PAST EXAM PAPER & MEMO N4

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(16030164)-1-T960(E)(A7)TAPRIL EXAMINATIONNATIONAL CERTIFICATEMATHEMATICS N4(16030164)7 April 2016 (X-Paper)09:00–12:00Scientific calculators may be used.This question paper consists of 5 pages and 1 formula sheet.Copyright reservedT (E)(A )T

(16030164)-2-T960(E)(A7)TDEPARTMENT OF HIGHER EDUCATION AND TRAININGREPUBLIC OF SOUTH AFRICANATIONAL CERTIFICATEMATHEMATICS N4TIME: 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 intermediate steps and simplify where possible.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.7.Use only BLUE or BLACK ink.8.Write neatly and legibly.Copyright reservedPlease turn over

(16030164)-3-T960(E)(A7)TQUESTION 11.11.1.1Draw the graph of y 2 cos ecx,0 0 x 2 (3)1.1.2Is the graph of y 2 cosecx in QUESTION 1.1.1 above a function or arelation?(1)Is the graph of y 2 cosecx in QUESTION 1.1.1 above symmetrical orasymmetrical about the X-axis?(1)1.2.1Draw the graph of the inverse of y 11x(3)1.2.2What is the domain of the graph of the inverse ofQUESTION 1.2.1 above?1.1.31.21.3y 11x in(2)Given:a b c 32a c 3 bc b a2Solve the value of 'a' by using Cramer's rule only.1.4(8)Given:120 113Calculate the value of the determinant.(2)[20]QUESTION 22.1Calculate the value of sin 2 A if cos A 2.2Solve for the value of 'A' if:24and A is in the first quadrant.25(3)5 cos 2 A 4 3 cos A;0 0 A 18002.3(5)Prove that:2 sin A 1 sec Acos A sin 2 ACopyright reserved(3)Please turn over

(16030164)2.4-4-Simplify: sin 1800 a tan 1800 a sin 900 atan 1800 a cos a . sin a 2.52.6T960(E)(A7)T (2)312and sin y and both 'x' and 'y' are acute angles, determine without135the use of a calculator the value of cos x y .If cos x (4)Prove that:2 2 cos 2 A 2 cos A(3)[20]QUESTION 33.1The perimeter of a field in the form of a right-angled triangle is 60 m. The length ofthe hypotenuse is 25 m.Calculate the lengths of the other two sides.3.2(5)Solve for 'x' if:4.4 x 293.3(3)Make 'c' the subject of the formula if:t i 0 1 e Rc 3.4Solve for 'x' if:x 3.5(3)7 5 0x(4)Rationalise:1cos j sin 3.6(3)Given:z 3 j6Convert z into polar form. The argument may only be positive.Copyright reserved(2)[20]Please turn over

(16030164)-5-T960(E)(A7)TQUESTION 44.1Determine the following limit:lim x 3 x 1 x x3 1 2 4.2(4)Given:y x 1 x 5 2Calculate, using differentiation, the co-ordinates of the maximum and minimumturning points.4.3Differentiate with regard to 'x':y 2 cot x ln x 6 4.4(7)32 e cos xx(5)Use the quotient rule to differentiate the following:y xx 4(4)[20]QUESTION 55.1Simplify: cos25.2 A sin 2 A dA(3)Evaluate:4 1 ln 3 x dx(3)25.35.3.15.3.25.4Sketch and indicate clearly the area enclosed by the graph of y x 2 2 ,x 0 and x 2 . Also indicate the representative strip used to calculatethe indicated area.(3)Calculate, using integration, the magnitude of the area indicated inQUESTION 5.3.1 above.(4)Integrate: sin 2 x cos x 21x 3 3 cos 3 4 x dxa3x (7)[20]TOTAL:Copyright reserved100

(16030164)T960(E)(A7)TFORMULA SHEETMATHEMATICS N4a x b log a x logbnn n x logexa bj c dj a c and b d(r ) r n sin(a b) sina cosb sinb cosacos(a b) cosa cosb sina sinbtan (a b) tan a tan b1 tan a tan byaxdydxn-1naxka xka x nak nxsin xkxcos xcos x-sin xtan xsec2xcot x-cosec2xsec xsec x tan xcosec x-cosec x cot xnn ax dx sin2x cos2x 11 cot2x cosec2x1 tan2x sec2xax n 1 Cn 1y u(x) · v(x) dyu x v 1 x u 1 x v x dxy u ( x)v( x)dy v x u 1 x u x v1 x dx v x 2dy dy du dx du dx sin xdx -cosx ca x dx a nx cx ka dx ka x c nabAox ydxuCopyright reserved cos xdx sinx c tan xdx nsec x c sec xdx n(sec x tan x) c

MARKING GUIDELINENATIONAL CERTIFICATEAPRIL EXAMINATIONMATHEMATICS N47 APRIL 2016This marking guideline consists of 8 pages.Copyright reservedPlease turn over

MARKING GUIDELINE-2MATHEMATICS N4T960(E)(A7)TQUESTION 11.11.1.1yy 2 cos ecx2 2 x-2 labelling .1yx 11 y1 x shapex 11ylabelling(3) 1.2.2Copyright reserved x : x 1, x R (2)Please turn over

MARKING GUIDELINE1.3-3MATHEMATICS N4 11 11 1 2112 12 1 1 11 1 1 11 1 122 1 1 1 2 1 3 2 1 521Do 211 11 10 123 1113 11 11 1 3 1 0 0 122 1 3 1 3 1 3 2 2 3Da 3T960(E)(A7)T 11 6 6a 1,095,51.4 1 1 . 0. 1 2 3 1 6Copyright reserved (8) (2)[20]Please turn over

MARKING GUIDELINE-4MATHEMATICS N4T960(E)(A7)TQUESTION 22.1sin 2 A 2 sin A cos A 7 24 2 . 25 25 336 6252.2 (3) 5 2 cos2 A 1 3 cos A 4 010 cos A 3 cos A 1 0 2 cos A 1 5 cos A 1 011 cos A j cos A 252A 600 101,53702.32.42.52.6 A 1800 cos 115 (5)2 sin A 1 sec Acos A 2 sin A cos A2 sin A 1 sec Acos A 1 2 sin A 1 sec Acos Asin a tan a cos atan a. cos a. sin a 1 (3) (2)cos x y cos x cos y sin x. sin y3 5 4 12 . .5 13 5 1315 48 65 6563 65 (4) 2 2 2 cos2 A 1 2 cos A2 4 cos2 A 2 2 cos A 4 cos2 A 2 cos A2 cos A 2 cos A Copyright reserved (3)[20]Please turn over

MARKING GUIDELINE-5MATHEMATICS N4T960(E)(A7)TQUESTION 3x y z 60 1 z 25ln 1 x y 35 2 3.1 x 2 y 2 625 3 n 3 x 2 35 x 6252x 2 35x 300 0 x 20 x 15 0x1 20, x 2 15 n 2 y1 : 20 y 35 y1 15y 2 : 15 y 35y 2 203.23.3x ln 4 ln 7,25ln 7,25x ln 4x 1,4291 ee tRCtRC(5) iI0 I0 iI0(3) I i ln 0 I0 I i t ln 0 RC I0 ln e tRCc Copyright reserved t I i R ln 0 I0 (3)Please turn over

MARKING GUIDELINE3.4-6MATHEMATICS N4x 2 5x 7 0x 5 5 2 4 1 7 2 5 3x 2 5 j 3x1 2 5 j 3x2 23.53.6T960(E)(A7)T (4)1cos j sin xcos j sin cos j sin cos j sin 2cos j sin cos j sin cos j 2 sin 2 cos j sin cos2 sin 2 cos j sin (3)z 3 j6r 3 2 6 2r 45 6,708 tan 1 2 63,4350 (2)[20]Copyright reservedPlease turn over

MARKING GUIDELINE-7MATHEMATICS N4T960(E)(A7)TQUESTION 44.1lim x x 1 x 1 1x x 1 x 2 x 1 2x x 1 2x x 11 1 1 2 2 2 1 1 1 2 2 3 7 4.2 (4)dy 3x 2 18x 15 dx2x 6x 5 0 x 1 x 5 0x1 1 x 2 5 y1 1 9 1 15 1 25 3232 y 2 5 9 1 15 5 253 02 d2y 6 x 18dx 2d2yx 1, 2 0mdxd2yx 5, 2 0Mdxm 1, 32 & M 5,0 4.34.4(7) dy62 2 cos ec 2 x 3 sec x tan x 0 2 dxxx (5)Let12u x ; u1 1 2 x v x 4; v 1 1dy v.u uv dxv21Copyright reserved1 x 4 1 x .12 x x 4 2 (4)[20]Please turn over

MARKING GUIDELINE-8MATHEMATICS N4T960(E)(A7)TQUESTION 55.1 cos 2 Asin 2 A c25.2dA ln 3x ln x 24(3) ln 3.4 ln 4 ln 3.2 ln 2 1,504 5.3 (3)5.3.1yy x2 2dx2 yx 2 shapeshaded arearep. strip(3)5.3.2Aox x 2 2 dx2 02 x3 2 x 3 0 23 2 2 0 3 2 6,667 u 5.4 2 cos x 4 x 3x xx 3 4 x 9 sin ca 4 3(4)(7)[20] TOTAL:Copyright reserved100

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