PhysicsAndMathsTutor.com

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2 physicsadmathstutor.com Jue f(x) = 3e x 1 l x, x > 0. (a) Differetiate to fid f (x). (3) The curve with equatio y = f(x) has a turig poit at P. The x-coordiate of P is α. 1 (b) Show that α = e α. 6 () The iterative formula 1 x x = + 1 x = 0 is used to fid a approximate value for α. 6 e, 1, (c) Calculate the values of x 1, x, x 3 ad x 4, givig your aswers to 4 decimal places. () (d) By cosiderig the chage of sig of f (x) i a suitable iterval, prove that α = correct to 4 decimal places. () 8 *3494B080*

3 physicsadmathstutor.com Jauary The poit P lies o the curve with equatio 1 y = l x. The x-coordiate of P is 3. 3 Fid a equatio of the ormal to the curve at the poit P i the form y = ax + b, where a ad b are costats. (5) 6 *N3495A060*

4 physicsadmathstutor.com Jauary (a) Differetiate with respect to x (i) x e 3x +, (4) 3 (ii) cos( x ). 3x (4) dy (b) Give that x = 4 si(y + 6), fid i terms of x. dx (5) 8 *N3495A080*

5 physicsadmathstutor.com Jue 006. Differetiate, with respect to x, (a) e 3x + l x, (b) (5 + x ). 3 (3) (3) Q (Total 6 marks) 4 *N3581A044*

6 physicsadmathstutor.com Jauary The curve C has equatio x = si y. π (a) Show that the poit P, lies o C. 4 dy 1 (b) Show that at P. dx = (1) (4) (c) Fid a equatio of the ormal to C at P. Give your aswer i the form y = mx + c, where m ad c are exact costats. (4) 6 *N3583A064*

7 4. (i) The curve C has equatio physicsadmathstutor.com Jauary 007 x y = 9 + x. Use calculus to fid the coordiates of the turig poits of C. (6) (ii) Give that 3 x y = (1+ e ), dy 1 fid the value of at x = l 3. dx (5) 8 *N3583A084*

8 physicsadmathstutor.com Jauary 007 Questio 4 cotiued *N3583A094* 9 Tur over

9 physicsadmathstutor.com Jue A curve C has equatio y = x e x. (a) Fid d y dx, usig the product rule for differetiatio. (3) (b) Hece fid the coordiates of the turig poits of C. (c) Fid d y dx. (d) Determie the ature of each turig poit of the curve C. (3) () () 6 *N6109A064*

10 physicsadmathstutor.com Jauary 008. A curve C has equatio x y = e ta x, x ( + 1) π. (a) Show that the turig poits o C occur where ta x = 1. (b) Fid a equatio of the taget to C at the poit where x = 0. (6) () 4 *H6315RB044*

11 physicsadmathstutor.com Jauary A curve C has equatio y = 3si x+ 4cos x, -π x π. The poit A(0, 4) lies o C. (a) Fid a equatio of the ormal to the curve C at A. (b) Express y i the form Rsi( x+ ), where R > 0 ad 0 Give the value of to 3 sigificat figures. π. (c) Fid the coordiates of the poits of itersectio of the curve C with the x-axis. Give your aswers to decimal places. (5) (4) (4) 18 *H6315RB0184*

12 physicsadmathstutor.com Jauary 008 Questio 7 cotiued *H6315RB0194* 19 Tur over

13 physicsadmathstutor.com Jue The poit P lies o the curve with equatio y e x + = 4 1. The y-coordiate of P is 8. (a) Fid, i terms of l, the x-coordiate of P. () (b) Fid the equatio of the taget to the curve at the poit P i the form y = ax + b, where a ad b are exact costats to be foud. (4) *N30745A04*

14 physicsadmathstutor.com Jue (a) Differetiate with respect to x, (i) e 3 x (si x+ cos x), (3) (ii) Give that 3x + 6x 7 y =, x 1, ( x + 1) (3) (b) show that (c) Hece fid dy 0 = dx ( x+ 1). 3 d d y x ad the real values of x for which d y 15 =. dx 4 (5) (3) 18 *N30745A0184*

15 physicsadmathstutor.com Jue 008 Questio 6 cotiued *N30745A0194* 19 Tur over

16 physicsadmathstutor.com Jauary 009 dy 1. (a) Fid the value of at the poit where x = o the curve with equatio dx y = x (5x 1). si x (b) Differetiate with respect to x. x (6) (4) *H3113A08*

17 physicsadmathstutor.com Jauary Fid the equatio of the taget to the curve π x= cos( y+ π ) at 0,. 4 Give your aswer i the form y = ax + b, where a ad b are costats to be foud. (6) 10 *H3113A0108*

18 physicsadmathstutor.com Jauary Sketch the graph of y = l x, statig the coordiates of ay poits of itersectio with the axes. (3) 1 *N35381A018*

19 physicsadmathstutor.com Jue 010. A curve C has equatio 3 y = ( 5 3 x), 5 x 3 The poit P o C has x-coordiate. Fid a equatio of the ormal to C at P i the form ax + by + c = 0, where a, b ad c are itegers. (7) 4 *H35385A048*

20 physicsadmathstutor.com Jue y C O x Figure 1 Figure 1 shows a sketch of the curve C with the equatio y = x x+ x ( 5 )e. (a) Fid the coordiates of the poit where C crosses the y-axis. (1) (b) Show that C crosses the x-axis at x = ad fid the x-coordiate of the other poit where C crosses the x-axis. (3) (c) Fid d y. dx (d) Hece fid the exact coordiates of the turig poits of C. (3) (5) 14 *H35385A0148*

21 physicsadmathstutor.com Jue 010 Questio 5 cotiued *H35385A0158* 15 Tur over

22 physicsadmathstutor.com Jauary The curve C has equatio y 3+ six = + cos x (a) Show that dy 6six+ 4cosx+ = dx cos ( + x) (4) (b) Fid a equatio of the taget to C at the poit o C where x = π. Write your aswer i the form y = ax + b, where a ad b are exact costats. (4) *H35404RA08*

23 8. (a) Give that physicsadmathstutor.com Jauary 011 d ( cos x) = si x dx show that ( ) d sec x = sec x ta x. dx (3) Give that x = sec y (b) fid d x dy i terms of y. () (c) Hece fid d y dx i terms of x. (4) 6 *H35404RA068*

24 physicsadmathstutor.com Jauary 011 Questio 8 cotiued Q8 (Total 9 marks) END TOTAL FOR PAPER: 75 MARKS 8 *H35404RA088*

25 physicsadmathstutor.com Jue Differetiate with respect to x (a) l ( x + 3x+ 5) (b) cos x x () (3) *P38159A04*

26 physicsadmathstutor.com Jue f ( x) = 4x 5 x, x ± 3, ( x+ 1)( x 3) x 9 x 1 (a) Show that f( x) = 5 ( x+ 1)( x+ 3) (5) 5 The curve C has equatio y = f (x). The poit P 1, lies o C. (b) Fid a equatio of the ormal to C at P. (8) 16 *P38159A0164*

27 physicsadmathstutor.com Jue 011 Questio 7 cotiued *P38159A0174* 17 Tur over

28 physicsadmathstutor.com Jue (a) Express cos 3x 3si 3x i the form R cos (3x + ), where R ad are costats, R 0 ad 0 < α < π. Give your aswers to 3 sigificat figures. (4) x f( x) = e cos3x (b) Show that f (x) ca be writte i the form f() x = Re x cos ( 3x + α ) where R ad are the costats foud i part (a). (5) (c) Hece, or otherwise, fid the smallest positive value of x for which the curve with equatio y = f (x) has a turig poit. (3) 0 *P38159A004*

29 physicsadmathstutor.com Jue 011 Questio 8 cotiued Q8 (Total 1 marks) TOTAL FOR PAPER: 75 MARKS END *P38159A034* 3

30 physicsadmathstutor.com Jauary Differetiate with respect to x, givig your aswer i its simplest form, (a) x l ( 3x) (4) (b) si 4x 3 x (5) *P40084A04*

31 physicsadmathstutor.com Jauary 01 π π 4. The poit P is the poit o the curve x= ta y+ with y-coordiate. 1 4 Fid a equatio of the ormal to the curve at P. (7) 8 *P40084A084*

32 3. physicsadmathstutor.com Jue 01 y P C O x Figure 1 Figure 1 shows a sketch of the curve C which has equatio x y = e 3 si 3x, π x π 3 3 (a) Fid the x coordiate of the turig poit P o C, for which x 0 Give your aswer as a multiple of. (6) (b) Fid a equatio of the ormal to C at the poit where x = 0 (3) 8 *P40686RA083*

33 physicsadmathstutor.com Jue (a) Differetiate with respect to x, 1 (i) x l( 3x) (ii) 1 10x ( x 1), givig your aswer i its simplest form. 5 (6) (b) Give that x = 3ta y fid d y dx i terms of x. (5) 4 *P40686RA043*

34 physicsadmathstutor.com Jauary The curve C has equatio y = ( x 3) 5 The poit P lies o C ad has coordiates (w, 3). Fid (a) the value of w, () (b) the equatio of the taget to C at the poit P i the form y = mx+ c, where m ad c are costats. (5) *P41486A08*

35 physicsadmathstutor.com Jauary (i) Differetiate with respect to x (a) y = x 3 l x (b) y = ( x+ si x) 3 (6) Give that x = cot y, (ii) show that d y dx = 1 1+ x (5) 14 *P41486A0148*

36 physicsadmathstutor.com Jue Give that x = sec 3y, 0 < y < 6 π (a) fid d x dy i terms of y. () (b) Hece show that dy dx = 1 6xx 1 1 ( ) (4) (c) Fid a expressio for d y dx i terms of x. Give your aswer i its simplest form. (4) 16 *P43016A0163*

37 Core Mathematics C3 Cadidates sittig C3 may also require those formulae listed uder Core Mathematics C1 ad C. Logarithms ad expoetials e x l a = a x 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(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 Formulae List: Core Mathematics C3 Issue 1 September 009

38 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C Issue 1 September 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 = = < = + x x r r x x x r 1, ( 1 1 ) ( 1 ) ( 1 1 ) ( 1 ) ( 1 K K K K R) Logarithms ad expoetials a x x 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 x y d 1 h{(y 0 + y ) + (y 1 + y y 1 )}, where a b h =

39 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 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C1 Issue 1 September 009

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com 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)