
(ii) Find the effective rate of interest that is equivalent to an annual nominal rate of
8 % compounded monthly. Express the answer as a percentage correct to one decimal place.
(iii) Find the co-ordinates of the point on the parabola y = 1 − x2 where the tangent
is parallel to the line y = x + 1 .
(iv) A function is defined as f (x) = 2x − 1 .
(a) Find the inverse function f − 1 (x) .
(b) Show that f 1() = f − 11() .
(c) Sketch the graphs of y = f (x) and y = f − 1(x) on the same diagram.
(d) Hence find the values of x for which f (x) ≥ f − 1(x).
Question 4 (12 marks) Use a SEPARATE book clearly marked Question 4
(i) Consider the arithmetic sequence ln 64, ln 48, ln 36 , … Constants p and q have values p = ln 2 and q = ln 3 .
(a) Find the common difference ofthe arithmetic sequence in terms of p
(b) Show that a9 = 8 q − 10p where a9 is the ninth term ofthe arithmetic sequence.
(c) Show that 9 ak = 18(2q − p) . k=1
(ii) Consider the curve given by the equation y = x4 − 4x3 + 28 .
(a) Find the stationary points on the curve and determine their nature.
(b) Find the values of x for which the curve is concave down.
(c) Hence sketch the curve y = x4 − 4x3 + 28 .
Question 5 (12 marks) Use a SEPARATE book clearly marked Question 5
(i) A piecemeal function is defined by:
2x + k, 0 ≤ x < 2
f (x) = 1 2
Find the value of k for which f (1) = f (3) .
(ii) A company produces and sells DVDs with a production level of
x thousand DVDs per month where 0 < x ≤ 18 .
The monthly cost of production C(x) thousand dollars is given by C(x) = 160 + 10x .
The selling price per DVD is p(x) dollars where p(x) = 1 (200 − 5x) .
(a) Write an expression in terms of x for the monthly revenue R(x) thousand
dollars received by selling x thousand DVDs.
(b) Show that the monthly profit P(x) thousand dollars is given by
P(x) = − 5 (x − 16)2 + 160 .
(c) Hence write down the maximum possible monthly profit and the production level required to achieve this.
(d) Find the break-even point.
(e) Sketch the graph of the profit function P(x) clearly indicating the point of
maximum profit and the break even point.
Question 6 (12 marks) Use a SEPARATE book clearly marked Question 6
(i) A parabola has its vertex at the point − 1 , − 3 1 and intersects the y-axis at the
point (0,−3) .
(a) Find the equation ofthe parabola in the form y = a(x + b)2 + c .
(b) If the parabola intersects the x-axis at the points P and Q, show that
PQ = 2 1 units.
(ii) In this question the following formulae may be used:
PV = PMT1 − (1+ i)−n FV = PMT (1+ i)n − 1
A house is purchased by a family for $1 000 000. The family makes a deposit of $200 000 and takes out a loan to cover the balance. The loan is to be repaid by making equal monthly payments over a period of 20 years. The annual interest rate on the loan is 6% compounded monthly on the unpaid balance.
(a) Find the monthly repayment.
(b) Find the unpaid balance after the family has been making payments for
exactly 15 years.
(c) Calculate the equity the family has in the house after making payments for 15 years ifthe net market value ofthe house is then $1 500 000.
2022-03-07