o)-j[a:l f,+, f.6118 Iz, l.6t 2L-- L 1 = l-( l?-6 -Oi0OrL66-<o b1 Srin G0rrraqr )o xn*t=lt-r.,), xo=1.5 1' f(x):2x3+v-19 ( t \t

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1 S14 ]AL C34 1' f(x):2x3+v-19 (a) Show that the equation f(x) : 0 has a root a in the interval lt.s, Z] Q) The only real root of f(x) : 0 is a The iterative formula ( t \t xn*t=lt-r.,), xo=1.5 can be used to find an approximate value for a (b) Calculate xl, xrand x' giving your answers to 4 decimal places. (c) By choosing a suitable interval, show that a: correct to 4 decimal places. (2) "*fr*)-:f'a +lz),e ;ra$ --- (O r Fr t.lt rt tn o)-j[a:l c) -s f,+, f.6118 Iz, l.6t 2L-- L 1 = l-( l?-6 -Oi0OrL66-<o b1 Srin G0rrraqr )o rjj z -

2 2. A curve C has the equation x3-3xy -x *!3-11 :0 Find an equation of the tangent to C atthe point (2, -l),giving your answer in the form ax + by + c:0, where a, b and c are integers. (6)..l- i,- t-z ).^ -n - - *--+ {g{- -3*) r#-' ++gg -3:#-

3 3. Given that cos20 y= I + sin20 7t^3t 44 show that dy- o de L + sin20, -I.'.* where a is a constant to be determined. --]L=lrr,3L u1 qtlslrlo -- \r r g--lul2-q V!,,,LCirtO '- U--E--:al tl t7, db -( t+!rn2-o)1 ;i eo) (4)

4 4. Find (a) Irr.+ 3;12dx (b) I#r* a) (2)

5 11 f(x) = x,)3, l*l. -; Find the first three non-zero terms of the binomial expansion of f(r) in ascending powers of x. Give each coefficient as a simplified fraction. --8t1LrE*I* =zt*c{?:lf J (s)

6 6. (a) Express 5-4x (2x-l)(x+1) in partial fractions. (b) (i) Find a general solution of the differential equation Given that y: 4 when x : 2, (2x -1)(x + 1)* (5-4x)y, x > 1 2 (ii) find the particular solution of this differential equation. Give your answer in the formy: f(r). - S.r.Y--!:lr* =?-+. {--JL- 12rffir+*)?*-1 7rt (7) {qq}- lntf = Ia -3ln 3 tl rl lrrt 9 i -ztn}tq

7 7. The function f is defined by ^ 3x-5 t:xr--+ x+1, xelr.,x*-1 (a) Find an expression for f-l(r) (b) Show that ff(x) =!::, x-l' r JR,.r * -1.x * I where a is an integer to be determined. (4) The function g is defined by g:x'-' x' -3x, x JR, 0 ( x ( 5 (c) Find the value of fg(2) (2) (d) Find the range of g 'L-- "+* ( - 0f{rr; : 3(*:)-s +l-'- 3 -S:) I.x s &+J {( th)'' t' {- I -.--!''a'l- -f'ae-a -ry 1x.-ts - S(x+dtx-y<\ iz-s + (Y-tr) 1< (>(t) 4w -?D 1>t- 'l+ d z-s -.e*7-\ d) t6 c) (t *, (z) g ( (ta-tt 1) ' (bt-) t -y g -l ).x (x-:) *L S Y g(r's). grs), -a * to sll 1tx-) S,l

8 8. The volum e V of a spherical balloon is increasin g at a constant rate of 250 cm3 s-r. Find the rate of increase of the radius of the balloon, in cm s-1, at the instant when the volume of the balloon is cm3. Give your answer to 2 significant figures' (s) lyou may assume that the volume V of a sphere of radius r is given by the 4 J'formula V = axrt.f J - N :-}So- $",' * trj^tul V-a lz(flx} s'\v da -W*-4.4l.: - il n@{n(3 *-*a1lc;,,tsez

9 9. Figure 1 Figure 1 shows a sketch of part of the curve with equation y = eg, x ) 0 The finite region R, shown shaded in Figure 1, is bounded by the curve, the x-axis and the linesx:4andx:9 (a) Use the trapeziumrule, with 5 strips of equal width, to obtain an estimate for the area of R, giving your answer to 2 decimal places. (b) Use the substitutiq!, = J; to find, by integrating, the exact value for the area of R. z^d h'l (7) rg') (4) t W'2rq v,u e l- rz?,o

10 10. (a) Use the identity for sin(l + B) to prove that sin2a =2sinA cosa (2) (b) Show that A curve C has the equation d- i [t"(t*1;r))] : cosec, djf' (4) y: tn(tan(]x)) - :slnr, o < x < it (c) Find the x coordinates of the points on C where $ = O dr Give your answers to 3 decimal places. (Solutions based entirely on graphical or numerical methods are not acceptable.) (6) 4\ let i:6-- Sr^ {&B) aj,n{f * 8 + ho *S,^B S'r*2I- t }&n*(na- & *-WA;y- _x@ wi - fura 31,^r talx=-t tl Smlt 2Srnx tiox r?_ S&A2t ;l{,n2rc *- =r-1i:,s{r:r(t'f-o-?l171...i Or4p. *-z 0,?,6s +

11 11. X'igure 2 Figure 2 shows a sketch of part of the curve c with equation where a is aconstant and a > ln4 y : ga_3x _ 3e_r,.X lr The curve c has a turning point p and crosses the x-axis at the Figure 2. poifi Q as shown in (a) Find, in terms of a, the coordinates of the point p. (6) (b) Find, in terms of a, thex coordinate of the point e. (c) Sketch the curve with equation y :ls'-z*- 3.,1,.r lr, a ) ln4 Show on your sketch the exact coordinates, in terms of a, ofthe points at which the Wcurve meets or cuts the coordinate axes. - j-- :-oc r-lf,;;3x:---

12 a -Sxe,- b) 1,0 = F' -7'r C) 5e- [n g^-r*? [n3;''. -xln 3 + l{r{, a -3r s & -lx-, ln3 *7,-. *6-,"") tr lgo'ix' -i<-" \.'' L* t 4-ln3 L z Lrt*-lY'3) 2- aso y= {-Z

13 t2. Figure 3 Figure 3 shows a sketch of part of the curve C with parametric equations x = tant, y = 2sin2t, 0 < I <: The finite region S, shown shaded in Figure 3, is bound "OO:the curve C, the line x = re and the x-axis. This shaded region is rotated through 2tr rudians about the x-axis to form a solid of revolution. (a) Show that the volume of the solid of revolution formed is given by., +o l' (tan2 r - sin2l)dr Jo' (b) Hence use integration to find the exact value for this volume. (6) (6)

14 lia) 7L du fla rc''[i ' Vo[uruo- n f,y"tr ' "J t' # oo tnnt ' di '' t'5 wa a bant 3', 4 S,n+k :, Serlt.'" r/oturua-, ArrJ:'^* v titt' *L E b"nt 'o ; t=o,q, Yrr t s@ dt't'?ttifgt*':"'ros2 )*t, r.lf Can't, t $br'(o't tl 3 wj otat{?-srr2t 'tu D H.-'E::*,-',-H,Ii,;1:, 3;i' : ;;Ir: E, t,*n J' Srr]* -t - tu+lc*zr al o5 ; t*..ie g.rr* +ltoscr!*? = 2n (zsdu+ cal'? -TdP t, Jo s,lrr[ Zt*^ -+LSr* ]r -{: v,?.rr[ c,*.ti ' rr ) -(o) J :rrrfq*[ l z +r (tj-s-4q

15 13. (a) Express 2sin0 * cos 0 inthe form Rsin (0 + a),where R and a are constants, R >0and0 <a< 90o. Giveyourvalueofato2decimalplaces. ( 5,o,fi>-t* ) z k",fcosol t kc,&&suro1, _( Srne{ = \,, lw tt?(e 4Etrl=i_ i t.rcr --L ( = o.+ 636q+- - 0r. * o 'sa" eh+zlye.6 {fs,,*[eto'k6, - ry 'G$h(q+r6.b+) 2m 4m ix 4m {S'nO Figure 4 Figure 4 shows the design for a logo that is to be displayed on the side of a large building. The logo consists of three rectangles, C, D and, E, each of which is in contact with two horizontal parallel lines /, and,lr. Rectangle D touches rectangles C and.e as shown in Figure 4. Rectangles C, D and' E eachhave length 4 m and width 2 m. The acute angle d between the line l, and the longer edge of each rectangle is shown in Figure 4. Given that l, and lrarc 4 m apart, (b) show that Given also that 0 < 0 < 45o, (c) solve the equation 2sin0*cos0:2 2sin0*cos0:2 (2) giving the value of d to 1 decimal place. Rectangles c and D atd, rectangles D and, E touch for a distance Figure 4. hrn as shown in Using your answer to part (c), or otherwise, (d) find the value of fr, giving your answer to 2 significant figures.

16 b) **w 4s'gtk F tksrrg+2corb=4 ZS,nB +CoB =L c) G Srr^ (g* o'tkcl s L g+ 06.sp.Srrr-- (fr) = 6 3 kr lr6.s6. - 'i: $t 36'qo ft ^).L :. h: *-Z 3 tnn0 3 A t r'^ 2 * l-3- -)

17 --j,t'rg--=- lqf-*.e--* '=:0" Relative to a fixed origin O,the line I has vector equation where,t is a scalar parameter. ' [-l].'[j] Points A and B lie on the line /, where A has coordinates (1, a, 5) and B has coordinates (b, -1, 3). (a) Find the value of the constant a and the value of the constant b. (b) Find the vector AB. (2) The point C has coordinates (4, -3,2) (c) Show that the size of the angle CAB is 30" (d) Find the exact area of the triangle CAB, giving your answer in the form k.6, where ft is a constant to be determined. (2) The point D lies on the line / so that the area of the triangle CAD is twice the area of the triangle CAB. (e) Find the coordinates of the two possible positions of D. -[ +?-\ 6, - t -q-&-.=1tlt3=tij

18 d) Arreo, f,ftt J-( SrnSo i tzg + 3G An* = 6'li frb,,[fr " 3{i,[' r)', 6L LAg (+), i -fax t-f,bl s,n3o 3 6,.8 L.: \[B],,S.6Ca (2*.-),'l-re \ \/l- I ffi, zxfd ("

C4 "International A-level" (150 minute) papers: June 2014 and Specimen 1. C4 INTERNATIONAL A LEVEL PAPER JUNE 2014

C4 International A-level (150 minute) papers: June 2014 and Specimen 1. C4 INTERNATIONAL A LEVEL PAPER JUNE 2014 C4 "International A-level" (150 minute) papers: June 2014 and Specimen 1. C4 INTERNATIONAL A LEVEL PAPER JUNE 2014 1. f(x) = 2x 3 + x 10 (a) Show that the equation f(x) = 0 has a root in the interval [1.5,