MATHEMATICS
PAPER 2
GRADE 12 
NSC PAST PAPERS AND MEMOS
FEBRUARY/MARCH 2018

INSTRUCTIONS AND INFORMATION
Read the following instructions carefully before answering the questions.

1. This question paper consists of 10 questions.
2. Answer ALL the questions in the ANSWER BOOK provided.
3. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining your answers.
4. Answers only will NOT necessarily be awarded full marks.
5. You  may   use   an   approved   scientific   calculator  (non-programmable   and non-graphical), unless stated otherwise.
6. If necessary, round off answers to TWO decimal places, unless stated otherwise.
7. Diagrams are NOT necessarily drawn to scale.
8. An information sheet with formulae is included at the end of the question paper.
9. Write neatly and legibly.

QUESTIONS 

QUESTION 1
An organisation decided that it would set up blood donor clinics at various colleges. Students would donate blood over a period of 10 days. The number of units of blood donated per day by students of college X is shown in the table below.

1.1Calculate:
1.1.1 The mean of the units of blood donated per day over the period of 10 days   (2)
1.1.2 The standard deviation of the data    (2)
1.1.3 How many days is the number of units of blood donated at college X outside one standard deviation from the mean? (3)
1.2  The number of units of blood donated by the students of college X is represented in the box and whisker diagram below.

1.2.1 Describe the skewness of the data.           (1)
1.2.2 Write down the values of A and B, the lower quartile and the upper quartile respectively, of the data set.   (2)
1.3 It was discovered that there was an error in counting the number of units of blood donated by college X each day. The correct mean of the data is 95 units of blood.
How many units of blood were NOT counted over the ten days?  (1) [11]

QUESTION 2
The table below shows the number of hours that a sales representative of a company spent with each of his nine clients in one year and the value of the sales (in thousands of rands) for that client.

2.1Identify an outlier in the data above. (1)
2.2  Calculate the equation of the least squares regression line of the data. (3)
2.3 The sales representative forgot to record the sales of one of his clients. Predict the value of this client's sales   (in thousands of rands) if he spent 240 hours with him during the year.   (2)
2.4  What is the expected increase in sales for EACH additional hour spent with a client? (2) [8]

QUESTION 3
In the diagram, P, Q(-7 ; -2), R and S(3 ; 6) are vertices of a quadrilateral. R is a point on the x-axis. QR is produced to N such that QR = 2RN. SN is drawn.  PTO = 71,57°  and SRN = θ.

Determine:
3.1The equation of SR (1)
3.2 The gradient of QP to the nearest integer (2)
3.3 The equation of QP in the form y = mx + c (2)
3.4 The length of QR. Leave your answer in surd form. (2)
3.5 tan(90° - θ) (3)
3.6 The area of ΔRSN , without using a calculator (6) [16]

QUESTION 4
In the diagram,  PKT is a common tangent to both circles at K(a ; b).  The centres of both  circles lie on the line y =  ½x.  The equation of the circle centred at  0  is  x² + y² =180. The radius of the circle is three times that of the circle centred at M.  

4.1 Write down the length of OK in surd form.   (1)
4.2 Show that K is the point (-12 ; - 6). (4)
4.3 Determine:
4.3.1 The equation of the common tangent, PKT, in the form y = mx + c  (3)
4.3.2 The coordinates of M (6)
4.3.3 The equation of the smaller circle in the form     (x — a)² + (y — b)² = r ²    (2)
4.4 For which value(s) of r will another circle, with equation x² + y² = r ², intersect the circle centred at M at two distinct points? (3)
4.5 Another circle,  x² + y² + 32x +16y + 240 = 0 ,  is drawn. Prove by calculation that this circle does NOT cut the circle with centre M(-16 ; -8). (5)  [24]

QUESTION 5
5.1 If  cos 2θ= - 5 , where 2θ ∈ [180°; 270º] , calculate, without using a calculator, the values in simplest form of:
                          6
5.1.1 sin2θ  (4)
5.1.2 sin²θ  (3)
5.2 Simplify sin(180° - x).cos(-x) + cos(90° + x).cos(x -180°) to a single trigonometric ratio.   (6)
5.3 Determine the value of sin 3x.cos y + cos 3x. sin y if  3x + y = 270°  (2)
5.4 Given:   2 cos x = 3 tan x
5.4.1 Show that the equation can be rewritten as     2 sin ² x + 3 sin x  - 2 = 0.   (3)
5.4.2 Determine the general solution of x if   2 cos x = 3 tan x.   (5)
5.4.3 Hence, determine two values of y,  144° ≤ y ≤ 216°, that are solutions of 2 cos 5y = 3 tan 5y.  (4)
5.5 Consider:  g(x) = - 4 cos(x + 30°)
5.5.1  Write down the maximum value of g(x).   (1)
5.5.2 Determine the range of g(x) + 1.   (2)
5.5.3  The graph of g  is shifted  60°  to the left and then reflected about the x-axis  to form a new graph  h.  Determine the equation of  h  in its simplest form.   (3) [33]

QUESTION 6
PQ and AB are two vertical towers.
From a point R in the same horizontal plane as Q and B, the angles of elevation to P and A are θ and 2θ respectively.
AQR = 90° + θ, QAR = θ and QR = x .

6.1 Determine in terms of x and θ :
6.1.1 QP (2)
6.1.2  AR (2)
6.2 Show that AB = 2x cos²θ (4)
6.3 Determine AB if θ = 12°. (2) [10]
                       QP

QUESTION 7
In the diagram, PQRT is a cyclic quadrilateral in a circle such that PT = TR. PT and QR are produced to meet in S. TQ is drawn.  SQP = 70°

7.1 Calculate, with reasons, the size of:
7.1.1 T₁ (2)
7.1.2  Q₁ (2)
7.2 If it is further given that PQ ΙΙ TR:
7.2.1 Calculate, with reasons, the size of T₂      (2)
7.2.2 Prove that TR = RQ (2) [8]
                          TS     RS                                                                                    

QUESTION 8
In the diagram, PR is a diameter of the circle with centre O. ST is a tangent to the circle at T and meets RP produced at S. SPT = x and S = y.

Determine, with reasons, y in terms of x.    [6]

QUESTION 9
In the diagram, DEFG is a quadrilateral with DE = 45 and GF = 80. The diagonals GE and DF meet in H.  GDE = FEG and DGE = EFG

9.1 Give a reason why ΔDEG III ΔEGF. (1)
9.2 Calculate the length of GE. (3)
9.3 Prove that ΔDEH III ΔFGH. (3)
9.4  Hence, calculate the length of GH. (3) [10] 

QUESTION 10
10.1 In the diagram,   0 is the centre of the circle with A, B and C drawn on the circle.

Prove the theorem which states that BOC = 2A.    (5)
10.2 In the diagram, the circle with centre F  is drawn. Points A, B, C and D  lie on the circle. Chords AC and BD intersect at E such that EC = ED. K is the midpoint of chord BD. FK, AB, CD, AF, FE and FD are drawn. Let B = x .

10.2.1 Determine,  with  reasons,  the  size  of EACH  of the  following  in terms of x:

1. F₁   (2)
2. C  (2)

10.2.2 Prove, with reasons, that AFED is a cyclic quadrilateral.    (4)
10.2.3  Prove, with reasons, that F₃ = x   (6)
10.2.4 If area ΔAEB = 6,25 x area ΔDEC, calculate AE  (5) [24]
                                                                                  ED

TOTAL:150