OCR Maths FP1. Topic Questions from Papers. Complex Numbers. Answers
|
|
- Natalie Weaver
- 6 years ago
- Views:
Transcription
1 OCR Maths FP1 Topic Questions from Papers Complex Numbers Answers PhysicsAndMathsTutor.com
2 . 1 (i) i Correct real and imaginary parts z* = i 1i Correct conjugate seen or implied Correct real and imaginary parts (iii) i Attempt to use w* Obtain correct answer in any form 7 (Q, June 00). x y =1 and xy = 10 ±( i ) Attempt to equate real and imaginary parts of (x + iy) and 1 0i Obtain each result Eliminate to obtain a quadratic in x or y Solve to obtain x = (±) or y =(±) Obtain correct answers as complex numbers. Show correct process for subtracting fractions (Q, June 00). (i) Circle Centre (0, ) Radius Straight line Through origin with positive slope Sketch(s) showing correct features, each mark independent 0 or 0 +0i and + i ftf Obtain intersections as complex numbers t 7 8. (a) (i) + = = Values stated (Q, June 00) 1. (i) + 1i i 8i 10 +1i 1 (10 + 1i) or + i ft Attempt to multiply correctly Obtain correct answer Multiply numerator & denominator by conjugate Obtain denominator Their part (i) or i derived again / (Q1, Jan 00)
3 7. (a) (i) 1 1 Obtain correct answer, decimals OK (b) 1 i Using tan 1 b / a, or equivalent trig allow + or - Obtain 0.9 Obtain correct answer Express LHS in Cartesian form & equate real and imaginary parts Obtain x = 1 and y = - Correct answer written as a complex number (c) Sketch of vertical straight line Through ( - 0., 0) 8. (i) 10 (Q7, Jan 00). (i) + i 1 Conjugate seen (Q, June 00). 7 (i) -7i Real part correct Imaginary part correct + i - + 1i iz stated or implied or i = -1 seen Real part correct Imaginary part correct (iii) 1 ( 7i) or equivalent 8 Multiply by conjugate Real part correct Imaginary part correct N.B. Working must be shown.. (i) Circle, Centre O radius Sketch showing correct features (Q, June 00).. 8 (i) Circle, Centre O radius One straight line Through O with +ve slope In 1 st quadrant only 1+i 7 Sketch showing correct features Attempt to find intersections by trig, solving equations or from graph Correct answer stated as complex number (Q, June 00)
4 . 9 Attempt to equate real and imaginary parts of (x +iy) and 1 x y = 1 and xy = ± ( + i) D +8i Obtain each result Eliminate to obtain a quadratic in x or y Solve to obtain x = ( ± ), or y = ( ± ) 1 Obtain only correct two answers as complex numbers. Expand to obtain r r (Q, Jan 007). 10 (i) Circle Centre (1, -1) Passing through (0, 0) Sketch a concentric circle Inside (i) and touching axes Shade between the circles. (i) 1 Show given answer correctly (Q, Jan 007). 11 (i) 1 Show given answer correctly 1 ± i (iii) 7 Attempt to solve quadratic equation or substitute x + iy and equate real and imaginary parts Obtain answers as complex numbers Obtain correct answers, simplified Correct root on x axis, co-ords. shown Other roots in nd and rd quadrants Correct lengths and angles or coordinates or complex numbers shown. (i) Correct expression for u (Q, Jan 007)
5 1 EITHER a = b = OR, a = b = Use trig to find an expression for a (or b) Obtain correct answer Attempt to find other value Obtain correct answer a.e.f. (Allow. ) State equations for a and b Attempt to solve these equations Obtain correct answers a.e.f. SR ± scores only Show result true for n = 1 (Q1, June 007) 1 8 (i) Circle, centre (, 0), y-axis a tangent at origin Straight line, through (1, 0) with +ve slope In 1 st quadrant only Inside circle, below line, above x-axis Bft 8 8 Deduce no solutions Sketch showing correct features N.B. treat diagrams asa MR Sketch showing correct region SR: ft for any correct features (Q8, June 007) 10 1 (i) x y =1 and xy =1 9 Attempt to equate real and imaginary parts of (x + iy) and 1+0i Obtain each result Eliminate to obtain a quadratic in x or y ±( + i ) z = 1± 1 + 0i + i, - - i * *dep ft 11 Solve to obtain x =(±) or y =(±) Obtain correct answers as complex numbers Use quadratic formula or complete the square Simplify to this stage Use answers from (i) Obtain correct answers (Q10, June 007)
6 1 (i) z* = + i 1 +1i PhysicsAndMathsTutor.com Conjugate seen or implied Obtain correct answer i -1 0i ft ft Correct z i or expansion of (z I) seen Real part correct Imaginary part correct (iii) 9 + i 1 8 Multiply by conjugate Numerator correct Denominator correct (Q, Jan 008) 1 (i) + i 8 7 (i) Use det A = 0 Horizontal straight line in quadrants Through (0, ) Straight line Through O with positive slope In 1 st quadrant only State or obtain algebraically that y = Use suitable trigonometry Obtain correct answer a.e.f. decimals OK must be a complex number (Q, Jan 008) 17 (i) Correct modulus 0.97 or.1 o Correct argument, any equivalent form (a) Circle centre A (, ) Through O, allow if centre is (, ) (b) A(, ) Half line with +ve slope Starting at (, 0) Parallel to OA, (implied by correct arg shown) O r (Q, June 008) 18 9 (i) Attempt to equate real and imaginary parts of (x + iy) and + 1i x y = and xy = Obtain both results Eliminate to obtain a quadratic in x or y ±( + i) Solve a term quadratic & obtain x or y Obtain correct answers as complex nos. 1i Correct real and imaginary parts (iii) Attempt to solve a quadratic equation x = ± 1i Obtain correct answers x =± ( ± i) Each pair of correct answers a.e.f. (Q9, June 008)
7 i. 17 Multiply by conjugate of denominator Obtain correct numerator Obtain correct denominator Both diagonals correct (Q1, Jan 009) 10 0 (i) x y =, xy = Attempt to equate real and imaginary parts Obtain both results a.e.f. x 8 x ( 10 + i = 0 x = ± 10, y = ± ± ) z = ± i 10 z ± ( ± i ) = ft Eliminate to obtain quadratic in x or y Solve to obtain x (or y) values Correct values for both x & y obtained a.e.f. Correct answers as complex numbers Solve quadratic in z Obtain correct answers Use results of (i) Obtain correct answers, ft must include root from conjugate (iii) ft 1 Sketch showing roots correctly (iv) ft ft 1 Sketch of straight line, Bisector toα (Q10, Jan 009). 1 (i) 11 9i Obtain correct answers. Correct real and imaginary parts 1 + 1i Correct real and imaginary parts. Either p q 1, pq 8 Both values stated or used (Q, June 009). (i), or o AEF State correct answers (a) (b) ft Circle, centre (, -), through O ft for (, ) only Straight line with +ve slope, through (, -) or their centre Half line only starting at centre (iii) ft ft ft Area above horizontal through a, below (b) Outside circle (i) Show that terms cancel (Q, in pairs June 009)
8 x iy Conjugate known Equate real and imaginary parts x + y = 1 x + y = 9 Obtain both equations, OK with factor of i Solve pair of equations z = + i Obtain correct answer as a complex number S.C. Solving z + iz = 1 + 9i can get max /, not first (Q, Jan 010) 8 (i) Attempt to equate real and imaginary parts of (x + iy) & 1i x y = and xy = Obtain both results, a.e.f Obtain quadratic in x or y Solve to obtain x = ( ± ) or y = ( ± ) ± ( i) Obtain correct answers as complex nos ft Circle with centre at their square root Circle passing through origin ft nd circle centre correct relative to 1 st Circle passing through origin 9 (Q8, Jan 010) (i) + 1i 1 7. o or 1.18 ft ft Correct real and imaginary parts Correct modulus Correct argument i 7 Multiply by conjugate Obtain correct numerator Obtain correct denominator (Q, June 010)
9 (i) (a) (b) Circle centre (, ), through origin Vertical line, clearly x = ft ft Inside their circle And to right of their line, if vertical (Q, June 010) 10 7 (i) Attempt to equate real and imaginary parts x y = xy = Obtain both results Eliminate to obtain quadratic in x or y Solve to obtain x or y value z = + i Obtain correct answer as a complex no. 1 Obtain given answer correctly (iii) w = ± 11i w = i 11 Attempt to solve quadratic equation Obtain correct answers Choose negative sign Relate required value to conjugate of (i) Obtain correct answer (Q10, June 010). 8(i) 1 +1i Real and imaginary parts correct z* seen Multiply by w* 7 1 i 7 7 Obtain correct real part or numerator Obtain correct imaginary part or denom. Sufficient working must be shown (Q, Jan 011)
10 9 (i) (a) * Vertical line dep Clearly through (, 0 ) (b) Sloping line with +ve slope Through ( 0, -) ft Half line starting on y-axis o shown convincingly ft Shaded to left of their (i) (a) ft Shaded below their (i) (b) must be +ve slope ft Shaded above horizontal through their (0, -) NB These marks are independent, but / only for fully correct answer. 8 (Q, Jan 011) 0 (i) a Correct modulus arg a 0,, 1.0 Correct argument Circle Centre ( 1, ) Through origin, centre ( 1, ) and another y intercept Vertical line * Through a or their centre, with +ve gradient D Correct half line 8 (Q, June 011) 1 9 (i) 1 + 0i 1 State correct value Use a = ( sum of roots ) a = Obtain correct answer Use b = product of roots b = 11 Obtain correct answer Substitute, expand and equate imag. parts Obtain a = - Equate real parts Obtain b = (iii) Attempt to equate real and imaginary parts of (p+iq) & 1 0i or root from p q 1 and pq 1 Obtain both results cao Obtain quadratic in p or q Solve to ob tain p ( ) or q ( ) Obtain correct answers as complex nos Attempt at all roots ( ± i) 7 State other two roots as complex nos 1 (Q9, June 011)
11 1 a 1 Use formula for modulus a = 1 Obtain correct answer tan a Use formula for argument 0.9 or. or 0.1π FT Obtain correct answer allow 0.9 [] 1 (Q1, Jan 01) Attempt to equate real and imaginary parts x y and xy Obtain both results x x 18 0 or y y 18 0 Eliminate to obtain quadratic in x or y Solve to obtain x or y value x or y Both values correct ( i ) Correct answers as complex numbers [] (Q, Jan 01) Circle Centre (,1) Passing through O and crosses y-axis again Line, with correct slope shown line starting at O 1 Completely correct diagram for both loci Ignore shading [] (Q, Jan 01) MT 1 (i) 1 +11i Real part correct Imaginary part correct [] 1 Multiply by conjugate of denominator or find a pair of simultaneous equations 9i Obtain correct numerator or real part 9 Obtain correct denominator or imaginary part i 1 1 [] (Q1, June 01) 7 (i) Circle, centre (, ) ft Touching x-axis, ft for ( as centre ft Crossing y-axis twice Horizontal line, y intercept [] 7 1 +i 7 + i State correct answers [] 7 (iii) ft Inside circle or above line Completely correct diagram [] (Q7, June 01)
12 7 (i) z Allow. argz. or 0. Allow -7 o or -0.() [] z* = + i stated or used Obtain two equations from real and imaginary parts a b, b a 8 Obtain correct equations Attempt to solve linear equations a =, b = Obtain correct answers [] (Q, Jan 01) 8 7 (i) (a) Circle Centre O and radius [] 7 (i) (b) Horizontal line (, 1 ) on their line ½ line to left i.e. horizontal [] 7 Shade only inside their circle or above their horizontal line Completely correct diagram [] 8 (i) Obtain correct numerator from addition or partial fractions (Q7, Jan 01) 9 1 Use correct trig expression Obtain correct answer Correct expression for modulus FT Obtain correct answer aef i FT Correct conjugate seen or implied i FT Correct answer [] (i) (Q1, June 01) MT 0 x y 11 and xy Attempt to equate real and imaginary parts of (x + iy) and Obtain both results cao * Obtain a quadratic in x or y ±( + i) D Solve a term quadratic to obtain a value for x or y Obtain 1 correct answer as complex number Obtain only the other correct answer [] Establish result true for n = 1 or n = (Q, June 01) 1 (i) Use arg (z a) = in equation for l condone missing brackets 1 arg( z i) Obtain correct answer Use z a k in equation for C, k must be real z i Obtain correct answer z i or e.g. x 1 1 [] ( y ) 9 Obtain correct inequality, or answer consistent with sensible (i) arg( z i) or y x, x 0 Each correct single inequality, or answer consistent with sensible (i) [] SC if < used consistently, but otherwise all correct, B (Q, June 01)
OCR Maths FP1. Topic Questions from Papers. Matrices. Answers
OCR Maths FP Topic Questions from Papers Matrices Answers PhysicsAndMathsTutor.com . (i) A = 8 PhysicsAndMathsTutor.com Obtain given answer correctly Attempt to find A, elements correct All elements correct
More informationOCR Maths FP1. Topic Questions from Papers. Summation of Series. Answers
OCR Maths FP Topic Questions from Papers Summation of Series Answers PhysicsAndMathsTutor.com . Σr +Σr + Σ Σr = n(n +)(n +) Σr = n(n +) Σ = n PhysicsAndMathsTutor.com Consider the sum of three separate
More informationOCR Maths FP1. Topic Questions from Papers. Proof by Induction. Answers
OCR Maths FP Topic Questions from Papers Proof by Induction Answers PhysicsAndMathsTutor.com (iv) Explicit check for n = or n = M k = k ( k ) 0. Induction hypothesis that result is true for M k Attempt
More informationCandidates sitting FP2 may also require those formulae listed under Further Pure Mathematics FP1 and Core Mathematics C1 C4. e π.
F Further IAL Pure PAPERS: Mathematics FP 04-6 AND SPECIMEN Candidates sitting FP may also require those formulae listed under Further Pure Mathematics FP and Core Mathematics C C4. Area of a sector A
More informationEquations (Linear, Quadratic, Simultaneous & Basic Algebra)
Equations (Linear, Quadratic, Simultaneous & Basic Algebra) Mark Scheme Level Subject Exam Board Maths Module Core 1 Topic Sub Topic A Level OCR - MEI Algebra Booklet Mark Scheme Equations (Linear, Quadratic,
More informationMark Scheme (Results) Summer Pearson Edexcel GCE In Further Pure Mathematics FP2 (6668/01)
Mark Scheme (Results) Summer 017 Pearson Edexcel GCE In Further Pure Mathematics FP (6668/01) Edexcel and BTEC Qualifications Edexcel and BTEC qualifications are awarded by Pearson, the UK s largest awarding
More informationEdExcel Further Pure 2
EdExcel Further Pure 2 Complex Numbers Section : Loci in the Argand diagram Multiple Choice Test Questions 1 are about the following loci: P: z i = 2 Q: z i = z R: arg( z i) = S: z i = 2 z 1) Which of
More informationMARK SCHEME for the October/November 2011 question paper for the guidance of teachers 9709 MATHEMATICS. 9709/32 Paper 3, maximum raw mark 75
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advanced Subsidiary Level and GCE Advanced Level MARK SCHEME for the October/November 0 question paper for the guidance of teachers 9709 MATHEMATICS
More informationFurther 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
More informationQ Scheme Marks AOs. Attempt to multiply out the denominator (for example, 3 terms correct but must be rational or 64 3 seen or implied).
1 Attempt to multiply the numerator and denominator by k(8 3). For example, 6 3 4 8 3 8 3 8 3 Attempt to multiply out the numerator (at least 3 terms correct). M1 1.1b 3rd M1 1.1a Rationalise the denominator
More informationFP2 Mark Schemes from old P4, P5, P6 and FP1, FP2, FP3 papers (back to June 2002)
FP Mark Schemes from old P, P5, P6 and FP, FP, FP papers (back to June 00) Please note that the following pages contain mark schemes for questions from past papers. The standard of the mark schemes is
More informationIntroduction. The first chapter of FP1 introduces you to imaginary and complex numbers
Introduction The first chapter of FP1 introduces you to imaginary and complex numbers You will have seen at GCSE level that some quadratic equations cannot be solved Imaginary and complex numbers will
More informationFP2 Mark Schemes from old P4, P5, P6 and FP1, FP2, FP3 papers (back to June 2002)
FP Mark Schemes from old P, P5, P6 and FP, FP, FP papers (back to June 00) Please note that the following pages contain mark schemes for questions from past papers. The standard of the mark schemes is
More information4751 Mark Scheme June fraction line; accept to power ½ with denominator appropriate brackets answer M1 for a triple decker fraction or for
1 Question Answer Marks Guidance A A 2 square root symbol must extend below condone missing end bracket in [ r ] or [ r ] as final fraction line; accept to power ½ with denominator x y x y appropriate
More informationNot drawn accurately
Q1. A trapezium has parallel sides of length (x + 1) cm and (x + 2) cm. The perpendicular distance between the parallel sides is x cm. The area of the trapezium is 10 cm 2. Not drawn accurately Find the
More informationMark Scheme (Results) January Pearson Edexcel Level 3 Award in Algebra (AAL30)
Mark Scheme (Results) January 017 Pearson Edexcel Level 3 Award in Algebra (AAL30) Edexcel and BTEC Qualifications Edexcel and BTEC qualifications are awarded by Pearson, the UK s largest awarding body.
More informationSeparate sum (may be implied) ( 1)(2 1) ( 1) 6 n n n n n A1,A1 1 mark for each part oe
4755 Mark Scheme June 04 n n n (i) (ii) 0 0 (iii) r( r ) r r Separate sum (may be implied) ( )( ) ( ) 6 n n n n n A,A mark for each part oe ( )[( ) 6] 6 n n n nn ( )(linear factor) ( )( 5) 6 n n n A Or
More informationl Advanced Subsidiary Paper 1: Pure Mathematics Mark Scheme Any reasonable explanation.
l Advanced Subsidiary Paper 1: Pure athematics PAPER B ark Scheme 1 Any reasonable explanation. For example, the student did not correctly find all values of x which satisfy cosx. Student should have subtracted
More information2. 5y 1 B B1. 2 B2 All correct with no extras (B1 at least 4 correct factors) 4. 1, 2, 4, 5, 8, 10, 20, 40. No with correct working
1. 5 hundredths 1 B1 2. 5y 1 B1 3. 680 000 1 B1 4. 1, 2, 4, 5, 8, 10, 20, 40 2 B2 All correct with no extras (B1 at least 4 correct factors) 5. 36 4 (= 144) 176 + 103 + 144 (= 423) 15 28 = 420 Or 423 28
More informationM A1. 4 M1 for listing at least three multiples for any two of 25, 12, (i) 24
MA Practice papers Set : Paper H (Regular) mark scheme Version.0. 3 M 500 (00 00) (=0.5) 87 A M 8 0.5. (i) 4 50 75 4 M for listing at least three multiples for any two of 5,, 8 M for listing at least three
More informationM1 for a complete method with relative place value correct. Condone 1 multiplication error, addition not necessary.
. 54 4 6 080 96 5 4 0 8 0 6 9 6 50 4 0 000 80 4 00 6 000 + 00 + 80 + 6 = 96 4.96 M for a complete method with relative place value correct. Condone multiplication error, addition not necessary. M for a
More informationAS Mathematics MPC1. Unit: Pure Core 1. Mark scheme. June Version: 1.0 Final
AS Mathematics MPC1 Unit: Pure Core 1 Mark scheme June 017 Version: 1.0 Final FINAL MARK SCHEME AS MATHEMATICS MPC1 JUNE 017 Mark schemes are prepared by the Lead Assessment Writer and considered, together
More informationALGEBRAIC LONG DIVISION
QUESTIONS: 2014; 2c 2013; 1c ALGEBRAIC LONG DIVISION x + n ax 3 + bx 2 + cx +d Used to find factors and remainders of functions for instance 2x 3 + 9x 2 + 8x + p This process is useful for finding factors
More informationMark Scheme (Results) January International GCSE Mathematics A 4MA0/3H
Mark Scheme (Results) January 2017 International GCSE Mathematics A 4MA0/3H Edexcel and BTEC Qualifications Edexcel and BTEC qualifications come from Pearson, the world s leading learning company. We provide
More informationA marks are for accuracy and are not given unless the relevant M mark has been given (M0 A1 is impossible!).
NOTES 1) In the marking scheme there are three types of marks: M marks are for method A marks are for accuracy and are not given unless the relevant M mark has been given (M0 is impossible!). B marks are
More informationDraft Version 1 Mark scheme Further Maths Core Pure (AS/Year 1) Unit Test 1: Complex numbers 1
1 w z k k States or implies that 4 i TBC Uses the definition of argument to write 4 k π tan 1 k 4 Makes an attempt to solve for k, for example 4 + k = k is seen. M1.a Finds k = 6 (4 marks) Pearson Education
More informationCore Mathematics C1 Advanced Subsidiary
Paper Reference(s) 666/0 Edexcel GCE Core Mathematics C Advanced Subsidiary Monday 0 January 0 Morning Time: hour 0 minutes Materials required for examination Mathematical Formulae (Pink) Items included
More informationMark Scheme (Results) Summer 2009
Mark (Results) Summer 009 GCE GCE Mathematics (666/01) June 009 666 Core Mathematics C1 Mark Q1 (a) ( 7) = 6 B1 (1) (b) (8 + )( ) = 16 + 8 = 11, 6 A1, A1 (a) B1 for 6 only (b) for an attempt to epand their
More informationQ Scheme Marks AOs Pearson Progression Step and Progress descriptor. and sin or x 6 16x 6 or x o.e
1a A 45 seen or implied in later working. B1 1.1b 5th Makes an attempt to use the sine rule, for example, writing sin10 sin 45 8x3 4x1 States or implies that sin10 3 and sin 45 A1 1. Solve problems involving
More informationAH Complex Numbers.notebook October 12, 2016
Complex Numbers Complex Numbers Complex Numbers were first introduced in the 16th century by an Italian mathematician called Cardano. He referred to them as ficticious numbers. Given an equation that does
More informationab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c
More informationMARK SCHEME for the October/November 2013 series 9709 MATHEMATICS. 9709/11 Paper 1, maximum raw mark 75
CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advanced Subsidiary Level and GCE Advanced Level MARK SCHEME for the October/November 013 series 9709 MATHEMATICS 9709/11 Paper 1, maximum raw mark 75 This mark
More informationAS Level Further Maths
AS Level Further Maths Bronze Set B, Core Pure (Edexcel version) 2018 crashmaths Limited AS Further Maths CM Core Pure 1 Practice Paper (for Edexcel) / Bronze Set B Question Solution Partial Marks Guidance
More informationJune Further Pure Mathematics FP1 (new) Mark Scheme
June 009 6667 Further Pure Mathematics FP (new) Mar Q (a) Mars B () (c) (d) z + ( ) 5 (or awrt.4) α arctan or arctan arg z 0.46 or 5.8 (awrt) (answer in degrees is A0 unless followed by correct conversion)
More informationThe number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 72.
ADVANCED SUBSIDIARY GCE UNIT 475/0 MATHEMATICS (MEI) Introduction to Advanced Mathematics (C) THURSDAY 7JUNE 007 Additional materials: Answer booklet (8 pages) MEI Examination Formulae and Tables (MF)
More informationChapter 1: Precalculus Review
: Precalculus Review Math 115 17 January 2018 Overview 1 Important Notation 2 Exponents 3 Polynomials 4 Rational Functions 5 Cartesian Coordinates 6 Lines Notation Intervals: Interval Notation (a, b) (a,
More information4751 Mark Scheme June Mark Scheme 4751 June 2005
475 Mark Scheme June 2005 Mark Scheme 475 June 2005 475 Mark Scheme June 2005 Section A 40 2 M subst of for x or attempt at long divn with x x 2 seen in working; 0 for attempt at factors by inspection
More informationPaper 1 (Edexcel Version)
AS Level / Year 1 Paper 1 (Edexcel Version) Set A / Version 1 017 crashmaths Limited 1 y = 3x 4 + x x +1, x > 0 (a) ydx = 3x 3 3 3 + x 3 / x + x {+c} Attempts to integrate, correct unsimplified integration
More informationSample Assessment Materials
Edexcel Awards Mathematics Sample Assessment Materials Edexcel Level Award in Algebra (AAL0) Edexcel Level 3 Award in Algebra (AAL30) For first teaching from October 01 Pearson Education Limited is a registered
More informationJune Further Pure Mathematics FP1 (new) Mark Scheme
June 009 6667 Further Pure Mathematics FP (new) Mar Question Q (a) Mars B () (c) (d) z! # (" ) 5 (or awrt.4)! ) & ) & *! arctan' $ or arctan ' " $ ( % ( % arg z! "0.46 or 5.8 (awrt) (answer in degrees
More informationA-Level Notes CORE 1
A-Level Notes CORE 1 Basic algebra Glossary Coefficient For example, in the expression x³ 3x² x + 4, the coefficient of x³ is, the coefficient of x² is 3, and the coefficient of x is 1. (The final 4 is
More informationPhysicsAndMathsTutor.com GCE. Edexcel GCE Core Mathematics C2 (6664) Summer Mark Scheme (Results) Core Mathematics C2 (6664) Edexcel GCE
GCE Edexcel GCE Core Mathematics C () Summer 005 Mark Scheme (Results) Edexcel GCE Core Mathematics C () June 005 Core Mathematics C Mark Scheme 1. dy = x 1 dx B1 x 1 = 0 x = M1 A1ft y = 18 A1 () d y M1:
More informationMark Scheme (Results) Summer Edexcel Level 3 Award (AAL30) Algebra
Mark Scheme (Results) Summer 203 Edexcel Level 3 Award (AAL30) Algebra Edexcel and BTEC Qualifications Edexcel and BTEC qualifications come from Pearson, the world s leading learning company. We provide
More informationMaths Higher Prelim Content
Maths Higher Prelim Content Straight Line Gradient of a line A(x 1, y 1 ), B(x 2, y 2 ), Gradient of AB m AB = y 2 y1 x 2 x 1 m = tanθ where θ is the angle the line makes with the positive direction of
More informationMark Scheme (Results) January Pearson Edexcel Level 3 Award In Algebra (AAL30)
Mark Scheme (Results) January 0 Pearson Edexcel Level 3 Award In Algebra (AAL30) Edexcel and BTEC Qualifications Edexcel and BTEC qualifications are awarded by Pearson, the UK s largest awarding body.
More informationJune Core Mathematics C1 Mark Scheme
June 006 6663 Core Mathematics C Mark Scheme Question number (+c) Scheme Marks = 3 n n for some attempt to integrate Total 4 marks st 6 3 A for either or or better 3 for all terms in correct. Allow and.
More information4751 Mark Scheme June Mark Scheme 4751 June 2005
475 Mark Scheme June 005 Mark Scheme 475 June 005 475 Mark Scheme June 005 Section A 40 subst of for x or attempt at long divn with x x seen in working; 0 for attempt at factors by inspection 6y [ x =
More informationPLC Papers. Created For:
PLC Papers Created For: Algebra and proof 2 Grade 8 Objective: Use algebra to construct proofs Question 1 a) If n is a positive integer explain why the expression 2n + 1 is always an odd number. b) Use
More information, B = (b) Use induction to prove that, for all positive integers n, f(n) is divisible by 4. (a) Use differentiation to find f (x).
Edexcel FP1 FP1 Practice Practice Papers A and B Papers A and B PRACTICE PAPER A 1. A = 2 1, B = 4 3 3 1, I = 4 2 1 0. 0 1 (a) Show that AB = ci, stating the value of the constant c. (b) Hence, or otherwise,
More informationFP1 PAST EXAM QUESTIONS ON NUMERICAL METHODS: NEWTON-RAPHSON ONLY
FP PAST EXAM QUESTIONS ON NUMERICAL METHODS: NEWTON-RAPHSON ONLY A number of questions demand that you know derivatives of functions now not included in FP. Just look up the derivatives in the mark scheme,
More informationGCSE. Edexcel GCSE Mathematics A 1387 Paper 5525/05. Summer Edexcel GCSE. Mark Scheme (Results) Mathematics A 1387.
GCSE Edexcel GCSE Mathematics A 87 Summer 005 Mark Scheme (Results) Edexcel GCSE Mathematics A 87 NOTES ON MARKING PRINCIPLES Types of mark M marks: method marks A marks: accuracy marks B marks: unconditional
More informationEdexcel GCSE. Mathematics 2540 Paper 5540H/3H. Summer Mark Scheme (Results) Mathematics Edexcel GCSE
Edexcel GCSE Mathematics 540 Paper 5540H/H Summer 008 Mark Scheme (Results) Edexcel GCSE Mathematics 540 5540H/H (a) 4 00 8 900 M for 4 8 oe or 00 oe or 00 + 00 + 00 or 7.5 seen 8 A for 900 (SC: B for
More informationabc Mathematics Further Pure General Certificate of Education SPECIMEN UNITS AND MARK SCHEMES
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
More informationCore A-level mathematics reproduced from the QCA s Subject criteria for Mathematics document
Core A-level mathematics reproduced from the QCA s Subject criteria for Mathematics document Background knowledge: (a) The arithmetic of integers (including HCFs and LCMs), of fractions, and of real numbers.
More informationTime: 1 hour 30 minutes
Paper Reference(s) 6667/0 Edexcel GCE Further Pure Mathematics FP Bronze Level B Time: hour 30 minutes Materials required for examination papers Mathematical Formulae (Green) Items included with question
More informationThe number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 72.
ADVANCED SUBSIDIARY GCE UNIT 755/0 MATHEMATICS (MEI) Further Concepts for Advanced Mathematics (FP) THURSDAY 8 JANUARY 007 Additional materials: Answer booklet (8 pages) Graph paper MEI Examination Formulae
More informationMEI STRUCTURED MATHEMATICS 4751
OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education MEI STRUCTURED MATHEMATICS 75 Introduction to Advanced Mathematics (C)
More information* * MATHEMATICS (MEI) 4755 Further Concepts for Advanced Mathematics (FP1) ADVANCED SUBSIDIARY GCE. Thursday 15 January 2009 Morning
ADVANCED SUBSIDIARY GCE MATHEMATICS (MEI) 4755 Further Concepts for Advanced Mathematics (FP) Candidates answer on the Answer Booklet OCR Supplied Materials: 8 page Answer Booklet Graph paper MEI Examination
More information5w 3. 1MA0 Higher Tier Practice Paper 2H (Set D) Question Working Answer Mark Notes 1 (a) 5w 8 = 3(4w + 2) 5w 8 = 12w = 12w 5w 14 = 7w
(a) 5w 8 = (4w + ) 5w 8 = w + 6 8 6 = w 5w 4 = 7w M for attempting to multiply both sides by as a first step (this can be implied by equations of the form 5w 8 = w +? or 5w 8 =?w + 6 i.e. the LHS must
More informationMark Scheme (Results) January Pearson Edexcel International GCSE Mathematics A (4MA0) Paper 3H
Mark Scheme (Results) January 016 Pearson Edexcel International GCSE Mathematics A (4MA0) Paper H Pearson Edexcel Certificate Mathematics A (KMA0) Paper H Edexcel and BTEC Qualifications Edexcel and BTEC
More informationPhysicsAndMathsTutor.com
PhysicsAndMathsTutor.com 47 Mark Scheme June 00 (i) u =, u =, u = 8 The sequence is an Arithmetic Progression B B B For the correct value of u For both correct values of u and u For a correct statement
More informationA2 HW Imaginary Numbers
Name: A2 HW Imaginary Numbers Rewrite the following in terms of i and in simplest form: 1) 100 2) 289 3) 15 4) 4 81 5) 5 12 6) -8 72 Rewrite the following as a radical: 7) 12i 8) 20i Solve for x in simplest
More informationMark Scheme (Results) Summer Pearson Edexcel IAL in Further Pure Mathematics 2 (WFM02/01)
Mark Scheme (Results) Summer 06 Pearson Edexcel IAL in Further Pure Mathematics (WFM0/0) dexcel and BTEC Qualifications dexcel and BTEC qualifications are awarded by Pearson, the UK s largest awarding
More information2x + 5 = 17 2x = 17 5
1. (i) 9 1 B1 (ii) 19 1 B1 (iii) 7 1 B1. 17 5 = 1 1 = x + 5 = 17 x = 17 5 6 3 M1 17 (= 8.5) or 17 5 (= 1) M1 for correct order of operations 5 then Alternative M1 for forming the equation x + 5 = 17 M1
More informationAS Level / Year 1 Edexcel Maths / Paper 1
AS Level / Year Edexcel Maths / Paper March 8 Mocks 8 crashmaths Limited 4x + 4x + 3 = 4( x + x) + 3 Takes out a factor of 4 from first two terms or whole expression = 4 x + + 3 4 Completes the square
More informationIntegers, Fractions, Decimals and Percentages. Equations and Inequations
Integers, Fractions, Decimals and Percentages Round a whole number to a specified number of significant figures Round a decimal number to a specified number of decimal places or significant figures Perform
More informationGCE. Mathematics. Mark Scheme for June Advanced GCE 4725 Further Pure Mathematics 1
GCE Mathematics Advanced GCE 475 Further Pure Mathematics Mark Scheme for June 00 OCR (Oxford Cambridge and RSA) is a leading UK awarding body, providing a wide range of qualifications to meet the needs
More informationA booklet Mathematical Formulae and Statistical Tables might be needed for some questions.
Paper Reference(s) 6663/01 Edexcel GCE Core Mathematics C1 Advanced Subsidiary Quadratics Calculators may NOT be used for these questions. Information for Candidates A booklet Mathematical Formulae and
More informationPhysicsAndMathsTutor.com
Question Answer Marks Guidance Question 1 (i) [radius =] 15 isw or 5 5 B1 [C =] (10, ) B1 condone x = 10, y = 1 (ii) verifying / deriving that (1, 0) is one of the intersections with the axes [] B1 using
More informationPure Mathematics Year 1 (AS) Unit Test 1: Algebra and Functions
Pure Mathematics Year (AS) Unit Test : Algebra and Functions Simplify 6 4, giving your answer in the form p 8 q, where p and q are positive rational numbers. f( x) x ( k 8) x (8k ) a Find the discriminant
More informationHigher Mathematics Course Notes
Higher Mathematics Course Notes Equation of a Line (i) Collinearity: (ii) Gradient: If points are collinear then they lie on the same straight line. i.e. to show that A, B and C are collinear, show that
More informationMARK SCHEME for the May/June 2012 question paper for the guidance of teachers 0580 MATHEMATICS
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 0 question paper for the guidance of teachers 0580 MATHEMATICS 0580/4
More informationGCE Mathematics. Mark Scheme for June Unit 4721: Core Mathematics 1. Advanced Subsidiary GCE. Oxford Cambridge and RSA Examinations
GCE Mathematics Unit 7: Core Mathematics Advanced Subsidiary GCE Mark Scheme for June 05 Oxford Cambridge and RSA Examinations OCR (Oxford Cambridge and RSA) is a leading UK awarding body, providing a
More informationJanuary Further Pure Mathematics FP1 Mark Scheme
January 6667 Further Pure Mathematics FP Mar Q z + 8i + i (a) z i + i + i + 8i 8 + 5i z (b) ( ) + 5 z (or awrt 5.8) () ft () (c) 5 5 tan α or z arg π.... z + i (a) and attempt to multiply out for + i -
More informationMark scheme Pure Mathematics Year 1 (AS) Unit Test 2: Coordinate geometry in the (x, y) plane
Mark scheme Pure Mathematics Year 1 (AS) Unit Test : Coordinate in the (x, y) plane Q Scheme Marks AOs Pearson 1a Use of the gradient formula to begin attempt to find k. k 1 ( ) or 1 (k 4) ( k 1) (i.e.
More informationPLC Papers Created For:
PLC Papers Created For: Josh Angles and linear graphs Graphs of Linear Functions 1 Grade 4 Objective: Recognise, sketch and interpret graphs of linear functions. Question 1 Sketch the graph of each function,
More informationor i 2 = -1 i 4 = 1 Example : ( i ),, 7 i and 0 are complex numbers. and Imaginary part of z = b or img z = b
1 A- LEVEL MATHEMATICS P 3 Complex Numbers (NOTES) 1. Given a quadratic equation : x 2 + 1 = 0 or ( x 2 = -1 ) has no solution in the set of real numbers, as there does not exist any real number whose
More information284 B1 cao (ii) (b(i) or B1 for showing addition of 71 and 95 or 91 and 75
1. 7.8 B1 cao 2. 7 B1 cao 3. 7.84 B1 cao 4. 25 B1 cao 5. (a) 2, 3, 6, 7, 8 2 B2 for 2, 3, 6, 7, 8 (b) 3,8 1 B1 cao 6. (a)(i) 23 2 B1 cao (B1 for any 3 or 4 correct, no extras or 2, 3, 6, 7, 8 seen with
More informationNational 5 Learning Checklist - Relationships
National 5 Learning Checklist - Relationships Topic Skills Extra Stud / Notes Straight Line Gradient Represented b m Measure of steepness of slope Positive gradient the line is increasing Negative gradient
More informationMark Scheme (Results) June Pearson Edexcel International GCSE Mathematics A (4MA0) Paper 4HR
Mark Scheme (Results) June 016 Pearson Edexcel International GCSE Mathematics A (4MA0) Paper 4HR Edexcel and BTEC Qualifications Edexcel and BTEC qualifications come from Pearson, the world s leading learning
More informationSOLUTIONS FOR PROBLEMS 1-30
. Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS - 0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).
More information1MA1 Practice papers Set 3: Paper 3H (Regular) mark scheme Version 1.0 Question Working Answer Mark Notes
1. (a) 1, 0, 1,, 3 B for all 5 values and no extras (ignore repeats) (B1 for 4 correct values and no extras or all 5 correct values and one incorrect value) (b) x + x + 9 < 60 x < 51 x < 5.5 5 3 M1 for
More information(b) M1 for a line of best fit drawn between (9,130) and (9, 140) and between (13,100) and (13,110) inclusive
1 4 3 M1.1 (= 4) or.1. (=.13 ) 1 4 3 4. 1 4 3 4 4 4 3 + 9 = 11 11 = 1MA1 Practice Tests: Set 1 Regular (H) mark scheme Version 1. This publication may only be reproduced in accordance with Pearson Education
More informationCambridge Assessment International Education Cambridge International Advanced Level. Published
Cambridge Assessment International Education Cambridge International Advanced Level MATHEMATICS 9709/ Paper 07 MARK SCHEME Maximum Mark: 7 Published This mark scheme is published as an aid to teachers
More informationSystems of Equations and Inequalities. College Algebra
Systems of Equations and Inequalities College Algebra System of Linear Equations There are three types of systems of linear equations in two variables, and three types of solutions. 1. An independent system
More informationQ Scheme Marks AOs. Attempt to multiply out the denominator (for example, 3 terms correct but must be rational or 64 3 seen or implied).
Pure Mathematics Year 1 (AS) Unit Test 1: Algebra and Functions 1 Attempt to multiply the numerator and denominator by k(8 3). For example, 6 3 4 8 3 8 3 8 3 Attempt to multiply out the numerator (at least
More informationGCE. Mathematics. Mark Scheme for June Advanced Subsidiary GCE Unit 4721: Core Mathematics 1. Oxford Cambridge and RSA Examinations
GCE Mathematics Advanced Subsidiary GCE Unit 7: Core Mathematics Mark Scheme for June 0 Oford Cambridge and RSA Eaminations OCR (Oford Cambridge and RSA) is a leading UK awarding body, providing a wide
More informationList of PreCalculus Algebra Mathematical Concept Practice Sheets (Updated Spring 2015)
List of PreCalculus Algebra Mathematical Concept Practice Sheets (Updated Spring 2015) MAT 155P MAT 155 1 Absolute Value Equations P 7 P 3 2 Absolute Value Inequalities P 9 P 4 3 Algebraic Expressions:
More informationGCE Mathematics. Mark Scheme for June Unit 4721: Core Mathematics 1. Advanced Subsidiary GCE. Oxford Cambridge and RSA Examinations
GCE Mathematics Unit 47: Core Mathematics Advanced Subsidiary GCE Mark Scheme for June 04 Oxford Cambridge and RSA Examinations OCR (Oxford Cambridge and RSA) is a leading UK awarding body, providing a
More informationPhysicsAndMathsTutor.com
GCE Edecel GCE Core Mathematics C(666) Summer 005 Mark Scheme (Results) Edecel GCE Core Mathematics C (666) June 005 666 Core Mathematics C Mark Scheme Question Number. (a) Scheme Penalise ± B Marks ()
More information4024 MATHEMATICS (SYLLABUS D)
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Ordinary Level MARK SCHEME for the May/June 00 question paper for the guidance of teachers 404 MATHEMATICS (SYLLABUS D) 404/ Paper, maximum raw mark
More informationMark Scheme Mock Paper
Mark Scheme Mock Paper GCSE GCSE in Mathematics Specification A Higher Tier Paper 1 (Non-Calculator) Edexcel Limited. Registered in England and Wales No. 4496750 Registered Office: One90 High Holborn,
More informationGCSE. Edexcel GCSE Mathematics A Summer Mark Scheme (Results)
GCSE Edexcel GCSE Mathematics A 7 Summer 006 Mark Scheme (Results) NOTES ON MARKING PRINCIPLES Types of mark M marks: method marks A marks: accuracy marks B marks: unconditional accuracy marks (independent
More informationMark Scheme (Results) June GCE Further Pure FP1 (6667) Paper 1
Mark Scheme (Results) June 0 GCE Further Pure FP (6667) Paper Edexcel is one of the leading examining and awarding bodies in the UK and throughout the world. We provide a wide range of qualifications including
More informationMEI STRUCTURED MATHEMATICS FURTHER CONCEPTS FOR ADVANCED MATHEMATICS, FP1. Practice Paper FP1-A
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
More informationStep 1: Greatest Common Factor Step 2: Count the number of terms If there are: 2 Terms: Difference of 2 Perfect Squares ( + )( - )
Review for Algebra 2 CC Radicals: r x p 1 r x p p r = x p r = x Imaginary Numbers: i = 1 Polynomials (to Solve) Try Factoring: i 2 = 1 Step 1: Greatest Common Factor Step 2: Count the number of terms If
More informationCore Mathematics 2 Trigonometry
Core Mathematics 2 Trigonometry Edited by: K V Kumaran Email: kvkumaran@gmail.com Core Mathematics 2 Trigonometry 2 1 Trigonometry Sine, cosine and tangent functions. Their graphs, symmetries and periodicity.
More informationFP1 Mark Schemes from old P4, P5, P6 and FP1, FP2, FP3 papers (back to June 2002)
FP1 Mar Schemes from old P4, P5, P6 and FP1, FP, FP papers (bac to June 00) Please note that the following pages contain mar schemes for questions from past papers which were not written at an AS standard
More informationTime: 1 hour 30 minutes
Paper Reference (complete below) Centre No. Surname Initial(s) Candidate No. Signature Paper Reference(s) 6663 Edexcel GCE Pure Mathematics C Advanced Subsidiary Specimen Paper Time: hour 30 minutes Examiner
More informationTwitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Simplify: a) 3x 2 5x 5 b) 5x3 y 2 15x 7 2) Factorise: a) x 2 2x 24 b) 3x 2 17x + 20 15x 2 y 3 3) Use long division to calculate:
More information