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T950(E)(A)TAPRIL EXAMINATIONNATIONAL CERTIFICATEMATHEMATICS N3(16030143)1 April (X-Paper)09:00–12:00This question paper consists of 6 pages and 1 formula sheet of 2 pages.

(16030143)-2-T950(E)(A1)TDEPARTMENT OF HIGHER EDUCATION AND TRAININGREPUBLIC OF SOUTH AFRICANATIONAL CERTIFICATEMATHEMATICS N3TIME: 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.Questions may be answered in any order but subsections of questions must NOT beseparated.5.Show ALL the calculations and intermediary steps.6.ALL final answers must be accurately approximated to THREE decimal places.7.ALL graph work must be done in the ANSWER BOOK. Graph paper is NOTsupplied.8.Diagrams are NOT drawn to scale.9.Write neatly and legibly.

(16030143)-3-T950(E)(A1)TQUESTION 11.11.21.3Factorise the following expressions as far as possible in prime factors:1.1.1x(3x 2) y (3 y 2)(4)1.1.24n 4 p 3n 2 p 1(2)Factorise the following expression completely:2 x3 x 2 5 x 2(5)Simplify the following expression:x 1 2 x 1 2 x 2 7 x 17 2x 1 3 xx 2x 3(6)[17]QUESTION 22.1Simplify the following:1x 2 x33x 22.22.3(4)Use logs to the base 2 and simplify the following WITHOUT using a calculator:log0,5 128(4)Solve for x:2.3.12.3.2Copyright reserved16 32 x 3x 2(log x 2) log( x 2) 0(2 4)Please turn over(8)[16]

(16030143)-4-T950(E)(A1)TQUESTION 33.13.23.33.4Solve for x by completing the square:4 x 48 2 x 2(4)Make ' b ' the subject of the formula:x bD x b(4)Make ' w ' the subject of the formula:log e t log e p log e w ds(3)Alex paid a deposit of R3x for a computer. He paid the rest in 9 monthlyinstalments. He paid a total of R33x . What is the payment of each monthlyinstalment in terms of x .QUESTION 44.1Consider FIGURE A below. ABC is an isosceles triangle with AB BC andvertices A(2;1), B(4;5) and C(0;k).FIGURE ACopyright reservedPlease turn over(4)[15]

(16030143)4.2-5-T950(E)(A1)T4.1.1Find the length of AB.(2)4.1.2Determine the value(s) of k.(4)4.1.3Show that AB is perpendicular to BC if k 7 .(3)4.1.4Calculate the area of ABC when k 7 .(3)P ( 2; 1) and Q (4; 7) are points in the plane with M as the midpoint of PQ.Determine the equation of the line parallel to the y-axis and passing through thepoint M.4.3(4)Consider FIGURE B below. The lines BA and CA with equations y x 2 andˆ where B and Cy 3 x 3 respectively, intersect at A. Determine the size of BACare the intercepts on the x-axis as shown.FIGURE B(5)[21]

(16030143)-6-T950(E)(A1)TQUESTION 55.1Draw the graph defined by the equation: 3x 2 3 y 2 275.2Given : y x3 6 x 2 9 x5.2.15.2.25.3Determine(2)Make use of differentiation to determine the coordinates of the turningpoints of the given equation.(5)Draw the graph of the given function. Show ALL values at the points ofintersection with the system of axes and the co-ordinates of the turningpoints.(3)1dyif y 2 x . Leave the answers with positive indices and inxdxsurd form.(4)[14]QUESTION 66.16.26.3Prove the following trigonometric identity:sin 2 A tan 2 A cos 2 A sec 2 A(4)Calculate the value(s) of which will satisfy the equation if 0o 270o :sin 1 cos 2 (5)Consider FIGURE C below. An observer, standing at a point A is watching the topof a vertical tower BC . The angle of elevation of the top of the tower, BC, is 25oand the angle of depression of the foot of the tower is 20o . If the height of thetower BC is known to be 30 m, determine the following:6.3.1The distance between the observer at A and the point B.(3)6.3.2The distance between the two towers.(3)

(16030143)-7-T950(E)(A1)TFIGURE C6.4Consider FIGURE D below. The sketch represents the graph of f ( x) a sin pxwhere 0 x . Determine the values of a and p .(2)[17]FIGURE DTOTAL:Copyright reservedPlease turn over100

(16030143)-1-T950(E)(A1)TFORMULA SHEETAny applicable formula may also be used.1. Factors2. Logarithmsa3 - b3 (a - b)(a2 ab b2)log ab log a log ba3 b3 (a b)(a2 - ab b2)loga log a log bblogc alogb a logc b3. Quadratic formulax b b 2 4aclog a m m log a2alogb a 4. Parabola1loga by ax 2 bx cloga a 1 1n e 14ac b 2y 4a bx 2aaloga t t e1n m m5. Circlex2 y2 r 2x2D h4h6. Straight liney y1 m(x x1 )Perpendicular: m1 m2 1Parallel lines: m1 m2x 4Dh 4h 2Distance: D (x 2 x1 )2 (y 2 y1 )2 x x 2 y1 y 2 Midpoint: P 1; 2 2 Angle of inclination: θ tan -1m

(16030143)7. Differentiationlim f x h f x dy dx h 0h d nx nx n - 1dxMax/MinFor turning points: f ' x 08. Trigonometrysinθ y1 r cosecθcosθ x1 r secθtanθ y1 x cotθsin 2 θ cos 2 θ 11 tan 2 θ sec 2 θ1 cot 2 θ cosec2 θtanθ sinθcosθcotθ cosθsinθsinA sinB sinC abca 2 b 2 c 2 2bc cosAArea of ABC ½ ac sin BCopyright reserved-2-T950(E)(A1)T

MARKING GUIDELINENATIONAL CERTIFICATEAPRIL EXAMINATIONMATHEMATICS N31 APRIL This marking guideline consists of 10 pages.

MARKING GUIDELINE-2MATHEMATICS N3T950(E)(A1)TQUESTION 11.11.1.1x (3x 2) y (3 y 2) 3x 2 2 x 3 y 2 2 y 3( x 2 y 2 ) 2( x y ) 3( x y )( x y ) 2( x y ) ( x y )[3( x y ) 2] ( x y )[3x 3 y 2] 1.1.24n 4 p 3n 2 p 1 (n 2 p 1)(4n 2 p 1)1.2(4) (2)f ( x) 2 x 3 x 2 5 x 2 f (1) 2(1)3 (1) 2 5(1) 2 2 1 5 2 0 x-1 is a factor of f ( x) 2 x 2 3x 2x-1 2 x3 x 2 5 x 22 x3 2 x 2 3x 2 5 x3x 2 3 x- 2x 2-2 x 2. . f ( x) ( x 1)(2 x 2 3 x 2) ( x 1)(2 x 1)( x 2) (5)

MARKING GUIDELINE-3MATHEMATICS N3T950(E)(A1)T1.3 x 1 2 x 1 2 x 2 7 x 17 x 1 x 3 ( x 1)( x 3)( x 1)(x 3) (2 x 1)(x 1) 2 x 2 7 x 17 ( x 1)(x 3)x 2 4 x 3 2 x 2 x 1 2 x 2 7 x 17 ( x 1)(x 3) 5( x 2 2 x 3) ( x 1)(x 3) (6)[17]

MARKING GUIDELINE-4MATHEMATICS N3T950(E)(A1)TQUESTION 22.1x 12 x3 3x 22x 1122x2x 1123 3x 2 12x3x2x 1 6 x22.2 32 (4) log 0,5 128log 2 1281log 22log 2 27 log 2 2 1 7 log 2 2 1log 2 2 72.3 2.3.1(4)16 32 x 3x 2 16 32 x 3x 2 2 16 32 x 32 x 4.3x 4 12 4.3x 3x 3 x 1TEST : If x 1 then LHS RHS 52.3.2(4)(log x 2) log( x 2) 0 log x 2 0 orlog( x 2) 0 log x 2 x 2 100 x 2 1x 3 x 102x 100 (4)[16]

MARKING GUIDELINE-5MATHEMATICS N3T950(E)(A1)TQUESTION 33.1 (4)3.2D2 x b x b (4)

MARKING GUIDELINE-6MATHEMATICS N3T950(E)(A1)T3.3logetw ds ptw e ds pw 3.4p dse tDeposit is R3x Total price is R33x Total amount for 9 instalments R30x30 x10 x R Each monthly instalment R93Copyright reserved(3) (4)[15]Please turn over

MARKING GUIDELINE-7MATHEMATICS N3T950(E)(A1)TQUESTION 44.14.1.1AB x2 x1 y2 y1 2 4 2 5 1 222 4 16 20 2 54.1.2(2)BC 2 5 0 4 k 5 2 16 k 5 22 20 20 k 5 42 k 5 2 k 5 2 or k 5 2 k 7 k 3 4.1.35 1 24 27 51 M CB 0 42M AB M CB 1 M AB therefore CB AB4.1.4Area of ΔABC (3)1 base height21AB BC21 20 202 10 units 2 (4) (3)Copyright reserved

MARKING GUIDELINE-8MATHEMATICS N3T950(E)(A1)T4.2 x x y y coordinates of M 1 2 ; 1 2 2 2 2 4 1 7 ; 2 2 (1;3) Equationx 1 (4)4.3M AB 1 tan 1 M AC 3 tan 3 45o 71,565oˆ 71,565o 45o 26,565oBAC (5)[21]QUESTION 55.13x 2 3 y 2 27Y3 -303 X-3Copyright reserved(2)Please turn over

MARKING GUIDELINE5.2.1-9MATHEMATICS N3y x3 6 x 2 9 xdy 3x 2 12 x 9dxLet y 0T950(E)(A1)T 3 x 2 12 x 9 03( x 2 4 x 3) 0( x 1)( x 3) 0x 1 or x 3 f ( 1) ( 1)3 6( 1) 2 9( 1) 4f ( 3) ( 3)3 6( 3) 2 9( 3) 0Turningpoints ( 3;0) and ( 1; 4) (5) 5.2.2 (3)5.3y 1 2 xx 1y x 2x12 1 dy 2 x 2(0,5)x 2dxdy11 2 dxxx Copyright reserved (4)[14]Please turn over

MARKING GUIDELINE-10MATHEMATICS N3T950(E)(A1)TQUESTION 66.1 6.2(4)sin 1 cos 2 sin sin 2 sin 2 sin 0 sin (sin 1) 0 sin 0 or sin 1or 90o or 180o 0 o6.36.3.16.3.2Copyright reservedIn ABC :AB30 osin 70sin 45o30sin 70o AB sin 45o 39,868m(5) (3) In ABD : ADo cos 25AB AD AB cos 25o 39,868 cos 25o 36,133m (3)Please turn over