Technical Mathematics P2 (English) 2021 QP_hlayiso.com_.pdf

Downloaded from hlayiso.com GAUTENG DEPARTMENT OF EDUCATION PREPARATORY EXAMINATION 2021 11092 TECHNICAL MATHEMATICS PAPER 2 TIME: 3 hours MARKS: 150 16 pages + 2 information sheets P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 2 (Paper 2) 11092/21 INSTRUCTIONS AND INFORMATION 1. This question paper consists of 12 questions. 2. Answer ALL the questions in the SPECIAL ANSWER BOOK provided. 3. Clearly show ALL calculations, diagrams, graphs, etc. that you used in determining your 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 answers to TWO decimal places, unless stated otherwise. 7. Diagrams are NOT necessarily drawn to scale. 8. Write neatly and legibly. P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 3 (Paper 2) 11092/21 QUESTION 1 The picture below indicates three locations, namely The Village, Hospital and the University of South Africa, where a student will be on a given day. The diagram alongside it, not drawn to scale, represents the different locations as V (-2; 2), U (2; -1) and P (4; 5) as vertices of ∆VUP on a Cartesian plane. VB is drawn perpendicular to UP with B (x; y) and VB = 4,74 km. 1.1 Calculate: 1.1.1 The co-ordinates of M, the mid-point of UP (2) 1.1.2 The length of UP, the distance between the university and the hospital. Leave your answer in simplified surd form. (2) 1.1.3 The area of ∆VPU, correct to ONE decimal place (2) 1.1.4 The gradient of UV (2) 1.1.5 Angle 𝛽 (2) 1.2 Determine the equation of the straight line passing through P parallel to UV. (3) [13] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 4 (Paper 2) 11092/21 QUESTION 2 2.1 O (0; 0) is the centre of the circle in the diagram below. A (−4; 3) and B (𝑡; 𝑘) are points on the circle. B is the x-intercept of the circle. RA is a tangent to the circle at A and RC is a tangent to the circle at C. A (- 4; 3) 𝑥 B (t; k) R C 2.1.1 Determine the equation of the circle in the form 𝑥 2 + 𝑦 2 = 𝑟 2. (2) 2.1.2 Hence, write the co-ordinates of B. (2) 2.1.3 Calculate the gradient of OA. (2) 2.1.4 Determine the equation of the tangent RA in the form of 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0. (2) 2.1.5 Give a reason why RC = RA. (1) P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 5 (Paper 2) 11092/21 𝑥² 𝑦² 2.2 The diagram below shows the sketch of an ellipse defined by + = 1. 16 4 2.2.1 Determine the co-ordinates of P, the y-intercept of the ellipse. (1) 2.2.2 Show, by means of calculations that the point (3; 1,5) will lie outside the ellipse. (2) 2.3 Sketch the graph of the ellipse defined by 𝑥 2 + 9𝑦 2 = 81 on the system of axes in the ANSWER BOOK. Clearly indicate ALL intercepts with the axes. (3) [15] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 6 (Paper 2) 11092/21 QUESTION 3 3.1 The diagram below shows the point A(𝑥 ; 𝑦) on a Cartesian plane. It is given that 2sin 𝜃 = √3 and 𝜃 ∈ [90°; 180°]. Determine the following WITHOUT the use of a calculator: 3.1.1 The co-ordinates of A (3) 3.1.2 cos 𝜃 (1) 3.1.3 cot 2 𝜃 − 2sec𝜃 (3) 3.1.4 𝜃 (2) 3.2 Determine the numerical value of cos 2𝛽 + sin2 (𝛼 − 𝛽) if 𝛼 = 1,052 radians and 𝛽 = 0,319 radians. (2) [11] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 7 (Paper 2) 11092/21 QUESTION 4 4.1 Complete the following trigonometric identity: sec 2 𝑥 = 1 + … (1) 4.2 Prove the following identity: 2 sin² 𝑥 (1 + tan² 𝑥) = 2tan𝑥 (4) tan 𝑥 4.3 Simplify the following WITHOUT the use of a calculator: 𝜋 sin( 180º − 𝑥). cot( 180º + 𝑥) tan(360º − 𝑥) . tan 4 sin( 180º + 𝑥) (6) 4.4 Solve for 𝑥 if sec2𝑥 = 3,4518 and 2𝑥 ∈ [0°; 360°]. (5) [16] QUESTION 5 Given: 𝑓(𝑥) = 𝑠𝑖𝑛2𝑥 and 𝑔(𝑥) = 2𝑐𝑜𝑠𝑥 for 𝑥 ∈ [0°; 360°] 5.1 Draw the graphs of 𝑓 and 𝑔 on the same set of axes in the ANSWER BOOK. Clearly show the intercepts with the axes as well as the turning points of the graphs. (6) 5.2 Use your graphs to determine the following: 5.2.1 The period of 𝑓 (1) 5.2.2 Give TWO values of 𝑥 for which 𝑓(𝑥) = 𝑔(𝑥). (2) 5.2.3 The amplitude of ℎ if ℎ(𝑥) = 2𝑔(𝑥) (1) 5.2.4 Give the new equation of 𝑔 if 𝑔 is shifted 30° to the right. (1) 5.2.5 Give the range of 𝑓(𝑥 ) − 2. (2) [13] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 8 (Paper 2) 11092/21 QUESTION 6 A light pole PS is perpendicular to the horizontal plane 𝑄𝑆𝑅. SR = 30 m, SR ̂ Q = 26°, SQ̂ R = 49° and the angle of elevation of P from Q is 40°. Calculate the following: 6.1 QŜR (1) 6.2 The length of QS, correct to two decimal places (3) 6.3 The height of the pole PS (3) 6.4 Area of ∆QSR (3) [10] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 9 (Paper 2) 11092/21 QUESTION 7 7.1 Complete the following: The angle subtended by a chord at the centre of a circle is … (1) 7.2 P, Q, R and S are points on a circle. PS and QR are extended and meet at T so that ST = SQ. PO ̂ Q = 100°. 2 2 Determine, with reasons, the size of the following: 7.2.1 Ŝ1 (2) ̂2 7.2.2 Q (3) ̂1 7.2.3 U (3) [9] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 10 (Paper 2) 11092/21 QUESTION 8 8.1 Complete the following statement: The angle between the tangent to a circle and the chord drawn from the point of contact is … (1) 8.2 In the diagram TCG is a tangent to the circle at C. AOC is a diameter of the circle with centre O. A, B, C, D and E are points on the circumference of the circle. Ĉ4 = 40° and ̂ 3 = 35°. D 1 2 3 1 1 2 2 3 4 2 3 1 4 8.2.1 If Ĉ4 = 40°, write down, with reasons, two other angles which are equal to 40°. (3) 8.2.2 Determine, with reasons, the size of: (a) Ĉ3 (2) (b) ̂ E (2) 8.2.3 Show, with reasons, that AB is not parallel to CD. (3) [11] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 11 (Paper 2) 11092/21 QUESTION 9 In the diagram below R and S are points on side AB of ∆ABC. ST // RC with P on AT. AR = 2 units RP = 1 unit RS = 4 units AP = 5 units SB = 𝑥 units TP = 𝑦 units BT = 2 units TC = 8 units ST = 3 units PC = 14 units 9.1 Calculate, with reasons, the numerical values of: 9.1.1 x (3) 9.1.2 y (2) 9.2 Hence, show using appropriate calculations that ∆𝐵𝑆𝑇 ||| ∆𝐵𝑅𝐶. (4) [9] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 12 (Paper 2) 11092/21 QUESTION 10 In the diagram below, KLMS is a cyclic quadrilateral and PK is a tangent to the circle at K with PK // LN. K1 = 36° and Ŝ1 = 57° ̂ 1 10.1 Determine, with reasons, another TWO angles equal to 36°. (3) 10.2 ̂ 2. Calculate the size of K (1) 10.3 Prove that ∆KLS ||| ∆MNS. (4) [8] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 13 (Paper 2) 11092/21 QUESTION 11 11.1 A fan with rotating plastic blades rotates at 4 500 rev/min. The diameter of the fan is 360 mm. Fan blade Calculate the following: 11.1.1 The angular velocity of the blades in radians per second (3) 11.1.2 The circumferential velocity of the blades in metres per second (3) P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 14 (Paper 2) 11092/21 11.2 The diagram below is a circle with centre O. OABC is a sector and OB is perpendicular to AC at the point of intersection at D. AO = 50 cm and BD = 20 cm. O D A C 20 cm B Calculate: 11.2.1 The diameter of the circle (1) 11.2.2 The length of chord AC (4) 11.2.3 ̂ C in radians The size of AO (3) 11.2.4 The perimeter of sector OABC (4) 11.2.5 The area of sector OABC (3) [21] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 15 (Paper 2) 11092/21 QUESTION 12 12.1 The picture shows a glass filled with water with a uniform diameter of 6 cm from the top of the glass to the bottom of the glass. The glass has a height of 18 cm. The bottom part consists of a solid piece of glass which is 1 cm thick. The following formulae may be used: Volume of a cylinder = 𝜋𝑟²ℎ Total surface area of a cylinder = 2𝜋𝑟 2 + 2𝜋𝑟ℎ 12.1.1 Show that the total outer surface area of the cylindrical glass is given by TSA = 117𝜋 = 367,57 cm2 . (3) 12.1.2 With what factor will the outer surface of the glass increase if the radius and the height of the glass are doubled? (2) 12.1.3 Determine the volume of the water in the glass. (4) P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 16 (Paper 2) 11092/21 12.2 The widths of a bell crank are measured in 2 cm intervals as shown in the diagram below. Determine the area of the bell crank if the connector holes are each 2,5 cm in diameter. AB = 3,5 cm CD = 6 cm EF = 7,6 cm GH = 10,8 cm IJ = 16,2 cm KL = 18,6 cm MN = 19 cm OP = 17,8 cm QR = 12,5 cm ST = 8,2 cm UV = 6,5 cm (5) [14] TOTAL: 150 END

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 17 (Paper 2) 11092/21 INFORMATION SHEET: TECHNICAL MATHEMATICS  b  b 2  4ac 𝑏 4𝑎𝑐−𝑏 2 x 𝑥 = − 2𝑎 𝑦= 2a 4𝑎 𝑎 𝑥 = 𝑏 ⇔ 𝑥 = 𝑙𝑜𝑔𝑎 𝑏 , 𝑎 > 0, 𝑎 ≠ 1 and 𝑏 > 0 A  P(1  ni) A  P(1  ni) A  P(1  i) n A  P(1  i) n 𝑖𝑚 𝑚 𝑖𝑒𝑓𝑓 = (1 + ) − 1 𝑚 𝑓(𝑥+ℎ)−𝑓(𝑥) 𝑥 𝑛+1 𝑓 / (𝑥) = 𝑙𝑖𝑚 ℎ ∫ 𝑥 𝑛 𝑑𝑥 = 𝑛+1 + 𝐶 , 𝑛 ≠ −1 ℎ→0 1 𝑎𝑥 ∫ 𝑥 𝑑𝑥 = 𝑙𝑛( 𝑥) + 𝐶, 𝑥 > 0 ∫ 𝑎 𝑥 𝑑𝑥 = 𝑙𝑛 𝑎 + 𝐶, 𝑎 > 0  x  x2 y1  y 2  d  ( x2  x1 ) 2  ( y2  y1 ) 2 M  1 ;   2 2  y 2  y1 y  mx  c y  y1  m( x  x1 ) m  m  tan  x 2  x1 𝑥2 𝑦2 + =1 𝑎2 𝑏2 a b c In ABC:   a 2  b 2  c 2  2bc. cos A sin A sin B sin C 1 area of ΔABC = 2 𝑎𝑏 ⋅ sin C sin2 θ + cos2 θ = 1 1 + tan2 θ = sec 2 θ cot 2 θ + 1 = cosec 2 𝜃 𝜋𝑟𝑎𝑑 = 180°

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 18 (Paper 2) 11092/21 Angular velocity = 𝜔 = 2𝜋𝑛 = 3600 𝑛 where n = rotation frequency Circumferential velocity = 𝜈 = 𝜋𝐷𝑛 where D = diameter and n = rotation frequency 𝑠 = 𝑟𝜃 where r = radius and 𝜃 = central angle in radians 𝑟𝑠 𝑟2𝜃 Area of 𝑎 sector = 2 = 2 where r = radius, s = arc length and 𝜃 = central angle in radians 4ℎ2 − 4𝑑ℎ + 𝑥 2 = 0 where h = height of segment, d = diameter of circle and x = length of chord 𝑜 +𝑜 𝐴𝑇 = 𝑎 ( 1 2 𝑛 + 𝑜2 + 𝑜3 + 𝑜4 +. . . +𝑜𝑛−1 ) where a = equal parts, 𝑜𝑛 = 𝑛𝑡ℎ ordinate and n = number of ordinates OR 𝑜 +𝑜 𝐴𝑇 = 𝑎(𝑚1 + 𝑚2 + 𝑚3 +. . . +𝑚𝑛 ) where a = equal parts, 𝑚1 = 1 2 2 and n = number of ordinates