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

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1 MEI Mathematics in Education and Industry MEI STRUCTURED MATHEMATICS FURTHER CONCEPTS FOR ADVANCED MATHEMATICS, FP Practice Paper FP-A Additional materials: Answer booklet/paper Graph paper MEI Examination formulae and tables (MF) TIME hour 0 minutes INSTRUCTIONS Write your Name on each sheet of paper used or the front of the booklet used. Answer all the questions. You may use a graphical calculator in this paper. INFORMATION The number of marks is given in brackets [] at the end of each question or part-question. You are advised that you may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used. Final answers should be given to a degree of accuracy appropriate to the context. The total number of marks for this paper is 7. MEI July 004

2 Section A ( marks) Solve the equation x + x + 4 = 0. [] Express + j in the form x + yj. [] 4 j The matrix A is given by Find A = 5 (i) A, [] A -. [] 4 The matrices A, B and C are given by A =, and 0 = = 0 B C (i) Use A and B to prove that matrix multiplication is not commutative. [] Use A, B and C to give an example in which matrix multiplication is associative. [] 5 (i) Show that 8 5( x 7) x+ x 4 ( x+ )( x 4) [] Hence solve the inequality 8 > 5. [5] x+ x 4 The cubic equation x + px + qx+ r = 0 has roots α, β and γ. Find (i) α + β + γ. [4] The equation which has roots α, β and γ. [] n 7 Prove by induction that r = nn ( + )( n+ ). [8] r = MEI July 004 MEI Structured Mathematics Practice Paper FP-A Page

3 Section B ( marks) 8 (i) You are given that ( x+ )( x ) C Ax+ B+, ( x ) ( x ) x. Show that B = 0 and find A and C. [] Fig. 8 shows a sketch graph of ( x+ )( x ) y = ( x ). Fig. 8 Write down the coordinates of P and Q, the points where the graph cuts the x axis, and the equation of the line l. [] (iii) The line m has equation y = x. Prove that the curve does not cross the line m. [] 9 The matrices 0 P = 0 and 0 Q = 0 define transformations in the (x,y)-plane. (i) (A) Under P, the point M (, ) is transformed to M. Find the coordinates of M. [] (B) Find P. [] (C) Hence describe the transformation represented by P. [] (A) Under Q, the point M (, ) is transformed to M. Find the coordinates of M. [] (B) Find Q 4. [] (C) Hence describe the transformation represented by Q. [] (iii) Find PQ and describe the transformation that it represents. [] MEI July 004 MEI Structured Mathematics Practice Paper FP-A Page

4 0 You are given the complex numbers z = π π + j and z = 4cos + jsin. (i) Find the modulus and argument of z. [] Write z in the form a + bj where a and b are to be given exactly. [] (iii) Illustrate z and z on an Argand diagram. [] (iv) Find z z and indicate this on your Argand diagram. [] (v) Describe the locus of the points, z, on the Argand diagram for which z z =. Sketch the locus on your diagram. [4] MEI July 004 MEI Structured Mathematics Practice Paper FP-A Page 4

5 Qu Answer Mark Comment Section A ± 9 x + x+ 4= 0 x= ± j 7 x = (+ j) (+ j) (4+ j) + j+ 8j =. = (4 j) (4 j) (4+ j) + 0+ j 0 j = = (i) A = = = 9I A = 9 I A = A = (i) A =, 0 B = AB=, BA = 9 A=,, 0 B= C= AB=, ( ) 9 AB C= BC=, A(BC) = 8 5 (i) 8 8( x 4) ( x+ ) = x+ x 4 ( x+ )( x 4) 5x 5 5( x 7) = = ( x+ )( x 4) ( x+ )( x 4) For AB For BC For both multiples the same Accept 5x - 5 MEI July 004 MEI Structured Mathematics Practice paper FP-A Mark Scheme Page

6 5x 5 > 5 ( x+ )( x 4) < x 7 ( x+ )( x 4) x x < x x x+ < ( x )( x ) < 0 < x< BUT only provided ( x+ )( x 4) > 0 x > 4or x< so never. IF ( x+ )( x 4) < 0 < x< 4 then ( x )( x ) > 0 x> or x< < x< and < x< 4 7 (i) α+ β + γ = p, αβ + βγ + γα = q, αβγ = r ( ) α + β + γ = α + β + γ + ( αβ + βγ + γα) ( ) α + β + γ = α + β + γ ( αβ + βγ + γα) = p q ( α) + ( β) + ( γ) = ( α + β + γ) = ( )( ) ( )( ) ( )( ) ( α)(. β)(. γ) = ( αβγ ) = r α. β + β. γ + γ. α = ( αβ + βγ + γα) = q x px + qx r = Assume true for r = 0 k k kk ( + )(k+ ) i.e.s k = r = r= kk ( + )(k + ) Then S k+ = + ( k+ ) = ( k+ )( k(k + ) + ( k+ ) ) = = = ( k+ )( k + ( ) k+ ) p ( k )( k k k ) ( k )( k 7k ) which is the form for S k where k+ replaces k. So if true for k then true also for k + But it is true for k = ; S = =... So it is true for S and so for S etc Therefore true for all k. 5 4 B, 8 MEI July 004 MEI Structured Mathematics Practice paper FP-A Mark Scheme Page

7 Section B 8 (i) C ( Ax+ B)( x ) + C x x Ax+ B+ = ( x ) ( x ) ( x ) P (-, 0), Q(, 0) l: x = Ax+ B x + C x x ( )( ) Equating coeffs: A=, B A=, C = B= 0 (iii) Meet when x = x 0 ( x ) ( x ) = which has no solution. 9 (i) (A) (,) (,) ( B) P =. = = I (C) Reflection in the line y = x (A) (,) (-,) ( B) Q =. 0 0 = Q =. = = I (C) Rotation through 90 0 clockwise about the origin (iii) PQ =. 0 = 0 0 Rotation through 90 0 clockwise about the origin followed by a reflection in the line y = x. E MEI July 004 MEI Structured Mathematics Practice paper FP-A Mark Scheme Page 4

8 0 (i) z = 7, argz = arctan 0.857radians z = 4 + j = + j Accept 49. o (iii) z correctly illustrated. z correctly illustrated. (iv) z z ( ) (v) = + j + j = Argand Diagram The locus is a circle, centre z, and radius 4 Correct illustration Circle Centre Touching imaginary axis Through z MEI July 004 MEI Structured Mathematics Practice paper FP-A Mark Scheme Page 5

Further Concepts for Advanced Mathematics (FP1) FRIDAY 11 JANUARY 2008

ADVANCED SUBSIDIARY GCE 4755/0 MATHEMATICS (MEI) Further Concepts for Advanced Mathematics (FP) FRIDAY JANUARY 008 Additional materials: Answer Booklet (8 pages) Graph paper MEI Examination Formulae and