Please Read the Following Instructions Carefully

JULY EXAMINATION 2014

MATHEMATICS GRADE 12

PAPER 2; LO 3 & LO 4

Time: 3 hours Total: 150

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY

1. This question paper consists of 8 pages, graph paper, and a separate formula sheet. Please check that your paper is complete.

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. Answers must be rounded off to the first decimal place, unless otherwise stated.

7. All the necessary working details must be clearly shown.

8. It is in your own interest to write legibly and to present your work neatly. Page 1 of 8

1 of 8 SECTION A

QUESTION 1:

Bob, a church warden, decides to investigate the lifetime of a particular manufacturer’s brand of beeswax candle. Each candle is 30 cm in length. From a box containing a large number of such candles, he selects one candle at random. He lights the candle and, after it has burned continuously for x hours, he records its length, y cm, to the nearest centimetre. His results are shown in the table.

x 5 10 15 20 25 30 35 40 45

y 27 25 21 19 16 11 9 5 2

a) Use your calculator to evaluate the correlation coefficient r, for this data to 4 decimal places. (2)

b) What does the value of r, found above, tell us about the strength of a straight-line relationship? (1)

c) 1) Determine the equation of the least squares regression line of y on x to 4 decimal places. (4)

2) Interpret the value that you obtain for b. (2)

3) It is claimed by the candle manufacturer that the total length of time that such candles are likely to burn for is more than 50 hours. Comment on this claim, giving a numerical justification for your answer. (3)

d) Draw the scatter diagram and the line of best fit for the above data on the given graph paper. (5) [17]

2 of 8 QUESTION 2: In the diagram below, A(-4 ; 5); B(4 ; 1) and C(–1 ; –4) are the vertices of a triangle in a Cartesian plane. CE  AB with E on AB. E is the midpoint of straight line CD.

Determine: a) The length of BC (3)

b) The equation of line CD (5)

c) The coordinates of E (5)

d) The equation of the straight line parallel to AC and passing through D. (4)  e) The acute angle of AC B (5) [22]

QUESTION 3: A circle with centre C has equation x2  y2  4x  6y  7 .

a) 1) Find the coordinates of C. (2)

2) Find the radius of the circle, leaving your answer in surd form. (2)

b) The line with equation y  mx  3 intersects the above circle 1) Show that the x-coordinates of any points of intersection satisfy the equation 1 m2 x2  41 3mx  20  0 (3)

3 of 8 2) Show that the quadratic equation 1 m2 x2  41 3mx  20  0 has equal roots when 2m2  3m  2  0 (3)

3) Hence find the values of m for which the line is a tangent to the circle. (2) [12] QUESTION 4  O is the centre of the circle. STU is a tangent at T. BC  CT , AT C  105o and  CT U  40o

Calculate, with reasons, the size of:  a) A2 (2)

b)  (2) C3

 c) A1 (2)

d)  (2) 01

e)  (4) C 2 [12] QUESTION 5: 2700o tan 2700o a) Calculate the value if tan  rounded to 3 decimal digits (2) 4 4  4 5 b) If SinP  and P   90o ;90o  and SinQ  and Q   90o;90o . Find the 5 13 value of SinP  Q (5) sin180o  x.2cosx  540o  c) Simplify without the use of a calculator. 1 (5) tan x [12]

4 of 8 SECTION B

QUESTION 6:

a) Points A(2; 0) and B(6 ; 8) lie on a Circle. CD  AB and C is the centre of the circle y

B(6 ; 8) C(x; y)

D x A(2 ;0)

1) Determine the equation of the circle above with centre C(x ; y) passing through B and touching the x-axis at A. (6)

2) Calculate the equation of the tangent at B (4)

b) Show that the equation of the line through the point ( a ; b), perpendicular to the line ax  by  1is bx  ay  0 (4) [14] QUESTION 7: 3cos2  cos  2 cos 1 a) 1) Prove the following identity:  (6) 3sin 2  5sin sin

2) For which values of  in the interval 180o ;180o is the above identity not valid? (4)

b) Given f x  cos2x 1 and gx  sin2x f x 1) Solve for x if  1 (6) gx

2) Sketch the graphs of f(x) and g(x) on the same set of axes for x   90o;150o . Clearly show all intercepts with the axes, coordinates of the turning points and points of intersection. (6)

3) Determine the values of x for which f x  gx where x  0 (2) [24]

5 of 8 QUESTION 8: a) In the diagram below, M, C and N lie on the circle with centre O.

C N Use the diagram on the diagram sheet, to prove the theorem which states that:   M O N  2C (6)

b) In the diagram below, PQ is a tangent at Q. PRS is a secant of circle RSTWQ.  RW cuts SQ at K and cuts QT at L. PS //QT . RS  TW . Let  and S  x Q4  y

1) Name 3 other angles equal to x, stating reasons (6)

6 of 8 2) Prove that KQ is a tangent to the circle LQW (3)

3) Prove that PRKQ is a cyclic quadrilateral (6) [21]

QUESTION 9:

In the diagram below, YX QR and XS RN. M is the midpoint of XR TS MN. PY = 4 units. PQ = 7 units P

S T N Y X

Write down the values of the following ratios, giving reasons:

a) PS:SN (4)

b) MN:TS (3)

c) PX:XM (3) [10]

7 of 8 QUESTION 10:

   In a ABC given below, AD, BE and FC are the bisectors A , B and C respectively. AD  and BE cut at F and CDFE is a cyclic quadrilateral. C and F are joined. F AB  x ,   F B A  y and DC F  z .

Calculate, giving reasons, the value of z [6]

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