<p> JULY EXAMINATION 2014</p><p>MATHEMATICS GRADE 12 </p><p>PAPER 2; LO 3 & LO 4</p><p>Time: 3 hours Total: 150 </p><p>PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY</p><p>1. This question paper consists of 8 pages, graph paper, and a separate formula sheet. Please check that your paper is complete.</p><p>2. Read the questions carefully.</p><p>3. Answer all the questions.</p><p>4. Number your answers exactly as the questions are numbered.</p><p>5. You may use an approved non-programmable and non-graphical calculator, unless otherwise stated.</p><p>6. Answers must be rounded off to the first decimal place, unless otherwise stated.</p><p>7. All the necessary working details must be clearly shown.</p><p>8. It is in your own interest to write legibly and to present your work neatly. Page 1 of 8 </p><p>1 of 8 SECTION A</p><p>QUESTION 1: </p><p>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.</p><p> x 5 10 15 20 25 30 35 40 45</p><p> y 27 25 21 19 16 11 9 5 2</p><p> a) Use your calculator to evaluate the correlation coefficient r, for this data to 4 decimal places. (2)</p><p> b) What does the value of r, found above, tell us about the strength of a straight-line relationship? (1)</p><p> c) 1) Determine the equation of the least squares regression line of y on x to 4 decimal places. (4)</p><p>2) Interpret the value that you obtain for b. (2)</p><p>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)</p><p> d) Draw the scatter diagram and the line of best fit for the above data on the given graph paper. (5) [17]</p><p>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.</p><p>Determine: a) The length of BC (3)</p><p> b) The equation of line CD (5)</p><p> c) The coordinates of E (5)</p><p> d) The equation of the straight line parallel to AC and passing through D. (4)  e) The acute angle of AC B (5) [22]</p><p>QUESTION 3: A circle with centre C has equation x2  y2  4x  6y  7 .</p><p> a) 1) Find the coordinates of C. (2)</p><p>2) Find the radius of the circle, leaving your answer in surd form. (2)</p><p> 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)</p><p>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)</p><p>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</p><p>Calculate, with reasons, the size of:  a) A2 (2)</p><p> b)  (2) C3</p><p> c) A1 (2)</p><p> d)  (2) 01</p><p> 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]</p><p>4 of 8 SECTION B</p><p>QUESTION 6:</p><p> a) Points A(2; 0) and B(6 ; 8) lie on a Circle. CD  AB and C is the centre of the circle y</p><p>B(6 ; 8) C(x; y)</p><p>E</p><p>D x A(2 ;0)</p><p>1) Determine the equation of the circle above with centre C(x ; y) passing through B and touching the x-axis at A. (6)</p><p>2) Calculate the equation of the tangent at B (4)</p><p> 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</p><p>2) For which values of  in the interval 180o ;180o is the above identity not valid? (4)</p><p> b) Given f x  cos2x 1 and gx  sin2x f x 1) Solve for x if  1 (6) gx</p><p>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)</p><p>3) Determine the values of x for which f x  gx where x  0 (2) [24]</p><p>5 of 8 QUESTION 8: a) In the diagram below, M, C and N lie on the circle with centre O.</p><p>M</p><p>O</p><p>C N Use the diagram on the diagram sheet, to prove the theorem which states that:   M O N  2C (6)</p><p> 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</p><p>1) Name 3 other angles equal to x, stating reasons (6)</p><p>6 of 8 2) Prove that KQ is a tangent to the circle LQW (3)</p><p>3) Prove that PRKQ is a cyclic quadrilateral (6) [21]</p><p>QUESTION 9:</p><p>In the diagram below, YX QR and XS RN. M is the midpoint of XR TS MN. PY = 4 units. PQ = 7 units P</p><p>S T N Y X</p><p>M Q R</p><p>Write down the values of the following ratios, giving reasons:</p><p> a) PS:SN (4)</p><p> b) MN:TS (3)</p><p> c) PX:XM (3) [10]</p><p>7 of 8 QUESTION 10:</p><p>   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 . </p><p>A</p><p> x</p><p>E F</p><p> y z</p><p>B D C</p><p>Calculate, giving reasons, the value of z [6]</p><p>8 of 8</p>