Foundation Program Mathematics C SAMPLE E
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Foundation Program
SAMPLE E
Mathematics C
Final Term 1 Examination
Question 1 (12 marks) Use a SEPARATE book clearly marked Question 1
(i) Write the number 0 ⋅ 002075 in scientific notation correct to 3 significant figures.
(ii) Simplify x × x
x
(iii) Find log2 8 .
(iv) Find the midpoint of the interval joining the points (1,8) and (− 3,6) .
(v) Find the remainder when the polynomial x3 − 9x2 + 4x + 3 is divided by (x − 2) .
(vi) Find the 7th term of the geometric progression 1, , 3, … .
(vii) Draw separate sketches of the graphs of each of the following functions showing their
essential features.
− 1
(b) y = .
(c) y = 1 − e − x .
(viii) Evaluate lim .
(ix) If y = x3 − 2x2 solve = −2 .
Question 2 (12 marks) Use a SEPARATE book clearly marked Question 2
(i) Differentiate each of the following functions with respect to x . (a) (3x − 7)5 .
x − 1
(ii) (a) Solve the equation x + 2 = 3x .
(b) Sketch on the same diagram the graphs of y = x + 2 and y = 3x .
(c) Hence solve the inequality x + 2 > 3x .
(iii) Find the equation of the tangent to the 3 at the point on it
x
where x = − 1 .
(iv) If f (x) = find f − 1(5) .
Question 3 (12 marks) Use a SEPARATE book clearly marked Question 3
(i) B is the midpoint of the interval AD where A and D are the points (0, 2) and (4,0) respectively. The line BC is perpendicular to AD and meets the y-axis at the point C.
(a) Show this information on a diagram.
(b) Find the coordinates of the point B.
(c) Find the equation ofthe line BC.
(d) Find the coordinates of the point C.
(e) Find the equation ofthe circle with centre C and radius CD.
(ii) A universal set U and subsets A and B are defined as:
U = { positive integers less than 60 }
(a) Find n(A) .
(b) Find A ∩ B′ .
Question 4 (12 marks) Use a SEPARATE book clearly marked Question 4
(i) Consider the curve y = x4 − 2x2 + 4 .
(a) Find the stationary points ofthe curve and determine their nature.
(b) Show that the curve is symmetrical about the y-axis .
(c) Find the x-coordinates ofthe points of inflection of the curve.
(d) Draw a sketch ofthe curve showing the above features. (e) State the values of x for which the curve is concave down.
(f) Using your sketch, or otherwise, find the number of real solutions to the
equation x4 − 2x2 = 1 giving reasons for your answer.
(ii) If log a = 3 and log a y = 1 ⋅ 6 evaluate:
y
(a) log a x .
(b) logx x .
y
Question 5 (12 marks) Use a SEPARATE book clearly marked Question 5
(i) Solve the equation e2x − ex − 30 = 0 .
(ii) The number of people at a music festival at time t hours after 12 noon is
N(t) thousand , where N(t) = 18t − 2t2 + 20 , for 1 ≤ t ≤ 10 .
(a) (α) Express N(t) in the form a(t − h)2 + k , where a , h and k are
real numbers.
(β) Hence or otherwise find the maximum number ofpeople at the
music festival.
(b) Find the rate at which people are arriving at the music festival at 3 pm .
(c) Find the times between which the number ofpeople at the music festival exceeds 50 thousand . (Answer to the nearest minute.)
(iii) Three numbers a , b , c are said to be in harmonic progression if c = b − c .
(a) Show that if y + z , z + x , x + y are in harmonic progression then x2 , y2 , z 2
are in arithmetic progression.
(b) Hence or otherwise find 3 positive integers in harmonic progression.
Question 6 (12 marks) Use a SEPARATE book clearly marked Question 6
(i) Evaluate k2 . k=2
(ii) Find the interest earned when $15 000 is invested for 3 years at an annual
interest rate of 8 ⋅ 2% compounded quarterly. Give your answer correct to the nearest dollar.
(iii) In this question the following formulae may be used:
PV = PMT1 − (1+ i)−n FV = PMT (1+ i)n − 1
Farmer Tang borrows $180 000 to buy a harvesting machine. He pays interest on the loan at an annual rate of 12% , compounded quarterly. He makes equal quarterly repayments for 5 years to repay the principal and interest.
(a) Calculate the amount ofthe quarterly repayment.
(b) Calculate the total interest charged.
(c) Calculate the amount owed at the end ofthe second year.
(d) Calculate the interest charged in the first two years.
(e) At the end of the second year, the harvesting machine has a resale value of
$160 000 . Calculate Farmer Tang’s equity in the harvesting machine at this time.
2022-03-07