IB Math – Standard Level Year 2: May ‘ 10, Paper 2, TZ 2
Alei - Desert Academy 2011-12
IB Practice Exam: 10 Paper 2 Zone 2 – 90 min, Calculator Allowed
Name: __________________________________________Date: _________________Class: _________
1.
The following table gives the examination grades for 120 students.
Grade
1
2
3
4
5
Number of students
9
25
35
q
11
(a)
Find the value of
(i)
p;
(ii) q.
(b)
Find the mean grade.
(c)
Write down the standard deviation.
Cumulative frequency
9
34
p
109
120
(4)
(2)
(1)
(Total 7 marks)
2.
An arithmetic sequence, u1, u2, u3, ..., has d = 11 and u27 = 263.
(a) Find u1.
(2)
(b)
(i)
(ii)
Given that un = 516, find the value of n.
For this value of n, find Sn.
(4)
(Total 6 marks)
3.
Jan plays a game where she tosses two fair six-sided dice. She wins a prize if the sum of her scores
is 5.
(a) Jan tosses the two dice once. Find the probability that she wins a prize.
(3)
(b)
Jan tosses the two dice 8 times. Find the probability that she wins 3 prizes.
(2)
(Total 5 marks)
5
4.
2

Find the term in x4 in the expansion of  3 x 2 −  .
x

(Total 6 marks)
5.
2
x
Consider f(x) = 2 – x , for –2 ≤ x ≤ 2 and g(x) = sin e , for –2 ≤ x ≤ 2. The graph of f is given below.
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IB Math – Standard Level Year 2: May ‘ 10, Paper 2, TZ 2
(a)
On the diagram above, sketch the graph of g.
(b)
Solve f(x) = g(x).
(c)
Write down the set of values of x such that f(x) > g(x).
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(3)
(2)
(2)
(Total 7 marks)
6.
Let f(x) = ex sin 2x + 10, for 0 ≤ x ≤ 4. Part of the graph of f is given below.
There is an x-intercept at the point A, a local maximum point at M, where x = p and a local
minimum point at N, where x = q.
(a) Write down the x-coordinate of A.
(1)
(b)
Find the value of
(i)
p;
(ii) q.
(2)
(c)
Find
∫
q
p
f ( x)dx . Explain why this is not the area of the shaded region.
(3)
(Total 6 marks)
7.
The number of bacteria, n, in a dish, after t minutes is given by n = 800e013t.
(a) Find the value of n when t = 0.
(2)
(b)
Find the rate at which n is increasing when t = 15.
(c)
After k minutes, the rate of increase in n is greater than 10 000 bacteria per minute. Find the
least value of k, where k ∈ .
(2)
(4)
(Total 8 marks)
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IB Math – Standard Level Year 2: May ‘ 10, Paper 2, TZ 2
8.
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The diagram below shows a circle with centre O and radius 8 cm.
diagram not to scale
The points A, B, C, D, E and F are on the circle, and [AF] is a diameter. The length of arc ABC is 6
cm.
(a) Find the size of angle AOC.
(2)
(b)
Hence find the area of the shaded region.
(6)
2
The area of sector OCDE is 45 cm .
(c) Find the size of angle COE.
(2)
(d)
Find EF.
(5)
(Total 15 marks)
9.
In this question, distance is in metres.
Toy airplanes fly in a straight line at a constant speed. Airplane 1 passes through a point A.
 − 2
 x  3 
   
 
Its position, p seconds after it has passed through A, is given by  y  =  − 4  + p 3 
z  0 
 1 
 
   
(a) (i)
Write down the coordinates of A.
(ii) Find the speed of the airplane in m s–1.
(4)
(b)
After seven seconds the airplane passes through a point B.
(i)
Find the coordinates of B.
(ii) Find the distance the airplane has travelled during the seven seconds.
(c)
Airplane 2 passes through a point C. Its position q seconds after it passes through C is given
 x   2   − 1
     
by  y  =  − 5  + q 2  , a ∈ .
z  8   a 
     
The angle between the flight paths of Airplane 1 and Airplane 2 is 40°. Find the two values
of a.
(5)
(7)
(Total 16 marks)
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IB Math – Standard Level Year 2: May ‘ 10, Paper 2, TZ 2
10.
Alei - Desert Academy 2011-12
Consider f(x) = x ln(4 – x2), for –2 < x < 2. The graph of f is given below.
(a)
Let P and Q be points on the curve of f where the tangent to the graph of f is parallel to the xaxis.
(i)
Find the x-coordinate of P and of Q.
(ii) Consider f(x) = k. Write down all values of k for which there are exactly two solutions.
(5)
3
2
Let g(x) = x ln(4 – x ), for –2 < x < 2.
− 2x 4
(b) Show that g′(x) =
+ 3x 2 ln(4 − x 2 ) .
4− x2
(4)
(c)
Sketch the graph of g′.
(d)
Consider g′(x) = w. Write down all values of w for which there are exactly two solutions.
(2)
(3)
(Total 14 marks)
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IB Math – Standard Level Year 2: May ’10 Paper 2, TZ 2: MarkScheme
Alei - Desert Academy 2011-12
IB Practice Exam: 10 Paper 2 Zone 2 – MarkScheme
1.
(a)
(b)
(c)
(i)
evidence of appropriate approach
e.g. 9 + 25 + 35, 34 + 35
p = 69
(ii) evidence of valid approach
e.g. 109 – their value of p, 120 – (9 + 25 + 35 + 11)
q = 40
evidence of appropriate approach
fx
e.g. substituting into
, division by 120
n
mean = 3.16
1.09
(M1)
A1
N2
(M1)
A1N2
(M1)
∑
A1N2
A1N1
[7]
2.
(a)
(b)
evidence of equation for u27
e.g. 263 = u1 + 26 × 11, u27 = u1 + (n – 1) × 11, 263 – (11 × 26)
u1 = –23
(i)
correct equation
e.g. 516 = –23 + (n – 1) × 11, 539 = (n – 1) × 11
n = 50
(ii) correct substitution into sum formula
50(−23 + 516)
50( 2 × (−23) + 49 ×11)
e.g. S50 =
, S 50 =
2
2
S50 = 12325 (accept 12300)
M1
A1
N1
A1
A1N1
A1
A1N1
[6]
3.
(a)
(b)
36 outcomes (seen anywhere, even in denominator)
(A1)
valid approach of listing ways to get sum of 5, showing at least two pairs
e.g. (1, 4)(2, 3), (1, 4)(4, 1), (1, 4)(4, 1), (2, 3)(3, 2) , lattice diagram
4  1
P(prize) =
= 
36  9 
recognizing binomial probability
3
5
 8  1   8 
 1


e.g. B  8,  , binomial pdf,     
 9
 3  9   9 
P(3 prizes) = 0.0426
(M1)
A1N3
(M1)
A1N2
[5]
4.
evidence of substituting into binomial expansion
 5
5
e.g. a5 +   a 4 b +   a 3 b 2 + ...
 1
 2
(M1)
identifying correct term for x4
evidence of calculating the factors, in any order
(M1)
A1A1A
1
2
5
4
−2
e.g.  , 27 x 6 , 2 ; 10(3x 2 ) 3 

x
 x 
 2
Note: Award A1 for each correct factor.
4
term = 1080x
Note: Award M1M1A1A1A1A0 for 1080 with working shown.
A1N2
[6]
5.
(a)
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IB Math – Standard Level Year 2: May ’10 Paper 2, TZ 2: MarkScheme
Alei - Desert Academy 2011-12
(b)
A1A1A1
x = –1.32, x = 1.68 (accept x = –1.41, x = 1.39 if working in degrees)
(c)
–1.32 < x < 1.68 (accept –1.41 < x < 1.39 if working in degrees)
(a)
(b)
2.31
(i)
1.02
(ii) 2.59
N3
A1A1
N2
A2N2
[7]
6.
(c)
∫
q
p
A1
N1
A1N1
A1N1
f ( x)dx = 9.96
A1N1
split into two regions, make the area below the x-axis positive
R1R1
N2
[6]
7.
(a)
(b)
(c)
n = 800e0
n = 800
evidence of using the derivative
n′(15) = 731
METHOD 1
setting up inequality (accept equation or reverse inequality)
e.g. n′(t) > 10 000
evidence of appropriate approach
e.g. sketch, finding derivative
k = 35.1226...
least value of k is 36
METHOD 2
n′(35) = 9842, and n′(36) = 11208
least value of k is 36
(A1)
A1
appropriate approach
e.g. 6 = 8θ
AÔC = 0.75
evidence of substitution into formula for area of triangle
1
e.g. area =
× 8 × 8 × sin(0.75)
2
area = 21.8…
evidence of substitution into formula for area of sector
1
e.g. area =
× 64 × 0.75
2
area of sector = 24
evidence of substituting areas
1
1
e.g. r 2θ − ab sin C , area of sector – area of triangle
2
2
area of shaded region = 2.19 cm2
(M1)
N2
(M1)
A1N2
A1
M1
(A1)
A1N2
A2
A2N2
[8]
8.
(a)
(b)
A1
N2
(M1)
(A1)
(M1)
(A1)
(M1)
A1N4
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IB Math – Standard Level Year 2: May ’10 Paper 2, TZ 2: MarkScheme
(c)
(d)
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attempt to set up an equation for area of sector
1
e.g. 45 =
× 82 × θ
2
CÔE = 1.40625 (1.41 to 3 sf)
METHOD 1
attempting to find angle EOF
e.g. π – 0.75 – 1.41
EÔF = 0.985 (seen anywhere)
evidence of choosing cosine rule
correct substitution
(M1)
A1N2
(M1)
A1
(M1)
A1
e.g. EF = 8 2 + 8 2 − 2 × 8 × 8 × cos 0.985
EF = 7.57 cm
METHOD 2
attempting to find angles that are needed
e.g. angle EOF and angle OEF
EÔF = 0.9853... and OÊF (or OF̂E) = 1.078...
evidence of choosing sine rule
correct substitution
EF
8
e.g.
=
sin0.985 sin 1.08
EF = 7.57 cm
METHOD 3
attempting to find angle EOF
e.g. π – 0.75 – 1.41
EÔF = 0.985 (seen anywhere)
evidence of using half of triangle EOF
0.985
e.g. x = 8 sin
2
correct calculation
e.g. x = 3.78
EF = 7.57 cm
A1N3
(M1)
A1
(M1)
(A1)
A1N3
(M1)
A1
(M1)
A1
A1N3
[15]
9.
(a)
(i)
(ii)
(3, –4, 0)
 − 2
 
choosing velocity vector  3 
 1 
 
finding magnitude of velocity vector
e.g.
(b)
(i)
(ii)
A1
N1
(M1)
(A1)
(−2) + 3 + 1 , 4 + 9 + 1
2
2
2
speed = 3.74 ( 14 )
substituting p = 7
B = (–11, 17, 7)
METHOD 1
A1N2
(M1)
A1N2
appropriate method to find AB or BA
(M1)
e.g. AO + OB , A – B
 − 14 
 14 




AB =  21  or BA =  − 21
 7 
 −7 




(A1)
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IB Math – Standard Level Year 2: May ’10 Paper 2, TZ 2: MarkScheme
Alei - Desert Academy 2011-12
distance = 26.2 (7 14 )
METHOD 2
evidence of applying distance is speed × time
e.g. 3.74 × 7
distance = 26.2 (7 14 )
METHOD 3
attempt to find AB2, AB
2
2
2
e.g. (3 – (–11)) + (–4 – 17) + (0 – 7) ,
2
AB = 686, AB =
(c)
A1N3
(M2)
A1N3
(M1)
(3 − (−11)) + (−4 − 17) + (0 − 7)
2
2
2
686
(A1)
distance AB = 26.2 ( 7 14 )
 − 2
 – 1
 
 
correct direction vectors  3  and  2 
 1 
a
 
 
A1N3
(A1)(A1)
−1
 − 2   − 1
   
2 = a + 5,  3  •  2  = a + 8
 1  a
a
   
substituting
a +8
e.g. cos 40° =
14 a 2 + 5
a = 3.21, a = –0.990
2
(A1)(A1)
M1
A1A1N3
[16]
10.
(a)
(b)
(i)
(ii)
–1.15, 1.15
recognizing that it occurs at P and Q
e.g. x = –1.15, x = 1.15
k = –1.13, k = 1.13
evidence of choosing the product rule
e.g. uv′ + vu′
derivative of x3 is 3x2
− 2x
derivative of ln (4 – x2) is
4− x2
correct substitution
− 2x
e.g. x 3 ×
+ ln(4 − x 2 ) × 3 x 2
4 − x2
− 2x 4
g′(x) =
+ 3x 2 ln(4 − x 2 )
4 − x2
A1A1
N2
(M1)
A1A1
N3
(M1)
(A1)
(A1)
A1
AGN0
(c)
(d)
w = 2.69, w < 0
A1A1
N2
A1A2
N2
[14]
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