Mathematics Grade 12 P2 Eastern Cape June 2014 QP

Downloaded from hlayiso.comJune 2014 Common Test  Mathematics/P2 1 MATHEMATICS P2 COMMON TEST JUNE 2014 NATIONAL SENIOR CERTIFICATE GRADE 12 Marks: 125 Time: 2½ hours N.B. This question paper consists of 9 pages, 2 diagram sheets and 1 information sheet.   Copyright Reserved Please Turn Over

Mathematics/P2 Downloaded from 2 hlayiso.comJune 2014 Common Test NSC   INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of SIX questions. 2. Answer ALL the questions. 3. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining your 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, answers should be rounded off to TWO decimal places, unless stated otherwise. 7. A diagram sheet for answering QUESTION 4.1, QUESTION 5.2, QUESTION 6.1, QUESTION 6.2 and QUESTION 6.3 is attached at the end of this question paper. Write your name on these diagram sheets in the spaces provided and insert the diagram sheets inside the back cover of your ANSWER BOOK. 8. Diagrams are NOT necessarily drawn to scale. 9. An information sheet with formulae is included at the end of this question paper. 10. Number the answers correctly according to the numbering system used in this question paper. 11. Write neatly and legibly.      Copyright Reserved Please Turn Over

Mathematics/P2 Downloaded from 3 hlayiso.comJune 2014 Common Test NSC   QUESTION 1 1.1 A(1; 4), B(–2; –2) and C(4; 1) are vertices of ¨ ABC in the cartesian plane as shown in the figure below. M is the midpoint of AB. y A(1; 4) M  C(4; 1) x B(–2, –2) 1.1.1 Calculate the length of AC (leave the answer in simplified surd form) (3) 1.1.2 Determine the co-ordinates of M. (2) 1.1.3 Find the equation of the perpendicular bisector of AB. (4) 1.2 The equation of a circle is given as (x – 1)2 + (y + 1)2 = 2(x – y). 2 2 1.2.1 Write down the equation of the circle in the form x  a  y  b r 2. (4) 1.2.2 Hence, write down the co-ordinates of the centre and the length of the radius of the circle. (2) [15] Copyright Reserved Please Turn Over

Mathematics/P2 Downloaded from 4 hlayiso.comJune 2014 Common Test NSC QUESTION 2 2.1 In the diagram below, J (–2; 1), K(p; –4), L(5; 0) and M(3; 2) are vertices of trapezium JKLM. Also JK // ML. y M(3 ; 2) J(–2 ; 1) D x x L(5 ; 0) K(p ; –4) 2.1.1 Show that p = 3. (4) 2.1.2 Calculate JK: LM in the simplest form. (5) 2.1.3 Q (x; y) on JK is such that JQLM is a parallelogram, determine the co-ordinates of Q. (5) 2.1.4 Determine the equation of the line passing through K and M. (2) 2.1.5 Write down the value of D , the inclination of line KM. (1) 2.1.6 If points, R (1; k), J and L are collinear, calculate the value of k. (4) Copyright Reserved Please Turn Over

Mathematics/P2 Downloaded from 5 hlayiso.comJune 2014 Common Test NSC 2.2 In the diagram alongside, y centre Q of the circle lies on the straight line 3x + 4y + 7 = 0. W The straight line cuts the  circle at W and Z (–1; –1). The circle touches the Q P (0; 2) y - axis at P (0 ; 2). x  Z (–1 ; –1) 2 2 2.2.1 Determine the equation of the circle in the form x  a  y  b r2 (5) 2.2.2 Determine the length of diameter WZ. (1) [27] QUESTION 3 3.1 Show without using a calculator that: 6 2 sin 15o (3) 4 3.2 Write the following as a single trigonometric ratio: tan 180 o  T . cos 360 o  T (9) sin 180 o  T . cos 90 o  T  cos 540 o  T . cos  T o o 1 3.3 Prove that: sin 45  x . sin 45  x cos 2 x (5) 2 sin 33 o cos 33 o 3.4 Show  = 2 (6) sin 11o cos 11o tan 3x 3.5 Determine the general solution of the equation: 1 0 (5) tan 24q [28] Copyright Reserved Please Turn Over

Mathematics/P2 Downloaded from 6 hlayiso.comJune 2014 Common Test NSC QUESTION 4 4.1 Given ¨ FGH with Ĝ obtuse. F G H sin G sin H Use the above diagram to prove that: (5) g h 4.2 In the diagram alongside, P, Q and R W are points on the same horizontal plane. WP is a vertical telephone mast. The angle of elevation of W from Q is D . PQ̂R y, PR̂Q x and P QR = 15 metres. D x y R Q 15 m 4.2.1 Express PW in terms of PQ and Į. (2) 15 sin y 4.2.2 Hence, show that PR . (4) sin x  y tan D 4.2.3 If x = y, show that PW 7,5 (3) cos y [14] Copyright Reserved Please Turn Over

Mathematics/P2 Downloaded from 7 hlayiso.comJune 2014 Common Test NSC QUESTION 5 5.1 Complete the following theorem statement: Opposite angles of a cyclic quadrilateral are … (1) 5.2 In the figure alongside, RS and RNP are tangents to the circle with centre O at the points S and N. Radius NO is produced and cuts the circle at M and meets RS produced at T. T 2M 1 1 S 2 4 3 3 2 O 1 2 1 2 1 3 R N P 5.2.1 Why is RŜO = 90o? (1) 5.2.2 Prove that RNOS is a cyclic quadrilateral. (4) 5.2.3 If Ŝ1 x , determine, with reasons, FOUR other angles in the figure which are equal to x. (8) 1 5.2.4 Prove that Ŝ 3 Ô 3 (4) 2 [18] Copyright Reserved Please Turn Over

Mathematics/P2 Downloaded from 8 hlayiso.comJune 2014 Common Test NSC QUESTION 6 6.1 Given ¨ ABC with DE // BC as shown in the figure below: A D E B C AD AE Prove that: (6) DB EC 6.2 In the diagram below, M is the midpoint of QR in ǻ PQR. T is a point on PQ such that PM and TR intersect at G. GH ӝ PQ with H on QR. PG : PM = 1 : 3. P T G Q R H M Determine, with reasons, the numerical value of: QH 6.2.1 (3) HM RG 6.2.2 (5) RT Copyright Reserved Please Turn Over

Mathematics/P2 Downloaded from 9 hlayiso.comJune 2014 Common Test NSC 6.3 In the diagram below, the chord XY is produced by its own length to A, and tangent AZ is drawn to touch the circle at Z. XZ is produced to meet the line perpendicular to XA at B and AZ = AB. Y A 2 X 3 1 1 2 3 2 1 Z 2 1 B 6.3.1 Prove that Ẑ 3 = 90o. (5) 6.3.2 Prove that ǻAXZ ӝӛ ǻAZY (3) 6.3.3 Prove that AZ2 = AY. XA. (1) [23] TOTAL: 125  Copyright Reserved Please Turn Over

Mathematics/P2 Downloaded from 10 hlayiso.comJune 2014 Common Test NSC INFORMATION SHEET: MATHEMATICS INLIGTINGSBLAD: WISKUNDE  b r b 2  4ac x  2a A P (1  ni )  A P (1  ni )   A P(1  i) n    A P(1  i) n  n Tn a  (n  1)d   Sn 2a  (n  1)d  2 a r n  1 ; a Tn ar n 1   Sn r z 1  Sf ;  1  r  1  r 1 1 r F > x 1 i n 1 @   P x[1  (1  i ) n ]  i i f ( x  h)  f ( x ) f ' ( x) lim  ho 0 h § x  x2 y1  y 2 · d ( x 2  x1 ) 2  ( y 2  y1 ) 2   M ¨¨ 1 ; ¸¸  © 2 2 ¹ y 2  y1 y mx  c   y  y1 m( x  x1 )   m   m tan T  x 2  x1 2 2 xa  y b r2  a b c 1 In'ABC:   a2 b 2  c 2  2bc. cos A  area 'ABC ab. sin C  sin A sin B sin C 2 sin D  E sinD. cosE  cosD.sin E   sin D  E sinD. cosE  cosD.sin E  cos D  E cosD. cos E  sinD.sin E   cos D  E cosD. cos E  sinD.sin E  ­cos 2 D  sin 2 D ° 2 cos 2D ®1  2 sin D    sin 2D 2 sin D . cosD  °2 cos 2 D  1 ¯ n 2 ¦ fx  ¦ xi  x x      V2 i 1  n n n ( A) P ( A)      P (A or B) = P (A) + P (B) – P (A and B) nS yˆ a  bx    b ¦ x  x ( y  y) ¦(x  x) 2 Copyright Reserved Please Turn Over

Mathematics/P2 Downloaded from 11 hlayiso.comJune 2014 Common Test NSC NAME: ______________________________________________ DIAGRAM SHEET 1 QUESTION 4.1 F G H PLEASE TEAR ON DOTTED LINES T QUESTION 5.2 2M 1 1 S 2 4 3 3 2 O 1 2 1 2 1 3 P QUESTION 6.1 A D E B C Copyright Reserved Please Turn Over

Mathematics/P2 Downloaded from 12 hlayiso.comJune 2014 Common Test NSC NAME: ______________________________________________ DIAGRAM SHEET 2 QUESTION 6.2 P T G Q R H M QUESTION 6.3 Y A 2 X 3 1 1 2 3 2 1 Z 2 1 B  Copyright Reserved Please Turn Over