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Mark Scheme Pure Mathematics Year 1 (AS) Unit Test 5: Vectors

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Mark schemePure Mathematics Year 1 (AS) Unit Test 5: VectorsQ1SchemeMakes an attempt to use Pythagoras’ theorem to find a .For example,MarksAOsPearsonProgression Stepand ProgressdescriptorM11.1b4th 4 7 seen.2265Displays the correct final answer.A11.1bA11.1bFind the unitvector in thedirection of agiven vector1 4i 7 j 65(3)(3 marks)Notes Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free.1

Mark schemeMarksAOsPearsonProgression Stepand ProgressdescriptorStates that AB a bM12.2a3rd33States PQ PO OQ or PQ a b55M13.1aA12.2aA12.1Q2aPure Mathematics Year 1 (AS) Unit Test 5: VectorsSchemeStates PQ 33 a b or PQ AB55Draws the conclusion that as PQ is a multiple of AB the twolines PQ and AB must be parallel.Understand thecondition for twovectors to beparallel(4)2bB13PQ 10 cm 6 cm cao53.1a3rdUnderstand anduse positionvectors(1)(5 marks)Notes Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free.2

Mark schemeQ3aPure Mathematics Year 1 (AS) Unit Test 5: VectorsSchemeEquates the i components for the equation a b mc o.e.MarksAOsPearsonProgression Stepand ProgressdescriptorB12.2a3rd2p 6 4mEquates the j components for the their equation a b mcB12.2aM11.1bA11.1b 5 3p 5mMakes an attempt to find p by eliminating m in some way.For example,Understand thecondition for twovectors to beparallel10 p 30 20m2p 64o.e. or o.e.20 12 p 20m 5 3 p5p 5(4)3bUsing their value for p from above, makes a substitution intothe vectors to form a bM1ft1.1b10i – 5j 6i – 15jCorrectly simplifies.A1ft1.1b16i – 20j2ndAdd, subtract andfind scalarmultiples ofvectors bycalculation(2)(6 marks)Notes3aAlternatively, M1: attempt to eliminate p first. A1: m 4 and p 53bAlternatively, M1ft: substitute their m 4 into their a b mc. A1ft correct simplification. Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free.3

Mark schemePure Mathematics Year 1 (AS) Unit Test 5: VectorsMarksAOsPearsonProgression Stepand ProgressdescriptorMakes an attempt to find the vector AB .For example, writingAB OB OA or AB 10i qj (4i 7 j)M12.2a3rdShows a fully simplified answer: AB 6i (q 7) jA11.1bUnderstand anduse positionvectors2.2a4thQ4aScheme(2)4bCorrectly interprets the meaning of AB 2 13 , by writing 6 2 M1Use vectors tosolve simplegeometricproblems q 7 2 13 o.e.22Correct method to solve quadratic equation in q (full workingmust be shown).M11.1bM11.1bq 11A11.1bq 3A11.1bFor example, q 7 16 or q 2 14q 33 02q – 7 4 or (q 11)(q 3) 0 or q 14 142 4 1 332 1(5)(7 marks)Notes Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free.4

Mark schemePure Mathematics Year 1 (AS) Unit Test 5: VectorsMarksAOsPearsonProgression Stepand ProgressdescriptorStates or implies that BC 13i 8 j o.e.M12.2a4thRecognises that the cosine rule is needed to solve for BACby stating a 2 b 2 c 2 2bc cos AM13.1aMakes correct substitutions into the cosine rule.M11.1bM11.1bA11.1bQ5aScheme 233 2cos A 45 2104 2 2 45 Use vectors tosolve simplegeometricproblems 104 cos A o.e.7or awrt 0.614 (seen or implied by correct130answer).A 127.9 cao(5)5bStates formula for the area of a triangle.M13.1a1Area ab sin C2Makes correct substitutions using their values from above.Area 12 45 M1ft1.1bA1ft1.1b4thUse vectors tosolve simplegeometricproblems 104 sin127.9.Area 27 (units2)(3)(8 marks)Notes Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free.5

Mark schemePure Mathematics Year 1 (AS) Unit Test 5: VectorsMarksAOsPearsonProgression Stepand Progressdescriptor22or θ tan 1 (if θ shown on diagram33sign must be consistent with this).M11.1b2ndFinds 33.7 (must be negative).A1Q6aSchemeStates that tan θ Find the directionof a vector usingtan1.1b(2)6bMakes an attempt to use the formula F maM13.1a4thFinds p 10 Note: 8 p 6 3 p 10A12.2aFinds q 2 Note: 10 q 6 2 q 2A12.2aUnderstand thelink that vectorshave withmechanics3.1a2ndUse themagnitude anddirection of avector to find itscomponents(3)6cAttempt to find R (either 6(3i 2 j) or8i 10 j '10' i ' 2' j ).M1Makes an attempt to find the magnitude of their resultantforce. For example,M11.1bA11.1b R '18'2 '12'2 468 Presents a fully simplified exact final answer.R 6 13(3)(8 marks)Notes Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free.6

Mark schemeQ7aPure Mathematics Year 1 (AS) Unit Test 5: VectorsSchemeShows how to move from M to N using vectors.MarksAOsPearsonProgression Stepand ProgressdescriptorM11.1b3rdUnderstand anduse positionvectors41MN MB BC CN b a b55or14MN MO OA AN b a b553MN a b5A11.1b(2)7bShows how to move from S to T using vectors.M11.1b3rdUnderstand anduse positionvectors14ST SB BO OT a b a55or41ST SC CA AT a b a553ST a b5A11.1b(2) Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free.7

Mark scheme7cPure Mathematics Year 1 (AS) Unit Test 5: VectorsM1*Finds OD travelling via M.3.1a4thUse vectors tosolve simplegeometricproblems13 OD OM MD b a b 55 Finds OD travelling via .24 3 OD OT TD a a b 5 5 Recognises that any two ways of travelling from O to D mustbe equal and equates OD via M with OD via T.13 4 3 b a b a a b 55 5 5 1 3 4 3 Or a b a b 5 5 5 5 Equates the a parts: 4 3 or 5 4 3 or 3 5 45 5Equates the b parts:1 3 or 1 3 5 or 5 3 15 5Makes an attempt to solve the pair of simultaneous equationsby multiplying.For example, 15 25 20 and 15 9 3or 9 15 12 and 25 15 5Solves to find 11and 22Either: explains, making reference to an expression for OD or,1for example, MD that implies that D is the midpoint of2MNor finds MD DN or MD 1MN o.e.2and therefore MN is bisected by ST.Uses argument (as above) for bisection of ST using 12 Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free.8

Mark schemePure Mathematics Year 1 (AS) Unit Test 5: Vectors(9)(13 marks)Notes7cEquating, for example, OD via M with OD via N, will lead to a pair of simultaneous equations that has infinitelymany solutions. In this case, providing all work is correct, award one of the first two method marks, togetherwith the 3rd, 4th, 5th and 6th method marks, for a maximum of 5 out of 9.Alternative Method(M1) Finds OD travelling via N.43 OD OA AN ND a b a b 55 (M1) Finds OD travelling via S.1 3 OD OB BS SD b a a b 55 (M1) Equates OD via N with OD via S.43 1 3 a b a b b a a b 55 5 5 (M1) Equates the a parts:1 1 3 or 5 5 1 3 or 3 5 45 5(M1) Equates the b parts:4 3 1 or 4 3 5 5 or 5 3 15 5Proceeds as above. Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free.9

vectors to be parallel Equates the j components for the their equation a b mc 5 3p 5m B1 2.2a Makes an attempt to find p by eliminating m in some way. For example, 10 30 20 20 12 20 pm pm o.e. or 2 6 4 5 3 5 p p o.e. M1 1.1b p 5 A1 1.1b (4) 3b Using their value for p from above, makes a substitution into the vectors to form a b