HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time)
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1 HIGHER SCHOOL CERTIFICATE EXAMINATION 999 MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/ UNIT (COMMON) Time llowed Two hours (Plus 5 miutes redig time) DIRECTIONS TO CANDIDATES Attempt ALL questios. ALL questios re of equl vlue. All ecessry workig should be show i every questio. my be deducted for creless or bdly rrged work. Stdrd itegrls re prited o pge 8. Bord-pproved clcultors my be used. Aswer ech questio i SEPARATE Writig Booklet. You my sk for etr Writig Booklets if you eed them. 585
2 QUESTION Use SEPARATE Writig Booklet. () Evlute d. 0 3 (b) Differetite si 3. (c) The itervl AB hs ed poits A(, 7) d B(8, 8). Fid the coordites of the poit P which divides the itervl AB iterlly i the rtio : 3. (d) Write dow the equtio of the verticl symptote of y =. 3 (e) Fid the remider whe the polyomil P() = 3 is divided by (f) Use the substitutio u = t to evlute t sec d. π 0 ( ) 3 QUESTION Use SEPARATE Writig Booklet. () The stff i office cosists of mles d 7 femles. How my committees of 5 stff c be chose which coti ectly 3 femles? (b) Fid ll vlues of θ i the rge 0 θ π for which cosθ + 3siθ =. (c) Let f() = + log e. 6 (i) (ii) Write dow the turl domi for f(). Show tht, for ll vlues of i the turl domi, y = f() is icresig. (iii) Show tht the curve y = f() cuts the is betwee = 0 5 d =. (iv) Use Newto s method with first pproimtio of = 0 5 to fid secod pproimtio to the root of + log e = 0.
3 3 QUESTION 3 Use SEPARATE Writig Booklet. () y y = 3 si O π The shded regio bouded by y = 3 si, the is d the lie = π is rotted bout the is to form solid. Clculte the volume of the solid. (b) A fir, si-sided die is throw seve times. Wht is the probbility tht 6 occurs o ectly of the 7 throws? (c) P A C B Q Two circles itersect t two poits A d B s show i the digrm. The dimeter of oe circle is CA d this lie itersects the other circle t A d P. The lie CB itersects the secod circle t B d Q. Copy or trce the digrm ito your Writig Booklet. Prove tht CPQ is right gle. (d) (i) By equtig the coefficiets of si d cos, or otherwise, fid costts A d B stisfyig the idetity A( si + cos )+ B( cos si ) si + 8cos. si + 8cos (ii) Hece evlute. si + cos d
4 QUESTION Use SEPARATE Writig Booklet. k () Evlute ( ) k. 5 k= (b) y = y 6 B P (p, p ) O A The digrm shows the grph of the prbol = y. The tget to the prbol t P(p, p ), p > 0, cuts the is t A. The orml to the prbol t P cuts the y is t B. (i) Derive the equtio of the tget AP. (ii) Show tht B hs coordites (0, p + ). (iii) Let C be the midpoit of AB. Fid the crtesi equtio of the locus of C. (c) (i) Evlute d. 5 (ii) Use Simpso s rule with 3 fuctio vlues to pproimte d. (iii) Use your results to prts (i) d (ii) to obti pproimtio for e. Give your swer correct to 3 deciml plces.
5 5 QUESTION 5 Use SEPARATE Writig Booklet. () Prove by iductio tht, for ll itegers, ( ) + [ ( )] + ( ) ( ) = 3. 3 (b) Cosider the fuctio f() = e. (i) Show tht the miimum of f() occurs t = 0. (ii) Deduce tht f() 0 for ll. (iii) O the sme set of es, sketch y = e d y =. (iv) Fid the iverse fuctio of g() = e. 9 (v) (vi) Stte the domi of g (). For wht vlues of is log e ( + )? Justify your swer. Plese tur over
6 6 QUESTION 6 Use SEPARATE Writig Booklet. () A prticle moves i stright lie d its displcemet metres from the origi fter t secods is give by = cos 3t, t > 0. 6 (i) Whe is the prticle first t = 3? (ii) I wht directio is the prticle trvellig whe it is first t = 3? (iii) Epress the ccelertio of the prticle i terms of. (iv) (v) Hece, or otherwise, show tht the prticle is udergoig simple hrmoic motio. Stte the period of the motio. (b) O 6 cm 0 cm C cm B NOT TO SCALE 6 F A E D The digrm shows hgig bsket i the shpe of hemisphere with rdius 0 cm. Let O be the cetre of the sphere d let OA be the cetrl is. Two verticl wire supports, AB d AC, re show o the digrm. The legth of the rc BC is cm. A horizotl wire support is plced roud the surfce of the bsket. This wire meets AB t D d AC t E. The ple through DE prllel to the ple OBC cuts OA t F. The legth OF is 6 cm. Note tht BOC = DFE. (i) (ii) (iii) Show tht the legth of FD is 8 cm. Fid DFE i rdis. Fid the size of the gle DOE i rdis, correct to 3 deciml plces.
7 7 QUESTION 7 Use SEPARATE Writig Booklet. () y 8 O 5 30 m s y = A cricket bll leves the bowler s hd metres bove the groud with velocity of 30 m s t gle of 5 below the horizotl. The equtios of motio for the bll re.. = 0 d y.. = 0. Tke the origi to be the poit where the bll leves the bowler s hd. (i) (ii) (iii) Usig clculus, prove tht the coordites of the bll t time t re give by = 30t cos( 5 ), d y = 30t si( 5 ) 5t. Fid the time t which the bll strikes the groud. Clculte the gle t which the bll strikes the groud. (b) By cosiderig ( ) +, or otherwise, epress ( ) i simplest form. Ed of pper
8 8 STANDARD INTEGRALS d + =, ; 0, if < 0 + d = l, > 0 e d e =, 0 cos d = si, 0 si d = cos, 0 sec d = t, 0 sec t d = sec, 0 d = t, 0 + d = si, > 0, < < ( ) > > d = l +, ( ) d = l NOTE : l = log, > 0 e 0 Bord of Studies NSW 999
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