MEI STRUCTURED MATHEMATICS FURTHER CONCEPTS FOR ADVANCED MATHEMATICS, FP1. Practice Paper FP1-C

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1 MEI Mathematics i Educatio ad Idustry MEI STRUCTURED MATHEMATICS FURTHER CONCEPTS FOR ADVANCED MATHEMATICS, FP Practice Paper FP-C Additioal materials: Aswer booklet/paper Graph paper MEI Examiatio formulae ad tables (MF) TIME hour 0 miutes INSTRUCTIONS Write your Name o each sheet of paper used or the frot of the booklet used. Aswer all the questios. You may use a graphical calculator i this paper. INFORMATION The umber of marks is give i brackets [] at the ed of each questio or part-questio. You are advised that you may receive o marks uless you show sufficiet detail of the workig to idicate that a correct method is beig used. Fial aswers should be give to a degree of accuracy appropriate to the cotext. The total umber of marks for this paper is 7. MEI July 00

2 Sectio A (6 marks) Fid the sum of the first terms of the series ( ) + ( ) + ( 5) +.+(+)(+). [5] Solve the iequality x ( x + ) <. [5] ( x ) (i) If α + β + γ = -, αβ + βγ + γα = 9, ad αβγ = 8, write dow the cubic equatio with roots α, β ad γ. [] (ii) Use the factor theorem to idetify oe root of the equatio [] (iii) Show that the other two roots are imagiary. [] You are give that z = a + bj where a ad b are real. z* is the cojugate of z. Fid all possible values of z if zz* jz = 7 j. [6] 5 If the roots of the equatio x 9x + x 9 = 0 are α, β ad γ, show that the equatio whose roots are α, β ad γ is x x 8 = 0 [7] 6 Give that 5 8 M =, prove by iductio that, for ay positive iteger, + 8 M =. [8] MEI July 00 MEI Structured Mathematics Practice Paper FP-C Page

3 Sectio B (6 marks) (x+ )( x+ 6) 7 A curve has equatio y =. ( x ) (i) Write dow the co-ordiates of the poits where the curve crosses the co-ordiate axes. [] 65x + (ii) Show that the equatio ca be writte as y = + ( x ) [] (iii) Hece write dow the equatios of the asymptotes of the curve. [] (iv) Show that whe x >, y >. [] (v) Sketch the curve, showig clearly the behaviour of the curve for large positive ad egative values of x. [] 7 8 The matrix M = 6 defies a trasformatio i the (x,y)-plae. A triagle S, with area 5 square uits, is trasformed by M ito triagle T. (i) Fid the area of triagle T. [] (ii) Fid the matrix that trasforms T ito S. [] Triagle U is obtaied by rotatig triagle S through 5 aticlockwise about the origi. (iii) Fid the matrix that trasforms triagle S ito triagle U, leavig the etries i surd form [] (iv) Fid the matrix that trasforms triagle T ito triagle U. [] 9 (i) Write dow the sum of the roots of the cubic equatio z z + 8z + 8 = 0 [] You are give that α = + j is a root of the equatio. (ii) Write dow aother complex root, β, ad hece solve the equatio. [] (iii) Describe the locus of poits i the Argad diagram represetig the complex umbers z for which z α =. Sketch this locus o a Argad diagram. [] (iv) Fid α β i the form a + bj ad show that the poit z = α β lies o the locus i (iii). [] MEI July 00 MEI Structured Mathematics Practice Paper FP-C Page

4 Qu Aswer Mark Commet Sectio A ( r+ )( r+ ) = ( r + r+ ) = ( + )(+ ) + ( + ) + 6 = + + = ( ) ( 6 ) 5 x ( x+ ) < ( x ) If x>, ( x+ )( x ) < x x 5x 6< 0 ( x 6)( x+ ) < 0 < x< 6 Ifx<, ( x 6)( x+ ) > 0 x< (i) x + x + 9x+ 8= 0 5 (ii) ( x) x x x ( ) f = ; f = 0 x = is a root. (iii) x + x + 9x+ 8= 0 ( x+ )( x + 9) = 0 The other two roots are jad j. ( a+ bj)( a bj) j( a+ bj) = 7 j a + b a+ b= ( ) ( ) j 7 j a + b + b ja= 7 j a + b + b = 7ad a= b + b = b+ b = b= or 0 ( )( ) 0 z = + j or z = j α+ β + γ = 9, αβ + βγ + γα=, αβγ = 9 ( α ) + ( β ) + ( γ ) = α + β + γ 9= 0 ( α )( β ) + ( β )( γ ) + ( γ )( α ) ( αβ βγ γα) ( α β γ ) = = ( α )( β )( γ ) x x = ( ) ( ) = αβγ αβ + βγ + γα + 9 α + β + γ 7 = = (for a) 5 (both) 6 E 7 Or (x ) The deal with cubic Or graphically Or replace x by (x + ) is the equatio ad multiply everythig out MEI July 00 MEI Structured Mathematics Practice paper FP-C Mark Scheme Page

5 6 Sectio B 7 (i) (ii) Assume true for = k 8 k + k k i.e. M = k k k + + k 8k 5 8 The M = k k 5+ 0k 6k 8+ k k = 0k 8k 6k k ( k + 8( k + ) = ( k + ) ( k+ ) This is of the same form as M k with k replaced by (k + ) Therefore if it is true for = k the it is also true for = k But it is true for = ; M = 5 8 = Ad so is true for ay positive iteger,.,0, ( 6,0, ) ( 0, ) 65x+ ( x ) + 65x+ ( x ) ( x ) + ( x+ )( x+ 6) x + 65x+ 6 ( x ) ( x ) = = (iii) Asymptotes: x =, x = -, y = (iv) Whe x >, (65x + ), (x ) ad (x + ) are all >0 So y > (v) E (x s) (y) 8 Oe mark for each brach MEI July 00 MEI Structured Mathematics Practice paper FP-C Mark Scheme Page

6 8 (i) M = = 0 Area of S = 5 sq uits Area of T = 50 sq uit (ii) - 6 M = 0 7 For / 0 For matrix (iii) Aticlockwise rotatio of cos5 si5 0 0 = si5 cos5 For matrix with si ad cos For surd form (iv) 6 T S U = 0 9 (i) Sum = (ii) Other complex root, β = j + j+ j+ γ = γ = i.e. z =,+ j, j Correct product For or equivalet. 0 (iii) Circle, cetre + j, radius Diagram (iv) α + j + j = = = + j β j + Distace from Cetre =. + = MEI July 00 MEI Structured Mathematics Practice paper FP-C Mark Scheme Page

MEI STRUCTURED MATHEMATICS FURTHER CONCEPTS FOR ADVANCED MATHEMATICS, FP1. Practice Paper FP1-B

MEI Mathematics i Educatio ad Idustry MEI STRUCTURED MATHEMATICS FURTHER CONCEPTS FOR ADVANCED MATHEMATICS, FP Practice Paper FP-B Additioal materials: Aswer booklet/paper Graph paper MEI Examiatio formulae