MATH P2 G11 QP NOV2020 ENG D_hlayiso.com_.pdf

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE GRADE 11 NOVEMBER 2020 MATHEMATICS P2 EXEMPLAR MARKS: 150 TIME: 3 hours *Imat2* This question paper consists of 10 pages and an answer book of 20 pages.

Downloaded from hlayiso.com 2 MATHEMATICS P2 (EC/NOVEMBER 2020) INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of 9 questions. 2. Answer ALL the questions in the SPECIAL ANSWER BOOK provided. 3. Clearly show ALL calculations, diagrams, graphs, etc. 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 answers to TWO decimal places, unless stated otherwise. 7. Diagrams are NOT necessarily drawn to scale. 8. Write neatly and legibly. Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/NOVEMBER 2020) MATHEMATICS P2 3 QUESTION 1 In the diagram below, straight line PS is defined by 3 y  2x 6 and cuts the x-axis at Q(3; 0). MQR is a straight line which meets PR at R(10; 4). N(6; - 2) is a point on PS and RN is drawn. PQRˆ =T. 1.1 Determine the gradient of PS. (2) 1.2 Calculate the inclination angle of MR. (4) 1.3 Determine the value of T . (3) 1.4 Prove that RN A PS . (3) 1.5 Calculate the area of ' RQN . (6) 1.6 Calculate the y-intercept of MR. (4) [22] Copyright reserved Please turn over

Downloaded from hlayiso.com 4 MATHEMATICS P2 (EC/NOVEMBER 2020) QUESTION 2 ABCD is a parallelogram with A(-2; -6), B(4; 0), C(-1; 0) and D(x ; y) as shown below. 2.1 Calculate the length of BC. (2) 2.2 Determine the gradient of AB. (2) 2.3 Determine the equation of CD. (3) 2.4 Determine the coordinates of M, the midpoint of BD. (3) 2.5 Hence or otherwise, determine the values of x and y. (3) [13] Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/NOVEMBER 2020) MATHEMATICS P2 5 QUESTION 3 3.1 If 12 tan B  5 0 and 90q d B d 360q , determine the value of sin B  cos B with the aid of a sketch. (5) 3.2 If sin 43q p , determine the values of the following in terms of p, without a calculator. 3.2.1 cos133q (2) 3.2.2 tan(43q) (3) 3.3 Simplify each of the following fully, WITHOUT using a calculator: sin(90q  x) 3.3.1 y tan( x  180q) sin(360q  x) (5) tan 205q.cos 315q.sin135q 3.3.2 sin 210q.cos150q.tan 25q (7) [22] QUESTION 4 4.1 Prove that: sin T  cos T .sin T tan T cos T  (1  sin 2 T ) (4) 4.2 Determine the general solution of 2sin x cos x  cos2 x 0 (6) 4.3 Solve for D if : 2. sin D 1, where D  >0q;360q@ (3) 4.4 § x y· 3 If x and y are acute angles such that tan ¨ ¸ 1 and cos( x  y) , find the © 2 ¹ 2 values of x and y. (5) [18] Copyright reserved Please turn over

Downloaded from hlayiso.com 6 MATHEMATICS P2 (EC/NOVEMBER 2020) QUESTION 5 § x· Sketched below is a graph of f ( x) cos ¨ ¸ , where x  > 180q;180q@ . ©2¹ 5.1 For f ( x) , write down the: 5.1.1 Range (2) 5.1.2 Period (1) 5.2 Draw a graph of g ( x) sin( x  30q) for x  > 180q;180q@ on the axes provided in the ANSWER BOOK. Clearly show all intercepts with the axes, turning point, starting and ending points. (4) 5.3 For which values of x  > 180q;180q@ , is f ( x).g ( x) t 0 ? (3) [10] Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/NOVEMBER 2020) MATHEMATICS P2 7 QUESTION 6 A pole, 8 m tall, is held in a vertical position by two steel cables of equal length. In the sketch, PQ is the pole and PS and PR are the steel cables. The angle of elevation of the top of the pole from the anchor points R and S is 60q in both cases. P 8m Q 60q 60q R S 6.1 Determine the length of cable PS. Leave your answer in simplified surd form. (3) 6.2 ˆ =100q . Give your answer correct Determine the distance between R and S, if RQS to TWO decimal digits. (7) [10] Copyright reserved Please turn over

Downloaded from hlayiso.com 8 MATHEMATICS P2 (EC/NOVEMBER 2020) QUESTION 7 7.1 In the diagram below, D is the centre of circle ABC with radius BD produced to O. B D C O A ˆ = 2 u ABC Use the diagram to prove the theorem that states ADC ˆ . (5) 7.2 In the figure, O is the centre of circle ABCDE. DB = DF. AODF, BOE and BCF are ˆ = 10q . straight lines. CFD C F B 10q 3 1 2 1 2 3 2 1 D 2 1 3 O4 A E Calculate, giving reasons, the size of the following angles: 7.2.1 D̂ 2 (3) 7.2.2 Â (4) 7.2.3 Ô 2 (2) 7.2.4 Ĉ1 (2) 7.2.5 Ê (2) 7.2.6 Ĉ 2 (2) 7.2.7 Ô 3 (2) [22] Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/NOVEMBER 2020) MATHEMATICS P2 9 QUESTION 8 8.1 In the diagram below, O is the centre of circle AEF with CFD a tangent at F. E A O C D F ˆ. ˆ =A Use the diagram to prove the theorem which states that EFD (5) 8.2 In the diagram below, ADF is a tangent to the circle with points E, B, C and D on the circumference of the circle. AB || DC and EC = DC. A E 1 2 3 B O 1 2 D 2 x 1 C F 8.2.1 ˆ = x , name with reasons, FIVE other angles equal to x. If CDF (10) 8.2.2 Prove that ABCD is a parallelogram. (4) [19] Copyright reserved Please turn over

Downloaded from hlayiso.com 10 MATHEMATICS P2 (EC/NOVEMBER 2020) QUESTION 9 9.1 Complete the following theorem statements: 9.1.1 The line drawn from the centre of a circle to the midpoint of a chord is … (1) 9.1.2 The exterior angle of a cyclic quadrilateral is equal to the … (1) 9.2 In the diagram, FH is a diameter of the circle FCH with centre O. FC is a chord and LCH is a secant. LF is a tangent to the circle at F. E is a point on CH such that CE = HE. F 1 x O 1 2 3 1 2 L C E H 9.2.1 Prove that FC || OE. (5) 9.2.2 Prove that OFLE is a cyclic quadrilateral. (3) 9.2.3 If F̂1 x , express Ô1 with reasons in terms of x. (4) [14] TOTAL: 150 Copyright reserved Please turn over