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2014 T2 Trial SSI JB Answer

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  SEK KEB MEN SULTAN ISMAIL, JOHOR BAHRU Pre-U 95412 MATIDMATICS T TERM 2 TRI,AL EXAMINATION 2OI4 MARKING SCHEME I l. The functions/is defined as : _ 1x+lxl , x*0 flx): I z 12, r=0 (a) Without using graphs, determine whether the function/is continuous atx: 0. (b) Skefth the graphs of the firnctions/. [5 marks][2 marks]Marks (a) (0' .fli =|'2, (r, r(0 r=0 r)0 I , -/(r) =,\3- o: o lim f(r) = lim r:0 f+0*'' ' r+0* lim f(x) = lim. f(x):0 x+O-'- ' x+Ol' frg/(r): o fQ)=2 1 II }gftr) +f(o);;fis not continuous at x : 0 I (b) 1+1  ) A cylinder is inscribed inside a sphere of radius r. Given that the volume of the cylinderis a maximum, show that the ratio (volume of sphere) : (volume of cylinder) is r/5 : 1. [9 marks] ,':r'-T V= ttxzh v:nh(rz -+) Y= n(r'n- T) 1 To:o( -T) IVolume ma:rimum, ff= O n(,'-lfJ = o h: *4r -v3  ' h> 0, h: h, I 1 o=h,,r=n(h4V-#) V = 4=TEr3 3V3 I d.zv 3 , -:--hhz 2 o:h',#:-:G') :-{3r(<o) .'. The mocimurn volume of cylinder is ftzr3 I I (volume of sphere) 4z -Tr 3 1 \tr (volume of cylinder) 4e -ftT V3 a : VS I II 3. Evaluate th following @I:YO, O) f 6x3e' dx [3 marksJ [5 marks]Marks (a) IiiUrolar:I%', I (ln e)2 (lnl)2 =- 22 I 1 2 1 z  3(b) fi ortr' a* : li 3x27zxe* 1d,x Letu=3x2,9*=6x n:t2xex'dx=er' =13x2e,'13- I:6xe, dx l+l : l3xz ex2 - ger lloI = 13(4)e4 - 3enl- [0 - 3eo] I =9e+ +3 1 4. (a) Find the general solution ofthe differential equation dv fr:kdx+fi where t is a constant, giving your answer in the formy = f(r). [5 marks] (b) The gradient at any point P(x, .y) of a curve is proportional to the surn of thecoordinates of P. The curve passes through the pornt (1, -2) and its gradient at(1, -2) is -4. Find the equation of the curye. [4 marksJ (a) *-nr= * I Integrating factor = sl -n*c = e-tu I e-k 9-kve-k*-lsr-kx dx d Ero -u) - k* -o' le-kx = f ftys-x*6a I le-kx=-xe-tu-f-e-tudx 1 le-kx = -xe-w - -t, *,  = -X - -Y gat'x I o)H <, + D,#= Kx + y\ (1, *2),#= *0, Kl -21=-4 k=4 I -2: -l - - I gs+(t) 4 3 u 4 n II   = -x -i-2t+1x-t1 I 5. By using the standard Maclaurin series, find the first fow tenns in the series expansion tt(g). State the set of the values ofx for which the expansion is valid. [4 marksl (ff):]n1r+r4-h(r -r) 1 1 =- ) 3x-ry+g-$+... -x-ry+g-$+ ' I =X* -lr, *lr, -Tr* * , +;*t **. :|, -lr' **r' -#rn*... I rtlr lr , -T < r < il I  6. Given ttrat the equation ln x + x2 = 8 (a) Show that the equation has only one real root. [2 marks](b) Verifi, by calculationthatthis root lies betweenx=2 andr = 3. [2 marks](c) Use the Newton-Raphson method to find the of the equation ln x * xz - 8o giving your answer correct to two decimal places. [4 marks] (a) lnx: 8 - x2 .y - ktx ; the equatisn has only one real root. I I(b) lnx+rz-8:0 Letf (x) = lnr + xz - 8 f (2)= ln 2 + 22 -8: -3.307 (< o) f(3) :ln 3 + 32 - 8 = 2.099 P o) /is continuous on [2, 3l,fQ) and/(3) have opposite signs, ; The root of this equation lies between x = 2 and x = 3. 1 I (c) -f (x) = lnx + x -8 .f (x):L.*u lnrr, * xnz - 8 Xn+L = Xn {* z,^ ro :3 xt = t - tt Jili t :2 6686 x 2 67 (2 d p ) xz = 2.6686 - ln 2.6686 + Z.668O2 -8 -*r* +2(2s6g6) :2.6506 x 2.65 (2 d.p.) h = 2.6505 x 2.65 (2 d. p.) :. x:2.65 (2 d.p.) I I 1 I +
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