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

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Mathematics P -KZN September 016 Preparatory Eamiatio Basic Educatio KwaZulu-Natal Departmet of Basic Educatio REPUBLIC OF SOUTH AFRICA MATHEMATICS P PREPARATORY

1 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 CERTIFICATE GRADE 1 MARKS: 150 TIME: 3 hours This questio paper cosists of 14 pages, 1 iformatio sheet ad a aswer book of 1 pages.

2 Mathematics P Preparatory Eamiatio September 016 INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios This questio paper cosists of 11 questios. Aswer ALL the questios i the SPECIAL ANSWER BOOK. Clearly show ALL calculatios, diagrams, graphs, et cetera that you have used i determiig your aswers. Aswers oly will ot ecessarily be awarded full marks. A approved scietific calculator (o-programmable ad o-graphical) may be used, uless stated otherwise. If ecessary, aswers should be rouded off to TWO decimal places, uless stated otherwise. Diagrams are NOT ecessarily draw to scale.

3 Mathematics P 3 Preparatory Eamiatio September 016 QUESTION 1 Two schools, M-cee-ai High ad Bee Vee high are i competitio to see which school performed better i mathematics i the Jue Eamiatio. The marks of the leaers at M-cee-ai High school are recorded below. The bo whisker diagram below illustrates the results of Bee Vee High School. Both schools have 5 learers. (Marks are give i %). Marks of M-cee-ai High School learers The bo ad whisker diagram for the Bee Vee High School Learers 1.1 Write dow the five-umber summary for M-cee-ai High School. (4) 1. Draw the bo ad whisker diagram that represets M-cee-ai High School marks. Clearly idicate ALL relevat values. (3) 1.3 Commet o the skewess of the data of M-cee-ai High School. (1) 1.4 Calculate the mea mark of M-cee-ai High School. () 1.5 Determie which school performed better i the Jue Eamiatio ad give reasos for your coclusio. (3) [13]

4 Mathematics P 4 Preparatory Eamiatio September 016 QUESTION A survey was coducted idicatig the umber of bees that visited flowers over a period of 1 days. The iformatio is represeted i the table ad i the scatter plot below. No. of Flowers No. of bees SCATTER PLOT Number of bees Number of flowers.1 Write dow the coordiates of the outlier. (). Determie the equatio for the least squares regressio lie. (4).3 Calculate the correlatio co-efficiet if the outlier is ecluded. () [8]

5 Mathematics P 5 Preparatory Eamiatio September 016 QUESTION 3 I the diagram below, A ( 5; 1), B(1; 6) ad C(7; ) are vertices of ABC with AB produced to D. BD forms a agle, β, with the egative ais ad BC forms a agle, α, with the positive ais. A Bˆ C = θ y B(1; 6) θ A( 5; 1) D β O α C(7; ) Determie: 3.1 the legth AC () 3. the equatio of lie BC (3) 3.3 A Bˆ C (5) 3.4 the midpoit P of AB () 3.5 the equatio of the lie parallel to AC ad passig through the poit ( 1; 3) (3) 3.6 Show that AB is perpedicular to 6 + 5y = 18 (3) [18]

6 Mathematics P 6 Preparatory Eamiatio September 016 QUESTION 4 I the diagram below, cetre W of the circle lies o the straight lie 3 + 4y + 7 = 0 The straight lie cuts the circle at V ad Z( 1; 1). The circle touches the y-ais at G(0; ) y V M M(a ; b) W G(0; ) Z ( 1; 1) Determie the equatio of the circle i the form ( a) + (y b) = r. (5) 4.1. Determie the legth of diameter VZ (1) Calculate the gradiet of GZ. () Write dow the coordiates of the midpoit of GZ. (1) Determie the equatio of the lie that is the perpedicular bisector of GZ. (3) Show that the lie i QUESTION ad straight lie VZ itersect at W. () 4. The circle defied by ( + ) + (y 1) = 5 has cetre M, ad the circle defied by ( 1) + (y 3) = 9 has cetre N Show that the circles itersect each other at two distict poits. (6) 4.. Determie the equatio of the commo chord. (3) [3]

7 Mathematics P 7 Preparatory Eamiatio September 016 QUESTION I the diagram below, P( 3; ) Q ÔP = α. Q is the poit o the ais so that is a poit i the cartesia plae, with refle agle O PˆQ = o 90 y α O β Q P ( 3; ) Calculate without measurig: β. (3) 5.1. the legth of OP. () the co-ordiates of Q. (3) 5. If cos α 3 si α = si ( α + β) + k. Calculate the values of k ad β. (5) [13]

8 Mathematics P 8 Preparatory Eamiatio September 016 QUESTION O the same system of aes, sketch the graphs of f () = 3 cos ad g () = ta 1 for Clearly show the itercepts with the aes ad all turig poits. (5) Use the graphs i 6.1 to aswer the followig questios. 6. Determie the period of g. (1) 6.3 Determie the co-ordiates of the turig poits of f o the give iterval. () 6.4 For which values of will both fuctios icrease as icreases for ? () 6.5 If the y ais is moved 45 o to the left, the write dow the ew equatio of f i the form y =.. (1) [11]

9 Mathematics P 9 Preparatory Eamiatio September 016 QUESTION Determie the geeral solutio of: cos 54 o. cos + si 54 o. si = si (5) 7. ABCD is a trapezium with AD BC, B ÂD = 90 o ad o B ĈD = 150. CD is produced to E. F is poit o AD such that BFE is a straight lie, ad The agle of elevatio of E from A is θ, BC = ad CE = C Bˆ E = α. E A θ F 18 3 D B α 150 o C 7..1 Show that: BE = ABcosθ si ( α θ) (5) 7.. Show that the area of Δ BCE = 9 3 (3) Determie, without the use of a calculator, the value of for which the area of ΔBCE will be maimum. (3) 7..4 Calculate the legth of BE if = 3. (3) [19]

10 Mathematics P 10 Preparatory Eamiatio September 016 QUESTION 8 I the diagram below, PQ is the chord of circle O. OR is perpedicular to PQ ad OR itersects PQ at T. If the radius of the circle is 13 cm ad PT = 1 cm. Q O T R Calculate the legth of: P 8.1 PQ () 8. PR (4) [6]

11 Mathematics P 11 Preparatory Eamiatio September 016 QUESTION Complete the statemet so that it is true: The eterior agle of a cyclic quadrilateral is equal to (1) 9. I the diagram below the circle with cetre O passes through L, N ad P. KLM is a taget to the circle at L. NP, NL ad LP are joied. N O P 1 K L M Usig the above diagram, prove the theorem that states that P LˆM = Nˆ. (5)

12 Mathematics P 1 Preparatory Eamiatio September I the diagram below, BAED is a cyclic quadrilateral with BA DE. BE = DE ad o AÊD = 70. The taget to the circle at D meets AB produced at C. D 1 70 o 1 E 1 3 C B A Calculate, with reasos the sizes of the followig agles Â () 9.3. Bˆ 1 () Dˆ () Bˆ () Dˆ 1 (3) [17]

13 Mathematics P 13 Preparatory Eamiatio September 016 QUESTION 10 I the diagram below, SP is a taget to the circle at P ad PQ is a chord. Chord QF produced meets SP at S ad chord RP bisects Q Pˆ S. PR produced meets QS at B. BC SP ad cuts the chord QR at D. QR produced meets SP at A. Let Bˆ =. Q 1 F B 3 1 R D C 1 S A 1 P 10.1 Name, with reasos, 3 agles equal to. (4) 10. Prove that PC = BC () 10.3 Prove that RCQB is a cyclic quadrilateral. () 10.4 Prove that Δ PBS Δ QCR. (5) 10.5 Show that PB.CR = QB.CP (4) [17]

14 Mathematics P 14 Preparatory Eamiatio September 016 K QUESTION 11 I the diagram alogside, Δ KLM, DE LM, L Dˆ M = MDˆ E=. KD = 9, EM = 8 ad EK = 6 Calculate, with reasos LM. D 9 6 E 8 L M (5) TOTAL: [150]

15 Mathematics P -KZN September 016 Preparatory Eamiatio INFORMATION SHEET: MATHEMATICS INLIGTINGSBLAD: WISKUNDE b ± = b 4 ac a A = P( 1+ i) A= P( 1 i) A = P( 1 i) A = P( 1+ i) T a+( 1) d = S = ( a+ ( 1d) ) 1 T = ar a( r 1) S = F = f [( 1+ i) 1] i f( + h) f( ) '( ) = lim h 0 h r 1 ; r 1 [1 (1 + i) ] P= i ( ) ( ) d = 1 + y y1 M 1+ y1+ y ; S a = ; 1 < r < 1 1 r y = m+ c y y = m ) 1 ( 1 y y1 m= m =taθ 1 ( a) + ( y b) = r I ABC: si a A b c = = a b c 1 = + bc. cosa area ABC= ab. sic sib sic ( α+ β) = siα.cosβ cosα. siβ si( α β) = siα.cosβ cosα. siβ si + cos ( α+ β) = cosα.cosβ siα. siβ cos ( α β) = cosα.cosβ+ siα. siβ cos α si α cosα = 1 si α si α = siα. cosα cos α 1 ( i ) = i= 1 σ = (A) P( A ) = P(A or B) =P(A) +P(B) - P(A adb) y ˆ = a+ b ( S) b ( ) ( y y) = ( )

GRADE 12 JUNE 2017 MATHEMATICS P2

GRADE 12 JUNE 2017 MATHEMATICS P2 NATIONAL SENIOR CERTIFICATE GRADE 1 JUNE 017 MATHEMATICS P MARKS: 150 TIME: 3 hours *JMATHE* This questio paper cosists of 14 pages, icludig 1 page iformatio sheet, ad a SPECIAL ANSWER BOOK. MATHEMATICS