Mathematics paper 2 English grade 12 , Cheat Sheet of Mathematics

Download Mathematics paper 2 English grade 12 and more Cheat Sheet Mathematics in PDF only on Docsity! basic education Department: Basic Education REPUBLIC OF SOUTH AFRICA SENIOR CERTIFICATE EXAMINATIONS/ NATIONAL SENIOR CERTIFICATE EXAMINATIONS MATHEMATICS P2 MAY/JUNE 2023 MARKS: 150 TIME: 3 hours This question paper consists of 13 pages and 1 information sheet, Copyright reserved Please turn over Mathematics/P2 2 DBE/May/June 2023 SCINSC INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. 2. This question paper consists of 10 questions. Answer ALL the questions in the SPECIAL ANSWER BOOK provided. Clearly show ALL calculations, diagrams, graphs, ete. which you have used in determining your answers. Answers only will NOT necessarily be awarded full marks. You may use an approved scientific caiculator (non-programmable and non-graphical), unless stated otherwise. If necessary, round off answers correct to TWO decimal places, unless stated otherwise. Diagrams are NOT necessarily drawn to scale. An information sheet with formulae is included at the end of the question paper. Write neatly and legibly. Copyright reserved Please turn over Mathematics/P2 3 SC/NSC QUESTION 3 DBE/May/fune 2023 In the diagram, the equation of line AF is y=—x—- 11. B, a point on the x-axis, is the midpoint of the straight line joining A(-1 ; 4) and C. The angles of inclination of AF and AC are a and 63,43° respectively. AF cuts the x-axisin E. D isa point such that DE|| AC. ¥ Cc 63,43° x ACI;9 3.1 Calculate the: 3.1.1 Value of ¢ (2) 3.1.2 Size of a (2) 3.1.3 Gradient of AC, to the nearest whole number (2) 3.2 Determine the equation of AC inthe form y= mx+k. (2) 3.3 Calculate the: 3.3.1 Coordinates of C @) 3.3.2 Size of FED G) 34 G isa point such that EAGC, in that order, is a parallelogram. Determine the equation of a circle centred at G and passing through the point B. Write your answer in the form (x—a) +(y-8) =r’. (4) [18] Copyright reserved Please turn over Mathematics/P2 6 DBE/May/June 2023 SC/NSC QUESTION 4 In the diagram, the equation of the circle centred at N@-12 ; 5) is x7 + y? +24x-10y+153=0. The equation of the circle centred at M is (x+6)’ +(y+3) =25. PS and PR are tangents to the circle centred at M at S and R respectively. PR is parallel to the x-axis. K(-17 ; -5) is a point on PS. The straight line joining N and M cuts the smaller circle at T and the larger circle at 8. y S \ oO x M K-17 ; -5) P R Al Write down the coordinates of M. (2) 4.2 Calculate the: 4.2.1 Length of the radius of the smaller circle (2) 4.2.2 Length of TS (4) 43 Determine the equation of the tangent: 43.1 PR (2) 4.3.2 PS, inthe form y=mxte (5) 44 Quadrilateral PSMR is drawn. Calculate the: 44.1 Perimeter of PSMR (5) 442 Ratioof ———eaof ANPS ___ 2) area of quadrilateral PSMR. [22] Copyright reserved Please turn over Mathematics/P2 7 DBE/May/June 2023 SC/NSC QUESTION 5 5.1 Without using a calculator, simplify ihe following expression to a single trigonometry ratio: 1- sin(- @)cos(90° + 8) cos(6 - 360°) © 5.2 Given that cos 20° =p Without using a calculator, write EACH of the following in terms p: 5.2.1 cos 200° 2) 5.2.2 sin(-70°) Q) 5.2.3 sin 10° (3) 5.3 Determine, without using a calculator, the value of: cos(A +55°)cos(A +10°)+sin(A +55°)sin(A +10°) GB) 5.4 Consider: cos dx + sin2x— cos? x =-sinx sinx —2cosx 5.4.1 Prove the above identity. GB) 2x +sin2x—cos” 5.4.2 Determine the value of aso 3) —3sin’ x+6sinxcosx 5.5 Given: 3tan4x =—2cos4x 5.5.1 Without using a calculator, show that sin4x=—0,5 is the only solution to the above equation. 4 5.5.2 Hence, determine the general solution of x in the equation 3tan4x = —2cos4x (3) B8] Copyright reserved Please turn over Mathematics/P2 10 DBE/May/ June 2023 SC/NSC QUESTION 8 8.1 In the diagram, O is the centre of the circle. K, T and L are points on the circle. KT, TL, KL, OK and OT are drawn. OT intersects KL at M. ST is a tangent to the circle at T. STK =36° and OKL =18°. 36° no SS L 8.1.1 Determine, giving reasons, the size of: @) f, 2) ) £ @) (c) KOT (2) 8.1.2 Prove, giving reasons, that KM = ML. GB) Copyright reserved Please turn over Mathematics/P2 i DBE/May/June 2023 SC/NSC 8.2 In the diagram, ARDS is drawn. A, B and C are points on RD, RS and DS respectively such that AB || DS and RB: BS =5: 3. DC = 20 units and CS = 12 units. R A B D 20 Cc 12 Ss 8.2.1 Prove, giving reasons, that BC |] AD. (3) 8.2.2 If it is further given that RD = 48 units, calculate, giving reasons, the value of the ratio AD : AB. )) [15] Copyright reserved Please turn over Mathematics/P2 12 DBE/May/June 2623 SCINSC QUESTION 9 9.1 In the diagram, O is the centre of the circle. ABCD is a cyclic quadrilateral. B Noy Use the diagram in the ANSWER BOOK to prove the theorem which states that the opposite angles of a cyclic quadrilateral are supplementary, that is prove that B+D=180°. (5) 9.2 In the diagram, DEFG is a cyclic quadrilateral such that EF || DG. H is another point on the circle such that DH || FG. Chord EH is drawn. Let DB, =x. S PS ESG Prove, giving reasons, that D, = D,. (4) 19] Copyright reserved Please turn over