Technical Mathematics Grade 12 P2 Gauteng Mock 2019 QP (Eng)

Downloaded from hlayiso.com GAUTENG DEPARTMENT OF EDUCATION PREPARATORY EXAMINATION 2019 TECHNICAL MATHEMATICS PAPER 2 11092 TIME: 3 hours MARKS: 150 16 pages + 2 information sheets P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 2 (Paper 2) 11092/19 GAUTENG DEPARTMENT OF EDUCATION PREPARATORY EXAMINATION – 2019 TECHNICAL MATHEMATICS (Paper 2) TIME: 3 hours MARKS: 150 INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of 13 questions. 2. Answer ALL the questions. 3. Number the answers correctly according to the numbering system used in this question paper. 4. Clearly show ALL calculations, diagrams, graphs, etc. that you used in determining your answers. 5. Answers only will not necessarily be awarded full marks. 6. You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise. 7. If necessary, round-off answers to TWO decimal places, unless stated otherwise. 8. Diagrams are NOT necessarily drawn to scale. 9. Write neatly and legibly. P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 3 (Paper 2) 11092/19 QUESTION 1 The diagram below shows a building site with vertices A(−2; 5), B(−5; 2), C(0; −2), and E(4; 0). P is the point where AE intersects the y-axis. A(−2; 5) P B −5; 2 S ● O E(4; 0) C(0; −2) 1.1 Calculate the co-ordinates of S, the midpoint of BE. (2) 1.2 Determine the length of AB, in simplified surd form. (2) 1.3 The equation of AE is =− + . Determine the value of k. (2) 1.4 Determine the length of PC. (2) 1.5 Calculate the size of AEB, if α = 167,47°. (3) [11] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 4 (Paper 2) 11092/19 QUESTION 2 2.1 In the diagram below, O(0 ; 0) is the centre of the circle. RS is a tangent to the circle at Q(5 ; −12). S is the x- intercept and R is the y- intercept of tangent RS. T is an x-intercept of the circle. PQ is a straight line through O. P T S O Q(5; −12) R 2.1.1 Write down the co-ordinates of P. (2) 2.1.2 Determine: (a) The equation of the circle, in the form + = (2) (b) The gradient of OQ (2) 2.1.3 Give a reason why PQ ⊥ RS. (1) 2.1.4 Hence, determine the equation of tangent RS. (3) P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 5 (Paper 2) 11092/19 ² ² 2.2 The shape of a cycling track can be given by the equation + = 1. The diagram below gives a schematic representation of the track. N(e ; – 4,8) is a point on the ellipse. The ellipse intersects the y-axis at K and M. K ● N(e ; −4,8) M 2.2.1 Is the orientation of the ellipse vertical or horizontal? (1) 2.2.2 Write down the co-ordinates of M. (2) 2.2.3 Determine the value of e. (3) [16] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 6 (Paper 2) 11092/19 QUESTION 3 3.1 The diagram below shows point K −2; −2√3 on a Cartesian plane. A is the angle between the -axis and OK. A O K(−2; −2√3) Determine the following WITHOUT the use of a calculator: 3.1.1 OK (2) 3.1.2 sin A (1) 3.1.3 tan(360° − A) (2) 3.1.4 sec A (1) 3.2 Convert 225° to radians. (1) 3.3 Calculate the numerical value of sin²A. tanB if A = 1,25 radians and B = radians. (Answer correct to three decimal places.) (3) [10] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 7 (Paper 2) 11092/19 QUESTION 4 4.1 Simplify the following WITHOUT the use of a calculator: cos (180°+ ). sec tan 135°− (5) cos 4.2 Complete: 1 − cos =⋯ (1) 4.3 Hence, prove: tan (1 − cos² )cosec = sec (4) 4.4 Simplify the following: sin + cot (5) 1 + cos 4.5 Solve for if 2tan ( + 20°) = −3 and 180°< + 20°< 360° (4) [19] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 8 (Paper 2) 11092/19 QUESTION 5 The diagram shows the graph of ( ) = 2cos for ∈ [0°; 360°]. 5.1 Draw, on the same set of axes in the ANSWER BOOK, the graph of ( ) = sin( + 45°) for ∈ [0°; 360°]. Clearly show all the turning points, starting and end points and intercepts with the axes. (3) 5.2 Use the graphs to determine the following: 5.2.1 The range of f (2) 5.2.2 The amplitude of g (1) 5.2.3 The period of g (1) 5.2.4 (180°) (1) 5.3 For which value(s) of x is f (x).g (x) < 0? (3) [11] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 9 (Paper 2) 11092/19 QUESTION 6 The diagram shows a rectangular television screen TPRS. Q is a point in front of the screen on the same horizontal plane as P and R. M is a point on TS so that TM = MS. SR = QR = 3 m PTQ = 55°, MQS = 35° and PQR = 84,82°. T M S 55° 3 P R 35° 84,82° Q 6.1 Show that the length of PQ = 4,28 m. (2) 6.2 Calculate the length, PR, of the screen. (4) 6.3 Calculate the length of SQ. (2) 6.4 Calculate the size of QMS. (4) [12] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 10 (Paper 2) 11092/19 QUESTION 7 7.1 Complete the following statement. The angle subtended by a chord at the centre of a circle is … (1) 7.2 In the diagram below, a circle with centre O passes through P, Q, R and S. PR is a diameter of the circle. POQ = 114° and SPR = 32°. S 1 32° O P R 2 114° 2 1 2 3 1 Q 7.2.1 Determine, with reasons, the size of the following angles: (a) PSQ (2) (b) Q (2) (c) PQS (3) 7.2.2 (a) Give a reason why PO = OQ. (1) (b) Calculate the size of Q . (3) [12] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 11 (Paper 2) 11092/19 QUESTION 8 In the diagram, EA is a tangent to the circle at point A. The circle passes through A, B, C and D. AB ǁ CD, DAE = 50°and CBD = 25°. B A 2 1 1 2 50° 25° 2 3 1 C E D 8.1 Determine, with reasons, the size of the following angles: 8.1.1 D (3) 8.1.2 D (2) 8.1.3 E (1) 8.2 Is ABCE a cyclic quadrilateral? Give a reason for your answer. (2) [8] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 12 (Paper 2) 11092/19 QUESTION 9 9.1 Write down what you know about the 9.1.1 angles, and 9.1.2 sides of similar triangles. (2) 9.2 In the diagram below, AB = BC = 4 units, CD = 6 units and ADC = 53°. ∆ABE ⦀ ∆ADC. C = 90° and BE ⊥ AD. C 4 B 6 4 53° A D E Determine, with reasons: 9.2.1 The size of ABE (2) 9.2.2 The length of AD (2) 9.2.3 The length of BE (3) 9.2.4 The length of AE (2) [11] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 13 (Paper 2) 11092/19 QUESTION 10 In ∆ABC below, BC is produced to G. EG ǁ FC, = and = A E F H B G C 10.1 Complete: =… (2) 10.2 Determine the length of EF. (2) 10.3 If EF = units, determine FB. (2) 10.4 Hence, show that cannot be more than three units. (2) [8] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 14 (Paper 2) 11092/19 QUESTION 11 The gear system of a bicycle consists of two gear discs and a chain. The pedals of the bicycle are connected to the larger disc. A schematic drawing of the larger disc is shown below. The radius of the disc is 8 cm and the length of the smaller arc, AB, is 22 cm. A chord, CD, is drawn so that the height of segment CED, namely EF = 2,5 cm. O is the centre of the circle. A 8 C O 22 2,5 F E B D 11.1 The disc turns so that point A moves to point B. Use = , to determine the size of , in degrees. (3) 11.2 Determine the area of the smaller sector AOB if = 0,876 radians. (3) 11.3 An engineer wants to determine the length of chord CD. 11.3.1 Write down the length of the diameter of the circle. (1) 11.3.2 Determine the length of CD. (3) 11.4 The disc turns at an angular velocity of 17,6 rad/s. Calculate: 11.4.1 The rotational frequency of the disc (2) 11.4.2 The circumferential velocity of the disc in kilometres per hour (3) [15] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 15 (Paper 2) 11092/19 QUESTION 12 A propane tank is constructed in the shape of a cylinder with a hemisphere at each end, as illustrated in the picture and sketch below. The diameter of the tank is 1,22 m and the length of the cylinder is 1,98 m. 1,98 m 1,22 m The formulae given below can be used to answer the questions that follow. Volume of cylinder = π ℎ Surface area of cylinder = 2 +2 ℎ Volume of a hemisphere = ³ Surface area of a sphere = 4 ² 12.1 Write down an expression for the total volume of the tank in terms of r and h. (2) 12.2 Calculate the total volume of the tank. (3) 12.3 A second tank has twice the length of the first tank and twice its diameter. Calculate the ratio, , between the volumes of the two tanks WITHOUT calculating the volume of the second tank. (4) 12.4 The cylindrical section of the first tank must be painted. The cost of the paint is R545,99 per square metre. Calculate the cost of the paint needed to paint this section of the tank. (3) [12] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 16 (Paper 2) 11092/19 QUESTION 13 A quantity surveyor must determine the area of a plot of land next to a river. The distance, BC, is divided into 5 sections of equal width. AB = 21,5 m ; BC = 58,5 m ; DC = 14,8 m The lengths of the ordinates are: 21,5 m ; 18,3 m ; 15,4 m ; 15 m ; 15 m ; 14,8 m 13.1 Calculate the width of each of the 5 sections. (1) 13.2 Determine the area of the plot of land by using the mid-ordinate rule. (4) [5] TOTAL: 150 END

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 17 (Paper 2) 11092/19 INFORMATION SHEET: TECHNICAL MATHEMATICS  b  b 2  4ac b 4ac  b2 x x y 2a 2a 4a = ⇔ = , > 0, ≠ 1 and > 0 A  P(1  ni) A  P(1  ni) A  P(1  i) n A  P(1  i) n m  im  ieff   1   1  m   f  x  h  f  x x n 1 f /  x   lim n  x dx  C , n  1 h0 h n 1 ∫ = ( )+ , >0 ∫ = + , >0  x  x y  y2  d  ( x2  x1 ) 2  ( y2  y1 ) 2 M  1 2 ; 1   2 2  y 2  y1 y  mx  c y  y1  m(x  x1 ) m m  tan  x 2  x1 x2 y 2  1 a 2 b2 a b c In ABC:   a 2  b 2  c 2  2bc. cos A sin A sin B sin C areaofΔ ABC = ⋅ sinC sin θ + cos θ = 1 1 + tan θ = sec θ cot θ + 1 = cosec

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 18 (Paper 2) 11092/19  rad  180 Angular velocity = =2 = 360 where n = rotation frequency Circumferential velocity = = where D = diameter and n = rotation frequency s  r where r = radius and  = central angle in radians Area of sector = = where r = radius, s = arc length and  = central angle in radians 4ℎ − 4 ℎ + =0 where h = height of segment, d = diameter of circle and x = length of chord  o o  AT  a  1 n  o2  o3  o4  . . .  on1  where a = equal parts, oi = ith ordinate and  2  n = number of ordinates OR AT  a  m1  m2  m3 . . .  mn  where a = equal parts, m1  o1  o 2 2 and n = number of ordinates