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(16030192)T (E)(A )T-1-T940(E)(A6)TAPRIL EXAMINATIONNATIONAL CERTIFICATEMATHEMATICS N2(16030192)6 April 2016 (X-Paper)9:00–12:00REQUIREMENTS:Two sheets of graph paperScientific calculators may be used.This question paper consists of 7 pages and 1 formula sheet of 2 pages.Copyright reservedPlease turn over

(16030192)-2-T940(E)(A6)TDEPARTMENT OF HIGHER EDUCATION AND TRAININGREPUBLIC OF SOUTH AFRICANATIONAL CERTIFICATEMATHEMATICS N2TIME: 3 HOURSMARKS: 100INSTRUCTIONS AND INFORMATION1.Answer ALL the questions.2.Read ALL the questions carefully.3.Number the answers according to the numbering system used in this question paper.4.Show ALL formulae and intermediate steps and simplify where possible.5.Questions may be answered in any order, but subsections of questions must be kepttogether.6.Use only BLUE or BLACK ink.7.ALL final answers must be rounded off to THREE decimal places.8.Write neatly and legibly.Copyright reservedPlease turn over

(16030192)-3-T940(E)(A6)TQUESTION 11.11.2Fully factorise the following expressions:1.1.12 x 2 a 8a 8 x 2 32(4)1.1.23x3 y 12 x 2 y 9 xy(3)Simplify the following fractions:1.2.11.2.21.2.32 2 x 3x x 32(3)3 a3a(3)1 2ax 4a1x 1 2x 2 x 3 4 x 12 4ax 8a(5)[18]QUESTION 22.12.2Solve the following equation by using the quadratic formula: 3 x 2 4 x 5Change the subject of the formulas to the symbol in brackets:2.2.12.2.2.2.3(4)B pq. ( x)Kx(3)I nE.(n)R nr(3)9 petrol and a can of oil cost R137.81 together. Peter buys 14 petrol and a can ofoil for R198.26. What is the price of one litre of petrol and one can of oil?Copyright reservedPlease turn over(4)[14]

(16030192)-4-T940(E)(A6)TQUESTION 33.13.23.3Simplify:a 5b 3 a 2b 43a18b36(4)Solve for x :3.2 x 24(3)Simplify with the aid of logarithmic laws:13,45 265x 12,8 54, 6 Show ALL the steps. (Answers only are not acceptable).3.4Prove that(5)log x 9 log x 81 log x 3 5 log x 21877(4)[16]QUESTION 44.14.24.3A pulley turns at 3 500 revolutions per minute.Calculate:4.1.1The angular velocity of the pulley in radians per second.4.1.2The diameter of the pulley if the circumferential velocity is 2 015 cm/s.(2 x 3)(6)A sector of a circle has an arc of 22,5 cm, which subtends an angle .If the diameter of the circle is 48 cm, calculate:4.2.1The magnitude of the angle in degrees.(3)4.2.2The area of the circle sector.(2)Consider FIGURE 1 below. A paperweight, in the shape of a cone, which containinga liquid has a slant height, AC 13,2 cm and diameter, BC 16 cm. Determine thefollowing:4.3.1AD, the perpendicular height of the paperweight.(2)4.3.2How many millilitres of liquid will the paperweight hold when full?(3)Copyright reservedPlease turn over

(16030192)-5-T940(E)(A6)TFIGURE 14.4An irregular figure with one horizontal straight side is divided into 7 vertical strips of8 mm wide each. The ordinates dividing these strips have the following lengthsmeasured in cm: 0; 12; 13; 21; 24; 23; 18; and 0.Calculate the area of the figure.Copyright reserved(4)[20]Please turn over

(16030192)-6-T940(E)(A6)TQUESTION 55.1Consider FIGURE 2 below. ABC and ACD are two right-angled triangles withAB 134,5 cm, AD 56,9 cm and BC 128 cm. Determine the following:5.1.1The magnitude of the angle .5.1.2The length of side AC.5.1.3The magnitude of the angle . Give your answer for in radians.(3 x 3)(9)FIGURE 25.25.3Calculate the value of A for the following:3,5 tan A 7 0 ; A [0ο ;360ο ](3)An observer is standing 25 m from the foot of the tree. The angle of elevation to thetop of a tree is 57 o. Calculate the height of the tree.Copyright reservedPlease turn over(4)[16]

(16030192)-7-T940(E)(A6)TQUESTION 66.16.2Draw the following graph on its own system of axes using graph paper:1y cos x ; 0o x 360o2(3)Given: function y x 4 x 5Determine the following:26.2.1Zero points of the function(2)6.2.2y -intercept of the function(1)6.2.3The axis of symmetry(1)6.2.4The coordinates of the turning point(2)Hence, draw the graph of y x 2 4 x 5 on a system of axes, using graph paper.(3)6.3Draw the graph of x y 5 on the same system of axes used for QUESTION 6.2.(2)6.4Read from the graph the coordinates of the point where the two graphs in QUESTION6.3 intersect.TOTAL:Copyright reserved(2)[16]100

(16030192)-1-T940(E)(A6)TMATHEMATICS N2 - FORMULA SHEETThe right coneVolume ⅓ r2hSurface area rh 2 r 2 πr 2 πr πr 2The right pyramid1Volume (area of base) (perpendicular height)3The prismVolume (area of base) (perpendicular height)The cylinderVolume r2hSurface area 2 r2 2 rhThe sphere4V r3 ; A 4 r23Degrees and radians180 radarcSector: - ; A ½r2 radiusAngular velocity and circumferential velocityAngular velocity: w 2 nCircumferential velocity: v Dnn rotation frequency (r/s revolution per second)Mid-ordinate rule First ordinate last ordinate Area sum of the ordinates distance between the ordinates2 (distance between ordinates) (sum of mid-ordinates)GraphsStraight line: y mx c bAxis of symmetry: x 2aRoots: x Parabola: y ax2 bx c b b 2 4acCopyright reserved2aPlease turn over

(16030192)-2-Trigonometry90 180 sin sin (180 – )T940(E)(A6)Tcos -cos (180 – )tan –tan (180 – )Reciprocal Identitiessec x 1cos xcosec x Segments of circlesChord length xD h 1sin xHeight of segment hcot x Diameter of circle Dx24hRegular polygonsAngle subtended at centre of circumscribed circle by one side: 360 number of sidesR radius of circumscribed circlex length of sidex 2R sinθ2 Annulus: A R 2 r 2Copyright reserved 1tan x

MARKING GUIDELINE-1MATHEMATICS N2T.(E)(J)TMARKING GUIDELINENATIONAL CERTIFICATEAPRIL EXAMINATIONMATHEMATICS N2APRIL 2016This marking guideline consists of 11 pages.Copyright reservedPlease turn over

MARKING GUIDELINE-2MATHEMATICS N2T.(E)(A )TINSTRUCTIONS AND INFORMATION1. equals 1 mark and . equals ½ mark.2.Half marks are not allocated, unless indicated otherwise.3.Where a formula is required, using the wrong formula is a principle error and NOmarks are allocated.4.Students should show ALL formulae and intermediate steps and simplify wherepossible.5.ALL final answers must be rounded off to THREE decimal places (unless indicatedotherwise).6.Questions may be answered in any order, but subsections of questions must be kepttogether. If subsections are separated, the student can be penalised by ONE mark.7.8.Where a student copied wrongly from the question paper, and the standard of thequestion is still the same, the student will be penalised by ONE mark.If the copying error simplifies the question and makes it easier, the student forfeits themarks.Questions must be answered in blue or black ink. Answers in PENCIL are notmarked as it is regarded as rough work.Copyright reservedPlease turn over

MARKING GUIDELINE-3MATHEMATICS N2T.(E)(A )TQUESTION 11.11.1.12 x 2 a 8a 8 x 2 322a( x 2 4) 8( x 2 4) 2(a 4)( x 2)( x 2)1.1.2 (4)3x3 y 12 x 2 y 9 xy 3xy( x 2 4 x 3) 3xy( x 3)( x 1) (3)1.21.2.11.2.21.2.32 2 x 3x x 32212 4 x 9 x 2 6x12 5 x 2 6x(3)3 a3a 3a 3 a 3a4a 3 3a1 (3)2ax 4a1x 1 2x 2 x 3 4 x 12 4ax 8a 2a( x 2)4( x 3)x 1 ( x 3)( x 1)14a( x 2) 2 Copyright reserved(5)[18]Please turn over

MARKING GUIDELINE-4MATHEMATICS N2T.(E)(A )TQUESTION 22.12.2x 0, 786or x 2,1202.2.1pq.( x)Kxpq B2 Kxpq x KB 2 2.2.2 (4)B I (3)nE.(n)R nrIR Inr nE IR nE Inr IR n( E Ir ) IR n E IR IRor n IR E2.39 c 137,81.(1)14 c 198, 26.(2)(1) (2) 5 60, 45 12,09From (1)Copyright reserved(3) 9(12, 09) c 137,81 c 137,81 108,81 c 29(4)[14]Please turn over

MARKING GUIDELINE-5MATHEMATICS N2T.(E)(A )TQUESTION 33.13.2a 5b 3 3 a18b36 2 4a ba7 a 6b12ba7 a 3b6b a10b5 (4)3.2 x 242x 8 2 x 23x 3 (3) 3.3.x Show ALL the steps.Answers only are notacceptable.13,4 26512,8 54, 6513,45 265 log x log12,8 54, 6Without rounding off:12 log x log13, 45 log 265 log12,8 log 54, 61 log x 5log13, 4 log 265 log12,8 log 54, 62 log x 4, 002744313 x 104,002744313 x 10 063,390 log x 4, 003 x 104,003 x 10 069,317 Copyright reserved(5)Please turn over

MARKING GUIDELINE3.4-6MATHEMATICS N2T.(E)(A )Tlog x 9 log x 81 log x 3 5 log x 21877log x 9 log x 81 log x 3LHS log x 2187 log x 32 log x 34 log x 3 log x 37 2 log x 3 4 log x 3 log x 37 log x 3 5log x 37 log x 3 5 RHS7 (4)[16]QUESTION 44.14.1.1The angular velocity of the pulley in radians per second.w 2 n3500)60w 366,519rad / sw 2 (4.1.2(3)V Dn2015 D(3500)6010,995 DD 10,955 cm4.24.2.1sr22,5 24 (3)Final round off: 0,9375 rad or 53,715o where s 22,5cm and r 24 cm 0,938 rad or 53,743o Copyright reserved(3)Please turn over

MARKING GUIDELINE4.2.2-7MATHEMATICS N2A or1 2r 24.3.11(24) 2 (0,9375)2 270 cm 2 1 (24) 2 (0,938)2 270,144 cm 24.3T.(E)(A )T (2)AD 2 AB 2 BD 2 (13, 2) 2 (8) 2 110, 24AD 10,5 cm (2)4.3.24.41V r 2h31 (8) 2 (10,5) 3 703, 717 cm3 703, 717 m 0 0A [ 12 13 21 24 23 18] 0,82 88,8cm 2or 703,7167544 cm3 703,7167544 m Follow up with height of 4.3(3)or A 8880mm 2 (4)[20]QUESTION 55.15.1.1128134,5cos 0,952cos 17,824 *without rounding off : 17,885 (3)Copyright reservedPlease turn over

MARKING GUIDELINE-8MATHEMATICS N2T.(E)(A )T5.1.2AC 2 134,52 1282orAC 1706,25AC 41,3072sin17.824 opp134,5 134.5. sin17,824 41,17 orIn ABCorIn ABCAC128 AC 128 tan tan AC 128 tan17,885 AC 41,306 cmAC134,5 AC 134,5sin sin o AC 134,5sin17,885o AC 41,306 cm OrIn ABCAC128 AC 128 tan tan AC 128 tan17,824o AC 41,155 cm 5.1.3sin 46,326o 0,808 rad41,30756,9 sin 1 0,726 46,552 (3) or 0,812radians Or 46,348o 0,809 rad (or 46,549 0,812radians)5.2 (3)3,5tanA 7 0 ; A [0 ;360 ]tan A 2A 180 63,435 116,565A 360 63,435 296,565Copyright reserved (3)Please turn over

MARKING GUIDELINE-9MATHEMATICS N2T.(E)(A )T5.3AC25 AC 25 tan 57o AC 38, 497mtan 57o (4)[16]QUESTION 66.1y cos x 12for 0 x 360 shape shape½ mark( )x-intercepts60 ,300 ½ mark& ½ mark y-intercept0,5½ mark turning points(180 ,-1,5) ½ mark (360 , 0,5) ½ mark 6 x ½ marks 3 marks6.26.2.1(3)x2 4x 5 0 x 1 ( x 5) 0x 1 or Copyright reservedx 5 (2)Please turn over

MARKING GUIDELINE6.2.26.2.36.2.4let x 0y 5-10MATHEMATICS N2T.(E)(A )T (1) b2a ( 4) 2 2x (1)If x 2 theny (2) 2 4(2) 5 9therefore, TP is (2; 9) (2) shape ofparabola shapeof straightline TP- ½ mark x-intercepts(½mark & ½ mark) Y-intercept(½ mark)Shape (1 mark) Copyright reserved(3) Please turn over

MARKING GUIDELINE6.36.4x-intercept ½ marky-intercept ½ markshape 1 mark (0;–5)-11MATHEMATICS N2T.(E)(J)T(2) and (5;0)(2)[16]TOTAL:Copyright reserved100

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MATHEMATICS N2 - FORMULA SHEET The right cone Volume ⅓ r2h Surface area r h 2 r 2 U 2 U" U2 The right pyramid Volume 1 3 (area of base) (perpendicular height) The prism Volume (area of base) (perpendicular height) The cylinder Volume r2h S