Core Mathematics Cl 2 Advanced Subsidiary

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1 Write your name here c:rname Pearson Edexcel nternational Advanced Level Centre Number Core Mathematics Cl 2 Advanced Subsidiary Wednesday 25 May 216 Morning ime: 2 hours 3 minutes Paper Reference... WMAOl1 You must have: Mathematical Formulae and Statistical Tables (Blue) Total Marks _ Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and iategration, or have retrievable mathematical formulae stored in them. nstructions Use black ink or ballpoint pen. f pencil is used for diagrams sketches graphs it must be dark (HB or B). Coloured pencils and highlighter pens must not be used. Fill in the boxes at the top of this page with your name, centre number and candidate number. Answer all questions and ensure that your answers to parts of questions are clearly labelled. Answer the questions in the spaces provided there may be more space than you need. You should show sufficient working to make your methods clear. Answers without working may not gain full credit. When a calculator is used, the answer should be given to an appropriate degree of accuracy. nformation The total mark for this paper is 125. The marks for each question are shown in brackets use this as a guide as to how much time to spend on each question. Advice Read each question carefully before you start to answer it. Try to answer every question. Check your answers if you have time at the end. Turn over P46713RA ll ll PEARSON Cl216 Pearson Education Ltd. P46 713RA

2 r. The first three terms in ascending po wees of X in the binomial expansion of (+ px) 8 are given by where p and q are constants. Find the value of p and the va lue of q. + 12x + qx 2 z f ;j (1rp>t)~ 1~1v,.,..1,g~(1}'6(px) :::. 1 (5) f :c vi > J> 2,,JJu~ : 1 (., (1"(p,l' :::<tpx ;rd tur"'\: CJ( (1 f {pj<),c,:: z&p~ t.. {o,..p,r~ zlld ~~ tp == 1Z p~jf [P~ ~] l,,..p,.rr,.._j 3rd *1~ \, t" nett i]3( 2 ll lllll lllll lllll lllll lllllll llll llll lllll P46713RA252 _

3 r 2. Find the,ange of va ues of x for which (a) 4(x 2) :(; 2x + (b) (2x 3)(x + 5) > (c) both 4(x 2) :(; 2x + and (2x 3)(x + 5) > (r,.) tx<j~ u. 1 (_b) (l) ~J(2lf s 1... i 2x s X. ~ l.~ [lt~)(,.t~j':)tj X;J.. >( =~ 3>1:'5] l;< <(~] (2) (3) (1) :2 t =E '1 :j m Z, ~? 3 Z. 1 D ' 2 ' \._ P R A O 4 5 2

4 ( 3. Answer this question without a calculator, showing all your working and giving your answers in their simplest form. (i) Solve the equation 4 2x+ l = 84x (3) (ii) (a) Express 3.Jsm in the form kj2, where k is an integer. (2) (b) Hence, or otherwise, solve {;) (zi.yzxr1 =( 23)4,x (2) z4xt2 ::. 71'2.v: '\ {b) ( = 6 ll lllll lllll lllll lllll lllllll llll llll lllll P R A O _

5 r 4. )' y=~ 2 6 X Figure 1 Figure shows a sketch of part of the curve with equation y = ~, x? 2 The finite region R, shown shaded in Figure, is bounded by the curve, the xax is and the line x = 6 The table below shows corresponding values of x and y for y = ~ : (a) Complete the tabl e above, giving the miss ing value of y to 4 decimal places. (1) (b) Use the trapezium rule, with all of the values of y in the completed table, to find an approximate value for the area of R, givin g your answer to 3 dec imal places. (3) Use your answer to part (b) to find approximate values of 6 (c) (i) J ~ d\" 2 2 c (ii) s:2 (2 + ~) dx!,) A t. z. [ o.,. u ,.z.. z.. 2.i:.t., s > _,_ 2. gzf?t] (4) CJ z A= 1l. ss~i :: 1t, ~ ~6' (.("(+11 'l ( L) (;) u...~~6= 1: in,z,.. z...j 8 {;,) 2{~) + 1t.~~6 = 3b. ~ 6 R i 16w... :.f., i. ""2. ll lllll lllll lllll lllll lllllll lll lll lllll P R A O

6 ull 5. (i) U +=, n ~ ull 3 Given U 1 = 4, find 1 (b) 2.u11 11 ;::::. ] (1) (2) z... l1 (i i) Given 2.(1 3r) < r = find the least value of the positive integer n. {;) { (1\) l( ~ 3 ~ 4 't ~?, 1 (b) 2_ Ur\ = 4)(1() ~ doo n~1 {::) "'~~:J d~3 s" ~ tr Z,. ~ {" _ 1) d J 6,,; {[2t~fJ)~rv1)(3D ~ t,, < 1 [ 1,4311t 3] ( (1l) (3) nl1~+3n] ~o n:.o n<o. J, 2;,,t(d..l 12 ll lllll lllll lllll lllll lllllll llll llll lllll lllll P46713RA1252 _

7 x (a) Show that _ 1 can be written in the form Ax"+ Bx'!, where A, B, p and q 2 '\ X are constants to be determined. (3) ;z \ (b) Hence find f x 2 4 ~dx, x > O giving your answer in its simplest form. 4 l X. 'Z "'} 1 (_.)( "2. (4) 16 ll lllll lllll lllll lllll lllllll llll llll lllll lllll P46713RA1652

8 r 7. '? f (x) = 3x" + ax + bx, where a and b are constants. Given that (x 2) is a factor of f(x), (a) use the factor theorem to show that 2a + b = 7 (2) Given also that when f(x) is divided by (x + ) the remainder is 36 (b) find the value of a and the va lue of b. (4) f(x) can be written in the form f(x) = (x 2)Q(x), where Q(x) is a quadratic function. (c) (i) Find Q(x). (ii) Prove that the equation f(x) = has only one real root. You must justify your answer and show all you r working. (6\) {41Z) is,. ~cfe,r. f:(2):: scz.?.. "Cz)'Z.. 1bC2)... 1o = o u t4ti\ + zb 1o == M\ + Zb = 14 2~ + b :: '1 (t>) : {~(l\d.u' ~ 36 ;. r:c 1); ~ 1..,(1)\ b(1) 1o :: 3 + "' b 1 = ~b::23 Q z.;:t~:~ 3"' ;: 3o ~ ~ 2o+ b=1 fb ::15] Rx)::. 3J 1o x\1~x1 o (~ Z) ~Q.f. (4) 18 ll lllll lllll lllll lllll lllllll llll llll lllll lllll P46713RA1852

9 . :c Question 7 continued {t_)(~) '$,_ ( ~ ~ ~:z.~' 1:,13x1o (;).,?~ 1 4 +?5x 1o Q 'l. ).. ef;~~ :. rrjl) = t 'JlzJ(3Lt111~) :. Q&.) = ~) 4Y. +.s {:;) d\;~ b::4 c~~ w b,z.4~c:: Ct)z. 44(3)(~):: 4 : 1"J:., d.scr:""':"'.~ l s vl..~ M (t.,. roots So {:(x) hla.> o:j.~ ~ f {oof ;,e ;l::2 ~ _ ll lllll lllll lllll lllll lllllll llll llll lllll lllll P46713RA Turn over

10 r 8. n th;s quesfon the angle 8 ;s measured ;n degrees thrnughout (a) Show that the equation 5 + sin(} 3 cos(} ) = 2 cosu, ":. (2n + )9, n E Z. l C) z $ may be rewritten as 6sin 2 + sino = (3) (b) Hence solve, for9 < < 9, the equation 5 + sin 8 = 2cosfJ 3 cos(} Give your answers to one decimal place, where appropriate. (4) (h) ) 22 ll lllll lllll lllll lllll lllllll llll llll lllll lllll lllll lllll llll llll P46713RA2252 _

11 r 9. The first term of a geometric series is 6 and the common ratio is. 92 For this series, find (a) (i) the 2S1" term, giving your answer to 2 significant figures, t, z (ii) the sum to infinity. The sum ton terms of this series is greater than 72 (4) ~ rn z (b) Calculate the smal lest possible value of n. (4) ::r. 24 ll lllll lllll lllll lllll lllllll llll llll lllll lllll P46713RA2452 _

12 ( 1. The curve Chas equation y = sin (x + ~), ( x ( 27f (a) On the axes below, sketch the curve C. (b) Write down the exact coordinates of all the points at which the curve C meets or intersects the xaxis and the yaxis. (2) (3) ;z (c) Solve, for O ( x ( 27f, the equation ::a m )> giving your answers in the form kn, where k is a rational number. (4) {.! l)'2.. ~ s ~ ~ l'tf 5. zt "" X 1 26 ll lllll lllll lllll lllll lllllll llll llll lllll lllll ? R A O _

13 Question 1 continued (c) 6r~(y+7l)~f L1 (} be Xtf $f1 9 = e=f Jf e~i1t <+f = trr (x"?irr] ll lllll lllll lllll lllll lllllll llll llll lllll lllll lllll lllll llll llll P R A O _l 27 Turn over

14 11. ' ' ' ' ' ' ' ' ' ' ',, r 1n,,' A 1.2m 1.2 m.4m p Diagram not drawn to scale z. r m ', ' ' ' ' Figure 2 Figure 2 shows the design for a sa il APBCA. ' ' ' ' ' B The curved edge APB of the sa il is an arc ofa circle centre O and radius rm. The straight edge ACB is a chord of the circle. The height AB of the sai l is The maximum width CP of the sai l is.4 m. (a) Show that r = 2 (2) (b) Show, to 4 decimal places, that angle AOB = radians. (2) (c) Hence calculate the area of the sa il, giving your answer, in 111 2, to 3 decimal places. (4) (tt.) n,,z 'L 2.. '2.. r:: 1,'Z.. (r,o4) A ::. 1.&.4 +o,~, + o.16 O, <Zr: 1.6 D r~o.4 L Jc=i\ y;iojd A (h) s,, g:::.!1_. ~ () = o. 6~ 3~ n.j tz.. 2 (_ [. LAoK 1.zno''i\~.. 3 ll lllll lllll lllll lllll lllllll llll llll lllll lllll P46713RA352 Q

15 < w Question 11 co4.nued ~"'"t,' ~;; ~ rr} f,\d. :: { "2 :C 7. ZZ1o ~ 2:,14 ~,z.. z Art" vf b :: ti 2.4x 1.6 Art,\ b~ 6qcl :: = 1. 3,Z,.., ~ 2. t14 1.,z ~ [= o.6?4,.,2 { j ll lllll lllll lllll lllll lllllll llll llll lllll lllll lllll lllll llll llll P46713RA Turn over

16 12. y C (a, ) X Figure 3 Figure 3 shows a circle C C touches the yaxis and has centre at the point (a, ) where a is a positive constant. (a) Write down an equation for C in terms of a (2) Given that the point P(4, 3) lies on C, (b) find the value of a (,.) (3) (b) P(4 1 3) (l~)1.j, (3)2;; ~ ~ 16iA ~~ ~:: $t>.. ~ 1..~ r~u 34 ll lllll lllll lllll lllll lllllll llll llll lllll lllll P46713RA3452 l

17 13. (a) Show that the equation 1 may be written in the form y2 = 1!:_ 2 log 2 y = 5 log 2 x x > O,y > z (b) Hence, or otherwise, solve the s imu ltaneous equations 9 where k is a constant to be found. ~,., X (3) 2log 2 y = 5 log 2 x log xy = 3 t m... ll > :: m > P l ("') for x > O,y > (1,) 1 36 g~ f>,,cb$+r.j,.jr,7~ Z ( '~ ~.= S ( 'ix o!:h <:ff s 1~ x 1<1,. ('j"').,_..'z x ~ ~ z 'Z {o, (). 'd'):. ~ ~ '2. ~ ;(j:2. [ m: 2 ~ 32. cj ;, [Pc" >aj X..., ~:: 3 Ji ~, _j X. :: C })z._ sz X. ;;[':; ~.>ls::: 31. $ 1 ~ "t j; (%)) lej (5) _L_j ll lllll lllll lllll lllll lllllll llll llll lllll lllll R A O _ t ~ V) ~ ~.A >

18 14. y P(3,4) Q(,4) A X 11 > ::: m l:> Figure 4 Figure 4 shows a sketch of the graph of y = g(x), 3 ~ x ~ 4 and part of the l. me l. h. 2 wit equation y = x The graph ofy = g(x) consists of three line segments, from P(3,4) to Q(,4), from Q(,4) to R(2,) and from R(2,) to S(4, 1). The line l intersects y = g(x) at the points A and Bas shown in Figure 4. (a) Use algebra to find the x coordinate of the point A and the x coordinate of the point B. Show each step of your working and give your answers as exact fractions. (6) G Q ;s: ::: ij: % V) i, J:i' (b) Sketch the graph with equation 3 y = 2 g(x), J ~ X ~ 4 On your sketch show the coordinates of the points to which P, Q, Rand S are transformed. "1'\ ::. o4 = z_ c.=4 1.o ~:: Zx f 4 ~:; l,x {x = Zxf4'.i. )( ::. 4 t?:e:? :1c«Jr,v1f<..FA 4 ll lllll lllll lllll lllll lllllll llll llll lllll lllll lllll lllll llll llll P R A O _

19 ~ ' :: <C V'l Question 14 continued R; ;> >"'\: ~ ~.t 42 ~:~)(.J.C.. :. ~(l) + l. c.::. 1o 'j:. ~)(1,) _ ll lllll lllll lllll lllll lllllll llll llll lllll lllll P46713RA Turn over

20 15. r hem!{ ~ :x, :::l Figure 5 shows a design for a water barrel. Figure f :c ~ )> ::a m > t is in the shape of a right circular cylinder with height h cm and radius r cm. The barrel has a base but has no lid, is open at the top and is made of material of negligible thickness. The barrel is designed to hold 6 cm 3 of water when full. (a) Show that the total external surface area, S cm 2, of the barrel is given by the formula S = 7rr r (b) Use calculus to find the minimum va lue of S, giving your answer to 3 significant figures. (c) Justify that the value of S you found in part (b) is a minimum. {(A) V: rr?h bf!) : rt {z.. h 7 L V 6 ootjo r,," (3) (6) (2) 44 ll lllll lllll lllll lllll lllllll llll llll lllll lllll P46713RA4452

21 ( Question 15 continued (b) $;: 1'(Lr 12orJt,of 1 bl ank U, :. ~::;. d, 2rr, 11t:iooo =.o r2 Zrr, :: rzl!)oo (1 Zrrr1; 1'Zoooo z11 z.rr (. 211) ('3 61> 1T, = Zf;. 13 CW\ s ~ 'fl(26.13)..?~ u.13 S:: 61s3YZ ~~15o c,,.,_j _ ll lllll lllll lllll lllll lllllll llll llll lllll lllll P R A O Turn over

22 16. y C m Figure 6 Figure 6 shows a sketch of part of the curve C with equation y = x(x )(x 2) The point P lies on C and has x coordinate ~ 2 The line, as shown on Figure 6, is the tangent to Cat P.. dy (a) Find dx (b) Use pa11 (a) to find an equation for l in the form ax+ by= c, where a, band c are integers. (2) (4) The finite region R, shown shaded in Figure 6, is bounded by the line, the curve C and the xaxis. The line l meets the curve again at the point (2, ) ( c) Use integration to find the exact area of the shaded region R. ~ ::. x. c5ll. + z) (j:. x > 3x'l.. 1Zt (6) 48 ll lllll lllll lllll lllll lllllll llll llll lllll lllll P46713RA4852 _

23 Question 16 continued {b) ~:; ljl1){12) LA_ ~ J~ 8 l b t <f\=1 r [h= 4( Y {~.o ',w g _ ll lllll lllll lllll lllll lllllll llll llll lllll lllll P46713RA Turn over