Marking Scheme for Mathematics Examination: Method and Answer Marks, Lecture notes of Mathematics

Download Marking Scheme for Mathematics Examination: Method and Answer Marks and more Lecture notes Mathematics in PDF only on Docsity!

M15/5/MATME/SP2/ENG/TZ2/XX/M

MARKSCHEME

May 2015

MATHEMATICS

Standard level

Paper 2

18 pages

This markscheme is the property of the International Baccalaureate

and must not be reproduced or distributed to any other person

without the authorization of the IB Assessment Centre.

Examples

Correct answer seen Further working seen Action 1.

(incorrect decimal value)

Award the final A (ignore the further working)

sin 4

x sin x Do not award the final A

3. log a − log b log ( a − b ) Do not award the final A

3 N marks

If no working shown, award N marks for correct answers – this includes acceptable answers (see accuracy booklet). In this case, ignore mark breakdown ( M , A , R ). Where a student only shows a final incorrect answer with no working, even if that answer is a correct intermediate answer, award N.

• Do not award a mixture of N and other marks.
• There may be fewer N marks available than the total of M , A and R marks; this is deliberate as it penalizes candidates for not following the instruction to show their working.
• There may not be a direct relationship between the N marks and the implied marks. There are times when all the marks are implied, but the N marks are not the full marks: this indicates that we want to see some of the working, without specifying what.
• For consistency within the markscheme, N marks are noted for every part, even when these match the mark breakdown.
• If a candidate has incorrect working, which somehow results in a correct answer, do not award the N marks for this correct answer. However, if the candidate has indicated (usually by crossing out) that the working is to be ignored, award the N marks for the correct answer.

4 Implied and must be seen marks

Implied marks appear in brackets eg (M1).

• Implied marks can only be awarded if the work is seen or if implied in subsequent working (a correct final answer does not necessarily mean that the implied marks are all awarded). There are questions where some working is required, but as it is accepted that not everyone will write the same steps, all the marks are implied, but the N marks are not the full marks for the question.
• Normally the correct work is seen in the next line.
• Where there is an ( M1 ) followed by A1 for each correct answer, if no working shown, one correct answer is sufficient evidence to award the ( M1 ).

Must be seen marks appear without brackets eg M.

• Must be seen marks can only be awarded if the work is seen.
• If a must be seen mark is not awarded because work is missing (as opposed to M0 or A for incorrect work) all subsequent marks may be awarded if appropriate.

5 Follow through marks (only applied after an error is made)

Follow through ( FT ) marks are awarded where an incorrect answer (final or intermediate) from one part of a question is used correctly in subsequent part(s) or subpart(s). Usually, to award FT marks, there must be working present and not just a final answer based on an incorrect answer to a previous part. However, if the only marks awarded in a subpart are for the final answer, then FT marks should be awarded if appropriate. Examiners are expected to check student work in order to award FT marks where appropriate.

• Within a question part, once an error is made, no further A marks can be awarded for work which uses the error, but M and R marks may be awarded if appropriate. (However, as noted above, if an A mark is not awarded because work is missing, all subsequent marks may be awarded if appropriate).
• Exceptions to this rule will be explicitly noted on the markscheme.
• If the question becomes much simpler because of an error then use discretion to award fewer FT marks.
• If the error leads to an inappropriate value ( eg probability greater than 1, use of r > 1 for the sum of an infinite GP, sin θ = 1.5, non integer value where integer required), do not award the mark(s) for the final answer(s).
• The markscheme may use the word “ their ” in a description, to indicate that candidates may be using an incorrect value.
• If a candidate makes an error in one part, but gets the correct answer(s) to subsequent part(s), award marks as appropriate, unless the question says hence. It is often possible to use a different approach in subsequent parts that does not depend on the answer to previous parts.
• In a “show that” question, if an error in a previous subpart leads to not showing the required answer, do not award the final A1. Note that if the error occurs within the same subpart, the FT rules may result in further loss of marks.
• Where there are anticipated common errors, the FT answers are often noted on the markscheme, to help examiners. It should be stressed that these are not the only FT answers accepted, neither should N marks be awarded for these answers.

6 Mis-read

If a candidate incorrectly copies information from the question, this is a mis-read ( MR ). A candidate should be penalized only once for a particular mis-read. Use the MR stamp to indicate that this is a misread. Do not award the first mark in the question, even if this is an M mark, but award all others (if appropriate) so that the candidate only loses one mark for the misread.

• (^) If the question becomes much simpler because of the MR , then use discretion to award fewer marks.
• If the MR leads to an inappropriate value ( eg probability greater than 1, use of r > 1 for the sum of an infinite GP, sin θ = 1.5, non integer value where integer required), do not award the mark(s) for the final answer(s).
• Miscopying of candidates’ own work does not constitute a misread, it is an error.

7 Discretionary marks (d)

An examiner uses discretion to award a mark on the rare occasions when the markscheme does not cover the work seen. In such cases the annotation DM should be used and a brief note written next to the mark explaining this decision.

8 Alternative methods

Candidates will sometimes use methods other than those in the markscheme. Unless the question specifies a method, other correct methods should be marked in line with the markscheme. If in doubt, contact your team leader for advice.

• Alternative methods for complete parts are indicated by METHOD 1 , METHOD 2 , etc.
• Alternative solutions for parts of questions are indicated by EITHER... OR. Where possible, alignment will also be used to assist examiners in identifying where these alternatives start and finish.

13. Diagrams

The notes on how to allocate marks for sketches usually refer to passing through particular points or having certain features. These marks can only be awarded if the sketch is approximately the correct shape. All values given will be an approximate guide to where these points/features occur. In some questions, the first A1 is for the shape, in others, the marks are only for the points and/or features. In both cases, unless the shape is approximately correct, no marks can be awarded (unless otherwise stated). However, if the graph is based on previous calculations, FT marks should be awarded if appropriate.

14. Accuracy of Answers

If the level of accuracy is specified in the question, a mark will be allocated for giving the final answer to the required accuracy. When this is not specified in the question, all numerical answers should be given exactly or correct to three significant figures.

Do not accept unfinished numerical final answers such as 3/0.1 (unless otherwise stated). As a rule, numerical answers with more than one part (such as fractions) should be given using integers ( eg 6/8). Calculations which lead to integers should be completed, with the exception of fractions which are not whole numbers.

Intermediate values do not need to be given to the correct three significant figures. But, if candidates work with rounded values, this could lead to an incorrect answer , in which case award A0 for the final answer.

Where numerical answers are required as the final answer to a part of a question in the markscheme, the markscheme will show

a truncated 6 sf value, the exact value if applicable, and the correct 3 sf answer.

Units (which are generally not required) will appear in brackets at the end.

Section A

1. (a) evidence of choosing sine rule (M1)

eg ( ) (^) ( )

AC BC

sin ABCˆ^ sin BACˆ

correct substitution (A1)

eg

AC 10

sin 80 sin 35

AC =17.

AC = 17.2 (cm) A1 N

[3 marks]

(b) ACBˆ = 65 ^ (seen anywhere) (A1)

correct substitution (A1)

eg

10 17.1695 sin 65

× × × 

area =77.

area = 77.8 (cm )^2 A1 N

[3 marks]

Total [6 marks]

2. (a) (i) correct substitution (A1)

eg 6 × 2 + 3 × 2 + 6 × 1

u v  = 24 A1 N

(ii) correct substitution into magnitude formula for u or v (A1)

eg 62 + 32 + 6 ,^2 2 2 + 22 + 12 , correct value for v

u = 9 A1^ N

(iii) v = 3 A1 N

[5 marks]

(b) correct substitution into angle formula (A1)

eg

9 × 3

0.475882, 27.26604^ A1 N

[2 marks]

Total [7 marks]

5. (a)

A1A1A1 N

Note: Curve must be approximately correct exponential shape (increasing and concave up). Only if the shape is approximately correct, award the following: A1 for right end point in circle,

A1 for y -intercept in circle,

A1 for asymptotic to y = 2 , (must be above y = 2 ).

[3 marks]

(b) valid attempt to find g (M1)

eg f (^) ( x − (^3) )− 1 , g x ( ) = e x + −1 3+ 2 − 1 , e x + −1 3^ , 2 − 1 , sketch

g x ( ) = e x −^2 + 1 A2 N

[3 marks]

Total [6 marks]

6. METHOD 1

recognize that the distance walked each minute is a geometric sequence (M1)

eg r = 0.9, valid use of 0.

recognize that total distance walked is the sum of a geometric sequence (M1)

eg

n n

r

S a

r

correct substitution into the sum of a geometric sequence (A1)

eg

 −^ n 

any correct equation with sum of a geometric sequence (A1)

eg

n

 − = − n =

attempt to solve their equation involving the sum of a GP (M1) eg graph, algebraic approach

n = 16.54290788 A

since n > 15 R

he will be late AG N

Note: Do not award the R mark without the preceding A mark.

continued ...

7. attempt to set up equation (M1)

eg f = g , kx^2^ + kx = x −0.

rearranging their equation to equal zero M

eg kx^2^ + kx − x + 0.8 = 0, kx^2 + x k ( − 1) + 0.8 = 0

evidence of discriminant (if seen explicitly, not just in quadratic formula) (M1)

eg b^2^ − 4 ac , ∆ = ( k −1) 2 − 4 k × 0.8, D = 0

correct discriminant (A1)

eg ( k − 1)^2 − 4 k × 0.8, k^2 − 5.2 k + 1

evidence of correct discriminant greater than zero R

eg k^2 − 5.2 k + 1 > 0 , ( k − 1)^2 − 4 k × 0.8 > 0 , correct answer

both correct values (A1)

eg 0.2, 5

correct answer A2 N

eg k < 0.2, k ≠ 0, k > 5

[8 marks]

Section B

8. Note: The values of p and q found in (a) are used throughout the question. Please check FT

carefully on their values.

(a) attempt to find intersection (M1)

eg f = g

p = 1, q = 3 A1A1 N

[3 marks]

(b) f ′(^ p ) = − 1 A2 N

[2 marks]

(c) (i) correct approach to find the gradient of the normal (A1)

eg m m 1 2 = − 1 ,

f ( p )

, correct value of 1

EITHER

attempt to substitute coordinates (in any order) and correct

normal gradient to find c (M1)

eg

c

f p

= − × +

, 1 = 1 × 3 + c

c = 2 (A1)

y = x + 2 A1 N

OR

attempt to substitute coordinates (in any order) and correct normal gradient into equation of a straight line (M1)

eg

y x

f p

, y − 1 = 1 × ( x −3)

correct working

eg y = ( x −1) + 3 (A1)

y = x + 2 A1 N

(ii) (0, 2) A1 N

[5 marks]

(d) appropriate approach involving subtraction (M1)

eg ( ) d

b ∫ a (^) L^ − g^ x ,^ (^ )

3 x^2 − ( x +2)

∫

substitution of their limits or function (A1)

eg 0 ( ) d

p ∫ L^ − g^ x ,^ ∫(^ (^ x^ +^ 2)^ −^3 x^2 )

area = 1.5 A1 N

[3 marks] Total [13 marks]

Question 9 continued

(d) (i) correct approach (A1)

eg from t to 50.1, P (48.7 < X <50.1) , 0.

recognize conditional probability (seen anywhere, including in correct working) R

eg P (A|B)

correct substitution (A1)

eg

P (48.7 50.1) 0.

P ( 48.7) 0.

X

X

0.360 A1 N

(ii) P ( X ≥ 2) (A1)

attempt to find P ( X ≥ 2) (M1)

eg 1 − P ( X = 0) − P ( X = 1), P ( X = 2) + P ( X = 3)+…

recognize binomial distribution (M1)

eg X B( n , p )

0.924 A1 N

[8 marks]

Total [16 marks]

10. (a) area of ABCD = AB^2 (seen anywhere) (A1)

choose cosine rule to find a side of the square (M1) eg a^2^ = b^2 + c^2 − 2 bc cos θ

correct substitution (for triangle AOB) A

eg r^2 + r^2 − 2 × r × r cos θ, OA 2 + OB^2 − 2 × OA ×OBcos θ

correct working for AB^2 A

eg 2 r^2 − 2 r^2 cos θ

area = 2 r^2 (1 − cos θ) AG N

Note: Award no marks if the only working is 2 r^2 − 2 r^2 cos θ.

[4 marks]

(b) (i) 2

α r (accept 2 r^2 (1 − cos α)) A1 N

(ii) correct equation in one variable (A1)

eg

2 (1 cos )

− α = α

α = 0.511 (accept θ = 0.511) A2 N

Note: Award A1 for α =^ 0.511and additional answers.

[4 marks]

continued…