OCR GCE A LEVEL 2022 MATHEMATICS A MARKSCHEMES H240-02 PAPER 2-H240/02: Pure Mathematics and Statistics

H240/02 Mark Scheme June 2022 5 • When a value is not given in the paper accept any answer that agrees with the correct value to 3 s.f. unless a different level of accuracy has been asked for in the question, or the ...

H240/02 Mark Scheme June 2022 5 • When a value is not given in the paper accept any answer that agrees with the correct value to 3 s.f. unless a different level of accuracy has been asked for in the question, or the mark scheme specifies an acceptable range. NB for Specification B (MEI) the rubric is not specific about the level of accuracy required, so this statement reads “2 s.f”. Follow through should be used so that only one mark in any question is lost for each distinct accuracy error. Candidates using a value of 9.80, 9.81 or 10 for g should usually be penalised for any final accuracy marks which do not agree to the value found with 9.8 which is given in the rubric. g Rules for replaced work and multiple attempts: • If one attempt is clearly indicated as the one to mark, or only one is left uncrossed out, then mark that attempt and ignore the others. • If more than one attempt is left not crossed out, then mark the last attempt unless it only repeats part of the first attempt or is substantially less complete. • if a candidate crosses out all of their attempts, the assessor should attempt to mark the crossed out answer(s) as above and award marks appropriately. h For a genuine misreading (of numbers or symbols) which is such that the object and the difficulty of the question remain unaltered, mark according to the scheme but following through from the candidate’s data. A penalty is then applied; 1 mark is generally appropriate, though this may differ for some units. This is achieved by withholding one A or B mark in the question. Marks designated as cao may be awarded as long as there are no other errors. If a candidate corrects the misread in a later part, do not continue to follow through. Note that a miscopy of the candidate’s own working is not a misread but an accuracy error. i If a calculator is used, some answers may be obtained with little or no working visible. Allow full marks for correct answers, provided that there is nothing in the wording of the question specifying that analytical methods are required such as the bold “In this question you must show detailed reasoning”, or the command words “Show” or “Determine”. Where an answer is wrong but there is some evidence of method, allow appropriate method marks. Wrong answers with no supporting method score zero. If in doubt, consult your Team Leader. j If in any case the scheme operates with considerable unfairness consult your Team Leader. H240/02 Mark Scheme June 2022 6 Sig figs: “0.348 (3 sf)” means “answer that rounds to 0.348”, ISW. eg 0.347652 = 0.35 scores A1, 0.348 = 0.35 scores A1, but 0.35 alone scores A0 Other forms for probabilities Allow eg 20% or 1 in 5, but not odds eg 1:4 Question Answer Mark AO Guidance 1 (a) DR ( 2) ( 1)( 1) ( 1)( 2) x x x x x x + − − + + + or 2 2 2 2 1 3 2 x x x x x + − + + + oe (= 0) M1 M1 1.1 1.1 M1 for x(x + 2) − (x + 1)(x − 1) oe Multiply out brackets. Allow one error Ignore denominator even if “= 0” x = 1 2 − A1 1.1 NB correct with no working: SC B1 Alternative method x(x + 2) = (x + 1)(x − 1) M1 M1 for attempt “cross-multiply”. x 2 + 2x = x 2 − 1 or 2x = −1 oe M1 Multiply out brackets. Allow one error x = 1 2 − A1 [3] 1 (b) DR Solve quadratic in 1 3 x or x 3 or u (= x 3 or 1 3 x ) M1 3.1a or cubic in x Condone quadratic in x with x = 1 3 x or x = x 3 using any correct method. Must see attempt at correct method for this mark Allow arithmetical errors 3 1 x (or u) = 1 & 1 8 − or x 3 (or u) = 1 & –8 B1 1.1 Can be scored without M1 Condone x = 1, 1 8 − or x = 1, −8 or correct factorisation of quadratic Ignore x 3 = 0, if seen, for this mark x = 1& x = –2 with no extras B1f 1.1 ft their x 3 or 3 1 x If also x = 0, B0 NB correct with no working: M0B0B1 [3] 1 (c) DR Condone incorrect or omitted brackets eg (x 2 – 7)ln3 = ln 1 243 or x 2 − 7 = log3 ( ) 1 243 M1 3.1a Any correct step after log(both sides) or 2 7 3 x − = 3−5 or x 2 – 7 = –5 or 2 3 x = 32 or ANY correct step using indices x = ± 2 or ±1.41 (3 sf) A1 1.1 NB correct with no working or T & I: SC B1 H240/02 Mark Scheme June 2022 7 Question Answer Mark AO Guidance [2] 2 (a) (4i + 2j −5k) − (3i + 2j) (= i − 5k) M1 1.1 b − a or a − b attempted, using i, j, k or column vectors or (3i + 2j) − (4i + 2j −5k) (= 5k − i) May be implied by calculation seen AB = 26 or 5.10 (3 sf) or 5.1 A1 1.1 www. Correct answer, no working: M1A1 [2] Mark(s) cannot be gained retrospectively in (b) 2 (b) '26' = (p – 3)2 + 4 + 9 +(p – 4)2 +4 + 4 M1 3.1a Attempt AB2 = BP2 + PA2 (involving p) ft their AB Alternative methods for M1 Attempt |PC| 2 = (their radius)2 M1 or ( 7 2 – p) 2 + 4 + 1 4 = 13 2 Attempt PA PB . = 0 M1 or ((3–p)i + 2j + 3k).((4–p)i + 2j – 2k) = 0 p 2 – 7p + 10 = 0 oe or (p − 7 2 ) 2 = 9 4 A1f 1.1 Correct simplified equation, ft their (a), ie: or p 2 – 7p + 2 46 their 2 − a = 0 or (p − 7 2 ) 2 = 2 their 17 4 a − p = 2 or 5 A1f 1.1 ft only their (a) [3] 3 (a) (No because) they differ only by a constant B1 1.2 oe, eg They may have different constants of integration or eg c2 = c1 + 1 3 , or 1 3 is part of Ben’s c Only the “c”s are different If definite integral found, answers are same Not “Both are correct” or “just different correct methods” If differentiate, answers same [1] 3 (b) (i) 1 (1 ) 1 1 x a − + −         or 1 1 u 2 a +   −   or 2 1 ( 1) 4 a u   + −    oe M1 1.1 Attempt integral, must be of form k(1 + x) -1 or ku−1 or ku−0.5 (if from substitution u = (1 + x) 2 ) Ignore limits = 1 (1 ) 1 1 2 a − + − + oe M1 1.1 Attempt substitute appropriate limits into their integral (= 1 1 2 1+a − ) = 1 2( 1) a a − + A1 1.1 cao oe si