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

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1 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 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) Solve the iequality x x> 0. [] You are give that Ax ( )( x ) + Bx( x ) + Cx( x ) x + x+. Fid the values of A, B ad C. [] You are give that the equatio x + px + qx + r = 0 has roots α, -α ad β. (i) Show that q = -α. [] Show that r = pq. Show that this is true for the equatio x + 7x + 9x + = 0 but that it has oly oe real root. [] (i) Express + j i the modulus-argumet form (r θ) where r ad θ are give to two decimal places. [] Sketch o a Argad diagram the locus z z = where z = + j. [] The matrices A ad B are give by = ad = 0 0 (i) Fid A - ad B -. [] Show that (AB) - = B - A -. []. 6 (i) Show that (r + ) r r (r ) = r. [] Hece fid r. [] r = 7 You are give the equatio x x + 7x = 0. (i) Show by substitutio that x = + j satisfies this equatio. [] Write dow a secod root of the equatio. [] (iii) Fid the third root of the equatio. [] MEI, July 00 MEI Structured Mathematics Practice Paper FP-B Page

3 Sectio B (6 marks) (x ) 8 A curve has equatio y =. ( x + ) (i) Write dow the equatio of the asymptote that is parallel to the y-axis. [] Fid the secod asymptote of the curve. Describe clearly the behaviour of the curve for large positive ad egative values of x. [] (iii) Fid the values of x for which y =. [] (iv) Sketch the curve, showig clearly where it cuts the x axis. [] 9 A reflectio i a lie l o the coordiate plae is represeted by the matrix A where A = (i) Fid the image of the poit (, 6). Hece write dow the equatio of the mirror lie, l. [] 0 The matrix T = represets a rotatio. By cosiderig the image of the 0 poit (, ), fid the cetre ad the agle of the rotatio. [] (iii) Fid TA. [] (iv) Show that uder the trasformatio TA the poit (, -) is ivariat. Hece state the equatio of the lie of ivariat poits uder the trasformatio TA. [] 0 The quadratic equatio z + 6z + = 0 has complex roots α ad β. (i) Fid the roots i the form p + qj. [] Fid the modulus ad argumet of each root. Illustrate both roots o a argad diagram. [] (iii) Fid the value of α + β. Hece fid the equatio with roots α ad β. [] MEI, July 00 MEI Structured Mathematics Practice Paper FP-B Page

4 Qu Aswer Mark Commet Sectio A x x> 0 x x > 0 ( ) x > x > x> 0 ad ad < 0 ad < < < 0 x x x i.e. x > ad < x< 0 x= 0 A= A= x= C = C = 7 x= B= 7 B= Alt: Equate coeffs A+ B+ C = A B C = A= A= B+ C =, B+ C = C =, B= for A for B for C (i) q = α αβ + αβ = α I the equatio p ( ) ad r = αβ r = pq. p= 7, q = 9, r = ad 7 9= q = = α α+ β = β α, α =-9 α is ot real. Sice a cubic always has at least oe root, β must be real. (i) + j = r( cosθ + jsiθ) where r = + =.6 ad cos θ= θ = 6. z z = is a circle, cetre z, radius cetre radius correct sketch MEI July 00 MEI Structured Mathematics Practice paper FP-B Mark Scheme Page

5 (i) A =, 0 0 B = AB =. 0 0 = 0 0 ( AB) = c.a.o. c.a.o. 0 E B. A = = ( ) AB 6 (i) (r + ) r r (r ) = r + r + r r + r r = r r = ( r+ ) r r ( r ) =.. 0. =.. r= r=. =..... ( ) ( ) ( ). = + Summig both sides: r = + 0 r = ( + ) 7 (i) (+j) (+j) + 7(+j) = + 6j 8j j j = j(6 8 + ) = 0 Other complex root is the complex cojugate, j (iii) k( j)( + j) = - k = - k = - So other root is MEI July 00 MEI Structured Mathematics Practice paper FP-B Mark Scheme Page

6 Qu Aswer Mark Commet Sectio B 8 (i) x = - (x ) Whe x, y = ( x+ ) 8x 8 0 for large x. x x So asymptote is x axis ( y = 0) 0 from above as x ad 0 from below as x (iii) (iv) Whe y =, = ( x+ ) = (x ) x + x+ = x x x+ = ( )( ) (x ) ( x+ ) x x = 0 x=, Large x Geeral shape Cuts x axis whe x = ½ 9 (i) x = y = 6 l is y = x 0 0 = rotatio about the origi, through+ 90 o (iii) = (iv) = 0.6. = Hece ivariat lie is x+ y = 0 Or aticlockwise MEI July 00 MEI Structured Mathematics Practice paper FP-B Mark Scheme Page

7 0 (i) 6± ± 6 z + 6z+ = 0 z = = z = ± j The roots are j ad + j z = j z =, argz = π + arcta =.radias z = + j z =, argz = arcta =.radias (both) Accept.07 Argad diagram (both) (iii) α+ β = 6, αβ = ( ) α β α β αβ + = + = 6 0= ( αβ ) Also αβ = = 6 x + x+ = 6 0 MEI July 00 MEI Structured Mathematics Practice paper FP-B Mark Scheme Page

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

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