STPM Trial 2009 MathT Q&A Kelantan

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I. Using alcebraic laws ofsct. prove that I(Au B) C]u[(Au B) C']- B = A-B. [6 marb] 2. .Ii Find 1 3x dx ZTx' +1 [4 marb] 3. On the same diagram, sketch the graphs of y = x: I and Y - 2x - J. [3 morks) A region R is bounded by the two graphs and the y-axis. Find the area of region R.(7 marh] A solid is fonned by rotating the region R completely about y-axis. Find the volume ofthc sol id fonned. [5 marb] 4. The functionfis defined by fIx) = { ~ . x ~
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  I. Using alcebraiclaws ofsct. prove that I (A u B) C ]u[ (A u B) C ']- B = A-B. [6 marb] 2. .Ii Find 1 3x dx ZTx+1 [4 marb] 3. On thesame diagram,sketchthegraphs of y = x: I and Y - 2x - J. [3 morks) A region R isbounded by thetwo graphs and they-axis. Find the area of region R.(7 marh] Asol id is fonnedbyrotating the region R com pl etely abouty-axis. Findthe volume ofthc sol id fonned. [5 marb] 4. The functionfis definedby fIx) = { ~ . x ~ 3. O  x =3 (i)Find ! ~ T - fIx) . ! ~ T - f<x) nndhencedctennine if the functionfis continuous atx -3. [3 marb] (ii)Sketch the graph o ff [2 marb] S. Prove that. for aU yaJues of k. the line kx + y - 8k is atangent to the hyperbola xy = 16 . Hence,find the coordinates of the point of contact. (S marks] 6.Sketchthe graphs of y= ....!-. andY =I 2x- llonlhesamediagram .Hence, so lv cthe Ixl inequality IZx -I I> ~ . [7 marb] Ixl 7. [Y-2 Fin dthevaluesofx.yandzif 4..1' ;4 x' 4< -1  x issymmetric. 2: -1 [5 marb) 8.Giventhatf(x) ... f + a:/ + 8x + b, where a and b areconstant. f '(x) - 0when x z: -2.When 1( ..1') is divided by x + 2,the remainder is2. 9.Dctenninethe\'alues of a and b. (8) Show that theequati onf(x )- 0 has on ly one real root and find theset of values of x suchthatf(x} > 0(b) Express x + 4 in partial fructions. fIx) (I I Given thematricesA- P 4 7-3Find the values of p.q and r. -I) (I and2 -II [5 marb] papercollection   954 /1 10 . Hence.so lv e the simultaneousequations x+y-z ::-2 2y + Z a. 5.r -9 7x - 3y -:- 14 Giventhat y z> , showthat (I+X2 )d 1y + 3 x ~ = O dx'dx 2009 [4 marks) ) I.Express _- 1 _- aspartialfractio n. (r + I )(r + 2) (4 marks) ,.J- I Hence or otherwise. find the value of ) --- .:-=, (r +l )(r +2) ,..2.. I Find alsothe value of lim L --- ....... . ... 41 (r + l )(r + 2) (6 marks) [4 marks) 12 .By usingsketchi ng graph. show that theupproximatevalue of the root can be derived by the following Newton-Raphson it erativefonnula X . .1 =- X . - f(x ,, ) . [4 marks] f(x .) Sh ow that the equation Ian x 2.r hasa root in the interval(;. ~). [3 marks] Hence. find the root of me equation for xe ( ~ . %). correcttothree decimalplaces. [7 mor.tsJ papercollection   1. Ex press 3sinO-cosO inthefonn Rsin{O-a) ,with R>O andO o <a < 90 °. Henccsol veth eequation 3sinO-cosB =I. for 0° < 0<360°. 18 J 2. Byusingthesubstitution y = vx. show that thegeneral solution forthe differentialequations ~ = x + 2}' is AX l = x _ y) l ,where A is a arbitrary constant I 5 I dx 3x 3. In a chemica1process to obtain substanceBfrom substance A. everymolecule of A is directlychange to molecule of B.Therate of increment of molecule Batany instantarc directly proportional to the number of molecule Aat thatinstant. Initially the number of molecule of A is /,1 andthe number of molecule Bat t minutesafterthe processbeganis x. By assuming x as the continuousvariable.Sbow that the differentialequation representing th t' chemical process is ~ = *(M - x) , withkasaeonstant dl If x Owhent = O.showthat x=M(1- e- b) Sketch thegraph ofx versus t. 18 I 4. Ship A moveswitbvclocity ( 30~+40j) kmh-' and shipB moveswith velocity (15~+12j ) kmh-'. Initially.the po sitionvector of shipBrelative to ship Ais (5 ~ -1 - 12;') km. Calculate themagnitudeanddirection of ship A relative to shipBandthe nearest distance bctwccp ship A andshipB 1 5 1 5. (a)If ! and !!... aretheposition vectors of twopoints A and B re spectively.andPis a poi nt no +mb which divides AB in the ratio m: n. Show thatP has th eposition vector r =-- -- m+n PJ (b) Findavectorof theequation oflhe straight linepassing through the point A with position vCClOr a = 2 i +.i and is parallel to thevector ~ =-5 ~ -2 j I'I papercollection   6. lbe figure shows aquadrilateralwhichinscribed two isosceles triangles ABC and ADE withbase AB and AErespectively. Each trianglcsh asthe base angles of75 o. Be and DE areparallcland equal in lengths. ~ , ~ - - - - - - - - - - - - - - - - - - - - - - ~ ;  Sh ow that (a) Le BE = LBED = 90' (b) ACD is an equilateral triangle[4) r 31 7. The random variableX is normally distributed with mean ã jJ ã and variance 400 . It iskn own that I'(X > 1159) ,; 0.123and P(X > 769) ~ 0.881.Determinethe range of the value of I'. 15)8. In a certain city .records forburgla ri es s how th at theprobability of an arrest in one week is 0.42.l bc probability of an arrest and co nviction in that week is0.35.(a)Findtheprobability thataperson arrested for aparticularweekforburglary will be co nvicted. (b) What is the probability thatthere wi ll be 3 wee ks of su ccessf ularrested and conviction inonc particular month. .[5} 9. The continuous randomvariable Xhasprobability dcnsityfunctiongivenby { k (x+I), O';x S2 lex) = 2k 2S x S b. () otherwise Where k and b arc constants.(a) Giventhat P(X > 2.5) = .!.. find the value of theconstants k and b. 6 (b) Sketch the graph ofthc probabilitydensityfunction [(x). (e)Find 1'(1 .5 S X S2.5) 16) 1 31 12 1 papercollection 
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