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IB Questionbank Mathematical Studies 3rd edition 1

IB Math Studies Year II – Midterm Review Questions KEY

1. (a) Area = 2

1× 14 × 8 sin110° (M1)

= 52.62278676 m2

= 52.6 m2 (3s.f) (A1)

(b) 18

110sin

8

sin C (or equivalent) (M1)

sin C = 18

110sin8

C = 24.68575369

C = 24.7° (3s.f.) (A1)

Note: Accept all answers obtained from all appropriate

methods, given to the correct degree of accuracy. [4]

2. (a) Compound interest (A1)

8% per year (A1)

(b)

Year Value at beginning

of year

Value at end of year

1st CHF 500 CHF 540

2nd CHF 540 CHF 583.20

3rd CHF 583.20 CHF 629.86

4th CHF 629.86 CHF 680.25

5th CHF 680.25 CHF 734.67 (A1)

6th CHF 734.67 CHF 793.44 (A1)

[4]

3. (a) BC = 117cos)57)(48(25748 22 (or equivalent) (M1)

89.7 m (3 s.f.) (A1)

(b) Area of ABC = 2

1ab sinC =

2

1(48)(57)sin117° (M1)

= 1220 m2 (3 s.f.) (A1)

[4]


4. (a) BAC ˆ = 180 – 2×23° (M1)

= 134° (A1) (C2)

(b) 134sin

15

23sin

AB (M1)

Note: Follow through with candidate’s answer from (a)

AB = 134sin

23sin15

AB = 8.147702831...

= 8.15 (3 s.f.) (A1) (C2) [4]

5. (a) III (A1)

(b) I (A1)

(c) II (A1)

(d) IV (A1) [4]

6. Note: Award (A1) for each pair of correct entries in parts (a) and (c).

A list of numbers with no set brackets is acceptable.

(a) A B = {1, 3, 4, 7, 8, 9} (A1)(A1)(A1) (C3)

(b) A B C = {9} (A1) (C1)

(c) A = {1, 3, 4, 7, 8, 9} (A1)

A C = {6, 7} (A1)

(A C) B = {3, 6, 7, 9} (A1)(A1) (C4) [8]

7. (a) u1 + 3d = 12 (A1)(A1)

u1 + 9d = 42 (A1)(A1) (C4)

Note: Award (A1) for left hand side correct, (A1) for right hand

side correct.


(b) 6d = 30 (A1)

d = 5 (A1)

u1 = –3 (M1)(A1) (C4)

Note: Follow through (ft) from candidate's equations. [8]

8. (a) (x – 3)(x + 1) (A1)(A1) (C2)

Note: Award (A0)(A1) if the signs are reversed.

(b) A(1, 0), B(3, 0) (A1)(A1) (C2)

(c) x = 1 or x = 2

)31( = 1 or x =

)1(2

)2( = 1 (A1)(A1) (C2)

Note: Award (A1) for x = and (A1) for 1.

(d) C(1, –4) (A1)(A1) (C2) [8]

9. (a) Angle A = 90 – 5 = 85°. (M1)(A1) (C2)

(b) BC2 = 6

2 + 8

2 – 2 × 8 × 6 cos(85°) (M1)(A1)

so BC = 6330487.91 = 9.57 (3 s.f.) (A1) (C3)

(c) (B)sin

AC

(A)sin

BC (M1)

sin (B) = 572515275.9

)85(sin6 = 0.6244093654 (A1)

Angle B = sin–1

(0.6244093654) = 38.6° (A1) (C3)

Note: Allow 38.7° if obtained using 9.57. [8]


10.

U U

A AB B

C C

(a) (b)

(A2)(A2)

U U

A B

C

(c) (d)

A B

C (A2)(A2)

Note: Award (A0), (A0), (A2) ft, (A2) ft if and are

consistently reversed. [8]

11. (a) I = 0.04 × 2000 × 18 = 1440 Euros (M1)(A1)

Total amount = I + 2000 = 3440 Euros. (M1)(A1) (C4)

(b) 2000

1218

12

036.01 (M1)(A1)

= 3819.72 (A1)

= 3820 Euros, to nearest Euro. (A1) (C4) [8]

12. (a) Put x = 0 to find y = –2 (M1)

Coordinates are (0, –2) (A1) (C2)

Note: Award (M1)(A0) for –2 if working is shown. If not, award

(M0)(A0).


(b) Factorise fully, y = (x – 2) (x + 1). (A1)(A1)

y = 0 when x = –1, 2. (A1)(A1)

Coordinates are A(–1, 0), B(2, 0). (A1)(A1) (C6)

Note: Award (C2) for each correct x value if no method shown

and full coordinates not given. If the quadratic formula is used

correctly award (M1)(A1)(A1)(A1)(A1)(A1). If the formula is

incorrect award only the last (A1)(A1) as ft. [8]

13. (a) R2 = 36 so R = 6 cm (M1)(A1) (C2)

(b) Use cosine rule. AB2 = 6

2 + 6

2 – 2 6 6 cos (110 ) (M1)(A1)(ft)

AB2 = 96.6

AB = 9.83 cm (A1)(ft)

OR

)(110sin

AB

)(35sin

6 (M1)(A1)(ft)

AB = 9.83 cm (A1)(ft)

OR

552

110

sin (55 ) = 6

AB2

1

(M1)(A1)(ft)

AB = 9.83 (A1)(ft) (C3)

Note: If this method is used, then the 2

1 AB must be evident to

obtain the (M1) and the

first (A1) requires the 55 and the 6 to be correct.

(c) L = 36 or 6 or 10.6 cm (A1) (C1)

[6]

14. (a) AC2 = 9 + 9=18 (M1)

AC = 18 (= 4.24) (A1) (C2)


(b) Area of triangle ACD = 0.5 18 4.5 sin 25 (M1)(A1)

= 4.03 (A1) (C3)

(c) Area of triangle ABC = 0.5 3 3 = 4.5 cm2

(M1)(A1)

Total area = 8.53 (A1) (C3) [8]

15. (a)

correct

correct

correct

incorrect

incorrect

incorrect25

24

35

24

34

14

(A2) (C2)

(b) (i) 4

2

5

3

4

3

5

2 (A1)(A1)

Note: Award (A1) for each correct product.

= 20

12 (= 0.6) (A1) (C3)

(ii) 4

1

10

1

10

34

1

5

2

= (0.25) (A1)(A1)(A1) (C3)

Note: Award (A1) for 4

1

5

2 seen and (A1) for

10

1

10

3 seen.

[8]


16. (a)

(A4) (C4)

Note: Award (A1) for some indication of scale on the y-axis.

Award (A1) for at least one asymptote drawn. Award (A1) for

each of the two (smooth) branches. The left hand branch must

pass through 0. One branch should be above the horizontal

asymptote and the other below but if the asymptote is not

drawn, then there should be little or no overlap in heights of the

branches. If this condition is not fulfilled, award (A1)(A0) for

the curve.

(b) (i) Horizontal asymptote y = 3 (A1)

(ii) Vertical asymptote x = 1 (A1)(ft) (C2)

Equations for x and y must be seen, (ft) if reversed. [6]

17. (a) 400

30sin50sin

AC (M1)(A1)

Note: Award (M1) for using sine rule with values from the

problem, (A1) for correct substitution.

AC = 613 (3 s.f.) (A1) (C3)


(b) Perimeter = 400 + 613 + 788 = 1801m

Time in seconds = 10008.1

1801 (A1)(ft)(A1)

Note: Award (A1) for the perimeter, (A1) for finding the time in

seconds, and last (A1)(ft) for finding the time in minutes. The

time in minutes follow through from the time in seconds.

Time in minutes = .).37.16(3

50

60

1000fsto (A1)(ft) (C3)

[6]

18. (a) 2

1

8

a

a = 4 (A1)

OR

2

12

a

a = 4 (A1) (C1)

(b) 0.06252

18

7

(M1)(A1)(ft)

OR

0.06252

12

5

(M1)(A1)(ft) (C2)

(c) 4095/256) ().3(0.16

12

1

12

18

12

fs (M1)(A1)(ft)

(A1)(ft) (C3)

Note: Award (M1) for using correct formula and correct

substitution, (A1) for correct answer (15.99...). (A1) for correct

answer to 3 s.f. [6]


19. (a) 6x + 3 – 6 + 2x = 13

8x = 16 (M1)

x = 2 (A1) (C2)

(b) (x + 3) (x – 1) (A1)(A1) (C2)

(c) x = 1.64575..

x = 1.65 (A2) (C2)

Note: If formula is used award (M1)(A1) for correct solution. If

graph is sketched award (M1)(A1) for correct solution. [6]

20. (a) 19 seedlings (A1) (C1)

(b) (i) median 88 cm (A1)

(ii) 1st quartile 78 cm, 3

rd quartile 103 cm (both correct) (A1) (C2)

(c) 112 63 = 49 cm (A1) (C1)

Note: Accept 63 and 112 both seen, if they appear in the answer space

for (c) or under working for (c) (but not just implied or written

on the box plot).

(d) (C2)

Notes: Box with correct median and quartiles marked. (A1)(ft)

Both correct whiskers joined to box with straight lines (A1)(ft) (C2)

Allow maximum errors of 2.

Perfectly ruled lines are not essential. [6]


21.

(A1) (A1)

(A1)

(A1)

(A1)

(A1) (C6)

Notes: For any number entered exactly once, in the correct position,

award (A1) if incorrect award (A0).

If all numbers entered in all regions award (A0).

If any number is entered in more than one region, penalize that

number as follows:

(i) If none of the regions is correct award (A0)

(ii) If one of the regions is correct but other appearances of

that number are in the COMPLEMENT of the correct set,

award (A0) the first time this is seen.

(iii) If one of the regions is correct but other appearances of

that number are in a SUBSET of the correct set award (A0) the

first time this is seen.

Apply each of (ii) and (iii) at most once and award ft marks

when the error is seen repeatedly, however, (ii) and (iii) may

not both be applied to the same number and if both these errors

are present with more than one number involved, follow

through cannot be used until both penalties have been applied. [6]

22. (a) d = –7 (A1) (C1)

(b) S50 = 2

50(2(124) + 49(–7)) (M1)

Note: (M1) for correct substitution.


= –2375 (A1)(ft) (C2)


(c) 124 – 7(k – 1) < 0 (M1)

k > 18.7 or 18.7 seen (A1)(ft)

k = 19 (A1)(ft) (C3)

Note: (M1) for correct inequality or equation seen or for list of

values seen or for use of trial and error. [6]

23. (a) 0.965 (A1) (C1)

(b) y = 1.15x + 0.976

(A1) for 1.15x (A1) for +0.976 (A1)(A1) (C2)

(c) y = 1.15 (7) + 0.976 (M1)

Chemistry = 9.03 (accept 9) (A1)(ft) (C2)

Note: Follow through from candidate’s answer to (b) even if no

working is seen. Award (A2)(ft).

(d) the correlation coefficient is close to 1

OR strongly correlated variables

OR 7 lies within the range of physics marks. (R1) (C1) [6]

24. (a) (x − 5) (x + 2) (A1)(A1)

Note: Award (A1) for (x + 5)(x−2), (A0) otherwise.

If equation is equated to zero and solved by factorizing

award (A1) for both correct factors, followed by (A0). (C2)

(b) (i) −3, −2, −1, 0, 1, 2, 3 (A1)(A1)

Notes: Award (A2) for all correct answers seen

and no others.

Award (A1) for 3 correct answers seen. (C2)

(ii) −26,−7, 0, 1, 2, 9, 28 (A1)(A1)

Notes: Award (A2) for all correct answers seen

and no others.

Award (A1) for 3 correct answers seen.

If domain and range are interchanged award

(A0) for (b)(i) and (A1)(ft)(A1)(ft) for (b)(ii). (C2) [6]


25. (a) To double, interest = 3000 (A1)

3000 = 100

43000 n (M1)

Note: For substituting into the simple interest formula

n = 25 years (A1)(ft)

Note: (A1) for 3000 on one side of equation if not seen

separately.

For interest of 6000 award (M1)(A1)(ft) for answers

of 50 years. (C3)

(b) 6000 = 3000

n2

200

5.31 (M1)(A1)

Note: (M1) for substituting values into a compound

interest formula,

(A1) for correct values with a variable for the power.

n = 20 years (A1)

Note: If n used in formula instead of 2n, can allow

as long as final answer is halved to get 20. (C3) [6]

26. (a) FV = 8000 (1.0125)60

(M1)(A1)

Note: (M1) for substituting in compound interest

formula, (A1) for correct substitution

€16857 only (A1) (C3)

(b) 8000 (1.0125)n = 9058.17 (M1)

Note: (M1) for equating compound interest formula

to 9058.17

n =10 correct answer only (A1)

So 30 months, (ft) on their n (A1)(ft)

Note: Award (C2) for 2.5 seen with no working (C3) [6]


27. (a) 20 = u1 + 3d (A1)

32 = u1 + 7d (A1)

Note: Award (A1) for each equation, (A1) for correct answer.

OR

d = 4

2032 (A1)(A1)

Note: Award (A1) for numerator, (A1) for denominator.

d = 3 (A1) (C3)

(b) 2

10(2 × 11 + 9 × 3) or

2

10(11 + 38) (M1)(A1)(ft)

Note: Award (M1) for correct substituted formula, (A1) for

correct substitution, follow through from their answer to part

(a).

OR

11 + 14 + ... + 38 (M1)(A1)(ft)

Note: Award (M1) for attempt at the sum of a list, (A1)(ft) for

all correct numbers, follow through from their answer to part

(a).

= 245 (A1)(ft) (C3) [6]

28. (a) 3 (A1) (C1)

(b) −1/3 (ft) from (a) (A1)(ft) (C1)

(c) Substituting (6, 7) in y = their mx + c or equivalent to find c. (M1)

y = 93

1x or equivalent (A1)(ft) (C2)

(d) (1.5, 8.5) (A1)(A1)(ft)

Note: Award (A1) for 1.5, (A1) for 8.5. (ft) from (c),

brackets not required. (C2) [6]


29. (a)

(A1)(A1)(A1) (C3)

Note: Award (A1) for a labeled Venn diagram with appropriate

sets.

(A1) for 7, (A1) for 8 and 5.

(b) P (Spanish / one language only) =

20

5

20

8

20

8

(M1)(A1)(ft)

Note: Award (M1) for substituted conditional probability

formula, (A1) for correct substitution. Follow through from

their Venn diagram.

= 13

8 (0.615, 61.5%) (A1)(ft)

OR

P (Spanish / one language only) = 58

8 (A1)(ft)(M1)

Note: Award (A1) for their correct numerator, (M1) for correct

recognition of regions.

Follow through from their Venn diagram.

= 13

8 (0.615, 61.5%) (A1)(ft) (C3)

[6]

30. Financial penalty applies in part (a)

(a) I = 1200 1200600

2.71

125

(M1)(A1)

FP I = 518.15 euros (A1) (C3)

Notes: Award (M1) for substitution in the compound interest

formula, (A1) for correct substitutions, (A1) for correct answer.

If final amount found is 1718.15 and working shown award

(M1) (A1)(A0).

(b) 518.15 = 100

51200 r (M1)(A1)(ft)

r = 8.64 % (% sign not required) (A1)(ft) (C3)


Note: Award (M1) for substitution in the simple interest

formula, (A1)(ft) for correct substitution, (A1)(ft) for answer. [6]

31. (a) = 150

91 (0.607, %6.60 , 60.7%) (A1)(A1) (C2)


(b) = %74,74.0,50

37

150

111 (A1)(ft)(A1) (C2)

Note: Award (A1)(ft) for their numerator in (a) +20 provided

the final answer is not greater than 1. (A1) for denominator.

(c) 91

16 (0.176, 17.6%) (A1)(A1)(ft) (C2)

Note: Award (A1) for numerator and (A1)(ft) for denominator.

Follow through from their numerator in (a) provided answer is

not greater than 1. [6]

32.

(A1)(A1)(A1)

(A1)(A1)(A1) (C6)

Note: Award (A1) for each number placed once in the correct

region. Accept equivalent forms for numbers. [6]

33. (a) (i) 8.5 (cm) (A1)

(ii) 120° (A1)


(iii) 30° (A1) (C3)

(b) 30sin

5.8

sin120

BC (M1)(A1)(ft)

Note: Award (M1) for correct substituted formula, (A1) for

correct substitutions.

BC = 14.7 2

317 (A1)(ft) (C3)

[6]

34. (a) (x + 8)2 = (x + 7)

2 + x

2 (A1)

Note: Award (A1) for a correct equation.

x2 + 16x + 64 = x

2 + 14x + 49 + x

2 (A1)

Note: Award (A1) for correctly removed parentheses.

x2 – 2x –15 = 0 (A1) (C3)

Note: Accept any equivalent form.

(b) x = 5, x = –3 (A1)(ft)(A1)(ft) (C2)

Notes: Accept (A1)(ft) only from the candidate’s quadratic

equation.

(c) 30 cm (A1)(ft) (C1)

Note: Follow through from a positive answer found in part (b). [6]

35. Financial penalty applies in parts (b) and (c).

(a) 0.88×16000 OR 0.12×16000 OR 1920 (M1)

14080 (A1) (C2)

(b) 1.6407×5.25×14080 (M1)

FP 121280.54 USD (A1)(ft) (C2)

Note: Follow through from their answer to part (a).

(c) 12 × 8739.0

1 (M1)

FP 13.73 AUD (A1) (C2)

Note: If division used in part (b) and multiplication used in part

(c), award (M0)(A0) for part (b) and (M1)(A1)(ft) for part (c). [6]


36. Unit penalty applies in part (b).

(a) sin 9

4DB̂A (M1)

100 + their D)B̂(A (M1)

126° (A1) (C3)

Notes: Accept an equivalent trigonometrical equation involving

angle ABD for the first (M1).

Radians used gives 100°. Award at most (M1)(M1)(A0) if

working shown.

BD = 8 m leading to 127°. Award at most (M1)(M1)(A0)

(premature rounding).

(b) AC2 = 10

2 + 9

2 – 2 × 10 × 9 × cos(126.38...) (M1)(A1)

Notes: Award (M1) for substituted cosine formula.

Award (A1) for correct substitution using their answer to part

(a).

UP AC = 17.0 m (A1)(ft) (C3)

Notes: Accept 16.9 m for using 126.

Follow through from their answer to part (a).

Radians used gives 5.08. Award at most (M1)(A1)(A0)(ft) if

working shown. [6]

37. (a) %2.43,432.0,125

54

250

108 (A1)(A1) (C2)


(b) 106

25 (0.236, 23.6%) (A1)(A1) (C2)


(c) 170

71 (0.418, 41.8%) (A1)(A1) (C2)

Note: Award (A1) for numerator, (A1) for denominator. [6]

38. (a) –4, –3, –2, –1, 0, 1, 2 (A1) (C1)

Note: Award (A1) for correct numbers, do not penalise if

braces, brackets or parentheses seen.

(b) 7

4 (0.571, 57.1%) (A1)(ft)(A1)(ft) (C2)


Notes: Award (A1)(ft) for numerator, (A1)(ft) for denominator.

Follow through from part (a).

Note: There is no further penalty in parts (c) and (d) for use of

denominator consistent with that in part (b).

(c) 7

1 (0.143, 14.3%) (A1)(ft) (C1)

Note: Follow through from part (a).

(d) 7

1 (0.143, 14.3%) (A1)(ft)(A1)(ft) (C2)

Note: Award (A1)(ft) for numerator, (A1)(ft) for denominator.

Follow through from part (a). [6]

39. (a) 80

192221160 (M1)

Note: Award (M1) for substituting correct values into mean

formula.

1.75 (A1) (C2)

(b) An attempt to enumerate the number of goals scored. (M1)

2 (A1) (C2)


(c) 75.1

75.12 × 100 (M1)

14.3 % (A1)(ft) (C2)

Notes: Award (M1) for correctly substituted % error formula.

% sign not required.

Follow through from their answer to part (a).

If 100 is missing and answer incorrect award (M0)(A0).

If 100 is missing and answer incorrectly rounded award (M1)

(A1)(ft)(AP). [6]

40. (a) 1 (one) (A1) (C1)

Note: 6, {6} or {1} earns no marks.

(b) 1, 3, 5, 7, 9, 11 (A1) (C1)

Note: Do not penalise if braces, parentheses or brackets are

seen.

(c)

(A1)(A1)(ft)(A1)(ft)(A1)(ft) (C4)

Notes: Award (A1) for the empty set CBA .

Award (A1)(ft) for the correct placement of 6, 5, 1 and 3.

Award (A1)(ft) for the correct placement of 2, 4, 12, 7, 9, 11, 8.

Award (A1)(ft) for the correct placement of 10.

Follow through from part (b). [6]


41. (a) x = 2

4 (M1)

x = 2 (A1)

OR

x

y

d

d = 4 – 2x (M1)

x = 2 (A1)

(2, 7) or x = 2, y = 7 (A1) (C3)

Notes: Award (M1)(A1)(A0) for 2, 7 without parentheses.

(b) (i) C labelled in correct position on graph (A1) (C1)

(ii) 3 = 3 + 4x – x2 (M1)

Note: Award (M1) for correct substitution of y = 3 into

quadratic.

(x =) 4 (A1) (C2)

OR

Using symmetry of graph x = 2 + 2 (M1)

Note: Follow through from their x-coordinate of the vertex.

(x =) 4 (A1)(ft) (C2) [6]


42. (a) r = 3

1

108

36 (A1) (C1)

Note: Accept 0.333.

(b) u1

7

3

1 = 36 (M1)

Note: Award (M1) for correct substitution in formula for nth

term of a GP. Accept equivalent forms.

u1 = 78732 (A1)(ft) (C2)

Notes: Accept 78700. Follow through from their common ratio

found in part (a). If 0.333 used from part (a) award

(M1)(A1)(ft) for an answer of 79285 or 79300 irrespective of

whether working is shown.

(c) 118096 =

3

11

3

1178732

k

(M1)(M1)

Notes: Award (M1) for correct substitution in the sum of a GP

formula, (M1) for equating their sum to 118096. Follow

through from parts (a) and (b).

OR

Sketch of the function y = 78732

3

11

3

11

k

(M1)

Indication of point where y = 118 096 (M1)

OR

78 732 + 26 244 + 8748 + 2916 + 972 + 324 + 108 + 36 + 12 + 4

= 118 096 (M1)(M1)

Note: Award (M1) for a list of at least 8 correct terms, (M1) for

the sum of the terms equated to 118096.

k = 10 (A1)(ft) (C3)

Notes: Follow through from parts (a) and (b). If k is not an

integer, do not award final (A1). Accept alternative methods.

If 0.333 and 79285 used award (M1)(M1)(A1)(ft) for k = 5.

If 0.333 and 79300 used award (M1)(M1)(A0). [6]


43. (a) 45000 + (5 – 1)1750 (M1)(A1)

Note: Award (M1) for substituted AP formula, (A1) for correct

substitutions.

= 52000 USD (A1) (C3)

Notes: If a list is used, award (M1) for recognizing AP, award

(A1) for seeing 52000 in their list, (A1) for final answer.

(b) 2

10(2(45000) + (10 – 1)(1750)) (M1)(A1)

Notes: Award (M1) for substituted AP formula, (A1)(ft) for

correct substitutions. Follow through from their common

difference used in part (a).

= 528750 USD (A1)(ft) (C3)

Notes: Accept 529000.

If a list is used, award (M1) for recognizing sum of AP, (A1) for

seeing 60750 included in the sum or 528750 in a cumulative

list. [6]

44. (a) (i) 06

20 (M1)

= 333.0,6

2

3

1 (A1) (C2)

(ii) y = 3

1x + 2 (A1)(ft) (C1)

Notes: Follow through from their gradient in part (a)(i).

Accept equivalent forms for the equation of a line.

(b) area = 2

5.16 (A1)(M1)

Note: Award (A1) for 1.5 seen, (M1) for use of triangle formula

with 6 seen.

= 4.5 (A1) (C3) [6]

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