Technical Mathematics Grade 11 P2 2023 QP (Eng)

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE GRADE 11 NOVEMBER 2023 TECHNICAL MATHEMATICS P2 MARKS: 150 TIME: 3 hours This question paper consists of 17 pages, including a 2-page information sheet.

Downloaded from hlayiso.com 2 TECHNICAL MATHEMATICS P2 (ECNOVEMBER 2023) INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of TEN 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/NOVEMBER 2023) TECHNICAL MATHEMATICS P2 3 QUESTION 1 In the diagram below, ∆PQR has a right angle at P. The coordinates of the vertices are P(2 ; 6); Q(−1 ; 2) and R(6 ; k).  is the angle of inclination of line PR. y P(2 ; 6) R(6 ; 𝑘) Q(−1 ; 2) O 𝜃 x 1.1 Complete the statement: “When two lines are perpendicular the … of the gradients must be equal to -1.” (1) Determine: 1.2 The gradient of PQ (3) 1.3 Show that the value of 𝑘 = 3 (3) 1.4 The coordinates of the midpoint of QR (3) 1.5 The coordinates of S, so that QPRS is a rectangle (4) 1.6 The equation of PR (4) 1.7 θ, the inclination angle of line PR (3) 1.8 ̂ If the length of PQ = 5 units, calculate the size of Q (5) [26] Copyright reserved Please turn over

Downloaded from hlayiso.com 4 TECHNICAL MATHEMATICS P2 (ECNOVEMBER 2023) QUESTION 2 2.1 Given: 𝑥 = 30,5˚ and 𝑦 = 130,5˚ Determine the following: 2.1.1 tan(𝑥 + 𝑦) (2) 2.1.2 cosec(𝑦 − 𝑥) (3) 2.2 If sin 36° = 𝑘, express the following in terms of 𝑘 . 2.2.1 cos 36° (4) 2.2.2 sin 216° (2) 2.3 Solve for θ, θ ∈ [0°; 360°] rounded off to ONE decimal digit: tanθ = 2 sin 38,1° (4) [15] Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/NOVEMBER 2023) TECHNICAL MATHEMATICS P2 5 QUESTION 3 3.1 Simplify: 1 cos(360° − 𝜃). .tan(360° + 𝜃) cot(180° + 𝜃) cos(180° + 𝜃).tan(180° − 𝜃) (6) 3.2 Prove that: 1 2 1 + 𝑠𝑖𝑛𝑥 (𝑡𝑎𝑛𝑥 + ) = (4) 𝑐𝑜𝑠𝑥 1 − 𝑠𝑖𝑛𝑥 [10] Copyright reserved Please turn over

Downloaded from hlayiso.com 6 TECHNICAL MATHEMATICS P2 (ECNOVEMBER 2023) QUESTION 4 Given 𝑓(𝑥) = cos 𝑥 and 𝑔(𝑥) = sin 𝑥 + 1; 𝑥 ∈ (0°; 360°) 4.1 On the same axes, given in your SPECIAL ANSWER BOOK, draw the graphs of 𝑓(𝑥) = cos 𝑥 and 𝑔(𝑥) = sin 𝑥 + 1. Clearly show the intercepts with the axes, turning points and endpoints. (7) 4.2 Write down the range of 𝑔 . (2) 4.3 Write down the period of 𝑓 . (1) 4.4 Use your graphs to determine for which values of 𝑥, is 𝑓(𝑥). 𝑔(𝑥) ≤ 0. (2) [12] Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/NOVEMBER 2023) TECHNICAL MATHEMATICS P2 7 QUESTION 5 A farm has the shape shown in the figure below with a fence dividing it into two triangles. The lengths of two adjacent sides of the farm are 1 600 m and 2 400 m and makes an angle of 90°, with one another at A. BÂ C = 40° and CB̂ A = 120°. C B 120° 1 600 m 40° A D 2 400 m 5.1 ̂ A, stating a reason. Determine the size of BC (2) 5.2 Determine the length of AC, to the nearest whole number. (3) 5.3 Determine the total area of the farm, ABCD. (6) [11] Copyright reserved Please turn over

Downloaded from hlayiso.com 8 TECHNICAL MATHEMATICS P2 (ECNOVEMBER 2023) QUESTION 6 6.1 Complete the following theorem: “The angle subtended by an arc at the centre of a circle is … the size of the angle subtended by the same arc at the circumference of the circle.” (1) 6.2 In the diagram, O is the centre of the circle passing through A, B and C. CÂ B = 48°, CO ̂ B = 𝑥 , and Ĉ2 = y A 48° O 𝑥 1 2 𝑦 1 C 2 B Determine, with reasons, the sizes of the following: 6.2.1 𝑥 (2) 6.2.2 𝑦 (2) Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/NOVEMBER 2023) TECHNICAL MATHEMATICS P2 9 6.3 Complete the following statement: “The exterior angle of a cyclic quadrilateral is … to the interior opposite angle.” (1) 6.4 O is the centre of the circle, with PS as the diameter. TP̂ S = 120°. S 𝑧 𝑥 R O 120° 1 𝑦 T Q P Determine, with reasons, the sizes of the following: 6.4.1 𝑥 (2) 6.4.2 𝑦 (2) 6.4.3 𝑧 (2) [12] Copyright reserved Please turn over

Downloaded from hlayiso.com 10 TECHNICAL MATHEMATICS P2 (ECNOVEMBER 2023) QUESTION 7 7.1 Complete the following theorem: “The angle between the tangent to a circle and the chord drawn from the point of contact is … to the angle in the alternate segment.” (1) 7.2 In the given diagram, O is the centre of the circle and tangent FC touches the circle at C. Diameter BD extended meets tangent FC in E. AB̂ O = 42° and OB ̂ C = 25° Determine, with reasons, the sizes of the following: 7.2.1 Ĉ4 (2) 7.2.2 Ĉ3 (2) 7.2.3 ̂1 D (4) Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/NOVEMBER 2023) TECHNICAL MATHEMATICS P2 11 7.3 Complete the following theorem: “Two tangents drawn to a circle from the same point … the circle are equal in length.” (1) 7.4 PS and PR are tangents. O is the centre of the cirle. POT is a straight line. 𝑂̂1 = 100° and 𝑅̂2 = 10°. S P K 1 4 2 3 3 2 M 1 4 O T 2 100° 1 3 R 10° Determine, with reasons, the sizes of the following: 7.4.1 𝑆̂2 (4) 7.4.2 𝑆̂4 (2) 7.4.3 𝑃̂ (4) [20] Copyright reserved Please turn over

Downloaded from hlayiso.com 12 TECHNICAL MATHEMATICS P2 (ECNOVEMBER 2023) QUESTION 8 8.1 Complete the following theorem: “The line drawn from the centre of a circle to the midpoint of a chord is … to the chord.” (1) 8.2 In the given diagram, AB is a chord of a circle with centre O. OE bisects AB. AD = 12 cm, ED = 8 cm and OD = x. B E 8 D 𝑥 12 O A 8.2.1 Determine the length of OE, in terms of 𝑥 . (1) 8.2.2 Complete, with a reason, the statement: OA2 = 122 + … and determine the length of OA, in terms of 𝑥 . (3) 8.2.3 Determine the length of 𝑥 . (4) 8.2.4 Hence, determine the length of the radius. (1) [10] Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/NOVEMBER 2023) TECHNICAL MATHEMATICS P2 13 QUESTION 9 9.1 A wheel rotates at 12 revolutions per second. Determine the angular velocity of the wheel. (3) 9.2 The radius of a circular spinning toy is 40 mm. It rotates at 20 revolutions per minute. Determine the circumferential velocity of the toy. (4) 9.3 In the diagram below, the circle with centre O, has a chord, DE, length of 500 mm and the diameter, AC, is 56,6 cm. The chord, DE, divides the circle into two segments. A D E B h C Calculate the height of the minor segment, h (BC), in cm. (6) 9.4 A metal plate is cut into the shape of a sector of a circle. The radius of the circle is 2 m and the arc length is 2,5 m. 2m 𝜃 2,5 m 9.4.1 Determine the central angle of the sector in radians. (3) 9.4.2 Determine the area of the sector. (3) 9.4.3 The sector is bent into a cone. Determine the perpendicular height of the cone. (5) Copyright reserved Please turn over

Downloaded from hlayiso.com 14 TECHNICAL MATHEMATICS P2 (ECNOVEMBER 2023) 9.5 A toymaker wants to make a toy in the shape of a hollow sphere. The hollow sphere has an internal radius of 3,5 cm and an external radius of 5,5 cm. One cubic centimetre of metal weighs 30 grams. 4 𝑉 = 3 𝜋𝑟 3 3,5 cm Determine the mass of the toy. (5) [29] Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/NOVEMBER 2023) TECHNICAL MATHEMATICS P2 15 QUESTION 10 The irregular shape below has an area of 329 cm2. The horizontal side is 25 cm in length and divided into five equal parts. The ordinates are 12 cm, x cm, 13 cm, 11 cm, 11,8 cm and 18 cm. 12 cm 𝑥 cm 13 cm 11 cm 11,8 cm 18 cm 25 cm Determine, using the mid-ordinate rule, the value of 𝑥 . (5) [5] TOTAL: 150 Copyright reserved Please turn over

Downloaded from hlayiso.com 16 TECHNICAL MATHEMATICS P2 (ECNOVEMBER 2023) INFORMATION SHEET −𝑏 ± √𝑏 2 − 4𝑎𝑐 𝑏 4𝑎𝑐 − 𝑏 2 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) 𝑛 𝑛 𝑖𝑚 𝑚 𝐴 = 𝑃(1 + 𝑖) 𝐴 = 𝑃(1 − 𝑖) 𝑖𝑒𝑓𝑓 = (1 + ) − 1 𝑚 𝑥1 + 𝑥2 𝑦1 + 𝑦2 𝑑 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 𝑀( ; ) 2 2 𝑦2 − 𝑦1 𝑦 = 𝑚𝑥 + 𝑐 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) 𝑚= 𝑥2 − 𝑥1 𝑥2 𝑦2 𝑚 = 𝑡𝑎𝑛𝜃 + =1 𝑎2 𝑏 2 𝑓(𝑥 + ℎ) − 𝑓(𝑥) 𝑓 ′ (𝑥) = lim ℎ→0 ℎ 𝑛 𝑘𝑥 𝑛+1 ∫ 𝑘𝑥 𝑑𝑥 = + 𝐶 , 𝑛, 𝑘 ∈ ℝ with 𝑛 ≠ −1 and 𝑘 ≠ 0 𝑛+1 𝑘 ∫ 𝑑𝑥 = 𝑘 ln(𝑥) + 𝐶, 𝑥 > 0 𝑎𝑛𝑑 𝑘 ∈ ℝ; 𝑘 ≠ 0 𝑥 𝑘𝑎𝑛𝑥 ∫ 𝑘𝑎𝑛𝑥 𝑑𝑥 = + 𝐶, 𝑎 > 0; 𝑎 ≠ 1 𝑎𝑛𝑑 𝑘, 𝑎 ∈ ℝ; 𝑘 ≠ 0 𝑛𝑙𝑛𝑎 𝜋𝑟𝑎𝑑 = 180° Angular velocity = 𝜔 = 2𝜋𝑛 = 360°𝑛 where 𝑛 = rotation frequency Circumferential velocity = 𝑣 = 𝜋𝐷𝑛 where 𝐷 = diameter and 𝑛 = rotation frequency Circumferential velocity = 𝑣 = 𝜔𝑟 where 𝜔 = angular velocity and 𝑟 = radius Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/NOVEMBER 2023) TECHNICAL MATHEMATICS P2 17 Arc length = 𝑠 = 𝑟𝜃 where 𝑟 = radius 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 𝑟𝑠 𝑟 2𝜃 Area of a sector = 2 = where 𝑟 = radius, 𝑠 = arc length and 𝜃 = central angle in radians 2 In ∆ABC: 𝑎 𝑏 𝑐 1 = = 𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐. 𝑐𝑜𝑠𝐴 𝐴𝑟𝑒𝑎 = 𝑎𝑏. 𝑠𝑖𝑛𝐶 sin 𝐴 sin 𝐵 sin 𝐶 2 𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1 1 + 𝑡𝑎𝑛2 𝜃 = 𝑠𝑒𝑐 2 𝜃 𝑐𝑜𝑡 2 𝜃 + 1 = 𝑐𝑜𝑠𝑒𝑐 2 𝜃 𝑂1 + 𝑂𝑛 where 𝑎 = width of equal parts, 𝑂𝑖 = 𝑖 𝑡ℎ ordinate and 𝐴𝑇 = 𝑎 ( + 𝑂2 + 𝑂3 + 𝑂4 +. . . +𝑂𝑛−1 ) 2 𝑛 = number of ordinates 𝑂𝑖 +𝑂𝑖+1 OR where 𝑎 = width of equal parts, 𝑚𝑖 = and 2 𝐴𝑇 = 𝑎(𝑚1 + 𝑚2 + 𝑚3 +. . . +𝑚𝑛−1 ) 𝑛 = number of ordinates; i = 1;2;3; …;n-1 Copyright reserved Please turn over