NATIONAL SENIOR CERTIFICATE GRADE 1 MATHEMATICS P NOVEMBER 01 MARKS: 150 TIME: 3 hours This questio paper cosists of 13 pages, 1 diagram sheet ad 1 iformatio sheet. Please tur over
Mathematics/P DBE/November 01 INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios. 1.. 3. 4. 5. 6. 7. 8. 9. 10. 11. This questio paper cosists of 13 questios. Aswer ALL the questios. Clearly show ALL calculatios, diagrams, graphs, et cetera which you have used i determiig the aswers. Aswers oly will ot ecessarily be awarded full marks. You may use a approved scietific calculator (o-programmable ad ographical), uless stated otherwise. If ecessary, roud off aswers to TWO decimal places, uless stated otherwise. Diagrams are NOT ecessarily draw to scale. ONE diagram sheet for QUESTION 3. ad QUESTION 7.3 is attached at the ed of this questio paper. Write your cetre umber ad eamiatio umber o this sheet i the spaces provided ad isert the sheet iside the back cover of your ANSWER BOOK. A iformatio sheet with formulae is icluded at the ed of this questio paper. Number the aswers correctly accordig to the umberig system used i this questio paper. Write eatly ad legibly. Please tur over
Mathematics/P 3 DBE/November 01 QUESTION 1 The scatter plot below shows the age (i years) ad the average height (i cetimetres) of boys betwee ad 15 years. Scatter Plot Average height (i cm) 180 175 170 165 160 155 150 145 140 135 130 15 10 115 110 105 100 95 90 85 80 75 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 Age of boys (i years) [Source: www.fpotebook.com/edo/eam/hgtmsrmichar.htm] 1.1 Use the scatter plot to determie the average height of a 7-year-old boy. (1) 1. Describe the tred i the scatter plot. (1) 1.3 What is the approimate icrease i the average height per aum betwee the ages of ad 15 years? (3) 1.4 Eplai why the observed tred CANNOT cotiue idefiitely. (1) [6] Please tur over
Mathematics/P 4 DBE/November 01 QUESTION Abe plays for his school's cricket team. The umber of rus scored by Abe i the eight games that he batted i, is show below. (Abe was give out i all of the games.) 1 8 19 7 15 3 14 1.1 Determie the average rus scored by Abe i the eight games. (). Determie the stadard deviatio of the data set. ().3 Abe's scores for the first three of the et eight games were, 35 ad respectively. Describe the effect of his performace o the stadard deviatio of this larger set havig 11 data poits. ().4 Abe hopes to score a average of 0 rus i the first 16 games. What should his average i the last five games be so that he may reach his goal? (3) [9] QUESTION 3 I a certai school 60 learers wrote eamiatios i Mathematics ad Physical Scieces. The bo-ad-whisker diagram below shows the marks (out of 100) that these learers scored i the Physical Scieces eamiatio. Physical Scieces 3.1 Write dow the rage of the marks scored i the Physical Scieces eamiatio. (1) 3. Use the iformatio below to draw the bo-ad-whisker diagram for the Mathematics results o DIAGRAM SHEET 1. Miimum mark = 30 Rage = 55 Upper quartile = 70 Iterquartile rage = 30 Media = 55 (4) 3.3 How may learers scored less tha 70% i the Mathematics eamiatio? () 3.4 Joe claims that the umber of learers who scored betwee 30 ad 45 i Physical Scieces is smaller tha the umber of learers who scored betwee 30 ad 55 i Mathematics. Is Joe's claim valid? Justify your aswer. () [9] Please tur over
Mathematics/P 5 DBE/November 01 QUESTION 4 As part of a evirometal awareess iitiative, learers of Greeside High School were requested to collect ewspapers for recyclig. The cumulative frequecy graph (ogive) below shows the total weight of the ewspapers (i kilograms) collected over a period of 6 moths by 30 learers. Weight of ewspaper collected (i kilograms) 4.1 Determie the modal class of the weight of the ewspapers collected. (1) 4. Determie the media weight of the ewspapers collected by this group of learers. (1) 4.3 How may learers collected more tha 60 kilograms of ewspaper? () [4] Please tur over
Mathematics/P 6 DBE/November 01 QUESTION 5 ABCD is a rhombus with A( 3 ; 8) ad C(5 ; 4). The diagoals of ABCD bisect each other at M. The poit E(6 ; 1) lies o BC. A( 3 ; 8) θ y B M E(6 ; 1) P Q R S T D C(5 ; 4) 5.1 Calculate the coordiates of M. () 5. Calculate the gradiet of BC. () 5.3 Determie the equatio of the lie AD i the form y = m + c. (3) 5.4 Determie the size of θ, that is B ÂC. Show ALL calculatios. (6) [13] Please tur over
Mathematics/P 7 DBE/November 01 QUESTION 6 A circle cetred at N(3 ; ) touches the -ais at poit L. The lie PQ, defied by the equatio 4 4 y = +, is a taget to the same circle at poit A. 3 3 y Q A N(3 ; ) K O L B P 6.1 Why is NL perpedicular to OL? (1) 6. Determie the coordiates of L. (1) 6.3 Determie the equatio of the circle with cetre N i the form ( a) + (y b) = r (3) 6.4 Calculate the legth of KL. (3) 6.5 Determie the equatio of the diameter AB i the form y = m + c. (4) 7 16 6.6 Show that the coordiates of A are ;. (3) 5 5 6.7 Calculate the legth of KA. (3) 6.8 Why is KLNA a kite? () 6.9 Show that A Bˆ K = 45. (3) 6.10 If the give circle is reflected about the -ais, give the coordiates of the cetre of the ew circle. (1) [4] Please tur over
Mathematics/P 8 DBE/November 01 QUESTION 7 Cosider the diagram below where A( 5 ; ), B( 4 ; 1) ad C( 3 ; 3) are the vertices of ABC. y 10 8 A C 6 4 B / A / C / B -1-10 -8-6 -4-0 4 6 8 10 1 - -4 7.1 Describe the sigle trasformatio of ABC to A / B / C /. () 7. Write dow the geeral rule of the trasformatio i QUESTION 7.1. () 7.3 A / B / C / is elarged by a scale factor of to form A // B // C //. Draw the elargemet o DIAGRAM SHEET 1. () 7.4 Write dow the geeral rule of the trasformatio i QUESTION 7.3. (1) 7.5 ABC is reflected about the -ais ad the it is reflected about the y-ais to form DEF. 7.5.1 Write dow the coordiates of D, where D is the image of A after the trasformatio described above. () 7.5. Write dow the geeral rule of this trasformatio i the form: ( ; y) ( ; ) ( ; ). () 7.5.3 Describe a sigle trasformatio that ABC udergoes to form DEF. () [13] Please tur over
Mathematics/P 9 DBE/November 01 QUESTION 8 Aswer this questio WITHOUT usig a calculator. 8.1 The poit P(k ; 8) lies i the first quadrat such that OP = 17 uits ad T ÔP = α as show i the diagram alogside. y 17 P(k ; 8) O α T 8.1.1 Determie the value of k. () 8.1. Write dow the value of cosα. (1) 8.1.3 If it is further give that α + β = 180, determie cos β. () 8.1.4 Hece, determie the value of si( β α). (4) 8. Cosider the epressio: 1 cos si si cos 1 cos si 8..1 Prove that: = ta si cos (4) 8.. The above epressio is udefied if si cos = 0. Solve this equatio i the iterval 0 360. (4) [17] QUESTION 9 9.1 Simplify as far as possible: si θ si(180 θ ).cos(90 + θ ) + ta 45 (5) 9. Simplify without the use of a calculator: si104 (cos ta 38.si 15 1) 41 (8) [13] Please tur over
Mathematics/P 10 DBE/November 01 QUESTION 10 The graphs of f ( ) = si( + 30 ) ad g( ) = cos for 90 180 are give below. The graphs itersect at poit P ad poit Q. y g 1 f Q -90-60 -30 0 30 60 90 10 150 180 P -1-10.1 Calculate f (0) g(0). (1) 10. Calculate the -coordiates of poit P ad poit Q. (7) 10.3 For which values of will f ( ) g( )? () 10.4 Graph h is obtaied by the followig trasformatio of f: h ( ) = f ( + 60 ). Describe the relatioship betwee g ad h. () [1] Please tur over
Mathematics/P 11 DBE/November 01 QUESTION 11 ABCD is a parallelogram with AB = 3 uits, BC = uits ad A Bˆ C = θ for 0 < θ 90. A 3 θ B 3 D C 11.1 Prove that the area of parallelogram ABCD is 6siθ. (3) 11. Calculate the value of θ for which the area of the parallelogram is 3 3 square uits. (3) 11.3 Determie the value of θ for which the parallelogram has the maimum area. () [8] Please tur over
Mathematics/P 1 DBE/November 01 QUESTION 1 A hot-air balloo H is directly above poit B o the groud. Two ropes are used to keep the hot-air balloo i positio. The ropes are held by two people o the groud at poit C ad poit D. B, C ad D are i the same horizotal plae. The agle of elevatio from C to H is. C Dˆ B = ad C Bˆ D = 90. The distace betwee C ad D is k metres. H θ B 90 C k D 1.1 Show that CB = k si. (5) 1. Hece, show that the legth of rope HC is k ta. (3) 1.3 If k = 40 m, = 3 ad HD = 31,8 m, calculate θ, the agle betwee the two ropes. (4) [1] Please tur over
Mathematics/P 13 DBE/November 01 QUESTION 13 The face of a stadard clock is positioed such that the cetre is at the origi. At a certai time, the ed of the miute had is at the poit P( ; 4). 37 miutes later, the ed of the miute had is at the poit P / (a ; b). y P D D / O P / 13.1 Determie the value of a ad b. (6) 13. OD is the positio of the hour had whe the miute had is at P ad OD / is the positio of the hour had whe the miute had is at P /. Calculate the agle betwee OD ad OD /. (4) [10] TOTAL: 150 Please tur over
Mathematics/P DBE/November 01 CENTRE NUMBER: EXAMINATION NUMBER: DIAGRAM SHEET 1 QUESTION 3. Physical Sciece Mathematics QUESTION 7.3 y 10 8 A C 6 4 B / A / C / B -1-10 -8-6 -4-0 4 6 8 10 1 - -4
Mathematics/P DBE/November 01 INFORMATION SHEET b ± b 4 ac = a A = P( 1+ i) A = P( 1 i) A = P( 1 i) A = P( 1+ i) i= 1 1 = i= 1 ( + 1) i = 1 T = ar a( r 1) S = F = f '( ) [( 1+ i) 1] i = lim h 0 f ( + h) f ( ) h r 1 = S = ( a + ( 1 d ) T a + ( 1) d ; r 1 ( ) ( ) 1 + y1 + y d = 1 + y y1 M ; y = m + c y y = m ) ( a) + ( y b) = r I ABC: si a A area ABC 1 ( 1 b c = = a = b + c bc. cos A si B si C 1 = ab. si C S ) a = ; 1 < r < 1 1 r y y1 m = m = taθ ( α + β ) = siα.cos β cosα. si β si( α β ) = siα.cos β cosα. si β si + cos ( α + β ) = cosα.cos β siα. si β cos ( α β ) = cosα.cos β + siα. si β cos α si α cos α = 1 si α si α = siα. cosα cos α 1 ( ; y) ( cosθ y siθ ; y cosθ + siθ ) ( i ) = σ = i= 1 f ( A) P( A) = P(A or B) = P(A) + P(B) P(A ad B) y ˆ = a + b ( S ) [1 (1 + i) ] P = i b ( ) ( ) 1 ( y y) =