Technical Mathematics Grade 12 IEB Paper 2 2019 Question Paper

Downloaded from hlayiso.com TOTAL MARKS NATIONAL SENIOR CERTIFICATE EXAMINATION NOVEMBER 2019 TECHNICAL MATHEMATICS: PAPER II EXAMINATION NUMBER Time: 3 hours 150 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question paper consists of 25 pages and an Information Sheet of 2 pages (i–ii). Please check that your question paper is complete. 2. Read the questions carefully. 3. Answer ALL the questions on the question paper and hand this in at the end of the examination. Remember to write your examination number in the space provided. 4. Diagrams are not necessarily drawn to scale. 5. You may use an approved non-programmable and non-graphical calculator, unless otherwise stated. 6. Round off your answers to one decimal digit where necessary, 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. 9. One blank page (page 25) is included at the end of the paper. If you run out of space for a question, use this page. Clearly indicate the question number of your answer should you use this extra space. FOR OFFICE USE ONLY: MARKER TO ENTER MARKS Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 TOTAL 11 15 26 12 12 23 16 16 19 150 IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 2 of 25 QUESTION 1 The picture below shows a side view design of a swing. 250 cm temporary brace can be removed when concrete hardens 60 cm 200 cm The diagram below, NOT drawn to scale, models the above swing design in a Cartesian ˆ = θ with A (100; 250) and B (200; 0). plane such that AO = AB and OAB y A (100; 250) 𝜃𝜃 M B (200; 0) x O Calculate: 1.1 the length of AB in simplified surd form (3) IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 3 of 25 1.2 the gradient of AB (2) 1.3 the midpoint M of line OA (2) 1.4 the size of 𝜃𝜃 (4) [11] IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 4 of 25 QUESTION 2 2.1 The diagram below shows line LP, with equation y =− x + 2. NR  LP and NL ⊥ LP. M (–4; 4) is the midpoint of line LN. P is a point on the x-axis and R is a point on the y-axis. 2.1.1 Determine the equation of line LN. (3) IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 5 of 25 2.1.2 Calculate the coordinates of N (show all working). (5) 2.1.3 Determine the coordinates of P. (2) 2.1.4 Determine the equation of the circle with midpoint O, passing through point P. (2) IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 6 of 25 2.2 Draw the graph defined by x2 y 2 + = 1 9 16 Clearly show ALL the intercepts with the axes. (3) [15] IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 7 of 25 QUESTION 3 3.1 ( ) In the diagram below, P 2 3; −2 is a point in the Cartesian plane with origin O. The angle of OP with the positive x-axis is 𝜃𝜃. y ϴ x O ( P 2 3; −2 ) Determine the following: 3.1.1 the length of OP (2) 3.1.2 the value of 𝜃𝜃 (3) 3.2 Calculate the numerical value of sec ( a − b ) if a = 2,695 and b = 1,112. (2) IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 8 of 25 3.3 Simplify without the use of a calculator and clearly show all working: sin 210° tan 45° cos 315° sin 45° cos 60° (5) 3.4 Prove the following identity: tan x ⋅ sin x = sec x − cos x (4) 3.5 Solve for x: = 3.5.1 cosec 2 x 2,114 ; 2 x ∈ [0°;180°] (4) IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 9 of 25 sin ( 360° − x ) ⋅ cos (180° − x ) ⋅ tan (180° + x ) 3.5.2 Simplify: 5  cos2 x ⋅ sin  π  6  (6) [26] IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 10 of 25 QUESTION 4 ) sin ( x − 30° ) for x ∈ [0°;360°] Given: f ( x ) = 2 cos x and g ( x= 4.1 Draw a sketch graph of f and g on the same set of axes on the grid below. Clearly indicate ALL turning points and intercepts with the axes. (6) 4.2 Write down the amplitude of f . (1) 4.3 Give the period of g . (1) 4.4 Determine the values of x for which f ( x ) > g ( x ) (4) [12] IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 11 of 25 QUESTION 5 The diagram below shows a vertical pole AD with points C and B on the same horizontal ˆ = 58° , CAB plane as point A, the base of the pole. If CDA ˆ= 108° , CBA ˆ = 30° , CD = 2 m and AB = 2,3 m. 58° 2m 108° Calculate: 5.1 the length of AC (2) 5.2 the area of ΔABC (3) 5.3 the length of BC (3) IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 12 of 25 5.4 ˆ , if BD = 2,5 m the size of CDB (4) [12] IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 13 of 25 QUESTION 6 = ˆ 120° In the figure below, O is the centre of circle SPR with PT = PR and O1 120° 6.1 Calculate, with reasons, the size of the following angles: 2 6.1.1 P (4) 1 6.1.2 R (3) IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 14 of 25 6.2 ABCD is a cyclic quadrilateral. MK is a tangent of the circle at C. CA bisects BĈD . If AC and BD intersect at E and ˆ ˆ= 110° , calculate, with reasons, = 50° and BEA BCM the size of each of the following: 6.2.1 D̂2 (2) 6.2.2 B̂1 (3) 6.2.3 D̂1 (5) IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 15 of 25 6.3 In the diagram, points P, Q, R and T lie on the circle. MW is a tangent to the circle at P. PT is produced to meet RU at U. MP̂R = 78° PQ̂T = 49° QT̂R = 32° Calculate, with reasons, the size of the following: 6.3.1 a (2) 6.3.2 b (3) IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 16 of 25 6.3.3 c (1) [23] IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 17 of 25 QUESTION 7 A picture of a shelf stand is shown below, which is modelled by the accompanying diagram. In ∆KLM, C and Q are points on KM B and P are points on KL BB LM and BQ B PM BC PQ KM = 20 units, PL = 4 units and KQ : QM = 3 : 1 7.1 Determine, stating reasons, the lengths of each of the following: 7.1.1 QM (3) 7.1.2 KP (3) IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 18 of 25 7.1.3 KB (3) 7.2 7.2.1 Prove that ∆KPM ||| ∆KBQ. (4) 7.2.2 Hence, or otherwise, determine the length of BQ if PM = 10 units. (3) [16] IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 19 of 25 QUESTION 8 8.1 A wheel has a radius of 25 cm. The circumferencial velocity of the rotating wheel is given as 8,75 cm per second. Calculate the angular velocity of the rotating wheel in radians per second. (5) IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 20 of 25 8.2 The picture and diagram below show one of the mechanisms of a machine in a textile factory where two pulleys with centres A and E are connected with a cross- belt. CP, CQ, CR and CT are tangents to the circles at points P, Q, R and T. cross belt pulley pulley Reflex angle Reflex on large angle on pulley small pulley Distance between centres The reflex angles on both pulleys are 240° The radius of the large pulley is 28 cm and the radius of the small pulley is 12 cm AB = BC and CD = DE 8.2.1 Calculate the length of the section of the belt that is in contact with the large pulley. (4) IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 21 of 25 8.2.2 Calculate the total length of the belt, if the length of the section of the belt that is in contact with the small pulley, is 50,3 cm. (7) [16] IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 22 of 25 QUESTION 9 9.1 A regular octahedron is given with lengths of all edges 3 2 units. ABCD is a square. E F 9.1.1 Calculate the height of ΔABE, the left face of the octahedron. (3) 9.1.2 Calculate the total outer surface area of the octahedron. (4) IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 23 of 25 9.2 The picture and diagram below show a solid shape that consists of a right cylindrical section with a hemispherical section at one end, with a right cylindrical section removed. Calculate the volume of this hollowed solid shape as illustrated in this diagram. 70 mm 30 mm radius 30 mm The following formulae may be used: 4 3 Volume of sphere = pr 3 Volume of cylinder = p r 2 h (8) IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 24 of 25 9.3 The irregular figure as shown in the picture below has one straight side of 6,5 m long. It is divided into 5 equal parts with ordinates 0,8 m, 1,3 m, 1,1 m and 0,5 m given. Calculate the area of the irregular figure using the mid-ordinate rule. 6,5 m 0,8 m 1,3 m 0,5 m 1,1 m (4) [19] TOTAL: 150 marks IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: TECHNICAL MATHEMATICS: PAPER II Page 25 of 25 ADDITIONAL SPACE (ALL questions) REMEMBER TO CLEARLY INDICATE AT THE QUESTION THAT YOU USED THE ADDITIONAL SPACE TO ENSURE THAT ALL ANSWERS ARE MARKED. IEB Copyright © 2019