Core Mathematics
All questions to be answered in the Answer Books provided
INSTRUCTIONS
1.This question paper consists of 20 pages and a Formula Sheet.
2.Please check that your paper is complete.
3. Read the questions carefully.
4. Number your answers exactly as the questions are numbered.
5. You may use an approved non-programmable and non-graphical calculator, unless otherwise stated.
6. All the necessary working details must be clearly shown.
7. It is in your own interest to write legibly and to present your work neatly.
SECTION A
QUESTION 1
In the diagram, K(-1 ; -2) is the midpoint of LN with N(3 ; 6).
PK LN with P on the x-axis.
The angle of inclination of NL is θ.
1.1 / Determine:1.1.1 the gradient of NK
1.1.2 the size of θ, correct to 1 decimal place
1.1.3 the co-ordinates of L
1.1.4 the length of NK, correct to 1 decimal place / (3)
(2)
(4)
(3)
1.2 / Determine the equation of the straight line parallel to PK and that passes through N. / (5)
[17]
QUESTION 2
In the diagram, A(4 ; 3), B(2 ; 7), C(-2 ; 5) and D(0 ; 1) are the vertices of a quadrilateral.
1.1 / Show that the co-ordinates of M, the midpoint of BD, are (1 ; 4) / (2)1.2 / Show that AM BD / (3)
1.3 / Prove that A, M and C are collinear / (3)
[8]
QUESTION 3
3.1Given below are four graphs. You are required to estimate a value for r between -1 and 1 to predict the correlation between x and y:
(i) (ii)
(iii)(iv)
For each graph, write down an approximate value for r:(i) ______
(ii) ______
(iii) ______
(iv) ______/ (4)
3.2 / Below is a table showing the relationship between the monthly rent (x) a person pays for an apartment and the person’s monthly income (y). Both are given in thousands of Rands.
Monthly Rent (x) / Monthly Income (y)
8.5 / 25
4.5 / 15
6.5 / 16.5
11.5 / 32
9.5 / 30
10.5 / 33
5.5 / 15.5
7.5 / 18
9.5 / 27.5
15 / 42
8.7 / 26
3.2.1 Determine the equation of the regression line that best fits the data. / (3)
3.2.2 Predict the individual’s monthly income if the rent per month is R9000. / (2)
3.2.3 Calculate the correlation coefficient r. / (3)
3.2.4 What does the correlation coefficient suggest about the relationship between
monthly rent and monthly income? / (2)
[14]
QUESTION 4
The diagram has points A, B, C and D on the circumference of the circle with centre O.
Use the diagram to prove the theorem that states that the opposite angles of a cyclic quadrilateral are supplementary (i.e. A + D = 180° or C + B = 180°)
[6]QUESTION 5
In the diagram, A, B, C and D are points on the circumference of the circle with centre O.
BOD is a diameter of the circle.
Calculate, giving reasons, the size of:
5.1 / / (1)5.2 / / (1)
5.3 / / (3)
[5]
QUESTION 6
6.1 / In the diagram, reflex angle = θP is the point (-12 ; y) and OP = 13 units
Calculate, without the use of a calculator:
6.1.1 y
6.1.2 sin θ
6.1.3 / (3)
(1)
(3)
6.2 / Prove the identity:
/ (4)
[11]
QUESTION 7
7.1Use the set of axes provided to sketch the graph of y = sin 3x on the domain x[0°;360°]
(3)
7.2 / In the diagram, the graphs of two trigonometric functions are shown for x[0°;360°].f and g intersect at A(63,4° ; p) and B.
7.2.1 / Complete:
a) ______
b) ______/ (2)
(2)
7.2.2 / Write down the range of f. / (2)
7.2.3 / Give the coordinates of B. / (2)
7.2.4 / Determine the value of p. / (1)
7.2.5 / For which values of x [0° ; 360°] is ? / (2)
[14]
TOTAL SECTION A = 75
SECTION B
QUESTION 8
8.1 / Building regulations require the provision of lightning protection for thatch roof structures.In the diagram, AD represents a vertical lightning mast constructed on the same horizontal plane as a thatch roof cottage.
B and C are two points on the ground that are in line with the front of the cottage.
B and C lie on the circumference of the lightning protection provided by the mast.
is the angle of elevation from B to the top of the mast at A.
AD is perpendicular to the ground at D that lies in the same horizontal plane as B and C.
8.1.1 Express BD, the radius of protection, in terms of p and α. / (3)
8.1.2 Hence show that:
BC = 2p.cos α. cos θ / (5)
8.1.3 If BC = 29,5 m, p = 21,2 m and α = 45°, calculate the size of θ correct to 1
decimal place. / (4)
8.1.4 Calculate, to the nearest metre, the shortest distance from D to the house. / (4)
[16]
QUESTION 9
The line RP with equation y – x – 2 = 0 is a tangent to a circle with centre (4 ; 4) at P.
PT is a diameter of the circle.
Determine:
9.1 / The equation of PT. / (4)9.2 / The coordinates of P, the point of contact. / (4)
9.3 / The equation of the circle. / (3)
9.4 / The coordinates of T. / (7)
9.5 / The length of VT if V is a point on the tangent at T such that MV = units.
Leave your answer in surd form. / (3)
[21]
QUESTION 10
Students were surveyed about how long it took them to get to school in the morning.
A histogram for the data collected is drawn below:
10.1 / Using the given histogram, complete the frequency distribution table below:
CLASS / FREQUENCY / MIDPOINT
/ 25 / 2,5
/ (4)
10.2 / Estimate the mean for this data. / (3)
10.3 / Estimate the standard deviation for this data. / (3)
10.4 / Which is the median class for this set of data? / (1)
10.5 / Would you suggest this data is normally distributed? Motivate your answer. / (2)
[13]
QUESTION 11
11.1 / In the diagram, what can be said about points A, B, C and DIf:
11.1.1 / (2)
11.1.2 / (2)
11.1.3 and and
(2)
11.2 / In each of the following, give a reason why :
11.2.1
/ (1)
11.2.2
/ (1)
11.2.3
/ (1)
[9]
QUESTION 12
In the diagram, O is the centre of the circle. A, E, C, W and N are points on the circle. Diameter BC is drawn to bisect chord AW at J. AN is a diameter of the circle. A tangent is drawn to the circle at N. Chord AW is extended to U and meets the tangent at P.
12.1 / Prove that ONPJ is a cyclic quadrilateral / (5)12.2 / Prove that OC bisects ACW / (3)
[8]
QUESTION 13
13.1 / In the figure below, ΔPMN has V as the midpoint of PM.RT || VN and
Determine the following (without giving reasons)
i)
ii)
iii)
/ (1)
(2)
(2)
13.2 / In the diagram,
DE // BC
DF // AC
PFE is a straight line.
In ΔABC, give a ratio for (no reasons needed):
13.2.1
13.2.2
/ (1)
(2)
[8]
TOTAL SECTION B = 75
Page 1 of 20
Please turn over