Grade 12 Mathematics Paper 2 Page 9 of 12
GRADE 12 EXAMINATION
JULY 2012
Mathematics: Paper 2
EXAMINER: Combined Paper / MODERATORS: JE; RN; SS; METIME: 3 Hours / TOTAL: 150
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY
1. This question paper consists of 11 questions, graph paper, and an Information Sheet.
2. Read the questions carefully.
3. Answer all the questions.
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. Make sure your calculator is in degree mode.
7. Round off your answers to TWO decimal places where necessary.
8. All the necessary working details must be clearly shown.
9. It is in your own interest to write legibly and to present your work neatly.
NAME______
Mark Allocation (for educator’s use only)
Section A:
Q1 / Q2 / Q3 / Q4 / Q5Marks Earned
Total / 15 / 17 / 21 / 10 / 15
Section B:
Q6 / Q7 / Q8 / Q9 / Q10 / Q11 / TOTALMarks Earned
Total / 7 / 7 / 37 / 7 / 4 / 10 / 150
SECTION A [78 MARKS]
QUESTION 1 [15]
a) Write a single rule for each of the transformations of the liberty life logo in the form
(x; y) …
(1) A to C (2)
(2) A to D (2)
(3) A to E (2)
(4) C to D (2)
(5) A to B (2)
b) The logo of a well-known bank in
South-Africa, Standard Bank,
is placed in the Cartesian Plane.
The logo in the first quadrant is rotated
through 180o about the origin and then it
is further rotated through 30o in an
anticlockwise direction about the origin.
Determine the image point of A(5; 2) after
(1) the first rotation, and (2)
(2) the second rotation. (3)
______
QUESTION 2 [17]
2.1 PQRS is a rectangular snooker table with PQ parallel to the x-axis and PS parallel to the
y-axis.
K is the position of the ball. ABCD is a parallelogram.
If the ball is hit hard from K to B(−6; 4), it hits C(0; −8) then D(3; −2) then A and returns to K.
a) Find the co-ordinates of A. (2)
b) Find the co-ordinates of K, the mid-point of AB. (2)
c) Show that BC is parallel to the diagonal, PR, of the snooker table. (3)
d) If the ball lands at T(−2; y) on line BC, find the value of y,
showing all your calculations. (3)
e) If T(−2; −4) find the distance of the ball from T to the hole S.
(Leave in simplest surd form) (3)
f) Give the equation of the straight line TS. (2)
g) Find the angle of inclination of the line BC. (2)
______
QUESTION 3 [21]
a) The individual masses (in kg) of 25 rugby players are given below:
MASS (kg) / FREQUENCY / CUMULATIVE FREQUENCY / MIDPOINT60 ≤ x < 70 / 5
70 ≤ x < 80 / 7
80 ≤ x < 90 / 7
90 ≤ x < 100 / 4
100 ≤ x < 110 / 2
(1) Complete the table (like the table shown above) on the DIAGRAM SHEET. (3)
(2) Draw an ogive (cumulative frequency curve) of the above information on the
grid provided on the DIAGRAM SHEET. (3)
(3) Using the information in the table, calculate the mean mass of the rugby
players. (2)
(4) Calculate the standard deviation of the grouped data. (3)
(5) How many rugby players have masses within one standard deviation of the
mean? (2)
(6) From your calculations, calculate the percentage of the rugby
players who have masses within one standard deviation of the mean. (2)
b)
pq r p r q p
q
r
Diagram 1 Diagram 2 Diagram 3
(1) Match one of the following descriptions with each of the above graphs:
1. negatively skewed.
2. positively skewed.
3. symmetrical distribution. (3)
(2) Which measure of central tendency does p, q and r represent? (3)
______
QUESTION 4 [10]
The graph shows the curves of: f(x) = a cos x and
g(x) = tan bx for x [−180o; 360o]
a) Determine the values of a and b in the given equations. (2)
b) What is the period of function g? (1)
c) What is the amplitude of function f? (1)
d) Determine the maximum value of f(x) – g(x) in the interval 0o ≤ x ≤ 90o. (1)
e) For which values of x will f(x) . g(x) ≤ 0 for x [90o; 270o). (2)
f) Calculate the length of AB (in surd form) for x = 120o. (3)
______
QUESTION 5 [15]
a) If = 250,4o and = 84,3o, calculate the values of the following, rounded off
to two decimal digits:
(1) sin (1)
(2) sin + sin (1)
b) Simplify: (7)
c) If 5 cos 2A + 3 = 0 and 0o £ 2A £ 180o , determine without using a calculator, the
value of: (Leave your answer in surd form if necessary.)
(1) sin 2A (3)
(2) cos A (3)
______
TURN OVER FOR SECTION B
SECTION B [72 MARKS]
QUESTION 6 [7]
In the grid below a, b, c, d, e, f and g represent values in a data set written in an increasing
order. No value in the data set is repeated.
a / b / c / d / e / f / gDetermine the value of a, b, c, d, e, f and g if
· The maximum value is 42
· The range is 35
· The median is 23
· The difference between the median and the upper quartile is 14
· The interquartile range is 22
· e = 2c
· The mean is 25 (7)
______
QUESTION 7 [7]
In the diagram below, A, B and C are the vertices of a triangle. AC is extended to cut the x-axis
at D.
(a) If G(a ; b) is a point such that A, G and M (the midpoint of BC) lie on the same straight line, show that b = 2a +1. (4)
(b) Hence calculate TWO possible values of b if . (3)
______
QUESTION 8 [37]
a) If sin 61o = , determine the following in terms of p:
(1) sin 241o (2)
(2) cos 61o (2)
(3) cos 122o (3)
(4) cos 73o cos 44o + sin 73o sin 44o (3)
b) Solve for ; if and 2sincos=0,5. (4)
c) Find the general solution of sin5A.cos3A – cos 5A.sin3A = cosA. (6)
d) Show that (5)
e) Given the identity = 2 cos x
(1) For which values of x in the interval x [−180o; 180o] is the identity
undefined? (4)
(2) Prove the identity. (6)
(3) Hence, calculate the value of: (2)
Note: you will not be awarded any marks for (3) if you just plug the expression into your calculator.
QUESTION 9 [7]
A surveillance camera, A, at the top of a security building shows two cars parked outside
the building. The angle of elevation of A from D is . Car C is equidistant from car D and
the building. Let x denote the distance DC and .
Prove that: AB = 2 x cos.tan
QUESTION 10 [4]
A pen stand made of wood is in the shape of a rectangular prism with four conical depressions
to hold pens.
VOLUME OF A CONE =
The dimensions of the rectangular prism are 15 cm × 10 cm × 3,5 cm.
The radius of each depression is 0,5 cm and its depth is 1,4 cm.
Determine the volume of wood left after the conical depressions have been drilled. (4)
______
QUESTION 11 [10]
a) A triangle PQR is enlarged by factor k, through the origin to form P’Q’R’. If the area of triangle PQR is 18 units 2 and the area of triangle P’Q’R’ is 50 units 2 and P = (−27; 15), find the co-ordinates of P’. (4)
b) The program selector of a washing machine
can be rotated in either direction. It stands
on point A (−5; 5) and you want to select
the program Cottons & Linens at point B(7; 1).
What is the smallest angle, rounded to two
decimal digits, through which the selector must
be turned?
(6)
NAME:______
Question 3(a) (1)
MASS (kg) / FREQUENCY / CUMULATIVE FREQUENCY / MIDPOINT60 ≤ x < 70 / 5
70 ≤ x < 80 / 7
80 ≤ x < 90 / 7
90 ≤ x < 100 / 4
100 ≤ x < 110 / 2
Question 3(a) (2)