Technical Mathematics Grade 12 P2 EC June 2022 QP (Eng)

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE GRADE 12 JUNE 2022 TECHNICAL MATHEMATICS P2 MARKS: 150 TIME: 3 hours This question paper consists of 13 pages, including a 1-page information sheet and a special answer book.

Downloaded from hlayiso.com 2 TECHNICAL MATHEMATICS P2 (EC/JUNE 2022) 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/JUNE 2022) TECHNICAL MATHEMATICS P2 3 QUESTION 1 In the diagram below ABFC is a parallelogram with vertices A(2; 3), B(−4; 2), F and C(4;−5). G is the midpoint of BC and θ is the inclination angle. Determine: 1.1 The length of AC (leave your answer in simplified surd form.) (2) 1.2 The equation of straight-line AC in the form y = … (4) 1.3 The size of  (3) 1.4 The coordinates of G (2) 1.5 Hence, the coordinates of F (3) 1.6 If BC ⊥ AG. Show ALL calculations (4) [18] Copyright reserved Please turn over

Downloaded from hlayiso.com 4 TECHNICAL MATHEMATICS P2 (EC/JUNE 2022) QUESTION 2 2.1 In the diagram below, the straight line y = x + 3 , intersect with the circle x 2 + y 2 = 29 at A and B. 2.1.1 Determine the coordinates of A and B. (7) 2.1.2 Given: the point C (−5; 2). (a) Show that C lies on the circle. (2) (b) Determine the equation of the tangent to the circle at point C in the form y = ... (4) x2 y2 2.2 Sketch the graph of + = 1 . Clearly indicate the intercepts. 40 64 (3) [16] Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/JUNE 2022) TECHNICAL MATHEMATICS P2 5 QUESTION 3 3.1 In the diagram below, P(-8; -15) is a point on the Cartesian plane. OP forms a reflex angle θ with the positive x-axis. Determine the following, WITHOUT using a calculator: 3.1.1 The length of OP (3) 3.1.2 tanθ (1) 3.1.3 cosec2 θ – 1 (3) 3.2 If a = 135,5 and b =15,7 , determine the numerical value of the following, rounded off to THREE decimal digits: 3.2.1   sin  − b  2  (2) 3.2.2 sec ( a + b ) (2) 3.3 Solve for x, rounded off to ONE decimal digit: 3.3.1 sin x + 1 = 0,587 for x 0;360 (4) 3.3.2 cot 2 x = 2,114 for x 90;360 (4) 3.4 If cos36 = a, determine the following in terms of a: 3.4.1 tan 36° (3) 3.4.2 sec2 144° (2) [24] Copyright reserved Please turn over

Downloaded from hlayiso.com 6 TECHNICAL MATHEMATICS P2 (EC/JUNE 2022) QUESTION 4 4.1 Simplify: sec2 2x − tan 2 2x (1) sec x tan 2 (180 − x ) 4.2 Simplify: + cos ( 360 − x ) sin (180 + x ) cosec (180 − x ) (8) 4.3 Prove that: sin ( 360 − x ) cot (180 − x ) = cos x (4) [13] Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/JUNE 2022) TECHNICAL MATHEMATICS P2 7 QUESTION 5 In the diagram below the graph of f ( x) = a tan x is given for the interval x  0;360 . 5.1 Write down the value of a. (1) 5.2 Write down the equations of the asymptotes of f. (2) 5.3 Write down the period of f. (1) 5.4 On the same axes given in your SPECIAL ANSWER BOOK draw the graph of g ( x) = cos ( x − 60) . Clearly show the intercepts with the axes, turning points and endpoints. (3) 5.5 Write down the amplitude of g. (1) 5.6 Use your graphs to determine for which values of x is: 5.6.1 f ( x).g ( x) = 0 (2) 5.6.2 f ( x)  0 (4) [14] Copyright reserved Please turn over

Downloaded from hlayiso.com 8 TECHNICAL MATHEMATICS P2 (EC/JUNE 2022) QUESTION 6 6.1 Complete the cosine rule for ΔPQR. (1) 6.2 In the diagram below, O is the centre of the circle. AB = 9 units, BC = 8 units and AC = 10 units. Determine: 6.2.1 The size of B̂ (4) 6.2.2 ˆ , stating a reason Hence, the size of AOC (2) 6.2.3 The length of the diameter of the circle (5) 6.2.4 The area of ∆ABC (3) [15] Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/JUNE 2022) TECHNICAL MATHEMATICS P2 9 QUESTION 7 7.1 ΔABC is a right-angled triangle in the diagram below. AB = (x + 2) units, AC = 4 units and BC = x units in length. 7.1.1 Determine the length of BC. (3) 7.1.2 Give a reason why AB is a diameter of the circle through A, B and C. (1) 7.2 ˆ = 140° In the diagram below, AC is a diameter of the circle with centre O. DOB and ACBˆ = 35°. 7.2.1 Determine, stating reasons, the size of Ô1 . (3) 7.2.2 Determine, stating reasons, the size of B̂1 . (3) 7.2.3 ˆ Show that AC bisect DCB. (2) [12] Copyright reserved Please turn over

Downloaded from hlayiso.com 10 TECHNICAL MATHEMATICS P2 (EC/JUNE 2022) QUESTION 8 8.1 Complete the following theorem statement: The exterior angle of a cyclic quadrilateral is equal to the … (1) 8.2 In the diagram below BCDF is a cyclic quadrilateral with BF extended to meet DG in G. ˆ = 51 = DBC FBD ˆ ˆ BDC = 44 BD = DG 8.2.1 Show, stating reasons, that D̂2 = 44 . (5) 8.2.2 Hence, show that GFD  BCD , stating reasons. (5) [11] Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/JUNE 2022) TECHNICAL MATHEMATICS P2 11 QUESTION 9 9.1 Complete the following theorem statement: The tangent to a circle is perpendicular to the … of the circle at the point of contact. (1) 9.2 In the diagram below, CH is a tangent to the circle at C. DG extended meets the tangent in H. CH ∥ DF CDGˆ = 40 CGDˆ = 70 Stating reasons, calculate the size of the following angles: 9.2.1 Ĉ 2 (2) 9.2.2 F̂ (3) 9.2.3 D̂1 (2) 9.3 In the diagram below, CK is a common tangent. Show, stating reasons, that EF ∥ JH. (4) [12] Copyright reserved Please turn over

Downloaded from hlayiso.com 12 TECHNICAL MATHEMATICS P2 (EC/JUNE 2022) QUESTION 10 10.1 Complete the following theorem statement: If a line divides two sides of a triangle in the same proportion, then the line is … (1) 10.2 In ΔABC below, E, D and F is on the sides of the triangle such that AB ∥ DE and AD ∥ FE. AE = 4 units , EC = 10 units and DF = 3 units. Calculate, giving reasons, the lengths of the following: 10.2.1 FC (3) 10.2.2 BD (4) 10.3 ˆ = C. In the diagram below, D is a point on BC such that BAD ˆ BD = 4 units and AB = 8 units. 10.3.1 Prove that ∆ABD ⫴ ∆CBA (3) 10.3.2 Calculate the length of DC. (4) [15] TOTAL: 150 Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/JUNE 2022) TECHNICAL MATHEMATICS P2 13 INFORMATION SHEET −b  b2 − 4ac x=− b 4ac − b 2 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 +y  d= ( x2 − x1 ) + ( y2 − y1 ) M 1 2 ; 1 2  2 2  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 + 1 + C , n  −1 n h →0 h 1 ax  x dx = ln( x) + C , x0  a dx = ln a + C , a  0 x  rad = 180 Angular velocity =  = 2 n = 360n where n = rotation frequency Circumferentialvelocity = =  Dn where D = diameter and n = rotation frequency s = r where r = radius and  = 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 2 Area of a sector = = where r = radius, s = arc length and  = central angle in radians 2 2 In ∆ABC: a b c 1 = = sin A sin B sin C a2 = b2 + c2 − 2bc.cosA Area = 2 ab.sin C sin 2  + cos2  = 1 1 + tan 2  = sec2  cot 2  + 1 = cos ec 2 o +o  AT = a  1 n + o2 + o3 + o4 + ... + on−1  where a = width of equal parts, 𝑜_𝑖= 𝑖^𝑡ℎ ordinate and n  2  = number of ordinates o +o OR where a = width of 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