2018 Mathematics. Advanced Higher. Finalised Marking Instructions

Transcription

1 National Qualifications Mathematics Advanced Higher Finalised Marking Instructions Scottish Qualifications Authority 08 The information in this publication may be reproduced to support SQA qualifications only on a noncommercial basis. If it is reproduced, SQA should be clearly acknowledged as the source. If it is to be used for any other purpose, written permission must be obtained from permissions@sqa.org.uk. Where the publication includes materials from sources other than SQA (secondary copyright), this material should only be reproduced for the purposes of examination or assessment. If it needs to be reproduced for any other purpose it is the centre s responsibility to obtain the necessary copyright clearance. SQA s NQ Assessment team may be able to direct you to the secondary sources. These ing instructions have been prepared by examination teams for use by SQA appointed ers when ing external course assessments. This publication must not be reproduced for commercial or trade purposes.

2 General ing principles for Mathematics Always apply these general principles. Use them in conjunction with the detailed ing instructions, which identify the key features required in candidates responses. For each question, the ing instructions are generally in two sections: generic scheme this indicates why each is awarded illustrative scheme this covers methods which are commonly seen throughout the ing In general, you should use the illustrative scheme. Only use the generic scheme where a candidate has used a method not covered in the illustrative scheme. (a) (b) (c) (d) (e) (f) (g) Always use positive ing. This means candidates accumulate s for the demonstration of relevant skills, knowledge and understanding; s are not deducted for errors or omissions. If you are uncertain how to assess a specific candidate response because it is not covered by the general ing principles or the detailed ing instructions, you must seek guidance from your team leader. One is available for each. There are no half s. If a candidate s response contains an error, all working subsequent to this error must still be ed. Only award s if the level of difficulty in their working is similar to the level of difficulty in the illustrative scheme. Only award full s where the solution contains appropriate working. A correct answer with no working receives no, unless specifically mentioned in the ing instructions. Candidates may use any mathematically correct method to answer questions, except in cases where a particular method is specified or excluded. If an error is trivial, casual or insignificant, for example 6 x 6 =, candidates lose the opportunity to gain a, except for instances such as the second example in point (h) below. page 0

3 (h) If a candidate makes a transcription error (question paper to script or within script), they lose the opportunity to gain the next process, for example This is a transcription error and so the is not awarded. This is no longer a solution of a quadratic equation, so the is not awarded. x x x x x 0 x The following example is an exception to the above This error is not treated as a transcription error, as the candidate deals with the intended quadratic equation. The candidate has been given the benefit of the doubt and all s awarded. x x x x x 0 ( x )( x) 0 x or (i) Horizontal/vertical ing If a question results in two pairs of solutions, apply the following technique, but only if indicated in the detailed ing instructions for the question. Example: x x y 5 y 7 Horizontal: 5 x and x Vertical: 5 x and y 5 6 y 5 and y 7 6 x and y 7 You must choose whichever method benefits the candidate, not a combination of both. (j) In final answers, candidates should simplify numerical values as far as possible unless specifically mentioned in the detailed ing instruction. For example 5 must be simplified to 5 or must be simplified to 5 0 must be simplified to must be simplified to 8* must be simplified to 5 *The square root of perfect squares up to and including 00 must be known. page 0

4 (k) (l) Commonly Observed Responses (COR) are shown in the ing instructions to help common and/or non-routine solutions. CORs may also be used as a guide when ing similar non-routine candidate responses. Do not penalise candidates for any of the following, unless specifically mentioned in the detailed ing instructions: working subsequent to a correct answer correct working in the wrong part of a question legitimate variations in numerical answers/algebraic expressions, for example angles in degrees rounded to nearest degree omission of units bad form (bad form only becomes bad form if subsequent working is correct), for example ( x x x )( x ) written as ( x x x ) x x 5x 8x 7x gains full credit repeated error within a question, but not between questions or papers (m) In any Show that question, where candidates have to arrive at a required result, the last is not awarded as a follow-through from a previous error, unless specified in the detailed ing instructions. (n) (o) (p) You must check all working carefully, even where a fundamental misunderstanding is apparent early in a candidate s response. You may still be able to award s later in the question so you must refer continually to the ing instructions. The appearance of the correct answer does not necessarily indicate that you can award all the available s to a candidate. You should legible scored-out working that has not been replaced. However, if the scored-out working has been replaced, you must only the replacement working. If candidates make multiple attempts using the same strategy and do not identify their final answer, all attempts and award the lowest. If candidates try different valid strategies, apply the above rule to attempts within each strategy and then award the highest. For example: Strategy attempt is worth s. Strategy attempt is worth. Strategy attempt is worth s. Strategy attempt is worth 5 s. From the attempts using strategy, the resultant would be. From the attempts using strategy, the resultant would be. In this case, award s. page 0

5 Detailed ing instructions for each question. (a) start differentiation x... apply chain rule and complete, differentiation 9x. For do not accept... unless subsequently corrected. x. For accept eg or. x x 9. For candidates who interpret sin x as sin x, is available for sin x cos x. (b) evidence use of quotient rule with denominator and one term of numerator correct complete differentiation 57x e 7x 5x... OR 5x 5x 5 7x e 7e 7x... 7e 5x 7x Alternative Method (Product Rule) 5x 5e 7x... 5x... 7e 7x Alternative Method (Logarithmic differentiation) dy ln y 5x ln 7x and... y dx dy 7 y5 dx 7 x page 05

6 . (c) start to differentiate product with one term correct complete differentiation of product differentiate remaining terms 5 dy cos x... OR ysin x... dx dy 6 cos x... dx 7 dy y 6 dx OR ysin x 8 express derivative explicitly in terms of x and y 8 dy 6 ysin x dx cos x y. 5 and 6 are not available where the differentiation of ycosx leads to only one term.. 8 is available only where dy appears more than once after completing differentiation. dx page 06

7 . state expression x 7 A B x x 5 x x 5 form equation and find one unknown x 7 Ax 5 Bx AND eg A find second unknown and write integral expression integrate,, B AND dx stated or x x5 implied by ln x ln x 5 c. and may be awarded where an attempt at factorisation leads to an incorrect linear denominator.. At disregard the omission of dx.. At disregard the omission of modulus signs.. is not available where the constant of integration has either been omitted or first appears after an incorrect logarithmic term. x 7 A B x x 5 x x 5 do not award x 7 A x 5 B x award AND eg A B AND x x5 stated or implied by dx award ln x ln x 5 c award page 07

8 . (a) state general term, 9 r 5 x r x 9 r simplify powers of x OR coefficients, 9 r 5 r 9 r OR x state simplified general term,, (complete simplification) 9 5 x r 9r r 9r. Where candidates write out a full binomial expansion,, and are not available unless the general term is identifiable in (b). 9. Candidates who write down 9 r 5 r 9 r x with no working receive full s. r. is unavailable to candidates who in (a) produce further incorrect simplification subsequent to a correct answer eg 9r 5 r becomes r or x 6r becomes x.. General term has not been isolated.. General term has been isolated. 9 9r x x r0 9 5 r x r 9 r 9 9r 5 r0 r x r r 9 r x x r r0 r r r 9 x r r0 r 9 r0 9 5 x r 9 5 x r 9 r r 9 r 9r r 9r Do not award. Award and. Disregard the incorrect use of the final equals sign. Award, and.. Binomial expression has been equated to general term. 5 x x 9 9 r 5 r x x 9 r Disregard the incorrect use of the equals sign. Award. page 08

9 (b) determine value of r r 5 evaluate term, Where candidates write out a full expansion may be awarded where this is complete and correct at least as far as the required term. 5 may be awarded only where the required term is clearly identified from the expansion.. 5 is not available to candidates who interpret the term independent of x as substituting x 0. Binomial expansion as far as required term. or x 5x 50x 500x x x x x x x x x x page 09

10 . (a) state conjugate z p 6i stated or implied substitute for z, z, expand and apply i. At accept p i pi 8., 8 p p i. To award i must be multiplied by another complex number. Candidates find zz : p pi i 8 do not award award (b) find value of p 5. start process obtain remainder of 7 express gcd in terms of 06 and 9 obtain a and b, (9 68 5) (5 7) 7 9 (06 9) a, b 5. At the gcd and the final line of working do not have to be stated explicitly.. The minimum requirement for is 06 9 (5) 7.. Do not accept where the values of a and b have not been explicitly stated. page 0

11 6. find dy dt complete differentiation and, relate derivatives dy dt t dy dx t dt and dy dt stated or dx dx dt implied at 5 evaluate gradient dy dx 9 find coordinates 0 x, y state equation of tangent. Evidence for could include eg. Where candidates evaluate dx dt dy t. dx t and dy dt 5 9 y x 5 before finding dy dx, award for evaluating the individual derivatives and award for evaluating dy dx.. At 5 9 accept eg y x 5 = 0, y 9x 0. Do not accept y 0... or Incorrect expression for dy. dt dy 5 leading to y x may be awarded a maximum of /5. dt t 9 0 y x. 9 page

12 7. (a) state transpose of C 0 0 stated or implied by obtain matrix 5 0 C D k 5 C D k award only (b) (i) begin to find determinant + 0 k simplify expression k Expansion about the first row 0 k + k k k 0 k k 6 k (ii) 5 state value of k 5 page

13 8. differentiate du cos d find limits for u u, u rewrite integral, / u du,5,6,7 integrate and evaluate 5 u 5 and / 80 (or 0 875). is available where candidates either omit limits or retain limits for.. Where candidates attempt to integrate an expression containing both u and, where is either inside the integrand or erroneously taken outside as a constant, only and may be available.. Where candidates do not change limits but who produce working leading to sin 5, may 5 6 be awarded.. For candidates who arrive at sin 5 by inspection full s are still available For candidates who integrate incorrectly, may be available provided division by zero does not occur. 6. is not available to candidates who write limits using degrees. 7. At accept decimal answers rounded to at least significant figures. Integration by parts π sin θ cosθ dθ sin sin 8sin cos sin d π 6 π 6 π 6 sin sin cos 5 d sin θ cosθ dθ sin 6 6 sin θ cosθ dθ 6 80 page

14 9. (a) form the sum of three consecutive integers,,,,5 n n n communication,5 n which is divisible by. Candidates may form the sum n n n leading to n at.. Withhold where candidates construct an expression of the form a and a. may be available. eg n n n leading to n which is divisible by. an an an, where. Where candidates equate an expression for the sum of consecutive integers to a multiple of see commonly observed responses.. At accept an expression such as n n n. 5. Withhold and where candidates form one (or more) sum of specific consecutive numbers eg. A. x x x k x k x k, where k x and k is divisible by. Award / B. x x x k x k x k, where k x. The candidate has defined k but has not communicated divisibility. Award but not. C. k k k k 6 k 6 k therefore statement is true The candidate has explicitly divided and there is no requirement to state that k is an integer. Award /. page

15 9. (b) appropriate form for odd number, decomposed into two consecutive integers,,. Where candidates write down k k k, k k k and omit k, award. k. Where candidates omit brackets and write down k k do not award unless the candidate demonstrates that k and k are two consecutive integers eg writing k, k.. Where candidates begin with consecutive integers, may be awarded only where k is associated with any odd integer and not by a restatement of the assertion in the question. A. k k k odd Do not award. The candidate has not defined a general odd integer. B. k k k and k represents any odd integer. Award. The candidate has defined a general odd integer. C. k k k k Award. The candidate has begun with the general form for an odd integer. page 5

16 0. substitute, collect real and imaginary parts and equate moduli process to obtain a linear equation in x and y sketch consistent with equation, x iy ( x ) ( y ) i eg y x complete sketch OR interpret equation OR line connecting 0,0 and, begin sketch of locus A straight line exhibiting any one of: bisection, perpendicularity, 0, or,0 complete annotations, A straight line exhibiting any two of: bisection, perpendicularity, 0, or,0 OR interpret separate loci interpret conditions OR two circles, centres, 0,0 and circles are congruent and intersect identify required locus, common chord extended beyond intersection points page 6

17 . At accept any line passing through an appropriate point, which has positive gradient.. Do not withhold for axes which are unlabelled. Accept x and y in lieu of Re and Im. Disregard any appearance of i in the diagram. page 7

18 . (a) π cos sin obtain A π π sin cos. Accept. (b) 0 obtain B 0 (c) correct order for multiplication (P BA) 0 0 multiplication completed and, appearance of exact values. Common factor not required for.. is unavailable to candidates who have incorrectly identified B 0 0. Incorrect order of multiplication 0 0 Do not award. is still available on follow through. page 8

19 (d) 5 valid explanation,,, 5 eg compare the elements of P with the general form of a rotation matrix. 5 may be awarded where a candidate s explanation makes reference to the specific entries of the leading diagonal of P ( they should be equal but are not ) or trailing diagonal ( one must be negative of the other but is not ). Withhold 5 for a statement such as Matrix P represents a reflection unless the reflection is specified eg Matrix P is a reflection in the line y x.. 5 may also be awarded where candidates investigate the images of at least two points, neither of which is O.. Candidates who respond with reference to the constituent transformations of P may be awarded 5 only where there is reference to both transformations and communication demonstrates understanding that an even number of reflections are required in order to produce a rotation. Examples of responses where 5 would be awarded cos sin P is not associated with rotation about the origin as it is in the form. sin cos (Explanation makes explicit reference to their form of P and implicitly refers to the form of a general rotation matrix.) Because P can no longer be expressed as a certain angle of rotation. cos cos cannot be expressed as an angle. cos sin and as the angles found are not all equal, this sin cos transformation is not a rotation about the origin. Examples of responses where 5 would not be awarded cos sin As matrix P doesn t exhibit. (No reference made to form of P). sin cos Because it gets reflected in the process. Since it is followed by a reflection. cos values different so do not relate to same angle. Rotation is around a point on graph not the origin. It would put the point back to where it started. a b Rotations must have the matrix. b a page 9

20 Question Generic Scheme Illustrative Scheme. show true for n LHS: RHS: 0 So true for n Mark 5 assume (statement) true for n k AND consider whether (statement) true for nk,7 Suitable statement and k r k r AND k r r correct statement for sum to k terms using inductive hypothesis combine terms in k k k k 5 express sum explicitly in terms of k or achieve stated aim/goal 5,6,7 AND communicate 5 ( ) k If true for n k then true for nk. Also shown true for n therefore, by induction, true for all positive integers n. page 0

21 Question Generic Scheme Illustrative Scheme Mark. RHS, LHS and/or True for n are insufficient for the award of. A candidate must demonstrate evidence of substitution into both expressions.. For acceptable phrases for n k contain: If true for... ; Suppose true for ; Assume true for. For unacceptable phrases for n k contain: Consider n k, assume n k and True for n k. An acceptable phrase may appear at 5. For, in addition to an acceptable phrase containing n k, accept: k r k Aim/goal:. r For unacceptable phrases for n k contain: Consider true for n k, true for n k ; k r k (with no further working) r. At accept k k or k k.. At accept... k is unavailable to candidates who write down the correct expression without algebraic justification. 6. Full s are available to candidates who state an aim/goal earlier in the proof and who subsequently achieve the stated aim/goal. 7. Following the required algebra and statement of the inductive hypothesis, the minimum acceptable response for 5 is Then true for n k, but since true for n, then true for all n or equivalent. page

22 . (a) Determine the relationship x h 500 between x and h h 500 x. Refer to general ing principle (m). h 5 x leading to correct answer award 5 h x leading to h 50 x do not award 5 h x leading to h 5 x or equivalent award h 5 x moving directly to correct answer award (b) interpret rate of change of x dx 0 dt 5 find dh dx dh x 500 x dx form relationship 6 dh dh dx stated or implied at 5 dt dx dt 5 multiply by dx dt,,6 5 dh dt 0x 500 x 6 evaluate dh dt,,5,6 dh 9 cms dt 0 6 dh. At 5 candidates may evaluate the derivatives separately eg dt. At 5 simplification is not required.. At 6 units are required. Accept decimal equivalent (0 5 cms ).. Where candidates produce an incorrect answer, accept a decimal rounded to at least significant figures. 5. Award 6 only where a candidate s final answer for dh is opposite in sign to that of dx dt dt. 6. For candidates who do not show evidence of related rates,, 5 and 6 are not available. page

23 . (a) (i) multiply first term by a power of the common ratio,, 80 find term, At accept any integer index other than 0 or.. Where candidates elect to repeatedly multiply, the minimum acceptable response for is At accept 0.. Award full s for a correct answer with no working. (ii) substitute,, 80 find sum to infinity,, 0. A correct answer without working receives no credit.. and are available only where a formula has been used.. At candidates may use either the formula for the sum to infinity or apply a limiting argument using the formula for the sum to n terms (of a geometric progression). page

24 . (b) (i) 5 substitute 6 find common difference. For a correct answer without working award 0/. 5 eg d (ii) 7 find simplified expression n (c) 8 set up equation n n 9 obtain quadratic equation in general form 9 6n 76n find values of n 0 n, n 9. At 9 ' 0 ' must appear.. At 0 candidates must obtain positive integer solutions. page

25 5. (a) start integration by parts,,,,6 x cosx... complete integration by parts,,,,6... cos xdx complete integration,,,,5,6 x cosx sinx c 9. When integrating, candidates who repeatedly multiply by cannot be awarded but and may still be available.. Candidates who communicate an intention to integrate sin x and differentiate x but who inadvertently produce the derivatives of both sin x and x cannot be awarded but and are still available.. For candidates who choose sin x as the function to differentiate and x as the function to integrate, is available only where this is processed correctly. and are not available.. An error in sign when differentiating or integrating a trigonometric function should not be treated as a repeated error. 5. Do not withhold for the omission of the constant of integration. 6. The evidence for, and may appear in (b). page 5

26 5. (b) identify integral form of, integrating factor 5 determine integrating factor, begin solution rewrite as integral equation integrate,5,6 evaluate constant,6,7,8 0 form particular solution,6,7,8 dx e x 5 x d sin y x x dx x x implied at y xsinxdx x 8 x y cosx sinx c x stated or π c π y x cosx x sinx x 9. Candidates who attempt to solve the equation using eg separation of variables or second order method receive 0/7.. Candidates who attempt to apply integration by parts to the entire differential equation receive 0/7.. At 5 accept an unsimplified integrating factor eg ln x e.. For candidates who omit the constant of integration, 8, 9 and 0 are not available is available only where a candidate integrates correctly based on their RHS at Where candidates obtain an integrating factor which is a constant, 5, 9 and 0 are not available. 7. For candidates who proceed from 8 by multiplying through by x : award 9 for multiplying through by x and award 0 for evaluating the constant of integration then stating the particular solution. 8. For candidates who proceed from 8 by multiplying through by x but who fail to multiply the constant of integration, 9 and 0 are not available. 7 page 6

27 6. (a) set up augmented matrix 5 7 a obtain two zeros a 7 9 complete row operations, a 8 0 obtain value for a a = 8. Only Gaussian elimination (ie a systematic approach using EROs) is acceptable for the award of and.. For accept any equivalent form.. is not available unless the candidate s augmented matrix exhibits redundancy. (b) 5 introduce parameter and substitute, 6 equation of line, 6 5 z t, y 5t x 9t, y 5t, z t. 5 and 6 are not available for substituting in either a numerical value or any expression in terms of a.. 5 is not available where the candidate substitutes into a row containing two other variables.. For 6 accept symmetric or vector form. 9 For the line, d n 5 5 n. eg Let z 0 so that x y. Award 5 Form second equation eg x 5y and solve x y z to give x, y leading to. Award page 7

28 6. (c) 7 8 write down normals start to find angle, 7, 5 stated or implied 8 cos OR cos find acute angle,, At 7 accept the use of Accept an answer in degrees which rounds to.. 9 is not available for incorrect working subsequent to a correct answer eg 90.. At 7 accept eg or 5 but not at For candidates who express an answer in degrees, the degree symbol must appear. Solution obtained by rearrangement of the vector product formula sin award 8 page 8

29 Question Generic Scheme Illustrative Scheme 6. (d) 0 explanation,, 0 Planes and are parallel because the normal of is a multiple of the normal of. Mark. For the award of 0 a statement must compare normal vectors or coefficients of x, y and z Accept eg or The normals are multiples of one another as justification for the planes being parallel.. Do not accept a plane equating to a vector eg π 5.. Withhold 0 from candidates who provide a correct description but who subsequently write π π or make reference to direction vectors. eg The planes are parallel because:. their normals are multiples of each other. Award 0 π π. Do not award 0.. their direction vectors are multiples of each other. Do not award 0 page 9

30 7. (a) Method first derivative and two evaluations OR all three derivatives OR all four evaluations obtain expression Method write down Maclaurin series for e x substitute. Simplification might not appear until (c) Method x f x e f f x e x f x e x f x 8e x 0 f 0 f 0 f 0 8 f x x x x... Method x x e x x...!! f x x x x... page 0

31 7. (b) (i) find g x evidence of product rule, 5 complete proof, g x sec xsec x tan x g x sec x tan x 5 g x sec x sec x sec x tan x tan x. Candidates can be awarded only where the product or quotient rule is required to differentiate their expression for g x.. At there must be clear evidence of the product rule (or quotient rule).. 5 is not available to candidates who obtain an incorrect answer at.. 5 can be awarded only where the candidate completes the differentiation correctly and shows clearly that the result is equivalent to the expression asked for in the question. (ii) 6 completes ALL evaluations 7 substitute 6 g 0 0 g0 g 0 0 g 0 7 g x x x is available only for powers of x with numerical coefficients. page

32 7. (c) 8 multiply expressions x x... x x multiply out and simplify 9 7 x x x.... For candidates who proceed via differentiation 8 is available for obtaining all three derivatives correctly.. 9 is available only for powers of x with numerical coefficients. (d) 0 write down terms 0 x 7x [END OF MARKING INSTRUCTIONS] page