Technical Mathematics Grade 12 P2 Eastern Cape Mock 2020 QP (Eng)

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE GRADE 12 SEPTEMBER 2020 TECHNICAL MATHEMATICS P2 MARKS: 150 TIME: 3 hours This question paper consists of 15 pages, including 1 information sheet and a special answer book.

Downloaded from hlayiso.com 2 TECHNICAL MATHEMATICS P2 (EC/SEPTEMBER 2020) INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of 11 questions. 2. Answer ALL the questions in the SPECIAL ANSWER BOOK provided. 3. Clearly show ALL calculations, diagrams, graphs, et cetera which you have used in determining the answers. 4. Answers only will NOT necessarily be awarded full marks. 5. You may use an approved scientific calculator (non-programmable and non-graphical) unless stated otherwise. 6. If necessary, round off your answers to TWO decimal places, unless stated otherwise. 7. Diagrams are NOT necessarily drawn to scale. 8. An information sheet with formulae is included at the end of the question paper. 9. Write neatly and legibly. Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/SEPTEMBER 2020) TECHNICAL MATHEMATICS P2 3 QUESTION 1 In the diagram below A(3; 2), B  6; 3 and C are the vertices of ABC. D  5;1 is the midpoint of BC. Straight line BC makes an angle  with the x-axis. 1.1 Calculate the length of AB. (2) 1.2 Determine: 1.2.1 The gradient of BC (2) 1.2.2 The size of , rounded off to ONE decimal digit (2) 1.2.3 The coordinates of C (4) 1.2.4 The equation of the line parallel to BC that goes through the point A in the form y = … (3) [13] Copyright reserved Please turn over

Downloaded from hlayiso.com 4 TECHNICAL MATHEMATICS P2 (EC/SEPTEMBER 2020) QUESTION 2 2.1 In the diagram below, the equation of the circle is given by x2  y 2  49. The point D  x;5 is a point on the circumference of the circle. EF and EG are tangents to the circle drawn from a common point, E, outside the circle. 2.1.1 Determine the x-coordinate of D. Leave your answer in simplified surd form. (3) 2.1.2 (a) Write down the equations of EF and EG. (4) (b) Hence, write down the coordinates of E. (1) 2.2 The equation of an ellipse is given by 16 x2  49 y 2  784 . 2.2.1 Rewrite the given equation into standard form. (2) 2.2.2 Hence, sketch the graph of the equation, clearly indicating the intercepts and the shape. (3) [13] Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/SEPTEMBER 2020) TECHNICAL MATHEMATICS P2 5 QUESTION 3 3.1 If   123 and B̂  65 , calculate the values of the following (rounded off to TWO decimal digits): 3.1.1 cosec A  tan B (2) 3.1.2 cot 2  A  2B (2)   3.2 Calculate: sin  sec2 6 4 (3) 3.3 If 12 cosec  13 and  [90;270] , with the aid of a sketch, calculate the value of cot   sec . (5) 3.4 Use fundamental identities and NOT a sketch to simply the following:  tan   11  cos   2 2 (4) 3.5 Simplify to a single trigonometric ratio of x: sin 180  x  .tan135 sec 180  x  .cos  360  x  (6) [22] Copyright reserved Please turn over

Downloaded from hlayiso.com 6 TECHNICAL MATHEMATICS P2 (EC/SEPTEMBER 2020) QUESTION 4 Given f ( x)  sin 3x and g ( x)   cos x for x [0;180] 4.1 Use the set of axes provided in the SPECIAL ANSWERBOOK to draw sketch graphs of the curves of f and g for x [0;180] . Clearly show ALL intercepts with the axes, coordinates of all turning points and end points of both curves. (5) 4.2 Use the graphs drawn in QUESTION 4.1, or otherwise, to determine the following: 4.2.1 The period of f (1) 4.2.2 The value(s) of x [0;180] for which: (a) g ( x)  0 (2) (b) f ( x).g ( x)  0 (4) (c) f ( x)  g ( x)  1 (2) [14] Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/SEPTEMBER 2020) TECHNICAL MATHEMATICS P2 7 QUESTION 5 In the diagram below ABC is given with AB = 8 units, BD = 6 units.  B̂  57  Ĉ  31  CADˆ  46 Calculate the following: 5.1 The area of ABD (3) 5.2 The length of AD (4) 5.3 The length of CD (4) [11] Copyright reserved Please turn over

Downloaded from hlayiso.com 8 TECHNICAL MATHEMATICS P2 (EC/SEPTEMBER 2020) QUESTION 6 6.1 Complete the following statement: The line segment joining the centre of the circle with the midpoint of a chord … (1) 6.2 In the diagram below, O is the centre of circle AEC.  OFE ⊥ AFC  AC = 48 units  OF = 7 units  EF = x units 6.2.1 Write AO in terms of x. (1) 6.2.2 Calculate, with reasons, the value of x. (5) Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/SEPTEMBER 2020) TECHNICAL MATHEMATICS P2 9 6.3 In the diagram below, DG is a chord of the circle DEHGC.  DAH  EAC  Ĥ1  24 ˆ  Ê  C 2 6.3.1 Name THREE angles each equal to 66, stating reasons. (4) 6.3.2 ˆ . Determine the size of HGC (2) 6.3.3 Prove that DG is a diameter of the circle. (3) [16] Copyright reserved Please turn over

Downloaded from hlayiso.com 10 TECHNICAL MATHEMATICS P2 (EC/SEPTEMBER 2020) QUESTION 7 7.1 Complete the following statement: If a line is drawn through the endpoint of a chord making an angle with the chord equal to an angle in the alternate segment, then the line … (1) 7.2 In the diagram below, O is the centre of the circle DFHE.  DE is a diameter to the circle.  GF is a tangent to the circle at F.  D̂1  16  F̂1  37 Determine, with reasons, the size of the following: 7.2.1 Ĥ 2 (2) 7.2.2 F̂2 (2) 7.2.3 Ô 2 (2) 7.2.4 Ê 2 (3) [10] Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/SEPTEMBER 2020) TECHNICAL MATHEMATICS P2 11 QUESTION 8 8.1 Complete the following statement: If a line is drawn … to one side of a triangle, then the line divides the other two sides proportionally. (1) 8.2 In the diagram below, ABC is a triangle with D the midpoint of BC.  H is a point on AB such that AD and HC intersect at G.  AB ∥ FG with F on BC  AG : AD = 1 : 3 Determine, with reasons, the numerical value of: BF 8.2.1 FD (3) CG 8.2.2 CH (3) Copyright reserved Please turn over

Downloaded from hlayiso.com 12 TECHNICAL MATHEMATICS P2 (EC/SEPTEMBER 2020) 8.3 In the diagram below, DE is the diameter of circle DHFE and HF = FE.  JE is a tangent to the circle at E.  DHJ is a straight line.  DF produced meets tangent EJ at G.  Ê3  x 8.3.1 Write down, with reasons, THREE other angles that are each equal to x. (5) 8.3.2 Prove that: ∆DFE /// ∆DEG (6) [18] Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/SEPTEMBER 2020) TECHNICAL MATHEMATICS P2 13 QUESTION 9 ˆ  60 and OD = 5 cm. In the diagram below, O is the centre of the circle with COD 9.1 Determine the arc length CD in cm. (4) 9.2 Determine the area of the sector. Leave the answer in radians and to the nearest integer. (4) 9.3 (a) Determine the length of the chord CD. (2) (b) Hence, determine the height of the segment between the chord CD and the arc CD. (5) [15] Copyright reserved Please turn over

Downloaded from hlayiso.com 14 TECHNICAL MATHEMATICS P2 (EC/SEPTEMBER 2020) QUESTION 10 10.1 A petrol lawn mower has a pull chord to enable the engine to start. In order for the engine to start, the pulley must turn at 180 rpm. The pulley has a radius of 6 cm. 10.1.1 How many radians per second must the pulley turn? (2) 10.1.2 How fast must the chord be pulled to start the mower? (3) 10.1.3 Hence, determine the angular velocity of the pulley. (3) 10.2 A right cylindrical container holds exactly one litre of water. What should the height of the container be if the radius is 12 cm? 4 V  bh V   r 2 h V   r 3 3 (4) [12] QUESTION 11 The area of an irregular metal sheet, with one side a straight side, is 256 m2. The ordinates are 2,2 m; 2,8 m; 3,1 m; 3,2 m; 2,9 m; 2,6 m; 2,1 m. Calculate: 11.1 The constant interval between the ordinates (4) 11.2 The length of the straight side (2) [6] TOTAL: 150 Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/SEPTEMBER 2020) TECHNICAL MATHEMATICS P2 15 INFORMATION SHEET b  b2  4ac b 4ac  b2 x x y 2a 2a 4a a x  b  x  log a b a  0 , a  1 and b  0 A  P(1  ni) A  P(1  ni) m  im  A  P(1  i) n A  P(1  i) n ieff  1    1  m  x  x y  y2  d  x2  x1    y2  y1  2 2 M 1 2 ; 1   2 2  y2  y1 y  mx  c y  y1  m  x  x1  m x2  x1 x2 y 2 m  tan   1 a 2 b2 f ( x)  lim f ( x  h)  f ( x ) x n 1  x dx  n C , n  1 h 0 h n 1 1 ax  x dx  ln( x)  C , x0  a dx  ln a  C x , a0  rad  180 Angular velocity    2 n  360n where n = rotation frequency Circumferential velocity     Dn where D = diameter and n = rotation frequency s  r where r = radius and  = central angle in radians 4h2  4dh  x2  0 where h = height of segment, d = diameter of circle and x = length of chord rs r 2 Area of a sector   2 2 where r = radius, s = arc length and  = central angle in radians In ∆ABC: a b c 1   a2  b2  c2  2bc.cos A Area  ab.sin C sin A sin B sin C 2 sin 2   cos 2   1 1  tan 2   sec 2  cot 2   1  cos ec 2 o o  where a = equal parts, 𝑜𝑖 = 𝑖𝑡ℎ ordinate and n = number AT  a  1 n  o2  o3  o4  ...  on1   2  of ordinates o o OR where a = equal parts, mi  i i 1 AT  a  m1  m2  m3  . . .  mn 1  2 and n = number of ordinates; i = 1;2;3; …;n-1 Copyright reserved Please turn over