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1 physicsadmathstuto.com

2 physicsadmathstuto.com Jue 005 5x 3 3. (a) Expess i patial factios. (x 3)( x ) (3) (b) Hece fid the exact value of logaithm. 6 5x 3 dx, givig you aswe as a sigle (x 3)( x ) (5) blak 4 *N03B044*

3 physicsadmathstuto.com Jauay blak 3x 6 A B C f( x), x 3. ( 3 x)( x) ( 3 x) ( x) ( x) (a) Fid the values of A ad C ad show that B = 0. (4) (b) Hece, o othewise, fid the seies expasio of f(x), i ascedig powes of x, up to ad icludig the tem i x 3. Simplify each tem. (7) 0 *N3553A000*

4 . physicsadmathstuto.com Jue 006 3x f( x) =, x <. ( x) blak Give that, fo x, 3x A B = ( x) ( x) ( x), whee A ad B ae costats, (a) fid the values of A ad B. (3) (b) Hece, o othewise, fid the seies expasio of f(x), i ascedig powes of x, up to ad icludig the tem i x 3, simplifyig each tem. (6) 4 *N3563A040*

5 physicsadmathstuto.com Jauay 007 x 4. (a) Expess i patial factios. ( x)(x3) (b) Give that x, fid the geeal solutio of the diffeetial equatio dy (x3)( x) (x) y. dx (3) (5) blak (c) Hece fid the paticula solutio of this diffeetial equatio that satisfies y = 0 at x =, givig you aswe i the fom y =f(x). (4) 8 *N356A080*

6 4. physicsadmathstuto.com Jue 007 ( 4 x ) B C A. ( x )( x ) ( x ) ( x ) blak (a) Fid the values of the costats A, B ad C. (4) (b) Hece show that the exact value of (4x ) dx is l k, givig the (x )(x ) value of the costat k. (6) 8 *N60A084*

7 physicsadmathstuto.com Jue (a) Expess i patial factios. 4 y (b) Hece obtai the solutio of dy cot x ( 4 y ) d x = (3) blak fo which y = 0 at x = π 3, givig you aswe i the fom sec x = g(y). (8) 0 *H3047A008*

8 physicsadmathstuto.com Jauay f( x 7x 3x 6 ) = ( 3x ) ( x), x < 3 blak Give that f (x) ca be expessed i the fom f( x A B C ) ( x ) ( x ) ( x), 3 3 (a) fid the values of B ad C ad show that A = 0. (4) (b) Hece, o othewise, fid the seies expasio of f (x), i ascedig powes of x, up to ad icludig the tem i x. Simplify each tem. (6) (c) Fid the pecetage eo made i usig the seies expasio i pat (b) to estimate the value of f (0.). Give you aswe to sigificat figues. (4) 8 *N303A088*

9 physicsadmathstuto.com Jue f(x) = 4 x A B C = ( x )( x )( x 3) x x x 3 blak (a) Fid the values of the costats A, B ad C. (b) (i) Hece fid f( x) dx. (4) (3) (ii) Fid f( x) dx i the fom k, whee k is a costat. 0 (3) 6 *H3465A068*

10 5. physicsadmathstuto.com Jue x x B C A ( x )( x ) x x blak (a) Fid the values of the costats A, B ad C. (b) Hece, o othewise, expad tem i x 5x 0 ( x )( x ) x. Give each coefficiet as a simplified factio. i ascedig powes of x, as fa as the (4) (7) 6 *H35386A063*

11 physicsadmathstuto.com Jauay 0 3. (a) Expess ( x )( 3x ) 5 i patial factios. (3) blak (b) Hece fid 5 dx, whee ( x )( 3x ) x. (3) (c) Fid the paticula solutio of the diffeetial equatio fo which y = 8 at y x 3x = 5 y, x, dx ( )( ) d x =. Give you aswe i the fom y f ( x) =. (6) 6 *H35405A064*

12 . physicsadmathstuto.com Jue 0 9x A B C = ( x ) ( x ) ( x ) ( x ) ( x ) blak Fid the values of the costats A, B ad C. (4) *P3860A04*

13 physicsadmathstuto.com Jauay 0 8. (a) Expess P( 5 P) i patial factios. (3) blak A team of cosevatioists is studyig the populatio of meekats o a atue eseve. The populatio is modelled by the diffeetial equatio dp dt = P( 5 P), t 0 5 whee P, i thousads, is the populatio of meekats ad t is the time measued i yeas sice the study bega. Give that whe t = 0, P =, (b) solve the diffeetial equatio, givig you aswe i the fom, P = a b c t e 3 whee a, b ad c ae iteges. (8) (c) Hece show that the populatio caot exceed 5000 () 4 *P40085A048*

14 physicsadmathstuto.com Jue 0. f( x) = A B C = x( 3x ) x ( 3x ) ( 3x ) blak (a) Fid the values of the costats A, B ad C. (4) (b) (i) Hece fid f( x) dx. (ii) Fid f( x ) dx, leavig you aswe i the fom a l b, whee a ad b ae costats. (6) *P4484A03*

15 3. Expess physicsadmathstuto.com Jauay 03 9x 0x 0 ( x )( 3x ) i patial factios. (4) blak 6 *P4860A068*

16 physicsadmathstuto.com Jue 03 (R)

17 Coe Mathematics C4 Cadidates sittig C4 may also equie those fomulae listed ude Coe Mathematics C, C ad C3. Itegatio ( costat) f(x) f( x) dx sec kx ta x cot x ta kx k l sec x l si x cosec x l cosec x cot x, l ta( x) sec x l sec x ta x, l ta( x 4 π ) dv du u dx = uv v dx dx dx Edexcel AS/A level Mathematics Fomulae List: Coe Mathematics C4 Issue Septembe 009 7

18 Coe Mathematics C3 Cadidates sittig C3 may also equie those fomulae listed ude Coe Mathematics C ad C. Logaithms ad expoetials e x l a = a x Tigoometic idetities si ( A ± B) = si Acos B ± cos Asi B cos( A ± B) = cos Acos B si Asi B ta A ± ta B ta ( A ± B) = ( A ± B ( k ) ta A ta B A B A B si A si B = si cos A B A B si A si B = cos si A B A B cos A cos B = cos cos A B A B cos A cos B = si si π ) Diffeetiatio f(x) ta kx sec x cot x cosec x f( x) g( x) f (x) k sec kx sec x ta x cosec x cosec x cot x f ( x )g( x) f( x)g ( x) (g( x)) 6 Edexcel AS/A level Mathematics Fomulae List: Coe Mathematics C3 Issue Septembe 009

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

20 Coe Mathematics C Mesuatio Suface aea of sphee = 4π Aea of cuved suface of coe = π slat height Aithmetic seies u = a ( )d S = (a l) = [a ( )d] 4 Edexcel AS/A level Mathematics Fomulae List: Coe Mathematics C Issue Septembe 009