abc General Certificate of Education Mathematics Further Pure SPECIMEN UNITS AND MARK SCHEMES ADVANCED SUBSIDIARY MATHEMATICS (56) ADVANCED SUBSIDIARY PURE MATHEMATICS (566) ADVANCED SUBSIDIARY FURTHER MATHEMATICS (57) ADVANCED MATHEMATICS (66) ADVANCED PURE MATHEMATICS (666) ADVANCED FURTHER MATHEMATICS (67)
General Certificate of Education Specimen Unit Advanced Subsidiary Eamination MATHEMATICS Unit Further Pure MFP abc In addition to this paper you will require: an 8-page answer book; the AQA booklet of formulae and statistical tables. You may use a graphics calculator. Time allowed: hour minutes Instructions Use blue or black ink or ball-point pen. Pencil should only be used for drawing. Write the information required on the front of your answer book. The Eamining Body for this paper is AQA. The Paper Reference is MFP. Answer all questions. All necessary working should be shown; otherwise marks for method may be lost. The final answer to questions requiring the use of tables or calculators should normally be given to three significant figures. Information The maimum mark for this paper is 75. Mark allocations are shown in brackets. Advice Unless stated otherwise, formulae may be quoted, without proof, from the booklet. []
Answer all questions. The equation 5 7 has a single real root α. Use the Newton Raphson method with first approimation to find the value of, giving your answer to significant figures. ( marks) The roots of the quadratic equation are α and β. (a) Without solving the equation, find the value of: (i) α β ; (ii) α β. (6 marks) β α Determine a quadratic equation with integer coefficients which has roots α β and β α ( marks) No credit will be given for simply using a calculator to find α and β in order to find the values in part (a). (a) Sketch the graph of y. State the coordinates of the points where the curve crosses the coordinate aes and write down the equations of its asymptotes. (6 marks) Hence, or otherwise, solve the inequality > ( marks) The comple number z satisfies the equation iz ( i) z * where * z is the comple conjugate of z. Find z in the form a ib, where a and b are real. (7 marks) []
5 Find the general solution in radians of the equation π tan 5 giving your eact answer in terms of π. (6 marks) 6 The matri A is and the matri B is. (a) Find the matri product AB. ( marks) The transformation T is given by M. y y Describe the geometrical transformation represented by T for each of the following cases: (i) M A; (ii) M B; ( marks) ( marks) (iii) M AB. ( mark) 7 (a) Find b a d, where b > a >. marks) Hence determine, where possible, the value of the following integrals, giving a reason if the value cannot be found. (i) d (ii) d ( marks) 8 (a) Epress n ( r)(r) in the form r constants a, b and c. n n a r b r cn, stating the values of the r r ( marks) Hence prove that n ( r)(r ) n ( n). ( marks) r Turn over []
9 A curve has equation 7 y. (a) (i) Prove that the curve crosses the line y k when ( k ) (k 7). ( mark) (ii) Hence show that if is real then either k or k 9. (5 marks) Use the results from part (a) to find the coordinates of the turning points of the curve. ( marks) A mathematical model is used by an astronomer to investigate features of the moons of a particular planet. The mean distance of a moon from the planet, measured in millions of kilometres, is denoted by, and the corresponding period of its orbit is P days. The model assumes that the graph of log P against log is the straight line drawn below. (a) Use the graph to estimate the period of the orbit of a moon for which.5. ( marks) The graph would suggest that P and are related by an equation of the form P k α where k and α are constants. (i) Epress log P in terms of log k, log and α. ( mark) (ii) Use the graph to determine the values of k and α, giving your answers to significant figures. ( marks) END OF QUESTIONS [5]
MFP Specimen abc Question Solution Marks Total Comments f ( ) 5 7 f ( ) 5 B f ( ) M f ( ) ( 5)/.77 (to SF) A Total (a)(i) α β ; αβ B Likely to be earned in (ii) α β ( α β ) αβ M 6 6 A (ii) α β ( α β ) 9 8 αβ Sum of roots α β α β B M A 6 Substitution into similar form as above α β ( α β ) αβ M Essentially this 7 A New equation y sum of new roots y product M Condone single sign error or missing ( ) 7 y y y 7y A (ft any variable fractional values) Must have Total [6]
MFP (cont) Question Solution Marks Total Comments, accept, y B (a) ( ) B, accept y, Asymptotes B asymptote is, y asymptote is B only y B, y B only y M One branch of hyperbola A 6 ft asymptotes Condone lack of symmetry to show second branch Appropriate method M Multiply both sides by ( ) Consideration of graph y > Considering ( ) > and ( ) < ()>( ) > only M Solution: < A > A Solution offered as < < unless ISW scores A, A Total 9 i( a i b) (i)( a i b) M ia b A aiaibb A Allow i b if cancelled Equating real parts a M a A Equating imaginary parts a a b M And attempt to find b b A i is comple number Total 7 [7]
MFP (cont) Question Solution Marks Total Comments 5 tan M Attempt at inverse tangent π A General solution of form nπ α M π nπ α A 5 m Making the subject nπ π π nπ π 6 A 6 5 Total 6 6 (a) AB M Clear attempt to multiply correctly A Correct (i) Rotation about origin M through 5 clockwise A oe (ii) Reflection in line y tan * m and attempt at M y tan A cosθ etc (iii) Reflection in -ais B Total 8 7(a) M Power.5 A correct Value of Integral a b A Or equivalent (i) as b b M Hence value of integral is A (ii) Integral does not eist/ cannot find value etc B Reason: as a a E Total 7 [8]
MFP (cont) Question Solution Marks Total Comments 8 (a) ( r)(r ) r 5r M n B Printed answer with a, b 5,c A r 5 r n Use of r and r formulae M n 5n ( n )(n ) ( n ) n 6 A ft m Factorising or multiplying out n ( n ) A ag Total 7 9 (a)(i) k( ) 7 leading to ( k ) (k 7) B ag (ii) Use of discriminant b ac M ( k ) 8(k 7) A Solving quadratic equation or factorising m Or use of formula ( k )( k 9) A k, k9 b ac. Hence k, or k 9 A 5 ag be convinced k : M Either k value and attempt to solve/factorise, A k 9: 5 5 A Coordinates of TPs (, ) and (5, 9 ) A Both Total (a) log.5.6 M From graph log P. m P days (to nearest day) A (i) log P log k α log B (ii) Intercept on vertical ais is.9 log k.9 M k 7.9 A Gradient of graph is given by α M α.5 A Total 8 TOTAL 75 [9]
General Certificate of Education Specimen Unit Advanced Level Eamination MATHEMATICS Unit Further Pure MFP abc In addition to this paper you will require: an 8-page answer book; the AQA booklet of formulae and statistical tables. You may use a graphics calculator. Time allowed: hour minutes Instructions Use blue or black ink or ball-point pen. Pencil should only be used for drawing. Write the information required on the front of your answer book. The Eamining Body for this paper is AQA. The Paper Reference is MFP. Answer all questions. All necessary working should be shown; otherwise marks for method may be lost. The final answer to questions requiring the use of tables or calculators should normally be given to three significant figures. Information The maimum mark for this paper is 75. Mark allocations are shown in brackets. Advice Unless stated otherwise, formulae may be quoted, without proof, from the booklet. []
Answer all questions. The cubic equation 5 k where k is real, has roots α, β and γ. (a) Write down the values of: (i) α β γ ; ( mark) (ii) αβ βγ γα. ( mark) (i) Show that α β γ 6. ( marks) (ii) Hence eplain why the cubic equation must have two non-real roots. ( marks) (c) Given that one root is (a) Given that i, find the value of k. (5 marks) sinh cosh ( > ) find the value of cosh. Hence obtain in the form ln p, where p is an integer to be determined. ( marks) ( marks) (a) Epress ( r )( r ) in partial fractions. ( marks) Hence find n ( r ) r giving your answer in the form B C A (5 marks) n n []
(a) Draw an Argand diagram to show the points A and B which represent the comple numbers i and 5 i respectively. ( mark) (i) The circle C has AB as a diameter. Find its radius and the coordinates of its centre. ( marks) (ii) Write down the equation of C in the form 5 (a) Use de Moivre s theorem to show that if z cosθ i sinθ, then z n z z k ( marks) cos nθ n ( marks) z (i) Write down the epansion of ( z ) in terms of z. ( marks) z (ii) Hence, or otherwise, show that (c) Solve the equation 8 sin θ cosθ cosθ (5 marks) 8 sin θ cosθ in the interval π < θ π, giving your answers in terms of π. ( marks) 6 (a) Epress i in the form r (cosθ i sinθ ), where r > and π < θ < π. ( marks) Obtain similar epressions for: (i) i (ii) i 7 Prove by induction that, for all positive integers n, ( marks) ( marks) n ( r ) n( n )( n ) r (7 marks) Turn over []
t t 8 (a) Use the definition t ( e e ) cosh to show that cosh t cosh t ( marks) A curve is given parametrically by the equations sinh t, y cosh t d (i) Show that dt dy dt cosh t (6 marks) (ii) Hence show that the length of arc of the curve from the point where t to the point where t is ( sinh ) ( marks) (c) Find the Cartesian equation of the curve. ( marks) END OF QUESTIONS []
MFP Specimen abc Question Solution Marks Total Comments (a)(i) α β γ B (ii) αβ βγ γα 5 B MA (i) α β γ ( α ) αβ 6 A (ii) Since sum of squares <, some of α, β, γ must be non-real As coefficients real, non real roots come in conjugate pairs E E (c) i is a root, so is i B p.i ( i)( i) and third root is αβγ 6 B B B Alternative solution substituting i into cubic ( i) 5 i ( i) 6 9i equation involving k k 6 k αβγ 6 B 5 /5 for slip (a) ( cosh ) cosh Total M ( cosh 5)( cosh ) A Or use of formula cosh E Some indication of rejection 5 cosh A 5 6 ln ln 9 M A ft provided p is an integer Total 6 []
[5] MFP (cont) Question Solution Marks Total Comments (a) r r r MA n r r MA n n n n A n n n S MA 5 Total 7 (a) Points plotted correctly B (i) The centre must be i i 5 i MA Accept ( ) ( ) i, but,, gets A The radius must be ( ) ( ) ( ) 5 MA ( ) ( ) i i M If diameter is taken as or radius taken as allow B (ii) equation is 5 i z M A Total 7
MFP (cont) Question Solution Marks Total Comments 5(a) n z cos z n ( nθ ) i sin ( nθ ) cos nθ i sin nθ cosnθ n z (i) ( ) z z 6 z z (ii) z z MA A Allow B only if cos nθ i sin nθ n z is quoted as MA M for attempt at epansion z isinθ MA z ( i sin ) cos θ 8cos θ 6 6 sin θ M Any form cos θ 8cos θ 6 θ A ft if i missing in ( ) i sinθ 8sin θ cos θ cos θ A 5 ag (no error) If M, allow B for cosθ and 8cosθ (c) cosθ M Allow B for any two correct answers π θ ±, ± 5π A 6(a) r π 5π θ ±, ± A 6 6 Total π θ MA 6 (i) r B ft wrong answer in (a) (ii) r π θ B ditto 6 B B ditto π θ B ditto 6 Total 7 [6]
MFP (cont) Question Solution Marks Total Comments 7 Assume result true for n k Then k ( ) k( k )( k ) ( k ) r M r ( k ( k ) k k A ( )( k 8 ) k k MA Any factor. If ( ) k taken out at start, all marks up to this point earned. ( )( k )( ) k k A Shown true for n B T T k and T k true E 7 Total 7 [7]
MFP (cont) Question Solution Marks Total Comments 8(a) t t ( ) cosh t e ( e t e t ) e M A t t ( e e ) cosht A ag Or cosh t t -t t t ( e e ) ( e e ) (i) d cosh t dt B d y cosh t sinh t dt B d dt dy dt cosh ( sinh t) t cosh t sinh t M A cosh t m Used relevantly cosh t A 6 (ii) s cosh t dt ( cosh t ) dt M m [ t sinh t] A sinh A (c) Use of cosh y sinh t t t sinh y M A A Total 6 TOTAL 75 [8]
General Certificate of Education Specimen Unit Advanced Level Eamination MATHEMATICS Unit Further Pure In addition to this paper you will require: an 8-page answer book; the AQA booklet of formulae and statistical tables. You may use a graphics calculator. Time allowed: hour minutes MFP abc Instructions Use blue or black ink or ball-point pen. Pencil should only be used for drawing. Write the information required on the front of your answer book. The Eamining Body for this paper is AQA. The Paper Reference is MFP. Answer all questions. All necessary working should be shown; otherwise marks for method may be lost. The final answer to questions requiring the use of tables or calculators should normally be given to three significant figures. Information The maimum mark for this paper is 75. Mark allocations are shown in brackets. Advice Unless stated otherwise, formulae may be quoted, without proof, from the booklet. [9]
Answer all questions. The diagram shows the curve C with polar equation r a sin θ, < θ π Find the area bounded by C. (6 marks) (a) Find the general solution of the differential equation d y y 6 cos (7 marks) d (i) Find the value of the constant λ for which λ sin is a particular integral of the differential equation d y y 6 cos d (ii) Hence find the general solution of this differential equation. (7 marks) []
The function y ( ) satisfies the differential equation d y f d (, y) where f (, y) y y and y (). 5 (a) Use the Euler formula with. places. r yr ( y ) y hf, h to obtain an approimation to (.) r r y giving your answer in four decimal ( marks) (i) Use the formula where k h f (, ) r y r y ( k k ) r yr and k h ( h, y ) with. f r k r h to obtain a further approimation to (.) y. (5 marks) (ii) Use the formula given in part (i), together with your value for (.) part (i), to obtain an approimation to y (.) places. y obtained in, giving your answer to three decimal (5 marks) Turn over []
(a) A point has Cartesian coordinates (, y) and polar coordinates ( r,θ ) origin., referred to the same Epress cos θ and sinθ in terms of, y, and r. ( mark) (i) Hence find the Cartesian equation of the curve with polar equation r cosθ sin θ ( marks) (ii) Deduce that the curve is a circle and find its radius and the Cartesian coordinates of its centre. ( marks) 5 (a) Using the substitution ln y, or otherwise, evaluate lim k where k >. Hence evaluate ln d ( marks) ( marks) 6 (a) The function f () is defined by f() e ( cos ) Use Maclaurin's theorem to show that when f() is epanded in ascending powers of : (i) the first two non-zero terms are (6 marks) (ii) the co-efficient of is zero. ( marks) Find e lim sin ( cos ) ( marks) []
5 7 (a) Show that the substitution dy u y d transforms the differential equation into the equation d d y dy y 5e d Show that e is an integrating factor of d u u 5e ( marks) d d u 5e d u Hence find the general solution of this differential equation, epressing u in terms of. (5 marks) (c) Hence, or otherwise, solve the differential equation y dy y 5e d d d dy completely, given that and y when. (9 marks) d END OF QUESTIONS []
MFP Specimen abc Question Solution Marks Totals Comments π A a sin θ dθ MA B π cosθ a dθ M M for d used r θ A if used correctly B for limits M if cos θ used π θ sinθ a A cao (a) π a A 6 Total 6 m ±i B C.F. is A cos B sin or A cos ( B) M A If m is real give M A ft is for m comple but incorrect but not Ae i Be i P.I. Try y p cos q sin M ( p cos q sin ) 6cos p cos q sin A p A GS y A cos B sin cos B 7 For adding their C.F. to their P.I. Must be constants (i) dy λ cos λ sin d MA d y λ cos λ sin λ cos d A If y λ sin µ cos used, then working at each stage must be correct for equivalent marks λ cos λ sin λ sin 6 cos MA cao λ A (ii) GS y A cos B sin B 7 Total []
MFP (cont) Question Solution Marks Totals Comments (a). 5 y. 5.. 57( 6 ) MA. 5 A (i) k. f (,. 5) 7( 6) M A. (.,.57). 886 k.f M A M for candidate s value from part (a). y. 5 (. 7 88)..568 A 5 (ii) k. f (.,. 568) 896 M A. (.,. 568 896) k. f. (a) (i).56 M A y. 5979.... 598 A 5 If answer not given to dp withhold this mark Total y cos θ, sinθ B r r y r M r r use of y r M y y A (ii) ( ) ( y ) 5 5(a) M A Centre (, ), radius 5 A Total 7 y k ln y k ln y ln y k y y MA k, y so lim y ln y A ln ln d MA A Total 6 [5]
MFP (cont) Question Solution Marks Totals Comments 6(a) cos f ( ) e f ( ) f B cos ( ) sin e f ( ) MA cos ( ) ( cos sin ) e f MA ( ) f A 6 cos ( ) ( ) f sin sin cos e MA ft - cos ( cos sin )( sin ) e ( ) f A (c) sin B [ Ignore higher power of ] 7(a) cos e lim MA ft sin Total du d y dy B d d d d y dy dy y 5e d M oe d d d u u 5e A ag d Integrating factor is d e e B ( ue ) 5e d d MA ue 5e A A u 5e Ae A 5 Provided A appears [6]
MFP (cont) Question Solution Marks Totals Comments 7 (c) d y y 5e Ae d M Integrating factor is e d e B d d 5 ( ye ) 5e Ae M A -5 ye e Ae B A y e Ae Be A 5 A, B B B Can be given at any stage y e e 5 e A 9 alt (c) Auillary equation is m m M m, A Complementary function is y Ae Be A Particular integral: try y ke M 6ke 8ke ke 5e A 5 k 5, k A y e Ae Be dy... d B 5 A, B y e 5 e e B,, 9 Total 7 TOTAL 75 [7]
General Certificate of Education Specimen Unit Advanced Level Eamination MATHEMATICS Unit Further Pure MFP abc In addition to this paper you will require: an 8-page answer book; the AQA booklet of formulae and statistical tables. You may use a graphics calculator. Time allowed: hour minutes Instructions Use blue or black ink or ball-point pen. Pencil should only be used for drawing. Write the information required on the front of your answer book. The Eamining Body for this paper is AQA. The Paper Reference is MFP. Answer all questions. All necessary working should be shown; otherwise marks for method may be lost. The final answer to questions requiring the use of tables or calculators should normally be given to three significant figures. Information The maimum mark for this paper is 75. Mark allocations are shown in brackets. Advice Unless stated otherwise, formulae may be quoted, without proof, from the booklet. [8]
Answer all questions. The matrices A and B are defined by A B Find the matrices: (a) AB; ( marks) T A T B. ( marks) The position vectors a, b and c of three points A, B and C are, and respectively. (a) Calculate ( b a) ( c a), ( marks) Hence find the eact value of the area of the triangle ABC. ( marks) The matrices A and B are defined by A B cos 5 sin 5 o o sin 5 cos 5 o o (a) Give a geometrical description of each of the transformations represented by the matrices A and B. (6 marks) For each of these transformations, find the line of invariant points. ( marks) [9]
[] (a) Factorise ( marks) It is given that and 9 B A Using your result in part (a), or otherwise, epress det(ab) in factorised form. ( marks) 5 The matri M is defined by M (a) Show that M has just two eigenvalues, and. (6 marks) Find an eigenvector corresponding to each eigenvalue. (5 marks) The matri M represents a linear transformation, T, of three dimensional space. (c) Write down a vector equation of the line of invariant points of T. ( mark) (d) Write down a vector equation of another line which is invariant under T. ( mark) TURN OVER FOR THE NEXT QUESTION Turn over
6 The planes Π and Π have equations y z and y z respectively. (a) Show that the plane Π is perpendicular to the plane Π. ( marks) Find the Cartesian equation of l, the line of intersection of the planes Π and Π. ( marks) (c) The line l meets the plane Π whose equation is at the point A. Find: y z 8 ( marks) (i) the coordinates of the point A ; (ii) the acute angle between the line l and the plane Π ; (iii) the direction cosines of l. ( marks) ( marks) ( marks) 7 Given that b c i and c a j epress in terms of i and j. ( a b) ( a b 5c) (6 marks) 8 A matri M is defined by (a) Find det M in terms of a. M 8 5 a ( marks) Find the value of a for which the matri M is singular. (c) Find M, giving your answer in tems of a. (d) Hence, or otherwise, solve y 8z y 5z y z END OF QUESTIONS ( mark) (6 marks) (5 marks) []
MFP Specimen abc Question Solution Marks Total Comments (a) MA (a) T T T B A ( AB) M A Total c a or b a seen B Either i j k M i j k or AA A correct (or all ve) ft A all correct ( b a) ( c a) M prev. result MA M A cao allowing ves in (a) (a) A Shear Parallel to y-ais,, ( ) ( ) Total 7 M A B e.g. (check suggested point) (, ) (, ) not sf B Rotation About -ais o of 5 M A A 6 Or in y z plane A y-ais B B -ais B Total 8 [] Or, Or y z, λ λ
MFP (cont) Question Solution Marks Total Comments (a) ( ) ( ) ( ) MA A ( )( ) A det A det B used M 5(a) det B 9 MA det AB ( )( )( )( ) λ A If any other method is used, it must be complete. λ λ Total 8 M ( λ )( 6 5λ λ ) ( λ) ( λ) M o.e. λ 7λ 5λ 9 A ( )( λ ) λ MA λ or A 6 ag, v v MM e.g. v, v MA A 5 MA for either (c) (d) r λ B r µ B Total []
MFP (cont) Question Solution Marks Total Comments 6(a). 6 MA 7 7 MA l is y z ( λ ) A oe (c)(i) Substitute λ, y λ, z λ into П 6 λ λ λ 8 λ Α is (,, ) M A A (ii) cosθ ± 6 6 M θ.9 required angle is 6. A A (iii) direction cosines are,, M 6 6 6 A,, Total 7 Sensible epansion M If a ab 5ac... used M unless some indication of understanding e.g. a. b j, a i, c k or similar. Cancelling out a a etc. M Cancelling out a b b a M a 5 c b 5c A 5i j AA 6 All A s depend on M Total 6 []
MFP (cont) Question Solution Marks Total Comments 8(a) ( ) ( a 5) 8( ) a MA M attempt; A correct unsimplified 5 5a A cao a B ft θ (c) a 6 a 5 a a 8 5 5 5 M A Finding determinants (co-factors) Any one correct row/column Use of B ft (a) wrong or correct having stated again det M a 5 a 5 5a 5 6 a a 8 5 (d) Realisation that a 5 5 5 5 5 M M A 6 B M A A Signs Transpose cao A any one correct (ft) A all cao Alt to (d) equations Answers M M A A equations equations Any one correct All cao Alt to (d) Cramer s Rule etc M, y, z AA A Alt to (d) Gaussian Elimination MA AA Total 5 TOTAL 75 [5]