Mark scheme. 65 A1 1.1b. Pure Mathematics Year 1 (AS) Unit Test 5: Vectors. Pearson Progression Step and Progress descriptor. Q Scheme Marks AOs

Transcription

1 Pure Mathematics Year (AS) Unit Test : Vectors Makes an attempt to use Pythagoras theorem to find a. For example, 4 7 seen. 6 A.b 4th Find the unit vector in the direction of a given vector Displays the correct final answer. 6 4i 7j A.b () ( marks) Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free.

2 Pure Mathematics Year (AS) Unit Test : Vectors a States that AB a b M.a rd States PQ PO OQ or States PQ a b or PQ a b PQ AB M.a A.a Understand the condition for two vectors to be parallel Draws the conclusion that as PQ is a multiple of AB the two lines PQ and AB must be parallel. A. (4) b PQ 0 cm = 6 cm cao B.a rd Understand and use position vectors () ( marks) Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free.

3 Pure Mathematics Year (AS) Unit Test : Vectors a Equates the i components for the equation a + b = mc o.e. p + 6 = 4m Equates the j components for the their equation a + b = mc p = m Makes an attempt to find p by eliminating m in some way. 0 p0 0m For example, o.e. or p 6 4 o.e. 0 p0m p B.a rd B.a Understand the condition for two vectors to be parallel p = A.b (4) b Using their value for p from above, makes a substitution into the vectors to form a + b 0i j + 6i j Correctly simplifies. 6i 0j Mft.b nd Aft.b () Add, subtract and find scalar multiples of vectors by calculation (6 marks) a Alternatively, M: attempt to eliminate p first. A: m = 4 and p = b Alternatively, Mft: substitute their m = 4 into their a + b = mc. Aft correct simplification. Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free.

4 Pure Mathematics Year (AS) Unit Test : Vectors 4a Makes an attempt to find the vector AB.For example, writing AB OB OA or AB 0 i qj (4i 7 j ) M.a rd Shows a fully simplified answer: AB 6 i ( q 7) j A.b Understand and use position vectors () 4b Correctly interprets the meaning of AB, by writing 6 q 7 o.e. Correct method to solve quadratic equation in q (full working must be shown). For example, q 7 6 or q 4q q 7 = ±4 or ( q)( q ) 0 or q M.a 4th Use vectors to solve simple geometric problems q = A.b q = A.b () (7 marks) Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free. 4

5 Pure Mathematics Year (AS) Unit Test : Vectors a States or implies that BC i8j o.e. M.a 4th Recognises that the cosine rule is needed to solve for by stating a b c bc cos A Makes correct substitutions into the cosine rule. BAC cos A o.e. M.a Use vectors to solve simple geometric problems 7 cos A or awrt 0.64 (seen or implied by correct 0 answer). A = 7.9 cao A.b () b States formula for the area of a triangle. Area = sin ab C Makes correct substitutions using their values from above. Area = 4 04 sin M.a 4th Mft.b Use vectors to solve simple geometric problems Area = 7 (units ) Aft.b () (8 marks) Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free.

6 Pure Mathematics Year (AS) Unit Test : Vectors 6a States that tan θ or θ tan (if θ shown on diagram sign must be consistent with this). Finds.7 (must be negative). A.b nd Find the direction of a vector using tan 6b Makes an attempt to use the formula F = ma M.a 4th Finds p = 0 Note: 8 p 6 p 0 A.a Finds q = Note: 0 q 6 q A.a () () Understand the link that vectors have with mechanics 6c Attempt to find R (either 6(i j) or 8i 0 j '0' i ' ' j). Makes an attempt to find the magnitude of their resultant force. For example, R '8' '' 468 Presents a fully simplified exact final answer. M.a nd A.b Use the magnitude and direction of a vector to find its components R 6 () (8 marks) Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free. 6

7 Pure Mathematics Year (AS) Unit Test : Vectors 7a 7b Shows how to move from M to N using vectors. 4 MN MB BC CN b a b or 4 MN MO OA AN b a b MN a b Shows how to move from S to T using vectors. 4 ST SB BO OT a b a or 4 ST SC CA AT a b a ST ab rd A.b () Understand and use position vectors rd A.b () Understand and use position vectors Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free. 7

8 Pure Mathematics Year (AS) Unit Test : Vectors 7c FindsOD travelling via M. OD OM MD b a b FindsOD travelling via T. OD 4 OT TD a a b M*.a 4th M*.a Use vectors to solve simple geometric problems Recognises that any two ways of travelling from O to D must be equal and equates OD via M with OD via T. 4 b a b a a b 4 Or a b a b Equates the a parts: 4 or 4 or 4 Equates the b parts: or or Makes an attempt to solve the pair of simultaneous equations by multiplying. For example, 0 and 9 or 9 and Solves to find and Either: explains, making reference to an expression for OD or, for example, MD that implies that D is the midpoint of MN or finds MD DN or MD MN o.e. and therefore MN is bisected by ST. Uses argument (as above) for bisection of ST using M*.a M*.a M*.a A.b B. B. Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free. 8

9 Pure Mathematics Year (AS) Unit Test : Vectors (9) ( marks) 7c Equating, for example, OD via M with OD via N, will lead to a pair of simultaneous equations that has infinitely many solutions. In this case, providing all work is correct, award one of the first two method marks, together with the rd, 4th, th and 6th method marks, for a maximum of out of 9. Alternative Method (M) FindsOD travelling via N. OD 4 OA AN ND a b a b (M) FindsOD travelling via S. OD OB BS SD b a a b (M) Equates OD via N with OD via S. a 4 b a b b a a b (M) Equates the a parts: or or 4 (M) Equates the b parts: 4 or 4 or Proceeds as above. Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free. 9