MATHEMATICS: PAPER II Page 1 of 24 HILTON COLLEGE TRIAL EXAMINATION AUGUST 2014 MATHEMATICS: PAPER II GENERAL INSTRUCTIONS

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1 MATHEMATICS: PAPER II Page of 4 HILTON COLLEGE TRIAL EXAMINATION AUGUST 04 Time: 3 hours MATHEMATICS: PAPER II GENERAL INSTRUCTIONS 50 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY.. This question paper consists of 3 pages. You are provided with a separate Answer Booklet for Geometry and an Information Sheet. Please check that your paper is complete.. Read the questions carefully. 3. This question paper consists of 6 questions. Answer all questions. 4. Number your answers exactly as the questions are numbered. 5. You may use an approved non-programmable and non-graphical calculator, unless a specific question prohibits the use of a calculator. 6. Round off your answers to one decimal digit where necessary, unless otherwise stated. 7. All necessary working details must be shown. 8. It is in your own interest to write legibly and to present your work neatly. 9. Please note that the diagrams are NOT necessarily drawn to scale. Please do not turn over this page until you are asked to do so

2 MATHEMATICS: PAPER II Page of 4 QUESTION SECTION A In the figure, A(-3;4), P(5;6), Q(3;3), R(4;-) and S 4 ; AQRT is a parallelogram. are given. (a) Determine the perimeter of AQP giving your answer correct to one decimal digit. (4) Dist. = ( x x ) + ( y y ) Dist. AQ = 37 Dist. QP = 3 Dist. PA = 7 Perimeter AQP 7,9 units.

3 MATHEMATICS: PAPER II Page 3 of 4 (b) Show that P, S and R are collinear. (3) Show that: mps = msr 6 m PS = = m SR = = m PS = m therefore, P, S and R are collinear. SR (c) Determine the midpoint of AR and hence or otherwise determine the coordinates of T. (5) Midpt AR = ; = ; Midpt QT = ; diagonals of a parallelogram bisect x + 3 y + 3 T = ; = T ; ( ) MARKS

4 MATHEMATICS: PAPER II Page 4 of 4 QUESTION The following information represents the amount of beef imported to South Africa over years in 000 tons (a) Calculate the mean amount of beef imported over years. () _ x 66,8 (b) Calculate the standard deviation and use this to comment on beef imports over the last years in South Africa. () sd 8,3 (c) Give the 5 number summary of the data set. (5) Minimum: 39 Lower Quartile: 48 Median: 68 Upper Quartile: 8 Maximum: 93 (d) Comment on the skewness of the data by making reference to mean and median. () The median is greater than the mean which is an indication that the data may be skewed left or negatively skewed. (e) If an outlier is a value of greater than Q3 +,5 ( Inter quartile range) or less than Q,5 ( Inter quartile range), show that there are no outliers in the data set. (3) Q +,5 ( Inter quartile range) 3 8 +,5( Q Q ) 3 8 +,5(34) = 33 Therefore there are no outliers in this data set. Q,5 ( Inter quartile range) 48,5(34) = 3 3 MARKS

5 MATHEMATICS: PAPER II Page 5 of 4 QUESTION 3 Gross Domestic Product (GDP) is the monetary value of all the finished goods and services produced within a country in a specific time period. The following table shows a comparison of the contribution (in %) that Agriculture and Manufacturing made to the country s GDP, over years. YEAR Agriculture (x) 4, 3,4 3,,7,9 3,0,9 3,0,6,5,6 Manufacturing (y) 9, 9,4 9, 8,5 7,5 7,0 6,8 5, 4,,8,4 (a) Determine the linear regression line to represent the data set. Give your answer correct to four decimal digits. (3) y = a + bx y = 5, , 6505x (b) By the process of extrapolation, use you answer 3(a) to determine the percentage (%) that manufacturing will contribute to GDP is Agriculture drops to,% in 06. () y = a + bx y = 5, , 6505(, ) y 0,0 Approximately 0% of GDP. (c) Comment on the reliability of making a prediction through extrapolation in this context. () This process of extrapolation is not reliable as many factors could influence future manufacturing within the country. (d) Determine and describe the correlation coefficient of the data set. () r 0,7 Fairly strong positive correlation 9 MARKS

6 MATHEMATICS: PAPER II Page 6 of 4 QUESTION 4 8 Given: sin β = and 90 β 70 7 With the aid of a sketch and without the use of a calculator, calculate: (a) tan β (-5;8) (3) 7 β x = 5 (pythag) 8 tan β = 5 (b) sin(90 + β ) () cos β 5 = 7 (c) cosβ (3) = sin β = cos β = cos β sin β 8 = 7 6 = 89 or 5 = 7 6 = 89 or 5 8 = = 89 8 MARKS

7 MATHEMATICS: PAPER II Page 7 of 4 QUESTION 5 Simplify without using a calculator: (a) sin(80 + α) sin(90 α).cos( α ) tan( α 80 ).cos(80 α) (6) ( sin α) = (cos α)(cos α) (tan α)( cos α) = cos α = sin α (b) sin90 cos5 tan 390 cos00 sin35 (6) sin0 ( cos45 ) tan 30 = sin0 sin 45. = 3 or = 3 sin0 ( cos45 ) tan 30 = sin0 sin 45 = tan 30 or = 3 cos80 ( cos45 ) tan 30 = cos80 sin 45 = tan 30 = 3 MARKS

8 MATHEMATICS: PAPER II Page 8 of 4 QUESTION 6 Given the equation: tan(5 θ ) = tanθ (a) Write down the general solution. (3) 5θ = θ + k80 ; k Ζ 4θ = k80 θ = k45 ; k Ζ (b) Write down the value(s) of θ [ 90 ;90 ] for which tanθ is undefined. () { 90 ;90 } (c) Hence or otherwise write down the values of θ [ 90 ;90 ] which satisfy the equation. (3) { 45 ;0 ;45 } 7 MARKS

9 MATHEMATICS: PAPER II Page 9 of 4 QUESTION 7 In the figure, O is the centre of the circle with diameter LN. V N M = x UV is a tangent to the circle at N. Prove that K = x Write down the letter (a) to (e) in your answer book and rewrite the missing part of the statements/reasons. (a) N = ; to tangent. (b) M + M = ; (c) L = ; (d) Therefore K = x ; (e) The angle formed between a tangent and chord. (4) (a) 90 x ; radius perpendicular (b) 90 ; angle in semi-circle (c) x ; int. angles of triangle add up to 80 deg (d) angle in same segment (e) is equal to the angle in the alternate segment or is equal to the angle subtended by that chord. 4 MARKS

() supplementary () A line from the centre of a circle to the midpoint of a chord, is to the chord.

10 MATHEMATICS: PAPER II Page 0 of 4 QUESTION 8 (a) Complete the following statements: () The opposite interior angles of a cyclic quadrilateral are. () supplementary () A line from the centre of a circle to the midpoint of a chord, is to the chord. () perpendicular (b) Refer to the diagram: O is the centre of the circle. LO N = 40 State whether the following statements are TRUE or FALSE: () M = 40 () False () Q = 70 () True (3) PNML is a cyclic quadrilateral. () True (4) P = M () True

11 MATHEMATICS: PAPER II Page of 4 (c) In the diagram below, O is the centre of the circle and TP is the diameter. PR is a tangent to the circle at P and T = 3 Find the size of R, giving all reasons. (4) O = T = 46 Angle at centre = times angle at circumference P = 90 Tangent perp. to radius R = 44 Interior angles of triangle OPR 0 MARKS TOTAL FOR SECTION A: 75 MARKS

12 MATHEMATICS: PAPER II Page of 4 QUESTION 9 SECTION B Refer to the diagram below to determine with reasons the value of x. (4) x + 4 = prop int. theorem; ED//CB,5 x x x + = 0 x x + 0 = 0 x = 5 x because x > 0 4 MARKS

13 MATHEMATICS: PAPER II Page 3 of 4 QUESTION 0 Prove the following identity: cos β cos β sin β = 3 (cos β + sin β ) + sin β cos β sin β LHS = 3 (cos β + sin β ) (cos β sin β )(cos β + sin β ) LHS = (cos β + sin β )(cos β + sin β ) (cos β sin β ) LHS = (cos β + sin β ) (cos β sin β ) LHS = cos β + sin β.cos β + sin β (cos β sin β ) LHS = + sin β.cos β (cos β sin β ) LHS = = RHS + sin β (5) 5 MARKS

14 MATHEMATICS: PAPER II Page 4 of 4 QUESTION A section of this picture of an irrigation system is represented in the figure below. BD = DM = x GD = 3x BGD = α G BD = θ D BM = β (a) Show that the area of BMD = x sin β (4) M = β Isosceles Triangle BD=DM B D M = 80 β Int. angles of triangle Area BMD = x. x.sin(80 β ) Area BMD = x sin β

15 MATHEMATICS: PAPER II Page 5 of 4 (b) Show that BG x α θ = [5 + 3cos( + )] (4) B DG = 80 ( α + θ ) Int. angles of triangle In BGD : Using the cosine rule BG x x x x = + (3 ) ( )(3 ).cos[80 ( α + θ )] BG = x x + BG 0 6 [ cos( α θ )] = + α + θ x [5 3cos( )] 8 MARKS

16 MATHEMATICS: PAPER II Page 6 of 4 QUESTION The following picture is represented on the figure below. The centre is represented by the letter E, H is the y-intercept of the circle and the x-intercept is at the origin. (a) Determine the centre (E) and radius of the circle if its equation is: x x y y = 0 (4) 7 7 x 4 x + ( ) + y 7 y + = ( ) + + y = ( x ) Centre 7 ; and radius 65 4,0 (b) Determine the coordinate H. (3) Y-intercept let x = 0 y 7 y = 0 y( y 7) = 0 y = 0 or y = 7 H (0;7)

17 MATHEMATICS: PAPER II Page 7 of 4 (c) Calculate E H O. (4) 7 H(0;7) and E ; m HE = = tanθ = 4 θ 60,3 E H O 80 ( , 3 ) E H O 9,7 (d) If IJ is the tangent to the circle at G (4;7), determine the equation of this tangent. (4) 7 E ; G(4;7) mge = = mij = radius tan 7 y = mx + c 33 c = y = x MARKS

18 MATHEMATICS: PAPER II Page 8 of 4 QUESTION 3 Given two circles with equations: x + y + mx 6y = and ( x 5) + ( y + n) = p (a) Find m and n if the circles are concentric i.e. they have the same centre. (4) x mx y y = ( x m) ( y 3) m = + + ( x + m) + ( y 3) = m + eq() ( x 5) + ( y + n) = p eq() m = 5 and n = 3 (b) Find two values of p it is further given that the radii of the circles differ by units. (3) m + p = or p m + = ( 5) + p = or p ( 5) + = p = 4 or p = 8 7 MARKS

19 MATHEMATICS: PAPER II Page 9 of 4 QUESTION 4 The picture is represented on the figure below. y L P T O Z x W The equation of the circle with centre P is: ( x 4) + ( y 3) = The equation of the circle with centre W is: ( x + 3) + ( y + ) = 0 L is the y-intercept of circle with centre P and Z is the x-intercept of the same circle. T is a point on the circle with centre P. The dashed lines represent the possible positions of circle with centre P. Circle with centre W is fixed. (a) Determine whether the two circles intersect, touch at a point or do not intersect. Show all working. (4) Centre P(4;3) and Centre W(-3;-) Dist. PW = 65 8, Adding Radii: + 0 7,7 Therefore, the circles do not intersect.

20 MATHEMATICS: PAPER II Page 0 of 4 (b) If Z = x, determine T in terms of x. (No reasons required) (3) L x radii PL PZ P x of = = = 80 int. T = 90 x at centre = at circumf (c) If it is further given that P Z O = 35 and Z ;0. Determine the numerical value of T. Round all answers correct to one decimal digit. (4) Z ;0 L is the y-intercept: let x=0 (0 4) + ( y 3) = y 6y = 0 y = , or y = 3 5 0,8 L(0 ; 5,) 5 m ZL = 0,4 5 tanθ = m L Z O 84,5 Z = 35 84,5 Z = 50,5 T = 90 50,5 T = 39,5 MARKS

21 MATHEMATICS: PAPER II Page of 4 QUESTION 5 In the diagram, IA and IE are tangents at the point A and E respectively to the circle with centre O. The diameter EN is produced to S. IA is also produced to S. IO and AE intersect at point P. S N O 3 A 3 4 P E I = Let A x (a) Prove that IEOA is a cyclic quadrilateral. (4) In IEOA: O A I = 90 tan radius O E I = 90 tan radius OEIA is a cyclic quad opp sup pl

22 MATHEMATICS: PAPER II Page of 4 (b) Name, with reasons, four other angles equal to x. (4) E = x tan chord theorem A3 = x OA = OE radii I x in same segment in cyclic quad IEOA = I x in same segment in cyclic quad IEOA = (c) Express S in terms of x. () In SIE : S+ x + 90 = 80 int. of S = 90 x (d) If SA = 00 units and x = 30, find the diameter of the circle NE. (4) In SAO : S AO = 90 tan rad S = 90 (30 ) S = 30 OA( radius) tan 30 = 00 OA 57,7units Diameter 5,5units 4 MARKS

23 MATHEMATICS: PAPER II Page 3 of 4 QUESTION 6 (a) Complete the statement: If two triangles are equiangular, then. () They are similar (b) In the diagram below: EHIJ is a parallelogram. F is on EH. JF produced meets IH produced at G. FJ intersects EI at K. Prove: () JK KF IK = (3) KE In IJK and EFK K I = K vertically opp. s 3 = E alt. s EH//JI J = F alt. s EF//JI (or third of ) IJK EFK JK IK = KF KE sides in proportion

24 MATHEMATICS: PAPER II Page 4 of 4 () GKI / / / JKE (3) In GKI and JKE K G I = K vertically opp. s 4 = J alt. s EJ//GI = E alt. s EJ//GI (or third of ) GKI JKE (3) JK = KF. KG (4) From 6(b): IK. KF JK = () KE From 6(b): JK KE = GK IK GK. KE JK = () IK () () : JK = JK = IK. KF GK. KE KE IK KF. GK MARKS TOTAL FOR SECTION B: 75 MARKS

I pledge that I have neither given nor received help with this assessment.

I pledge that I have neither given nor received help with this assessment. CORE MATHEMATICS PII Page 1 of 4 HILTON COLLEGE TRIAL EXAMINATION AUGUST 016 Time: 3 hours CORE MATHEMATICS PAPER 150 marks PLEASE READ THE FOLLOWING GENERAL INSTRUCTIONS CAREFULLY. 1. This question paper