THOMASMORECOLLEGE
Matric Trials.Time : 3 Hours. / CORE
MATHEMATICS
PAPER I
/ August 2011.Total : 150.
INSTRUCTIONS :
1.The examination paper consists of 8 pages (including this page) and 1 separate Diagram Sheet.Please check that your paper is complete.
2.This examination consists of FOUR SECTIONS.
SECTION A, SECTION B, SECTION C and SECTION D.
3.EACH SECTION must be written on a SEPARATE Answer Sheet.
4.Write your NAME and your MATHS TEACHER'S NAME on each new section.
5.Show all working clearly and label each answer clearly.
6.An approved calculator may be used.
7.Unless otherwise stated, round off all answers to 2 decimal places.
SECTION A:
Write your name and your Maths Teacher's name on your Answer Sheet.
Question 1
Solve for x: correct to 1 decimal place if necessary:
1a.(3)
b.(4)
c.(3)
[10]
Question 2
a.Given: . Determine the following in simplest form:
(4)
b.Simplify fully, without the use of a calculator:
(5)
c.Given the following arithmetic series: –1 + 2 + 5 + 8 ......
i.Determine the 52nd term.(3)
ii.Is 822 a term in this sequence? Explain mathematically.(3)
d.Determine, in terms of x, the value of:
(4)
e.The first two terms of a geometric series are given:
Determine the value(s) of x for which the given series will converge.(3)
f.The following formula is the sum of an arithmetic series: .
Determine the value of the 18th term.(4)
[26]
SECTION B:
Write the answers to this section on a new Answer Sheet.
Write your name and your Maths Teacher's name on your Answer Sheet.
Question 3
a.A bouncing ball is dropped out of the window of a block of flats. The ball fell 16m until it hit the ground for the first time. After each time it hits the ground, it rises ¾ of the distance of the previous bounce. Calculate the total distance that the ball bounces up and down. (5)
b.Determine in each of the following:
i.(3)
ii.(4)
iii.(3)
c.Determine the equation of the tangent to the curve of at the
point where x = 2.(6)
[21]
Question 4
Refer to the diagrams below. The total numbers of triangles in each diagram have been recorded below each diagram.
a.If this pattern was continued in the same way, determine the total number of triangles that would be formed in diagram number 4. (2)
b.If this pattern was continued in the same way, determine a formula (Tn) that would represent the total number of triangles formed each time. (4)
c.Determine which diagram in the sequence would yield 861 triangles.(3)
[9]
Question 5
a.Given the function .
i.Determine the equation of the inverse of f, in the form (3)
ii.The graph of f is shifted 3 units down to form the graph of g; determine the equation of
the graph of g in the form g(x) = ……(2)
iii.The graph of f is reflected around the Y – Axis to form the graph of h; determine the equation of the graph of h in the form h(x) = …… (2)
b.Refer to the diagram of f alongside.
i.Write down the domain and range
of f.(2)
ii.Will the graph of be a function or a
non–function? Explain.(2)
[11]
SECTION C:
Write the answers to this section on a new Answer Sheet.
Write your name and your Maths Teacher's name on your Answer Sheet.
Question 6
The graph alongside represents the graphs of f and g. The equation of g is: . The
x–intercepts of g are A (2 ; 0) and B (–2 ; 0) and the
y –intercept of g is C (0 ; –4). f and g intersect at B and C.
a.Determine theequation of the graph of g.
(You do not need to simplify your answer.)(4)
b.If it is given that the equation of g is: , then
determine the co–ordinates of D, the
turning point of g.(6)
c.Now, using the graph, determine the value(s) of:
i.x if (2)
ii.x for which (2)
iii.(2)
d.i.If h(x) = g(x) – 2; then write down the co–ordinates of the turning
points of h.(2)
ii.Hence, or otherwise, determine the number of real roots to the
equation: h(x) = 0.(2)
[20]
Question 7
The diagram below represents the graphs of and . f has
x–intercepts at (–1 ; 0) and (3 ; 0) and passes through the point (2 ; -6). g passes through the point
(–2 ; 1) and has a horizontal asymptote of y = –3 and a vertical asymptote of x = –1.
a.Determine the equations of f and g.(8)
b.Write down a restriction on the domain of f so that the graph of is a function.(2)
c.Determine the values of x for which the graph of f is decreasing.(2)
d.Determine the equation of the function that would be created by shifting the graph
of g 2 units to the left and 5 units upwards.(2)
[14]
SECTION D:
Write the answers to this section on a new Answer Sheet.
Write your name and your Maths Teacher's name on your Answer Sheet.
Question 8
Jo wants to build a kite as shown in the figure.
He uses a piece of dowel rod 400 cm long, that he cuts into two to
make diagonals AC and BD. BO = DO and .
a.If BO = x, show that the surface area of ABCD can be
represented by the formula(3)
b.Find the value of x for which the surface area of the kite
is a maximum.(4)
c.Calculate the maximum surface area of the kite.(2)
d.If the material to cover the kite costs R54 per m2, how
much will it cost Jo to cover his kite.(2)
[11]
Question 9
The equation of a function is defined by: . Determine the co-ordinates
of the turning point of g, and hence determine if the turning point is a local minimum or
a local maximum.[8]
Question 10
a.A bottle of medicine is to contain x units of vitamin C and y units of folic acid.
i.The total number of units of both must not exceed 300 units. Express this
as an inequality inx and y.(1)
ii.The medicine must contain at least 30 units of Vitamin C. Express this as
an inequality.(1)
iii.The number of units of folic acid must be at the least 20 but not more
than 100. Express thisas an inequality.(2)
b.A company manufactures a substance that requires two ingredients, A and B. x units of ingredient A and y units of ingredient B are used in this process.
The production manager begins to draw a mathematical model of this process.
The sketch below represents his model.
i.Only one of the constraints has been drawn. Determine this constraint that has been drawn
in terms of x and y.(2)
ii.The following are a further three constraints:
Sketch these constraints onto the Cartesian Plane (in the diagram sheet)and shade in
the feasible region.(4)
iii.The cost (C) of a unit of A is R 40 and a unit of B is R 80. Determine an equation
representing the cost (C) of production of the substance in terms of x and y.(2)
iv.Determine the minimum cost of the production of the substance.(4)
[16]
Question 11
The world population P, T years after the year 2000, can be predicted by the equation:
Calculate in which year the population will be 12 billion. (1 billion = 109)[4]
Answer Sheet for Question 11b. (Please hand this in with Section D)
Name:Maths Teacher:
1