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1 physicsadmathstuto.com

2 physicsadmathstuto.com Jauay a 7. Give that X = 1 1, whee a is a costat, ad a 2, blak (a) fid X 1 i tems of a. Give that X + X 1 = I, whee I is the 2 2 idetity matix, (b) fid the value of a. 14 *N34694A01428*

3 physicsadmathstuto.com Jauay A = , B = , C = blak (a) Descibe fully the tasfomatios descibed by each of the matices A, B ad C. (4) It is give that the matix D = CA, ad that the matix E = DB. (b) Fid D. (c) Show that E = (1) The tiagle ORS has vetices at the poits with coodiates (0, 0), ( 15, 15) ad (4, 21). This tiagle is tasfomed oto the tiagle OR S by the tasfomatio descibed by E. (d) Fid the coodiates of the vetices of tiagle OR S. (4) (e) Fid the aea of tiagle OR S ad deduce the aea of tiagle ORS. 24 *N34694A02428*

4 physicsadmathstuto.com a 2 5. R =, whee a ad b ae costats ad a > 0. a b (a) Fid R 2 i tems of a ad b. Give that R 2 epesets a elagemet with cete (0, 0) ad scale facto 15, Jue 2009 blak (b) fid the value of a ad the value of b. (5) 14 *M35146A01424*

5 physicsadmathstuto.com a 2 7. A =, whee a is a costat. 1 4 (a) Fid the value of a fo which the matix A is sigula. 3 2 B = 1 4 (b) Fid B 1. The tasfomatio epeseted by B maps the poit P oto the poit Q. Give that Q has coodiates (k 6, 3k + 12), whee k is a costat, Jue 2009 blak (c) show that P lies o the lie with equatio y = x *M35146A02024*

6 physicsadmathstuto.com Jauay 2010 a 5 5. A =, whee a is eal. 2 a + 4 blak (a) Fid det A i tems of a. (b) Show that the matix A is o-sigula fo all values of a. Give that a = 0, (c) fid A *N35143A01024*

7 physicsadmathstuto.com Jauay M = blak (a) Descibe fully the geometical tasfomatio epeseted by the matix M. The tasfomatio epeseted by M maps the poit A with coodiates (p, q) oto the poit B with coodiates (3 2, 4 2). (b) Fid the value of p ad the value of q. (c) Fid, i its simplest sud fom, the legth OA, whee O is the oigi. (d) Fid M 2. (4) The poit B is mapped oto the poit C by the tasfomatio epeseted by M 2. (e) Fid the coodiates of C. 22 *N35143A02224*

8 physicsadmathstuto.com Jue a 3 2. M =, whee a is a eal costat. 6 a blak (a) Give that a 2, fid M 1. (b) Fid the values of a fo which M is sigula. 4 *N35387A0428*

9 physicsadmathstuto.com Jue Wite dow the 2 2 matix that epesets blak (a) a elagemet with cete (0, 0) ad scale facto 8, (b) a eflectio i the x-axis. (1) (1) Hece, o othewise, (c) fid the matix T that epesets a elagemet with cete (0, 0) ad scale facto 8, followed by a eflectio i the x-axis. 6 1 A = ad 4 2 (d) Fid AB. k 1 B =, whee k ad c ae costats. c 6 Give that AB epesets the same tasfomatio as T, (e) fid the value of k ad the value of c. 14 *N35387A01428*

10 physicsadmathstuto.com Jauay A = B 2 0 = 3 1, blak (a) Fid AB. Give that C = (b) descibe fully the geometical tasfomatio epeseted by C, (c) wite dow C 100. (1) Q2 (Total 6 maks) *N35406A0332* 3 Tu ove

11 physicsadmathstuto.com Jauay A = 1 3 blak (a) Fid det A. (b) Fid A 1. (1) The tiagle R is tasfomed to the tiagle S by the matix A. Give that the aea of tiagle S is 72 squae uits, (c) fid the aea of tiagle R. The tiagle S has vetices at the poits (0, 4), (8, 16) ad (12, 4). (d) Fid the coodiates of the vetices of R. (4) 22 *N35406A02232*

12 physicsadmathstuto.com Jue (a) Give that 1 2 A = 2 1 blak (i) fid A 2, (ii) descibe fully the geometical tasfomatio epeseted by A 2. (4) (b) Give that B = descibe fully the geometical tasfomatio epeseted by B. (c) Give that k C = k 9 whee k is a costat, fid the value of k fo which the matix C is sigula. 8 *P38168A0832*

13 physicsadmathstuto.com Jue A = 4 a whee a ad b ae costats. b 2, blak Give that the matix A maps the poit with coodiates (4, 6) oto the poit with coodiates (2, 8), (a) fid the value of a ad the value of b. (4) A quadilateal R has aea 30 squae uits. It is tasfomed ito aothe quadilateal S by the matix A. Usig you values of a ad b, (b) fid the aea of quadilateal S. (4) 14 *P38168A01432*

14 physicsadmathstuto.com Jauay 2012 blak 4. A ight agled tiagle T has vetices A (, 11), B ( 21, ) ad C ( 24, ). Whe T is tasfomed by the matix P = , the image is T. (a) Fid the coodiates of the vetices of T. (b) Descibe fully the tasfomatio epeseted by P. The matices Q = ad R = epeset two tasfomatios. Whe T is tasfomed by the matix QR, the image is T. (c) Fid QR. (d) Fid the detemiat of QR. (e) Usig you aswe to pat (d), fid the aea of T. 8 *P40086A0824*

15 physicsadmathstuto.com Jauay A = blak (a) Show that A is o-sigula. (b) Fid B such that BA 2 = A. (4) 20 *P40086A02024*

16 physicsadmathstuto.com Jue (a) Give that blak A = B ad = fid AB. (b) Give that ad C = 3 2 D = 8 6, 5 2k 4 k, E = C + D whee k is a costat fid the value of k fo which E has o ivese. (4) 4 *P40688A0432*

17 physicsadmathstuto.com Jue M = blak (a) Fid det M. (1) The tasfomatio epeseted by M maps the poit (2a 7, a 1), whee a is a costat, oto the poit (25, 14). (b) Fid the value of a. The poit has coodiates (6, 0). Give that O is the oigi, (c) fid the aea of tiagle ORS. Tiagle ORS is mapped oto tiagle by the tasfomatio epeseted by M. (d) Fid the aea of tiagle OR'S'. Give that A = (e) descibe fully the sigle geometical tasfomatio epeseted by A. The tasfomatio epeseted by A followed by the tasfomatio epeseted by B is equivalet to the tasfomatio epeseted by M. (f) Fid B. (4) 24 *P40688A02432*

18 physicsadmathstuto.com Jauay The tasfomatio U, epeseted by the 2 2 matix P, is a otatio though 90 aticlockwise about the oigi. blak (a) Wite dow the matix P. (1) The tasfomatio V, epeseted by the 2 2 matix Q, is a eflectio i the lie y = x. (b) Wite dow the matix Q. (1) Give that U followed by V is tasfomatio T, which is epeseted by the matix R, (c) expess R i tems of P ad Q, (d) fid the matix R, (1) (e) give a full geometical desciptio of T as a sigle tasfomatio. 8 *P41485A0828*

19 physicsadmathstuto.com Jauay X = 1 a 3 2, whee a is a costat. blak (a) Fid the value of a fo which the matix X is sigula. 1 1 Y = 3 2 (b) Fid Y 1. The tasfomatio epeseted by Y maps the poit A oto the poit B. Give that B has coodiates (1, 7 2), whee is a costat, (c) fid, i tems of, the coodiates of poit A. (4) 14 *P41485A01428*

20 physicsadmathstuto.com Jue blak M = x x 2 3x 6 4x 11 Give that the matix M is sigula, fid the possible values of x. (4) 2 *P43138A0232*

21 A 1 = 1 2 (A 7I) physicsadmathstuto.com Jue blak A = ad I is the 2 2 idetity matix. (a) Pove that (b) Hece show that A 2 = 7A + 2I The tasfomatio epeseted by A maps the poit P oto the poit Q. Give that Q has coodiates (2k + 8, 2k 5), whee k is a costat, (c) fid, i tems of k, the coodiates of P. (4) 24 *P43138A02432*

22 physicsadmathstuto.com Jue 2013 R

23 physicsadmathstuto.com Jue 2013 R

24 Futhe Pue Mathematics FP1 Cadidates sittig FP1 may also equie those fomulae listed ude Coe Mathematics C1 ad C2. Summatios = 1 = = = ( + 1)(2 + 1) 2 ( +1) 2 Numeical solutio of equatios The Newto-Raphso iteatio fo solvig f( x ) = 0 : x + 1 f( x ) = x f ( x ) Coics Paabola Rectagula Hypebola Stadad Fom y 2 = 4ax xy = c 2 Paametic Fom (at 2, 2at) ct, c t Foci (a, 0) Not equied Diectices x = a Not equied Matix tasfomatios Aticlockwise otatio though θ about O: cosθ siθ siθ cosθ Reflectio i the lie cos 2θ si 2θ y = (taθ ) x : si 2θ cos 2θ I FP1, θ will be a multiple of Edexcel AS/A level Mathematics Fomulae List: Futhe Pue Mathematics FP1 Issue 1 Septembe 2009

25 Coe Mathematics C1 Mesuatio Suface aea of sphee = 4π 2 Aea of cuved suface of coe = π slat height Aithmetic seies u = a + ( 1)d S = 2 1 (a + l) = 2 1 [2a + ( 1)d] 4 Edexcel AS/A level Mathematics Fomulae List: Coe Mathematics C1 Issue 1 Septembe 2009

26 Edexcel AS/A level Mathematics Fomulae List: Coe Mathematics C2 Issue 1 Septembe Coe Mathematics C2 Cadidates sittig C2 may also equie those fomulae listed ude Coe Mathematics C1. Cosie ule a 2 = b 2 + c 2 2bc cos A Biomial seies 2 1 ) ( b b a b a b a a b a = + K K ( N) whee )!!(! C = = < = + x x x x x 1, ( 2 1 1) ( 1) ( 2 1 1) ( 1 ) (1 2 K K K K R) Logaithms ad expoetials a x x b b a log log log = Geometic seies u = a 1 S = a 1 ) (1 S = a 1 fo < 1 Numeical itegatio The tapezium ule: b a x y d 21 h{(y 0 + y ) + 2(y 1 + y y 1 )}, whee a b h =