PhysicsAdMathsTutor.com
physicsadmathstutor.com Jue 005 3. The fuctio f is defied by (a) Show that 5 + 1 3 f:, > 1. + + f( ) =, > 1. 1 (4) (b) Fid f 1 (). (3) The fuctio g is defied by g: + 5, R. 1 4 (c) Solve fg() =. (3) 6 *3494B060*
physicsadmathstutor.com Jauary 006 8. The fuctios f ad g are defied by f: + l, R, g: e, R. (a) Prove that the composite fuctio gf is gf: 4e 4, R. (4) (b) I the space provided o page 19, sketch the curve with equatio y = gf(), ad show the coordiates of the poit where the curve cuts the y-ais. (1) (c) Write dow the rage of gf. (1) d (d) Fid the value of for which [gf ( )] = 3, givig your aswer to 3 sigificat d figures. (4) 18 *N3495A0180*
physicsadmathstutor.com Jauary 006 Questio 8 cotiued *N3495A0190* 19 Tur over
physicsadmathstutor.com Jue 006 7. For the costat k, where k > 1, the fuctios f ad g are defied by f: l ( + k), > k, g: k, R. (a) O separate aes, sketch the graph of f ad the graph of g. O each sketch state, i terms of k, the coordiates of poits where the graph meets the coordiate aes. (5) (b) Write dow the rage of f. k (c) Fid fg i terms of k, givig your aswer i its simplest form. 4 (1) () The curve C has equatio y = f(). The taget to C at the poit with -coordiate 3 is parallel to the lie with equatio 9y = + 1. (d) Fid the value of k. (4) 18 *N3581A0184*
physicsadmathstutor.com Jue 006 Questio 7 cotiued *N3581A0194* 19 Tur over
physicsadmathstutor.com Jauary 007 6. The fuctio f is defied by f: l(4 ), < ad. (a) Show that the iverse fuctio of f is defied by f : e 1 1 ad write dow the domai of f 1. (b) Write dow the rage of f 1. (4) (1) (c) I the space provided o page 16, sketch the graph of y =f 1 (). State the coordiates of the poits of itersectio with the ad y aes. (4) The graph of y = + crosses the graph of y =f 1 () at = k. The iterative formula + 1 0 is used to fid a approimate value for k. 1 = e, = 0.3 (d) Calculate the values of 1 ad, givig your aswers to 4 decimal places. (e) Fid the value of k to 3 decimal places. () () 14 *N3583A0144*
physicsadmathstutor.com Jauary 007 Questio 6 cotiued 16 *N3583A0164*
physicsadmathstutor.com Jue 007 5. The fuctios f ad g are defied by 1 f : l( 1),, >, g:, 3, 3. (a) Fid the eact value of fg(4). (b) Fid the iverse fuctio f 1 (), statig its domai. () (4) (c) Sketch the graph of y = g( ). Idicate clearly the equatio of the vertical asymptote ad the coordiates of the poit at which the graph crosses the y-ais. (3) (d) Fid the eact values of for which 3 =3. (3) 10 *N6109A0104*
physicsadmathstutor.com Jue 007 Questio 5 cotiued *N6109A0114* 11 Tur over
physicsadmathstutor.com Jauary 008 8. The fuctios f ad g are defied by 3 f: 1, 3 g: 4, > 0, -1 (a) Fid the iverse fuctio f. () (b) Show that the composite fuctio gf is gf : 8 3 1. 3 1 (4) (c) Solve gf ( ) = 0. () (d) Use calculus to fid the coordiates of the statioary poit o the graph of y = gf(). (5) *H6315RB04*
physicsadmathstutor.com Jauary 008 Questio 8 cotiued Q8 END (Total 13 marks) TOTAL FOR PAPER: 75 MARKS 4 *H6315RB044*
physicsadmathstutor.com Jue 008 4. The fuctio f is defied by ( ) f: 1 1, 3. 3 > 3 (a) Show that 1 f( ) =, > 3. + 1 (4) (b) Fid the rage of f. (c) Fid f 1 (). State the domai of this iverse fuctio. () (3) The fuctio g is defied by g: 3,. (d) Solve fg( ) = 1. 8 (3) 1 *N30745A014*
physicsadmathstutor.com Jue 008 Questio 4 cotiued *N30745A0134* 13 Tur over
physicsadmathstutor.com Jauary 009 5. The fuctios f ad g are defied by (a) Write dow the rage of g. (b) Show that the composite fuctio fg is defied by fg : + 3e, R. (c) Write dow the rage of fg. d (d) Solve the equatio fg e ( ) ( + ). d = (1) () (1) (6) 1 *H3113A018*
physicsadmathstutor.com Jauary 009 Questio 5 cotiued *H3113A0138* 13 Tur over
5. physicsadmathstutor.com Jue 009 y O B A Figure Figure shows a sketch of part of the curve with equatio y = f(),. 1 The curve meets the coordiate aes at the poits A(0,1 k) ad B ( l k,0 ), where k is a costat ad k > 1, as show i Figure. O separate diagrams, sketch the curve with equatio (a) y = f( ), (3) (b) y = 1 f ( ). () Show o each sketch the coordiates, i terms of k, of each poit at which the curve meets or cuts the aes. Give that f( ) = e k, (c) state the rage of f, (1) (d) fid 1 f ( ), (3) (e) write dow the domai of 1 f. (1) 14 *H3464A0148*
physicsadmathstutor.com Jue 009 Questio 5 cotiued *H3464A0158* 15 Tur over
physicsadmathstutor.com Jue 009 7. The fuctio f is defied by 8 f( ) = 1 +, ( + 4) ( )( + 4) 4, 3 (a) Show that f( ) = (5) The fuctio g is defied by e 3 g( ) =, e, l e (b) Differetiate g( ) to show that g( ) = (e ) (3) (c) Fid the eact values of for which g( ) = 1 (4) *H3464A08*
physicsadmathstutor.com Jue 009 Questio 7 cotiued *H3464A038* 3 Tur over
physicsadmathstutor.com Jue 009 8. (a) Write dow si i terms of si ad cos. (b) Fid, for 0 < <, all the solutios of the equatio cosec 8cos = 0 (1) givig your aswers to decimal places. (5) 6 *H3464A068*
physicsadmathstutor.com Jauary 011 6. The fuctio f is defied by f: 3 5,, 5 (a) Fid 1 f ( ). (3) y 4 1 O 8 6 9 Figure The fuctio g has domai 1 8, ad is liear from ( 1, 9) to (, 0) ad from (, 0) to (8, 4). Figure shows a sketch of the graph of y = g(). (b) Write dow the rage of g. (c) Fid gg(). (d) Fid fg(8). (e) O separate diagrams, sketch the graph with equatio (1) () () (i) y = g( ), (ii) y = 1 g ( ). Show o each sketch the coordiates of each poit at which the graph meets or cuts the aes. (4) 1 (f) State the domai of the iverse fuctio g. (1) 18 *H35404RA0188*
physicsadmathstutor.com Jauary 011 Questio 6 cotiued *H35404RA0198* 19 Tur over
physicsadmathstutor.com Jue 011 4. The fuctio f is defied by f : 4 l ( + ),, 1 (a) Fid f 1 ( ). (b) Fid the domai of f 1. (3) (1) The fuctio g is defied by g: e, (c) Fid fg (), givig your aswer i its simplest form. (3) (d) Fid the rage of fg. (1) 8 *P38159A084*
physicsadmathstutor.com Jauary 01 7. The fuctio f is defied by ( ) f: 3 + 1 1 1,, + 7 4 R > + 4 1 (a) Show that f ( )= 1 (4) (b) Fid f 1 ( ) (3) (c) Fid the domai of f 1 (1) g( )= l + 1 ( ) (d) Fid the solutio of fg ( )= 1 7, givig your aswer i terms of e. (4) 16 *P40084A0164*
physicsadmathstutor.com Jauary 01 Questio 7 cotiued *P40084A0174* 17 Tur over
physicsadmathstutor.com Jue 01 6. The fuctios f ad g are defied by f : e +, g: l, 0 (a) State the rage of f. (b) Fid fg( ), givig your aswer i its simplest form. (c) Fid the eact value of for which f( + 3) = 6 (d) Fid f 1, the iverse fuctio of f, statig its domai. (1) () (4) (3) (e) O the same aes sketch the curves with equatio y = f( ) ad y = f 1 ( ), givig the coordiates of all the poits where the curves cross the aes. (4) 0 *P40686RA003*
physicsadmathstutor.com Jue 01 Questio 6 cotiued *P40686RA013* 1 Tur over
physicsadmathstutor.com Jauary 013 7. h( ) = + 4 + + 18, 0 5 ( + 5)( + ) (a) Show that h( ) = + 5 (4) (b) Hece, or otherwise, fid h( ) i its simplest form. (3) y y = h() O Figure Figure shows a graph of the curve with equatio y (c) Calculate the rage of h( ). = h( ). (5) *P41486A08*
physicsadmathstutor.com Jauary 013 Questio 7 cotiued *P41486A038* 3 Tur over
physicsadmathstutor.com Jue 013 (R)
physicsadmathstutor.com Jue 013 (R)
physicsadmathstutor.com Jue 013 7. The fuctio f has domai 6 ad is liear from (, 10) to (, 0) ad from (, 0) to (6, 4). A sketch of the graph of y = f() is show i Figure 1. y 10 O 6 Figure 1 (a) Write dow the rage of f. (b) Fid ff(0). (1) () The fuctio g is defied by g : 4 + 3, 5, (c) Fid g 1 () (3) (d) Solve the equatio gf() = 16 (5) 4 *P43016A043*
physicsadmathstutor.com Jue 013 Questio 7 cotiued *P43016A053* 5 Tur over
Core Mathematics C3 Cadidates sittig C3 may also require those formulae listed uder Core Mathematics C1 ad C. Logarithms ad epoetials e l a = a Trigoometric idetities si ( A ± B ) = si A cos B ± cos A si B cos( A ± B ) = cos A cos B m si A si B ta A ± ta B ta ( A ± B ) = ( A ± B ( k + ) 1m 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 1 π ) Differetiatio f() ta k sec cot cosec f( ) g( ) f () k sec k sec ta cosec cosec cot f ( )g( ) f( )g ( ) (g( )) 6 Edecel AS/A level Mathematics Formulae List: Core Mathematics C3 Issue 1 September 009
Edecel AS/A level Mathematics Formulae List: Core Mathematics C Issue 1 September 009 5 Core Mathematics C Cadidates sittig C may also require those formulae listed uder Core Mathematics C1. Cosie rule a = b + c bc cos A Biomial series 1 ) ( 1 r r b b a r b a b a a b a + + + + + + = + K K ( N) where )!!(! C r r r r = = < + + + + + + = + r r r 1, ( 1 1 ) ( 1 ) ( 1 1 ) ( 1 ) ( 1 K K K K R) Logarithms ad epoetials a b b a log log log = Geometric series u = ar 1 S = r r a 1 ) ( 1 S = r a 1 for r < 1 Numerical itegratio The trapezium rule: b a y d 1 h{(y 0 + y ) + (y 1 + y +... + y 1 )}, where a b h =
Core Mathematics C1 Mesuratio Surface area of sphere = 4π r Area of curved surface of coe = π r slat height Arithmetic series u = a + ( 1)d S = 1 (a + l) = 1 [a + ( 1)d] 4 Edecel AS/A level Mathematics Formulae List: Core Mathematics C1 Issue 1 September 009