IB Math – Standard Level Year 2: May ‘ 11, Paper 2, TZ 1 Alei -­‐ Desert Academy 2011-­‐12 IB Practice Exam: 11 Paper 2 Zone 1 – 90 min, Calculator Allowed Name:__________________________________________Date: ________________ Class: ________ 1.
The following diagram shows triangle ABC.
diagram not to scale
AB = 7 cm, BC = 9 cm and
(a)
= 120°.
Find AC.
(3)
(b)
Find
.
(3)
(Total 6 marks)
2.
Let f(x) = 3x2. The graph of f is translated 1 unit to the right and 2 units down.
The graph of g is the image of the graph of f after this translation.
(a)
Write down the coordinates of the vertex of the graph of g.
(2)
(b)
Express g in the form g(x) = 3(x – p)2 + q.
(2)
The graph of h is the reflection of the graph of g in the x-axis.
(c)
Write down the coordinates of the vertex of the graph of h.
(2)
(Total 6 marks)
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IB Math – Standard Level Year 2: May ‘ 11, Paper 2, TZ 1 3.
Alei -­‐ Desert Academy 2011-­‐12 In an arithmetic sequence u1 = 7, u20 = 64 and un = 3709.
(a)
Find the value of the common difference.
(3)
(b)
Find the value of n.
(2)
(Total 5 marks)
4.
A random variable X is distributed normally with a mean of 20 and variance 9.
(a)
Find P(X ≤ 24.5).
(3)
(b)
Let P(X ≤ k) = 0.85.
(i)
Represent this information on the following diagram.
(ii)
Find the value of k.
(5)
(Total 8 marks)
5.
A box holds 240 eggs. The probability that an egg is brown is 0.05.
(a)
Find the expected number of brown eggs in the box.
(2)
(b)
Find the probability that there are 15 brown eggs in the box.
(2)
(c)
Find the probability that there are at least 10 brown eggs in the box.
(3)
(Total 7 marks)
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IB Math – Standard Level Year 2: May ‘ 11, Paper 2, TZ 1 6.
Alei -­‐ Desert Academy 2011-­‐12 Let f(x) = cos(x2) and g(x) = ex, for –1.5 ≤ x ≤ 0.5.
Find the area of the region enclosed by the graphs of f and g.
(Total 6 marks)
7.
A company uses two machines, A and B, to make boxes. Machine A makes 60 % of the boxes.
80 % of the boxes made by machine A pass inspection.
90 % of the boxes made by machine B pass inspection.
A box is selected at random.
(a)
Find the probability that it passes inspection.
(3)
(b)
The company would like the probability that a box passes inspection to be 0.87.
Find the percentage of boxes that should be made by machine B to achieve this.
(4)
(Total 7 marks)
8.
The following diagram shows a waterwheel with a bucket. The wheel rotates at a constant rate in an
anticlockwise (counterclockwise) direction.
diagram not to scale
The diameter of the wheel is 8 metres. The centre of the wheel, A, is 2 metres above the water level.
After t seconds, the height of the bucket above the water level is given by h = a sin bt + 2.
(a)
Show that a = 4.
(2)
The wheel turns at a rate of one rotation every 30 seconds.
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IB Math – Standard Level Year 2: May ‘ 11, Paper 2, TZ 1 (b)
Show that b =
Alei -­‐ Desert Academy 2011-­‐12 .
(2)
In the first rotation, there are two values of t when the bucket is descending at a rate of
0.5 m s–1.
(c)
Find these values of t.
(6)
(d)
Determine whether the bucket is underwater at the second value of t.
(4)
(Total 14 marks)
9.
The following diagram shows the graph of f(x) =
.
The points A, B, C, D and E lie on the graph of f. Two of these are points of inflexion.
(a)
Identify the two points of inflexion.
(2)
(b)
(i)
Find f′(x).
(ii)
Show that f″(x) = (4x2 – 2)
.
(5)
(c)
Find the x-coordinate of each point of inflexion.
(4)
(d)
Use the second derivative to show that one of these points is a point of inflexion.
(4)
(Total 15 marks)
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IB Math – Standard Level Year 2: May ‘ 11, Paper 2, TZ 1 10.
Let f(x) = log3
(a)
Alei -­‐ Desert Academy 2011-­‐12 + log3 16 – log3 4, for x > 0.
Show that f(x) = log3 2x.
(2)
(b)
Find the value of f(0.5) and of f(4.5).
(3)
The function f can also be written in the form f(x) =
(c)
.
(i)
Write down the value of a and of b.
(ii)
Hence on graph paper, sketch the graph of f, for –5 ≤ x ≤ 5, –5 ≤ y ≤ 5, using a scale of
1 cm to 1 unit on each axis.
(iii)
Write down the equation of the asymptote.
(6)
(d)
Write down the value of f–1(0).
(1)
The point A lies on the graph of f. At A, x = 4.5.
(e)
On your diagram, sketch the graph of f–1, noting clearly the image of point A.
(4)
(Total 16 marks)
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IB Math – Standard Level Year 2: May ’11 Paper 2, TZ 1: MarkScheme Alei -­‐ Desert Academy 2011-­‐12 IB Practice Exam: 11 Paper 2 Zone 1 – MarkScheme 1.
(a)
(b)
evidence of choosing cosine rule
e.g. a2 + b2 – 2ab cos C
correct substitution
e.g. 72 + 92 – 2(7)(9) cos 120º
AC =13.9 (=
)
METHOD 1
evidence of choosing sine rule
(M1)
A1
A1
N2
3
N2
3
N2
3
(M1)
e.g.
correct substitution
A1
e.g.
A1
METHOD 2
evidence of choosing cosine rule
(M1)
e.g.
correct substitution
A1
e.g.
$
A1
[6]
2.
(a)
(b)
(c)
(1, – 2)
g (x) = 3(x – 1)2 – 2 (accept p =1, q = –2)
(1, 2)
(a)
th
A1A1 N2 2
A1A1
N2 2
A1A1
N2 2
[6]
3.
evidence of choosing the formula for 20 term
e.g. u20 = u1 + 19d
correct equation
(M1)
A1
e.g.
(b)
d=3
correct substitution into formula for un
e.g. 3709 = 7 + 3(n – 1), 3709 = 3n + 4
n = 1235
A1
A1
N2
3
A1
N1
2
σ=3
evidence of attempt to find P(X ≤ 24.5)
(A1)
(M1)
[5]
4.
(a)
e.g. z =1.5,
(b)
P(X ≤ 24.5) = 0.933
(i)
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A1
N3
3
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IB Math – Standard Level Year 2: May ’11 Paper 2, TZ 1: MarkScheme (ii)
Alei -­‐ Desert Academy 2011-­‐12 A1A1
Note: Award A1 with shading that clearly extends to right of the
mean, A1 for any correct label, either k, area or their value of k
z = 1.03(64338)
(A1)
attempt to set up an equation
(M1)
N2
e.g.
k = 23.1
A1
N3
5
[8]
5.
(a)
(b)
correct substitution into formula for E(X)
e.g. 0.05× 240
E(X) =12
evidence of recognizing binomial probability (may be seen in part (a))
e.g.
(c)
(A1)
A1 N2
(M1)
2
A1 N2
(A1)
(M1)
2
A1
3
(0.05)15 (0.95)225, X ~ B(240,0.05)
P(X =15) = 0.0733
P(X ≤ 9) = 0.236
evidence of valid approach
e.g. using complement, summing probabilities
P(X ≥10) = 0.764
N3
[7]
6.
evidence of finding intersection points
(M1)
e.g. f (x) = g (x), cos x2 = ex, sketch showing intersection
x = –1.11, x = 0 (may be seen as limits in the integral)
A1A1
evidence of approach involving integration and subtraction (in any order)
(M1)
e.g.
area = 0.282
A2
evidence of valid approach involving A and B
e.g. P(A ∩ pass) + P(B ∩ pass), tree diagram
correct expression
e.g. P(pass) = 0.6 × 0.8 + 0.4 × 0.9
P(pass) = 0.84
(M1)
N3
[6]
7.
(a)
(b)
(A1)
A1
N2
3
evidence of recognizing complement (seen anywhere)
(M1)
e.g. P(B) = x, P(A) = 1 – x, 1 – P(B), 100 – x, x + y =1
evidence of valid approach
(M1)
e.g. 0.8(1 – x) + 0.9x, 0.8x + 0.9y
correct expression
A1
e.g. 0.87 = 0.8(1 – x) + 0.9x, 0.8 × 0.3 + 0.9 × 0.7 = 0.87, 0.8x + 0.9y = 0.87
70 % from B
A1 N2
4
[7]
8.
(a)
METHOD 1
evidence of recognizing the amplitude is the radius
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(M1)
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IB Math – Standard Level Year 2: May ’11 Paper 2, TZ 1: MarkScheme Alei -­‐ Desert Academy 2011-­‐12 e.g. amplitude is half the diameter
A1
(b)
a=4
METHOD 2
evidence of recognizing the maximum height
e.g. h = 6, a sin bt + 2 = 6
correct reasoning
e.g. a sin bt = 4 and sin bt has amplitude of 1
a=4
METHOD 1
period = 30
AG N0
2
(M1)
A1
AG N0
2
(A1)
A1
AG N0
METHOD 2
correct equation
e.g. 2 = 4 sin 30b + 2, sin 30b = 0
30b = 2π
(A1)
A1
AG N0
(c)
recognizing h′(t) = –0.5 (seen anywhere)
attempting to solve
e.g. sketch of h′, finding h′
correct work involving h′
2
2
R1
(M1)
A2
e.g. sketch of h′ showing intersection, –0.5 =
(d)
t = 10.6, t = 19.4
METHOD 1
valid reasoning for their conclusion (seen anywhere)
e.g. h(t) < 0 so underwater; h(t) > 0 so not underwater
evidence of substituting into h
A1A1
N3 6
R1
(M1)
e.g. h(19.4),
correct calculation
e.g. h(19.4) = –1.19
correct statement
e.g. the bucket is underwater, yes
METHOD 2
valid reasoning for their conclusion (seen anywhere)
e.g. h(t) < 0 so underwater; h(t) > 0 so not underwater
evidence of valid approach
e.g. solving h(t) = 0, graph showing region below x-axis
correct roots
e.g. 17.5, 27.5
correct statement
e.g. the bucket is underwater, yes
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A1
A1
N0
4
N0
4
R1
(M1)
A1
A1
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IB Math – Standard Level Year 2: May ’11 Paper 2, TZ 1: MarkScheme Alei -­‐ Desert Academy 2011-­‐12 [14]
9.
(a)
B, D
(b)
(i)
A1A1 N2 2
f′(x) =
A1A1
Note: Award A1 for
(ii)
N2
and A1 for –2x.
finding the derivative of –2x, i.e. –2
evidence of choosing the product rule
(A1)
(M1)
e.g.
A1
2
(c)
(d)
f ′′(x) = (4x – 2)
valid reasoning
e.g. f ′′(x) = 0
attempting to solve the equation
e.g. (4x2 – 2) = 0, sketch of f ′′(x)
AG N0
R1
p = 0.707
A1A1
5
(M1)
evidence of using second derivative to test values on either side of POI M1
e.g. finding values, reference to graph of f′′, sign table
correct working
A1A1
e.g. finding any two correct values either side of POI,
checking sign of f ′′ on either side of POI
reference to sign change of f ′′(x)
R1 N0
N3 4
4
[15]
10.
(a)
combining 2 terms
e.g. log3 8x – log3 4, log3
(A1)
x + log3 4
expression which clearly leads to answer given
A1
e.g.
(b)
(c)
f(x) = log3 2x
attempt to substitute either value into f
e.g. log3 1, log3 9
f(0.5) = 0, f(4.5) = 2
(i)
a = 2, b = 3
(ii)
AG N0
(M1)
2
A1A1
A1A1
N3 3
N1N1
A1A1A1
N3
Note: Award A1 for sketch approximately through
(0.5 ± 0.1, 0 ± 0.1)
A1 for approximately correct shape,
A1 for sketch asymptotic to the y-axis.
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IB Math – Standard Level Year 2: May ’11 Paper 2, TZ 1: MarkScheme (iii)
x = 0 (must be an equation)
Alei -­‐ Desert Academy 2011-­‐12 A1
N1
[6]
(d)
(e)
–1
f (0) = 0.5
A1
N1
1
A1A1A1A1N4 4
Note: Award A1 for sketch approximately through (0 ± 0.1,
0.5 ± 0.1),
A1 for approximately correct shape of the graph
reflected over y = x,
A1 for sketch asymptotic to x-axis,
A1 for point (2 ± 0.1, 4.5 ± 0.1) clearly marked and
on curve.
[16]
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