Cetre No. Cadidate No. Surame Sigature Paper Referece(s) 6667/0 Edexcel GCE Further Pure Mathematics FP Advaced/Advaced Subsidiary Moday 28 Jauary 203 Morig Time: hour 30 miutes Materials required for examiatio Mathematical Formulae (Pik) Paper Referece 6 6 6 7 0 Items icluded with questio papers Nil Cadidates may use ay calculator allowed by the regulatios of the Joit Coucil for Qualificatios. Calculators must ot have the facility for symbolic algebra maipulatio or symbolic differetiatio/itegratio, or have retrievable mathematical formulae stored i them. Iitial(s) Examier s use oly Team Leader s use oly Questio Number Blak 2 3 4 5 6 7 8 9 Istructios to Cadidates I the boxes above, write your cetre umber, cadidate umber, your surame, iitials ad sigature. Check that you have the correct questio paper. Aswer ALL the questios. You must write your aswer to each questio i the space followig the questio. Whe a calculator is used, the aswer should be give to a appropriate degree of accuracy. Iformatio for Cadidates A booklet Mathematical Formulae ad Statistical Tables is provided. Full marks may be obtaied for aswers to ALL questios. The marks for idividual questios ad the parts of questios are show i roud brackets: e.g.. There are 9 questios i this questio paper. The total mark for this paper is 75. There are 28 pages i this questio paper. Ay pages are idicated. Advice to Cadidates You must esure that your aswers to parts of questios are clearly labelled. You should show sufficiet workig to make your methods clear to the Examier. Aswers without workig may ot gai full credit. This publicatio may be reproduced oly i accordace with Pearso Educatio Ltd copyright policy. 203 Pearso Educatio Ltd. Priter s Log. No. P4485A W850/R6667/57570 5/5/5/5/6/ *P4485A028* Total Tur over
. Show, usig the formulae for r ad r= r= r 2, that 32 ( r ) 2 = ( 2+ )( 2 ), for all positive itegers. r= (5) 2 *P4485A0228*
2. Fid, i the form a+ i b where ab,, z = + 50 3 4i (a) z, (b) z 2. Fid (c) z, (d) arg z 2, givig your aswer i degrees to decimal place. 4 *P4485A0428*
3. f( x) = 2x + x 2 2 5, x 0 (a) Fid f() x. The equatio f( x ) = 0 has a root i the iterval [4.5, 5.5]. (b) Usig x 0 = 5 as a first approximatio to, apply the Newto-Raphso procedure oce to f( x ) to fid a secod approximatio to, givig your aswer to 3 sigificat figures. (4) 6 *P4485A0628*
4. The trasformatio U, represeted by the 2 2 matrix P, is a rotatio through 90 aticlockwise about the origi. (a) Write dow the matrix P. () The trasformatio V, represeted by the 2 2 matrix Q, is a reflectio i the lie y = x. (b) Write dow the matrix Q. () Give that U followed by V is trasformatio T, which is represeted by the matrix R, (c) express R i terms of P ad Q, (d) fid the matrix R, () (e) give a full geometrical descriptio of T as a sigle trasformatio. 8 *P4485A0828*
5. 2 2 f( x) = ( 4x + 9)( x 6x+ 34) (a) Fid the four roots of f( x ) = 0 Give your aswers i the form x= p+i q, where p ad q are real. (5) (b) Show these four roots o a sigle Argad diagram. 2 *P4485A0228*
6. X = a 3 2, where a is a costat. (a) Fid the value of a for which the matrix X is sigular. Y = 3 2 (b) Fid Y. The trasformatio represeted by Y maps the poit A oto the poit B. Give that B has coordiates (, 7 2), where is a costat, (c) fid, i terms of, the coordiates of poit A. (4) 4 *P4485A0428*
7. The rectagular hyperbola, H, has cartesia equatio xy = 25 The poit P 5p, 5 p, ad the poit Q 5q, 5 q, where p, q p q, are poits o the rectagular hyperbola H. (a) Show that the equatio of the taget at poit P is 2 py+ x=0 p (b) Write dow the equatio of the taget at poit Q. (4) () The tagets at P ad Q meet at the poit N. Give p+ q 0, 0 pq 0 (c) show that poit N has coordiates, p+ q p+ q. (4) The lie joiig N to the origi is perpedicular to the lie PQ. (d) Fid the value of pq 2 2. (5) 8 *P4485A0828*
8. (a) Prove by iductio that, for +, rr ( + 3) = 3 ( + )( + 5) r= (6) (b) A sequece of positive itegers is defied by u u = u + ( 3+ ), + =, Prove by iductio that u = 2 ( ) +, + (5) + 22 *P4485A02228*
9. y y 2 = 36x P O S N x Figure Figure shows a sketch of part of the parabola with equatio y The poit P (4, 2) lies o the parabola. 2 = 36x. (a) Fid a equatio for the ormal to the parabola at P. (5) This ormal meets the x-axis at the poit N ad S is the focus of the parabola, as show i Figure. (b) Fid the area of triagle PSN. (4) 26 *P4485A02628*