St Joh s College Prelmar Eamatos Jul 04 Mathematcs Paper Eamer: G Evas Tme: 3 hrs Moderator: D Grgoratos Marks: 50 PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY. Ths questo paper cossts of pages, cludg a Aswer Sheet (pages 9 ad 0) ad a Iformato Sheet (pages ad ). Please check that our paper s complete.. Read the questos carefull. 3. Aswer the questos o the separate paper provded, ecept Questo 4(a) ad ad Questo 6 whch must e aswered o the Aswer Sheet. Wrte our ame o the Aswer Sheet ad dcate our teacher s tals. 4. Numer our aswers eactl as the questos are umered ad aswer the questos the correct order. 5. You ma use a approved o-programmale ad o-graphcal calculator, uless otherwse stated. 6. Roud off our aswers to oe decmal dgt where ecessar. 7. All the ecessar workg detals must e clearl show. Equatos ma ot e solved solel wth a calculator. 8. It s essetal that ou preset our work eatl ad logcall. 3 4 5 6 7 8 9 0 Out of 0 6 5 4 7 4 9 Mark TOTAL Page of
SECTION A QUESTION (a) Smplf the epresso:.9 (3) 3.6 Solve for, wthout the use of a calculator: () log 4 (3) 3 () 9 3 54 (6) (c) Solve for ad : 3 3 (6) (d) Cosder the quadratc equato: p p 6 0. () Solve the equato, terms of p, usg the quadratc formula. () () Hece, state for whch value(s) of p the equato wll have real roots. Gve a reaso for our aswer. () [] QUESTION (a) Gve f ( ) solve for : f( ) f 0 (5) Gve: g'( ) ad g(4) 3 Fd the equato of the perpedcular to the graph of g at = 4. (5) Page of
(c) () Determe f '( ) f 3 3 f ( ) (5) () Fd a epresso for d d f 3 ad state a restrctos. (5) [0] QUESTION 3 (a) Cosder the followg pcture patter, whch cotues the same sese deftel:.. 3. 4. Calculate the umer of square locks the 40 th pcture. (3) The sum to ft of a certa geometrc sequece s 4 tmes the frst term. Calculate the commo rato. (3) (c) I a arthmetc sequece, the th term s gve as T ad the sum of the frst terms s S. It s gve that: T 0 T9 6 S 0 S9 57 Fd the value of T. (5) [] QUESTION 4 (Aswer ths questo o the Aswer Sheet) (a) Sketch the graph of f( ) o the gve set of aes o the 3 Aswer Sheet, dcatg asmptotes ad tercepts wth the aes. (6) State the equato of the as of smmetr whch has a postve gradet ad add ths le to our sketch. (3) Page 3 of
(c) The world populato s the total umer of lvg humas o Earth. As of 03, t s estmated at 7.66 llo (7 66 000 000) the Uted States Cesus Bureau. The world populato has epereced cotuous growth sce the ed of the Great Fame ad the Black Death 350, whe t was ear 370 000 000. [Wkpeda - adapted] Usg ths formato, the world populato ca e modelled the formula: 370,00448 where s mllos ad s the umer of ears sce the ear 350. () Estmate the world populato, to the earest mllo, the ear 800. (3) () I whch decade dd the populato frst eceed 5 000 mllo? (4) [6] QUESTION 5 The proalt that Loel Mess scores a game of soccer s whlst the proalt that he does ot score s 3. (a) Determe the value of, showg that ts value s 3. (3) I how ma of the et 60 games s Mess lkel to score? () [5] 74 Marks Page 4 of
SECTION B QUESTION 6 (Aswer ths questo o the Aswer Sheet) A surve was coducted o a group of 00 studets to see who would watch the followg World Cups: Soccer, Rug ad Crcket. The results are summarsed as follows: Soccer: 70 Rug: 60 Crcket: 46 Ol Soccer: 5 Ol Rug: 0 All three: 0 Soccer ad Rug: 4 (a) Represet the gve formato usg the Ve Dagram o the aswer sheet. (7) (c) What s the proalt that a radoml selected studet wll ot watch a of the World Cups? () Are the evets Soccer ad Rug depedet? Gve a mathematcal motvato for our aswer. (3) [] QUESTION 7 I order to u ther frst cars, Coor ad Lam oth take out loas for R50 000 wth a terest rate of % per aum over a perod of 5 ears wth mothl stalmets startg oe moth s tme. However, Coor s loa s wth smple terest ad Lam s loa s wth mothl compoudg. (a) Determe the total value of Coor s loa, cludg terest. (3) Calculate the percetage omal terest rate applcale to Lam s loa, gvg the aswer correct to decmal places. (3) (c) Calculate Lam s mothl repamets. (4) (d) Who pas more terest - Coor or Lam? Epla wh ths s so. (4) [4] Page 5 of
QUESTION 8 (a) For the graph defed a > 0, > 0 f ( ) a c the followg formato s gve: 4ac 0 () Make a eat rough sketch of f (), showg ts correct oretato ad posto wth respect to the aes. (3) () Wrte dow the coordates of the turg pot, usg the parameters a, ad c, where ecessar. () The graph of a cuc fucto g() s show elow (ot to scale). A s the -tercept, B ad D are -tercepts ad B ad C are statoar pots. C A(0; 6) B(; 0) D(3; 0) 3 () Fd the equato of the graph, provg that g ( ) 0 4 6, (N.B. ou ma ot assume ths result our workg) (6) () Fd the value(s) of for whch: () g () s creasg (5) () g'( ) s creasg (3) (3) For whch values of k, wll g( ) khave eactl three real solutos? () Page 6 of
(c) A sem-crcle of dameter s cut from a rectagle wth a legth of 80 cm. Calculate the value of whch gves the mamum possle (shaded) area. Gve our aswer terms of. (6) QUESTION 9 [7] (a) Ad s a arthmetc ug ad Gerald s a geometrc ug ad the are havg a maratho ug-race over a dstace of 6 metres. Ad hops the sequece: 50 cm, 48 cm, 46 cm, 44 cm, 4 cm...etc. Gerald starts wth a hop of 00 cm ad each of hs susequet hops s 85% of the legth of the prevous hop. Assume oth ugs take the same tme to make oe hop. Arthmetc Ad Geometrc Gerald 6 metres () How ma hops does Ad take to cross the fshg le? (4) () Whch ug crosses the fshg le frst? (4) The followg result s gve: r r 6 () Calculate the sum of the square umers: + 4 + 9 +... + 0 000 (3) () Determe the value of: 40 r (3) r3 [4] Page 7 of
QUESTION 0 Cosder the area, A, cotaed etwee a paraola ad a horzotal le. h Let e the horzotal ase of the area ad h the heght,.e. h s the vertcal dstace from the turg pot to the ase. The area of A s gve the formula: A h. 3 I oth cases elow the equato of the curve s shaded area. 6. You are requred to fd the (a) (4) 76 Marks (5) [9] TOTAL: 50 marks Page 8 of
ANSWER SHEET Name: Teacher: KJ GE WLY DG BT SM JJ Questo 4(a) [6] Questo 4 [3] Page 9 of
Questo 6(a) World Cup Surve Soccer Rug Crcket Questo 6 Questo 6(c) [7] [] [3] Page 0 of
Page of INFORMATION SHEET a ac 4 ) ( ( ) T a d ( ) S a d T ar ( ) ; ar S r r ; a S r r h f h f f h 0 lm P A P A P A P A F P ) ( ) ( d ; M c m ) ( m m ta m ) ( ) ( r a
I ABC : a s A s B c s C a c c. cos A area ABC a. s C s ( ) s.cos cos. s s ( ) s.cos cos. s cos ( ) cos.cos s. s cos ( ) cos.cos s. s cos cos s s cos s s. cos f ( ) P ( A) ( A) ( S) P ( A or B) P ( A) P ( B) P ( A ad B) ( )( ) ŷ a ( ) Page of