This paper consists of 10 pages with 10 questions. All the necessary working details must be shown.

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1 Mathematics - HG Mar 003 Natioal Paper INSTRUCTIONS This paper cosists of 0 pages with 0 questios. A formula sheet is icluded o page 0 i the questio paper. Detach it ad use it to aswer the questios i this questio paper. Aswer ALL the questios. All the ecessary workig details must be show. Clearly umber all the aswers correctly. The diagrams are ot draw to scale. A diagram sheet is icluded. Detach it ad place it iside the ANSWER BOOK. No-programmable calculators may be used, uless the questio states otherwise. The umber of decimal digits to which aswers must be rouded off will be stated i the questio where ecessary.

2 ANALYTICAL GEOMETRY NOTE: - USE ANALYTICAL METHODS IN THIS SECTION. - CONSTRUCTION AND MEASUREMENTS ARE NOT TO BE USED. QUESTION. I the diagram alogside k is the lie x + 3 y + 3 = 0 ad A( 3; 0) is a poit i ( 3; a 0) A Cartesia plae. Y k B k O k X.. Determie the equatio of lie k if k k ad k passes through poit A. (3).. Determie the equatio of lie k 3 if k3 k ad k 3 passes through poit A. (4)..3 Calculate the distace AB betwee the lies k ad k. Leave the aswer i surd form if ecessary...4 If P(x; y) is a poit o k such that BP = AB, calculate the possible co-ordiates of P. (5) (6). I the diagram alogside, P, R(4; 4), S ad T(0; 4) are the vertices of a rectagle. P ad S lie o the x-axis. The diagoals itersect at W. P Y T (0; 4) O W S X R (4; -4).. Show that the co-ordiates of S are ( 5 ; 0) + (5).. Determie the gradiet of TS rouded off to two decimal digits. ()..3 Calculate R Tˆ S rouded off to two decimal digits. (4) [9] QUESTION

3 . The poit P(x; y) is twice as far from the poit A (4; ) as it is from the origi. Prove that the equatio of the locus of P is 3x + 3y + 8x 4y 0 = 0 (4)... Show that the equatio of the taget to the circle x + y 4x + 6y + 3 = 0 at the poit (5; ) is y = 3x If T(x; y) is a poit o the taget i QUESTION.., such that its distace from the cetre of the circle is 0 uits, determie the values of x ad y (8) (7).3 I a attempt to fid the coditio that the lie y = mx + c, m 0, is a taget to the graph of y + 4x = 0, the followig solutio was give. State the lie i which a error appears ad give the correct aswer for that lie..3. y c The lie meets the graph where y + 4 = 0 m.3. That is where my + 4y 4c = If the lie is a taget to the graph, this equatio will have real uequal roots..3.4 The required coditio therefore is mc = (3) []

4 TRIGONOMETRY QUESTION 3 3. k If cosec θ = k + (0 < k < ) ad 90 θ 70, determie, with the aid of a diagram, the value of cosec θ + cot θ i terms of k (7) 3. Simplify to oe trigoometric ratio of α: ta(80 α) cot(α 80 ) cosec(α 90 ) si(360 α) (9) 3.3 Determie the values of x [ 90 ; 90 ] for which si 600 ta( 300 ) ta x = cos( 0 ) (6) [] QUESTION 4 4. Solve for x i ta x = si x if x [ 80 ; 80 ]. () 4. Use the set of axes provided o the diagram sheet to draw sketch graphs of f ( x) = ta x ad g( x) = si x for x [ 80 ; 80 ]. Idicate the itercepts with the axes as well as the co-ordiates of ay turig poits of the graphs. 4.3 Use the graphs i QUESTION 4. as well as the aswers to QUESTION 4. to determie the values of x [ 80 ; 80 ] for which ta x si x (6) (5) [] QUESTION 5 5. Determie the geeral solutio of x, rouded off to TWO decimal digits, if: (8) 3 si x 4 cosec x + 4 = 0 5. If si( θ α) = k si ( θ + α), k, determie ta θ i terms of k ad ta α (6) Prove the idetity: ta A cos A = ta A cos A (6) 5.3. For which values of A [0 ; 90 ] is the idetity ot valid? (3) [3] QUESTION 6 C

5 I ABC, 90 < A < 80 (5) Redraw this diagram i the aswer book, or use the diagram o the diagram sheet to prove that a = b + c bc cos A 6.. Hece write cos A i terms of a, b ad c () 6..3 Deduce that (a b + c)(a + b c) cos A = bc (3) 6. I the diagram alogside, P, Q ad R represet three poits alog two walls of a room. R is a poit i the corer such that PRˆ Q = 90 T is a poit vertically above R. QR = PR = 00 uits. q R T p P t Q 6.. Prove that PQ = 00 uits. () 6.. Use the result of QUESTION 6..3, or otherwise, to prove i PTQ that: t cos PTˆ Q = (5) p 6..3 Calculate the size of P Tˆ Q, rouded off to the earest iteger, if p = t (3) [9]

6 EUCLIDEAN GEOMETRY NOTE: - DIAGRAMS FOR PROVING THEORY MAY BE USED ON THE DIAGRAM SHEET OR REDRAWN IN THE ANSWER BOOK. - DETACH THE DIAGRAM SHEET FROM THE QUESTION PAPER AND PLACE IT INSIDE THE ANSWER BOOK. - GIVE A REASON FOR EACH STATEMENT. QUESTION 7 7. I the diagram alogside, DB is a chord of the circle such that D Bˆ C = DÊB. ABC is a straight lie. Redraw this diagram i the aswer book or use the diagram o the diagram sheet, to prove the theorem which states that ABC is a taget to the circle at B. A E B O D C (6) 7. I the diagram alogside, P, Q ad R are poits o a circle. YR bisects PRˆ Q with Y o PQ. X PQ produced meets RS at S such that SR = SY. QX SR P R 3 Y 3 Q Prove that: S 7.. SR is a taget to the circle (6) 7.. QR is a taget to the circle through Q, X ad P (3) [5]

7 QUESTION 8 I the diagram alogside, two circles itersect at A ad C. The larger circle passes through O, the cetre of the smaller circle. AD is a taget to the larger circle ad meets the smaller circle at D. DC produced meets the larger circle at B. BO produced meets AD at E. Let Â = x. A 3 E 3 4 O 3 C D B Prove that: 8. OA bisects D ÂC (5) 8. Dˆ = 90 x (3) 8.3 AE = ED (5) 8.4 BA = BD (4) [7]

8 QUESTION 9 I the diagram alogside, YX QR ad XS RN M is the midpoit of XR TS MN PY = 4 uits PQ = 7 uits Y P T X S N Q M R Write dow the values of the followig ratios, givig reasos: 9. PS:SN (4) 9. MN:TS (3) 9.3 PX:XM (3) [0]

9 QUESTION 0 0. I the diagram alogside, KLM ad PQR are two triagles such that Kˆ = Pˆ, Lˆ = Q ˆ ad Mˆ = Rˆ. K P (7) Redraw this diagram i the aswer book, or use the diagram o the diagram sheet, to prove the theorem which states that Q R KL KM = PQ PR L M 0. I the diagram alogside, two circles itersect at B ad C. A is a poit o the larger circle. AB ad AC itersect the smaller circle at F ad E respectively. D is a poit o the larger circle. A F 4 3 E D Prove that: B C 0.. AD = AC.AE (8) 0.. AFE ACB (4) 0..3 AD = AB.AF () [] TOTAL: 00

10 Mathematics Formula Sheet (HG ad SG) Wiskude Formuleblad (HG e SG) b ± b 4ac x = a T = a + ( )d S ( a + l T a ( r ) = a. r S = ) = [ a + ( ) d] r S a ( r ) = S = r S a = r A = P r + 00 A = P r 00 f' ( x ) = lim h 0 f ( x + h ) h f ( x ) d = ( x x ) + ( y y ) y = mx + c y y = m( x x ) m = y x y x m = taθ x + x y + y ; x + y = r ( x p ) + ( y q ) = r I ABC: a b = si A si B = c si C a = b + c bc.cos A area ABC = ab.si C

September 2016 Preparatory Examination NSC-KZN. Basic Education. KwaZulu-Natal Department of Basic Education REPUBLIC OF SOUTH AFRICA MATHEMATICS P2

September 2016 Preparatory Examination NSC-KZN. Basic Education. KwaZulu-Natal Department of Basic Education REPUBLIC OF SOUTH AFRICA MATHEMATICS P2 Mathematics P -KZN September 016 Preparatory Eamiatio Basic Educatio KwaZulu-Natal Departmet of Basic Educatio REPUBLIC OF SOUTH AFRICA MATHEMATICS P PREPARATORY EXAMINATION SEPTEMBER 016 NATIONAL SENIOR