Technical Mathematics Grade 12 P2 NSC June 2021 Question Paper

basic education Department: Basic Education REPUBLIC OF SOUTH AFRICA SENIOR CERTIFICATE EXAMINATIONS/ NATIONAL SENIOR CERTIFICATE EXAMINATIONS TECHNICAL MATHEMATICS P2 2021 MARKS: 150 TIME: 3 hours This question paper consists of 14 pages and a 2-page information sheet. Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics/P2 2 DBE/2021 SC/NSC INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. 2. This question paper consists of 11 questions. Answer ALL the questions in the SPECIAL ANSWER BOOK provided. Clearly show ALL calculations, diagrams, graphs, etc. that you have used to determine your answers. Answers only will NOT necessarily be awarded full marks. If necessary, round off answers to TWO decimal places, unless stated otherwise. Diagrams are NOT necessarily drawn to scale. You may use an approved: scientific calculator (non-programmable and non-graphical), unless stated otherwise. An information sheet with formulae is included at the end of the question paper. Write neatly and legibly. Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics/P2 3 DBE/2021 SC/NSC QUESTION 1 The diagram below shows a side view of a slanted ladder KL against a vertical wall KZ. K, L and Z lie in the same vertical plane. The vertices of the right-angled triangle are K(1 ; 7), L(-3 ;-1) and Z(a; 6). The angle formed by KL and the x-axisis @. Ay Ka; >) 5 fa L(-3; Z(a;0) Determine: Ll The numerical values of a and b 1.2 The length of KL 13 The coordinates of the midpoint of KL 1.4 The gradient of KL 1.5 The size of @ (rounded off to ONE decimal place) 1.6 The equation of the straight line parallel to KL and passing through the point wait the equation in the form y=... 1.7 Whether point (—4; —2) lies on straight line parallel to KL Copyright reserved Please turn over (2) (2) (2) 2) (2) (3) (2) (15]

Downloaded from hlayiso.com Technical Mathematics/P2 DBE/2021 SC/NSC QUESTION 3 3.1 In the diagram below, P(3;m) isa point ina Cartesian plane with OP = V13 B isan acute angle. Py P(3; m) a Determine, WITHOUT using a calculator, the numerical value of: 3.1.1 m (1) 3.1.2 sec? 8 + tan? B (4) 3.2 If cosé = 5 where 0°<@<90° and tana =—-1 where 0° < a < 180° Calculate: 3.24 The size of @ (1) 3.2.2 The size of @ (3) 3.2.3 The value of cos (a — @) Q) 3.3 Solve for x: 2tanx + 0,924=0 for xe[0° ; 360°] (4) {15} Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics/P2 SC/NSC QUESTION 2 2.1 2.2 DBE/2021 In the diagram below, A, B(—5; 12) and C(t;0) lie ona circle with centre O at the origin. A line is drawn to intersect the circle at B and C. Ais on the x-axis and AB is drawn. Determine: 2.1.1 The equation of the circle 2.1.2 The numerical value of 2.1.3 The equation of the tangent to the circle at B inthe form y= ... Draw, on the grid provided in the ANSWER BOOK, the graph defined by: x? y + 16 35 Clearly show ALL the intercepts with the axes. Copyright reserved Please turn over (2) (1) (4) G3) [10]

Downloaded from hlayiso.com Technical Mathematics/P2 DBE/2021 SC/NSC QUESTION 4 Simplify the following to a single trigonometric ratio: 41 cos @ (tan? + cot) (5) 42 sin? (180° + B) -cosec(z — B) : sec(27 ~ B)-cos(180° — B) (7) [12] QUESTION 5 Given: f(x) =tanx and g(x) =cos(x— 45°) for x €[0°; 360°] 5.1 Draw sketch graphs of f and g on the same set of axes on the grid provided in the ANSWER BOOK. Clearly indicate ALL turning points, end points, asymptotes and intercepts with the axes. (8) 5.2 Use your graphs to write down the value(s) of x for which: 5.2.1 f is undefined (2) 5.2.2 f(x)-g(x) <0 where xe[90° ; 180°] (2) [12] Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics/P2 DBE/2021 SC/NSC QUESTION 6 The diagram below shows farmland in the form of a cyclic quadrilateral PQRS. The land has the following dimensions: PQ=1200m QR = 750m A Q =60° A Ry = 40,5° P, Q, R and § lie on the same horizontal plane. Determine: 6.1 The length of PR (3) 62 Thesizeof § (1) 6.3 The length of PS (3) 6.4 The area of AQPR (3) [10] Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics/P2 8 DBE/2021 SC/NSC QUESTION 7 7A Complete the following theorem statement: Angles subtended by a chord of the circle, on the same side of the chord ... qd) 7.2 In the diagram below, circle PTRS, with centre O, is given such that PS = TS. POR isadiameter, OT and OS are radii. Ri =56° a e - : a yh “ \ RO + > / \ a ra \ f\ J. vy \ Se / | | oN / \ / see \ / py? —_ NB T 721 Determine, stating reasons: (a) Three other angles each equal to 56° (5) (b) Thesizeof Pr 6) (c) Thesize of $5 G) 7.2.2 Prove, stating reasons, that OT is NOT parallel to SR. (3) (15) Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics/P2 9 DBE/2021 SC/NSC QUESTION 8 The diagram below shows circle LMNP with KL a tangent to the circle at L. LN and NPK are straight lines. Ni=27° and M =98° 8.1 Determine, giving reasons, whether line LN is a diameter. (2) 8.2 Determine, stating reasons, the size of: 8.2.1 P, (2) B22 PB 2) 823 La Q) 8.3 Prove, stating reasons, that: 8.3.1 AKLP ||| AKNL (3) 8.3.2 KL’ = KN-KP 2) 8.4 Hence, determine the length of KP if it is further given that KL =6 units and KN = 13 units. (2) 8.5 Determine, giving reasons, whether KLMN isa cyclic quadrilateral. (G3) [18] Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics/P2 1 DBE/2021 SC/NSC QUESTION 9 The diagram below is a picture of a triangular roof truss, as shown. Triangle ABC has AB =AC. DE || BC and F is the midpoint of BC. AE: EC=1:2 and AB=1,8m. A D, E 1 B F ' Cc 9.1 Determine the length of: 9.1.1 DB, giving reason(s) 912 DF if DF= > AD 9.2 Determine, giving reasons, whether EF is parallel to AB. Copyright reserved Please turn over (2) (2) (3) [7]

Downloaded from hlayiso.com Technical Mathematics/P2 il DBE/2021 SC/NSC QUESTION 10 10.1 A double-headed rotating sprinkler (as shown in the picture below) is used to irrigate a circular vegetable garden. The diagram below shows the circular areas covered by the sprinkler heads. The top sprinkler head covers a circular area with a radius of 6,95 m. The bottom sprinkler head covers a circular area with a radius of 4 m. O represents the location of the rotating sprinkler. OC represents the radius of the larger circular area and OB is the radius of the smaller circular area. Chord EBD is a tangent to the smaller circle at B. DOUBLE-HEADED SPRINKLER TOP VIEW Circular area covered by bottom sprinkler head Bottom Sprinkler 10.1.1 Determine: (a) The length of BC qd) (b) The length of chord ED (4) Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics/P2 12 SC/NSC DBE/2021 10.1.2 The bottom sprinkler head is set to irrigate only a sector which is 20% of the circular garden, as shown in the shaded part of the diagram below. -» ed Calculate: (a) The size of acute angle FOG in radians (3) (b) The area of the sector that is irrigated (3) 10.2 A Ferris wheel with a radius of 10 m rotates at 18 revolutions per hour. 10.2.1 Determine the rotational frequency of the rotating wheel in radians per second. (2) 10.2.2 Calculate the circumferencial velocity (in metres per second) of the rotating wheel. (3) 10.2.3 Calculate the angular velocity of the rotating chairs (in radians per second). (3) [19] Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics/P2 13 DBE/2021 SC/NSC QUESTION 11 11.1 A rectangular carpet was damaged by a chemical substance. The diagram below shows the damaged irregular surface of the carpet. The total surface area of the rectangular carpet is 187,5 m?. The ordinates of the undamaged part of the carpet are: p ; 12m;18m; 15 m; 11m and 7m. The breadth of the rectangular carpet is 7,5 m. | 1om><— p ——> F - 2m Damaged 18m — area Te im 7,5 m hp Um en Im 11.4.1 Determine the length of the rectangular carpet. (2) 11.1.2 Determine the value of p. (1) 11.13 It takes 0,25 hours to repair 1 m’. Calculate the minimum time required to repair the damaged area. HINT: Use the mid-ordinate rule. (6) Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics/P2 14 SC/NSC DBE/2021 11.2 A rectangular container with a volume of 1,5 € is filled with liquid. The liquid in the rectangular container must be poured into a number of identical, small, square-based, right, pyramid-shaped containers. The interior measurements of a pyramid, as in the diagram below, are as follows: e The perpendicular height is 3,5 cm e The length of the sides of the square base is 3 cm e The slant height is 3,81 cm CONTAINER INTERIOR MEASUREMENTS OF THE PYRAMID-SHAPED CONTAINER The following formula may be used: Volume of a rectangular prism = length x breadth x height Volume of a right pyramid = 3 (area of the base)x height Total surface area of a square-based pyramid= (3 (side length of base) (slant sin) + (side length)? 11.2.1 Determine the volume of the rectangular container in cm* if 1 £=1000 cm’. (1) 11.2.2 Calculate the total surface area of the pyramid-shaped container. G) 11.2.3 Determine how much milk (in cm’) will be poured into the last pyramid container. (4) [17] TOTAL: 150 Copyright reserved

Downloaded from hlayiso.com Technical Mathematics/P2 DBE/2021 SC/NSC INFORMATION SHEET: TECHNICAL MATHEMATICS ~b+b? —4ac b 4ac —b* x= x=-— y= 2a 2a 4a a =b&x=log,b, a>0,a#l1 andb>0 A=P(l+ni) A= P(1-ni) A= P(-i)" A=P(+i)" + \m ieg =(1+ 4) -1 h)- (0) tin LEAL) h0 h tl fx"ax= +C , n#-1 n+l a fiae=inx+e, x>0 fata =—+C > a>0 x Ina OF x, +x. + d=\(x, -x) +O, -—vy M2 272) y=mxte YY =mMx-x,) m=22 91 m=tan@ 2 ~* 2 2 +4 =1 a bp in adpe; 4- =? = © sinA sinB sinC Area of SABC =*ab-sinC sin’ @+cos’@=1 Copyright reserved @ = Bb +c —2be-cosA 1+tan? @ = sec? 6 cot” 6 +1=cosec’@

Downloaded from hlayiso.com Technical Mathematics/P2 DBE/2021 SC/NSC mrad =180° Angular velocity =@ = 27n = 360°n where v = rotation frequency Circumferential velocity = v = Dn where D = diameter and n = rotation frequency s=r0 where r = radius and @ = central angle in radians rs_ 70 : Area of a sector= FZ = > where r = radius, s = arc length and 6 = central angle in radians 4h? —4dh+x =0 where A= height of segment, d= diameter of circle and x= length of chord oO, +02 Ap =a(m+my+m3+...+m,) where a= equal parts, m = and n= number of ordinates OR o +o, . Ar af 1 - 2 +05 +03 +04+...+ on} where a = equal parts, 0, = i ordinate and n= number of ordinates Copyright reserved