Preparing To Study A Level Maths And A Level Further
Preparing to StudyA Level MathsAndA Level Further Maths(Edexcel)Miss CarpenterSubject Leader for Maths
Introduction to those considering A Level Mathematics/ FurtherMathematics next yearDue to the coronavirus pandemic it has been a while since you have studiedsome Maths. So to bridge the gap the following worksheets have been puttogether to enable the transition from GCSE to A Level to go smoothly.These are topics that you would have studied for GCSE Maths and so youshould be able to complete them. There are many sites to help you eg MyMathsA Level, Maths Genie A Level etc.The order the worksheets should be completed is as follows:1.2.3.4.5.6.7.8.9.10.AlgebraEquations 1Algebraic FractionsSimultaneous EquationsSurds and IndicesQuadratic FunctionsInequalitiesFunctions 1Functions 2Algebra RevisionPlease ensure this work is completed.I look forward to seeing you all in SeptemberKind regardsM CarpenterSubject Leader Mathematics
THE CAMPION SCHOOLALGEBRALV11. Expand and simplify the followinga) (3 4x)(4x 3)b) (7x 5y)(7x - 5y)c) (2x - 3) 2d) (2x - 1)(x - 3)(3x 2)e) 5(x - 3) 2 (x 2)f) (x 2) 22. Factorise completely the followinga) x 2 7x 10b) x 2 12x 20e) x 2 3xf) 2x 2 6xi) 2x 2 108j) 2x 2 x 3c) x 2 x 12d) x 2 3x 18g) x 2 25h) x 2 169k) 6x 2 x 1l) 6x 2 x 122x 4a)215xy 10y 2d)5yg)6y 9xb)3x 2 y 2xy 2e)xyh)62x 4i)3. Simplifyc)x2 xxf)12n8n 2 4n8xz4x 2 z 12xz 29k18k 27kp 3nk4. Simplify the followinga)x 12x 2e)2p 312 8pi)2x 2x 5x 6x 3b)7x 21x x2f)x 1x 2 3x 4j)x 12n 3c)8n 124n 2 8ng)2n 4x 2 5x 6k)x 3d)k 3k 2 3kh)x 1x 2 2x 1x 2 x 12l)x 4m)n)o)p)n 5n 2 25x 2 4x 3x 2 2x 3x 2 7x 18x 2 5x 6x 2 x 20x 2 9x 20
THE CAMPION SCHOOLEQUATIONS 1YR121. Solve the following equations by factorising.a)b)c)d)x2 6x 8 0x2 2x - 3 0x2 - 5x - 14 0x2 - 9x 14 0e)f)g)h)2x2 3x 1 03x2 - 13x 4 07x2 13x - 2 08x2 10 x - 3 02. Solve the following equations.a) x2 - 49 0b) x2 - 9 0c) 16x2 - 25 0d) x2 9x 0e) 5x2 6x 0f) 3x2 - 4x 03. Solve the following.a)b)c)d)x2 - 3x 412 - 7x x2 0x(x 7) 6 0x(4x 5) -1e)f)g)h)x(4x 1) 3xx(3x - 2) 84 x2 2(x 2)1 - x2 x(1 x)4. Solve the following by "completing the square". Give answers in surd forma)b)c)d)x2 8x 1x2 - 2x - 2 0x2 4x 2x2 x - 1 0e)f)g)h)x2 3x 1 02x2 - x - 2 0x2 - x 32x2 4x 75. Find, in surd form, the roots of the following equations.a) 5x2 9x 2 0b) 2x2 - 7x 4 0c) 4x2 - 7x - 1 0d) 1 5x - 5x26. By forming an equation solve the following.a) The sides of a right - angled triangle are x cm, (x 7) cm and (x 8) cm. Findthe lengths of the sides.b) The length of a rectangle is 5cm more than its width. If the area of the rectangleis 36cm2 find its dimensions.7.where a andx 2 8x 29 (x a)2 bb are constantsa) Find the value of a and the value of b.b) Hence, or otherwise, show that the roots ofx 2 8x 29 0are c d 5 , where c and d are integers to be found.
MATHEMATICSALGEBRAIC FRACTIONSSimplify:1.x 1 x 2 343.4x 3 x 2 525.3x 2 2x 1 342.2x 1 x 3 454.x 2 x 3 766.5x 2 3x 2 657.2x 3 3x 4 638.5x 2 x 5 124Simplify9.12.2 5 x 4x10.12 5x 13x11.32 x 3 x 424 x 4 x 313.25 x 3 x 114.38 x 7 x 416.12 2x 2 x 417.2x 5 3x x(x 3)19.1x 2 x 1 2x 2 x 120.Simplify15.18.23 2x 1 x 16x 2 2x 8 1x 2102x 2 3x 2 Simplify21. (x 2) 241x2 4yy2 y 3 y 2 4y 3Solvex 1 x 526. 34 6a2 922. 2a2 6a 9125.27.223.x 2 3xx 3x 10x 2551 1 x 3 x 2y2 y y 1xx 2 5xx 2 3x2x 2
THE CAMPION SCHOOLSIMULTANEOUS EQUS1. Solve the following simultaneous equations.a)c)2x y 18x 2y 12p 3q 53p 5q 21b)d)7p 3q 55p q 22x 7y 573y 11x 62. Solve the following pairs of equations.a)c)e)g)i)k)m)x2 y 2 5y x 13x 2 y 2 32x y 1xy 2x y 3 04x y 14x 2 y 0y 4x 2y 2x 21 3xy 0x 6y 1xy y 2 22x y 3b)d)f)h)j)l)n)y 2 x2 8x y 2y 2 xy 32x y 1y x 4y 2 5x 2 20x2 4y 2 22y x 2 02x y 2x2 y 5x 2 xy 0x y 1xy x 32x 5y 8LVI
THE CAMPION SCHOOLSURDS & 6010.45108Simplify11.8 18 2 212.75 2 12 13.28 175 14.1000 15.512 128 16.24 3 6 19.2363322740 90216 294Rationalise the denominators17.1518.22.12 123.1712 324.20.26 225.3621.15 326.12 213 2 2Find the values of127. 7014 232. 9 37.164 329. 3 128. 27 3133. 2 111 238. 2 4 234. 3 230. 2 3 228 339. 27 35.31. 27 3 34 240. ( 0 04)36. 32 227 3141. 16 32Solve the following quadratic equations leaving your answers in surd form.42. x 2 6x - 2 043. 2x 2 4 5x
THE CAMPION SCHOOLQUADRATIC FUNCTIONS1. Given that3x 2 12x 2 a(x b)2 c , find the values of a, b and c2. Given that4x 2 8x 7 p(x q)2 r , find the values of p, q and r.3. By "completing the square " find the least value of the following. In each casestate the value of x when the function takes this minimum value.b) x 2 7x 1a) x 2 8x 3c) 2x 2 4x 74. By "completing the square " find the maximum value of the following. In eachcase state the value of x when the function takes this maximum value.b) x 2 6 x 4a) x 2 x 6c) 2x 2 6 x 35. For the following graphs finda) the point where each curve cuts the y-axisb) the points where the curve cuts the x-axisc) any maximum or minimum pointsHence sketch each of the graphs.a) y x 2 8x 15b) y x 2 4x 12c) y x 2 2xd) y 3x x 2e) y 9 x 2f) y x 2 5x 6
THE CAMPION SCHOOLINEQUALITIES1. Find the range of values of x satisfying the following inequalities.a) 3x 18d) 2x 36g) -3x 12b) 5x 20xe) 62h) 7x 28c) 3x 27f) x 52. Solve the following inequalities.a) 3x 2 10d) 25 4x 3b) 5x 3 -2e) 9x 1 8c) 6 2x 2f) 3 5x 23. Solve the following inequalitiesa) 3x 2 2x 5d) 5 x 2x 1b) 4x 3 6 2xe) 8 3x x 4c) 4x 3 x 2f) 7 2x 4 x4. Solve the following inequalities.a) x 2 4x 5 0d) 9x 4 2x 2 0b) 3x 2 10x 8 0e) (x 4) 2 25c) x 2 8x 9f) x 2 20 8x5. A rectangular tile has length x cm and breadth (6 - x) cm. Given that the area ofthe tile must be at least 5 cm2, form a quadratic inequality in x and hence findthe set of possible values of x.6. Find the set of values of x for which2x(x 3) (x 2)(x - 3)7. Find the set of values for which (2x 1)2 9(4 - x)8. Find the set of values of x for whicha) 3(2x 1) 5 2xb) 2x 2 7x 3 0c) both 3(2x 1) 5 2x and 2x 2 7x 3 09. Find the set of values of m for which the equationhas real roots.x 2 (m 3)x 4m 010. Determine the nature of the roots of the following equationsb) 9 x 2 6 x 1 0c) 3 x 2 7 x 1 0a) 2x 2 7 x 15 0
THE CAMPION SCHOOLof1. Given the functions f : x 3x - 2,FUNCTIONS 1g : x 4 - 5x,YR12h : x x2 find the valuea) f(1)b) g(-2)c) h( -3)d) fg(1)e) hf(2)f) gf( -2)g) gg( -3)h) fh( -1)i) fff(2)j) fgh( -2)2. For the functions in Qu. 1, solve the following equations.a) f(x) 1b) g(x) -6c) h(x) 25d) f(x) x 3e) h(x) f(x)f) f(x) - g(x) 3g) h(x) - g(x) 10.3. Given f : x x, x ℜ,x 2Find a) ff(3)4. Given f : x x 2b) fff( -1)x 3, x ℜ,x 2c) Solve f(x) 2x 2 , find x if f(x) x5. Given f : x x 3, g : x 2x - 1, find the following functionsa) fg(x)b) gf(x)c) gg(x)6. Given f : x x - 2, g : x x2 and h : x a) fg(x)b) gf(x)c) fgh(x)d) fff(x)1find the following functionsxSolve the following equations.d) fg(x) 2e) gf(x) 9f) fg(x) gf(x)g) fg(x) - gh(x) 5
THE CAMPION SCHOOLFUNCTIONS 2YR121. Given f : x 1 - x , g : x 3x 2 , h : x 2x finda) fg(3)b) gfh(2)c) ggg(1)d) fg(x)e) hg(x)f) fgh(x)2. Given f : x x - 3 , g : x x2 solve the following equations.a) fg(x) f(x)3. Given f(x) b) gf(x) g(x)c) fg(x) gf(x)1, g(x) x 2 solve the following equationsxa) fg(x) g(x)b) gf(x) g(x)4. Given f : x 2x , g : x x 1, express the following in terms of f and g.a) x 2x 1b) x 4xc) x 2(x 1)5. Find the inverse function f 1 (x) for each of the followinga) f(x) 2x 5b) f(x) x3c) f(x) 12x 36. Find the inverse function of each of the following.a) f(x) 3x 45c) f(x) x 3,2x 1x 21e) f(x) x 4,3x 2x 23b) f(x) 2x 1,x 3x 3d) f(x) 3x 2,x 4x 4f) f(x) x 3,x 1x 1g) What do you notice about your answer to part (f)7. Find the inverse function of the following.a) f : x x 2 x 6x 12b) f : x x 2 7x 12 x 3 21d) x x 3
THE CAMPION SCHOOLALGEBRA REVa) x2 9x 20c) 9 6x x2e) 4x2 - 9g) 12 7x - 12x21. Factorise2. Solve the following equations.YR 12 R1b) 2x2 - 3x 1d) x2 - 16f) 9x2 - 3xh) 4x2 - 4xy - 3y2a) x2 - 3x 4b) 2 - x 3x2c) x2 10xd) 1 - x2 x(1 x)3. Solve the following equations leaving answers in surd forma) 4x2 - 7x - 1 0b) 2x 5 - 4x24. Solve the simultaneous equations.a)5. In each of the following findi) where the curve cuts the y-axisiii) any maximum or minimum pointsa) y x 2 3 x 186. Find the value of7. Simplifya)2x y 1 28 3b)b)182b)xy 2x y 3 0ii) where the curve cuts the x-axisiv) Hence sketch the curveb) y 2x 2 7 x 4a)48y 2 xy 3c)( 56 ) 213 1c) y x 2 4 x 4c)d)172(1 9 )3 23 2d)116 2( 25 )e)2 3 52 5 18. Find the set of values of x for whicha) ( x 1)( x 4) 2( x 4)b) 2( x 1)( x 4) ( x 2) 29. Without using a calculator, solve the equation10. Solve the following simultaneous equations3 ( 2 x 1) 9 x2x 2 y 2 2y 3x 9 02x y 1
A Level Maths . And . A Level Further Maths (Edexcel) Miss Carpenter . Subject Leader for Maths. Introduction to those considering A Level Mathematics/ Further Mathematics next year Due to the coronavirus pandemic it has been a while since you have studied some Maths. So
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