OCR Maths FP1. Topic Questions from Papers. Matrices. Answers

Transcription

1 OCR Maths FP Topic Questions from Papers Matrices Answers PhysicsAndMathsTutor.com

2 . (i) A = 8 PhysicsAndMathsTutor.com Obtain given answer correctly Attempt to find A, elements correct All elements correct A = 8 A =A I A - =I - A Use correct matrix A Obtain given answer correctly Multiply answer to (i) by A - or obtain A - or factorise A A Obtain given answer correctly. (i) i Correct real and imaginary parts (Q, June 00) (i) det(b) = 0 a - + = 0 Attempt to solve their det(b)=0 Obtain given answer Show correct process for adjoint entries Obtain at least correct entries in adjoint Divide by their determinant Obtain completely correct answer (iii) a = - Attempt product of B - C Obtain correct answer (Q7, June 00)

3 9. (i) 0 0 Each column correct Shear, e.g. (0,) transforms to (,) One example or sensible explanation (iii) M = 0 Attempt to find DC (not CD ) Obtain given answer (Q9, June 00). (i) + x x + x - ft Show correct expansion process, allow sign slips Obtain correct (unsimplified) expression Obtain correct answer State that M is non-singular as det M non-zero, their determinant ft. (Q, Jan 00)

4 . (i) 8 Either Or 0 ft Transpose leading diagonal and negate other diagonal Divide by determinant State or imply (AB) - = B - A - Use this result and obtain B - = C - A, or equivalent matrix algebra Matrix multn., two elements correct, for any pair All elements correct ft their (i) B = A - C Find A - Or B = 0 0 ft Premultiply by A - stated or implied Matrix multn. Two elements correct All elements correct Correct B - AB = a + c b + d a +cb +d a = 0, c =, b = 0., d = Find AB Solve one pair of simultaneous equations Each pair of answers Correct B - (Q, Jan 00)

5 8. (i) For correct vertex (, -) For all vertices correct For correct diagram Either 0 0, Reflection, in x-axis Correct matrix 0 0, Enlargement, centre O s.f. Correct matrix Or 0 0, Reflection, in the y-axis Correct matrix 0 0, Enlargement, centre O s.f. Correct matrix Or 0 0, Stretch, in x-direction s.f. Correct matrix 0 0, Stretch, in y-direction s.f. - Correct matrix 9 (Q8, Jan 00). 7 i) 7 0 Two elements correct All four elements correct 0 0 A B correctly found k = Find k (i) For other correct vertices (Q, June 00)

6 8 (i) For other correct vertices For completely correct diagram 0 Each column correct. Conjugate seen (Q, June 00) 7. 9 (i) Attempt at matrix multiplication A = 0 A = Correct A Correct A (Q7, June 00) 0 8. (i) Correct expansion process shown a a a Obtain correct unsimplified expression a a Obtain correct answer Solve their det M = 0 a = 0 or a = ft Obtain correct answers (iii) (a) (b) Solution, as inverse matrix exists or M nonsingular or detm 0 Solutions, eqn. is multiple of eqn 0 (Q8, June 00)

7 . (i) a = - State correct value a = 7 or a = 9 Sensible attempt at multiplication a = Obtain correct answer. Attempt to equate real and (Q, Jan 007) 9. (i) For at least two correct images For correct diagram, co-ords.clearly written down 90 0 clockwise, centre origin Or equivalent correct description 0 0 Correct matrix, not in trig form (iii) Stretch parallel to x-axis, s.f. Or equivalent correct description, but must be a stretch for nd Each correct column (Q9, Jan 007)

8 0. (i) Show correct expansion process for x Δ = det D = a - Correct evaluation of any x det Obtain correct answer Show correct process for adjoint D - = Δ a a a a 7 entries Obtain at least correct entries in adjoint Δ a 9 a ft all Divide by their determinant Obtain completely correct answer Attempt product of form D - C, or eliminate to get equations and solve Obtain correct answers, ft their inverse (Q0, Jan 007) (i) Transpose leading diagonal and negate other diagonal or solve sim. eqns. to get st column Divide by the determinant or solve nd pair to get nd column 0 (indep) ft Attempt to use B - A - or find B Attempt at matrix multiplication One element correct, a.e.f, All elements correct, a.e.f. NB ft consistent with their (i) ft (Q, June 007)

9 7 (i) a(a )+ det M = non-singular (iii) EITHER ft Show correct expansion process Show evaluation of a x determinant Obtain correct answer a.e.f. Substitute a = in their determinant OR Obtain correct answer and state a consistent conclusion det M = 0 so non-unique solutions Attempt to solve and obtain inconsistent equations Deduce that there are no solutions Substitute a = and attempt to solve Obtain correct inconsistent equations 8 Deduce no solutions 8 (i) Circle, centre (, 0), Sketch showing correct features (Q7, June 007) 9 (i) 0 0 Correct matrix Rotation (centre O), 0, clockwise (iii) Sensible alternatives OK, must be a single transformation (iv) Matrix multiplication or combination of transformations For at least two correct images For correct diagram (v) det C = area of square has been doubled State correct value State correct relation a.e.f. 9 0 (i) Attempt to equate real and (Q9, imaginary June 007) 7 (i) (, -) For other correct vertices seen, correct direction of shear seen For completely correct diagram, must include scales 0 Each column correct Consider sum as two separate parts (Q, Jan 008)

10 8 (i) B seen or implied or elements correct Obtain correct answer Obtain a x matrix Each row (or column) correct (iii) () 8 8 Obtain a single value Obtain correct answer, must have matrix (Q, Jan 008) 79 (i) a = - A - = a+ a ft Use det A = 0 Obtain correct answer Both diagonals correct Divide by det A x a =, y = a+ a+ ft ft Premultiply column by A -, no other method Obtain correct answers from their A (i) Obtain next terms (Q7, Jan 008) 0 (i) Two elements correct All four elements correct EITHER Both diagonals correct Divide by determinant OR Solve sim. eqns. st column correct nd column correct (i) Correct modulus (Q, June 008)

11 7 (i) Enlargement (centre O) scale factor Reflection Mirror line is y = x (iii) Stretch in y direction Scale factor, must be a stretch (iv) Rotation.9 o clockwise or equivalent (Q7, June 008) 0 (i) Find value of det AB Correct value seen Show correct process for adjoint entries Obtain at least correct entries in adjoint Divide by their determinant 0 ( AB ) = 0 a a Obtain completely correct answer (iii) EITHER State or imply (AB) - = B - A - B 0 0 = Obtain B - = (AB) - A Correct multiplication process seen Obtain three correct elements All elements correct OR Attempt to find elements of B All correct Correct process for B - elements correct All elements correct (Q0, June 008) 0 Both diagonals correct (i) 0 Divide by correct determinant a Two elements correct a Remaining elements correct Express as sum of terms (Q, Jan 009)

12 State or use correct result Combine matrix and its inverse 0 0 Obtain I or I but not Obtain zero matrix but not S.C. If 0/, for AA - = I Either Consider determinant of coefficients (Q, of Jan LHS 009) Either k k = ft Consider determinant of coefficients of LHS Sensible attempt at evaluating any det Obtain correct answer a.e.f. unsimplified Equate det to 0 Obtain k =, ft provided all M s awarded Or (i) Either D Reflection, in x-axis Eliminate either x or y Obtain correct equation Eliminate nd variable Obtain correct linear equation Deduce that k = (Q, Jan 009) (i) Either Or D D Reflection, in x-axis Stretch parallel to y-axis, s.f. (iii) (iv) 0 0 D 9 Reflection, in y = -x Each column correct Rotation, 90 o,clockwise about O S.C. If (iii) incorrect, for identifying their transformation, all details correct (Q, Jan 009). 7 a +b =, a + b = Obtain a pair of simultaneous equations a = -, b = Attempt to solve Obtain correct answers.. (i) 9i Correct real and imaginary (Q, parts June 009)

13 8. 8 (i) 0 (iii) Either 0 ft 0 Find coordinates (0, 0) (, ) (, ) (, ) found Accurate diagram sketched Each column correct Correct inverse for their stated Post multiply C by inverse of Correct answer found Or Aft Set up equations for elements from correct matrix multiplication All elements correct, - each error Shear, x axis invariant or parallel to x-axis eg image of (, ) is (, ) SR allow s.f. or shearing angle of correct angle to appropriate axis (Q8, June 009) 9. 9 (i) a a a a a a = 0 or ft ft Correct expansion process shown Obtain correct unsimplified expression Obtain correct answer Equate their det to 0 Obtain correct answers, ft solving a quadratic (iii) (a) (b) Equations consistent, but non unique solutions Correct equations seen & inconsistent, no solutions 0 0. i) Attempt to find next terms (Q9, June 009) 0 (i) a 0 Two elements correct Remaining elements correct a Correct determinant Equate det A to 0 and solve a = Obtain correct answer a. e. f. (Q, Jan 00)

14 (i) Rotation 90 o (about origin) Anticlockwise Either Show image of unit square after reflection in y = x 0 0 Deduce reflection in x-axis ft Each column correct ft ft for matrix of their transformation Or Post multiply by correct reflection matrix Obtain correct answer State reflection, in x-axis S.C. If pre-multiplication, M0 but Available for correct description of their matrix (Q, Jan 00) 9 (i) Show correct expansion process for or multiply adjoint by A Correct evaluation of any at any stage det A = = a Obtain correct answer A - = a a + a a 7 Attempt product of form A - C or a eliminate to get equations and solve Obtain correct answer ft all S.C. if det now omitted, allow max A ft Show correct process for adjoint entries Obtain at least correct entries in adjoint Divide by their determinant 7 Obtain completely correct answer (Q9, Jan 00)

15 (i) ( 7) Obtain a single value Obtain correct answer as a matrix BA = 0 Obtain a matrix All elements correct C seen or implied by correct answer ft Obtain correct answer, ft for a slip in BA (Q, June 00) (a) 0 Each column correct 0 SC B use correct matrix from MF Can be trig form (b) (i) Stretch, in x-direction sf Rotation, 0 o clockwise (Q, June 00) 9 (i) det A = a a Show correct expansion process for x Correct evaluation of any x Obtain correct answer (a) (b) (c) 9 Find a pair of inconsistent equations State inconsistent or no solutions Find a repeated equation State non unique solutions State that det A is non-zero or find correct solution State unique solution SC if deta incorrect, can score marks for correct deduction of a unique solution, but only once (Q9, June 00)

16 (i) ( 7 9 ) Each element correct SC (7,9) scores ( 8 ) * Obtain correct value dep Clearly given as a matrix (iii) Obtain matrix Obtain correct elements Obtain other correct elements 7 (Q, Jan 0) 7 (A - ) - = A seen or implied Use product inverse correctly A cao Obtain correct answer (Q, Jan 0) 8 (Q7, Jan 0)

17 9 9 (i) Show correct expansion process for x Correct evaluation of any x det M = a 7 a + correct answer (i) Each column correct 0 Solve detm = a = or * Obtain Enlargement correct or answer, stretch ft in their x and (i) y axes dep Scale factor (iii) Attempt to eliminate one variable (iii) (a) Obtain (,0),(,) correct indicated equations in unknowns Justify (8, ) seen infinite number of solutions SC Accurate / if diagram, unique solution including conclusion unit square consistent with their (i) or (b) detc = Correct value found 9 Scale factor for area 9 (Q9, Jan 0) 0 (i) a 0 B seen or implied elements correct Other elements correct, a.e.f., including brackets a a Sensible attempt at matrix multiplication for AB or BA Obtain correct answer (Q, June 0) k Obtain correct det Equate their det to 0 k Obtain correct answers (Q, June 0)

18 Show correct expansion process for or multiplication of C and adjc Correct evaluation of any det C = a Obtain correct answer Show correct process for adjoint entries a a a a Obtain at least correct entries in adjoint Obtain completely correct adjoint Divide their adjoint by their determinant 7 7 (Q, June 0) 8 (i) ( 0, ) seen (, 0 ) seen Square with A B and C positioned correctly 0 0 or 0 0 * Reflection in y = x or y = -x D Correct matrix, dep on stating reflection 0 0 or 0 0 * Enlargement scale factor or s.f. - D Correct matrix, dep on stating enlargement S.C. B for a pair of transformations consistent with their diagram. 7 (Q8, June 0) 0 State identity matrix is 0 Find a pair of simultaneous equations p q, pq, pq 0 Correct pair of distinct equations Attempt to solve p and q Obtain correct answers [] (Q, Jan 0)

19 (a) 0 0 [] Each column correct (b) (i) Stretch D Scale factor in the y direction Not in the y-axis [] (b) Correct value of determinant Scale factor for area Allow scale factor of stretch or eqiv. [] (Q, Jan 0) 9 (i) Show correct expansion process for Correct evaluation of any det X = 0 9a a Obtain correct answer aef [] 9 Their det X = 0 a = or 0 FT Obtain correct answers from their (i) FT [] 9 (iii) Show correct process for adjoint entries Obtain at least four correct entries in adjoint Obtain completely correct adjoint a 9 a ft Divide by their determinant a 9 8 a a a [] (Q9, Jan 0) 7 (i) Multiplication attempt, elements correct All elements correct [] EITHER B A ( AB) Stated or used ft Divide by correct determinant ft Both diagonals correct [] OR Either inverse correct Two elements correct in final answer, both inverses must be correct All elements correct (Q, June 0)

20 8 9 (i) * Shear dep eg image of (0, ) is (, ) or parallel to the x- axis [] 9 Either State Z = YX Obtain Y = ZX - 0 Or [] State or use correct inverse Matrix multiplication, elements correct Obtain completely correct simplified exact matrix Correct order for matrix multiplication 9 (iii) * Rotation dep 0 clockwise [] Obtain correct elements Obtain other correct elements Equate elements, correct Obtain completely correct simplified exact matrix (Q9, June 0) 9 0 (i) Show correct expansion process for Correct evaluation of any a -a Obtain correct answer [] 0 (a) det D = so unique sol n or solve to find correct solution (-/,, /) [] 0 (b) Their det D = 0, so non-unique solutions Attempt to solve equations with a = Explain inconsistency with correct working [] 0 (c) Their det D = 0, so non-unique solutions Attempt to solve equations with a = 0 Explain consistency with correct working [] SC once if unique solution following their incorrect det D non zero (Q0, June 0) 0 (i) a I or I seen or used elements correct Other elements correct [] a a or equivalent Divide by correct determinant Both diagonals correct [] (Q, Jan 0)

21 Show correct expansion process for correct x Correct evaluation of any x 7 Obtain correct term quadratic * Equate their det to 0 or D Attempt to solve a quadratic equation Obtain correct answers [] (i) Each column correct (Q, Jan 0) [] (i) Each column correct 0 [] Either Or Either Or 0 D Stretch, s.f. in y direction Shear, x-axis invariant e.g. (0,) (,) P: 0 0 Correct matrix (iii) 0 Q: 0 0 PQ: 0 0 D Shear, x axis invariant e.g. (0, ) (, ) Stretch, s.f. in y direction, Correct matrix [] N.B. in the x/y axis is incorrect Attempt at matrix multiplication of two x matrices from Obtain correct result cao [] (Q, Jan 0) (i) 7 Each element correct, missing brackets only [] Obtain matrix 0 Obtain correct elements Obtain other correct elements det CB = 0 FT Obtain their det CB, must be a matrix singular FT Correct conclusion from their det CB [] (Q, June 0) 7 (i) [] [] FT Each column correct Each column correct Attempt at matrix multiplication in correct order Obtain correct answer from their (i) and (iii) 0 0 [] (iv) Reflection, in y = x Correct description of their (iii) only [] (Q7, June 0)

22 0 (i) Show correct expansion process for Correct evaluation of any a + Obtain correct answer Use det A = 0 a = FT Obtain correct answer from their det A [] Show correct processes for adjoint entries 7 a a Obtain at least correct entries in adjoint a 8 a a Obtain completely correct adjoint Divide adjoint by their det A a Pre-multiply column matrix by their A - a 7 a A Obtain correct answer, for element correct [7] (Q0, June 0)