IB Math – Standard Level Year 2: May ‘ 10, Paper 1, TZ 1
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IB Practice Exam: 10 Paper 1 Zone 1 – 90 min, No Calculator
Name:_________________________________________ Date: ________________ Class:_________
1.
Let f(x) = 8x – 2x2. Part of the graph of f is shown below.
(a)
Find the x-intercepts of the graph.
(4)
(b)
(i)
Write down the equation of the axis of symmetry.
(ii)
Find the y-coordinate of the vertex.
(3)
(Total 7 marks)
2.
 1 3 2


Let W =  2 0 1  and P =
 0 1 3


(a)
 2
 
 3 .
1
 
Find WP.
(3)
(b)
 26 
 
Given that 2WP + S =  12  , find S.
 10 
 
(3)
(Total 6 marks)
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IB Math – Standard Level Year 2: May ‘ 10, Paper 1, TZ 1
3.
(a)
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Expand (2 + x)4 and simplify your result.
(3)
(b)
1 

Hence, find the term in x2 in (2 + x)4 1 + 2  .
 x 
(3)
(Total 6 marks)
4.
The straight line with equation y =
(a)
3
x makes an acute angle θ with the x-axis.
4
Write down the value of tan θ.
(1)
(b)
Find the value of
(i)
sin 2θ;
(ii)
cos 2θ.
(6)
(Total 7 marks)
5.
Consider the events A and B, where P(A) = 0.5, P(B) = 0.7 and P(A ∩ B) = 0.3.
The Venn diagram below shows the events A and B, and the probabilities p, q and r.
(a)
Write down the value of
(i)
p;
(ii)
q;
(iii)
r.
(3)
(b)
Find the value of P(A | B′).
(2)
(c)
Hence, or otherwise, show that the events A and B are not independent.
(1)
(Total 6 marks)
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IB Math – Standard Level Year 2: May ‘ 10, Paper 1, TZ 1
6.
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The graph of f(x) = 16 − 4 x 2 , for –2 ≤ x ≤ 2, is shown below.
The region enclosed by the curve of f and the x-axis is rotated 360° about the x-axis.
Find the volume of the solid formed.
(Total 6 marks)
7.
Let f(x) = log3 x , for x > 0.
(a)
Show that f–1(x) = 32x.
(2)
(b)
Write down the range of f–1.
(1)
Let g(x) = log3 x, for x > 0.
(c)
Find the value of (f –1 ° g)(2), giving your answer as an integer.
(4)
(Total 7 marks)
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IB Math – Standard Level Year 2: May ‘ 10, Paper 1, TZ 1
8.
Let f(x) =
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1 3
x − x 2 − 3 x . Part of the graph of f is shown below.
3
There is a maximum point at A and a minimum point at B(3, –9).
(a)
Find the coordinates of A.
(8)
(b)
Write down the coordinates of
(i)
the image of B after reflection in the y-axis;
(ii)
 − 2
the image of B after translation by the vector   ;
 5 
(iii)
the image of B after reflection in the x-axis followed by a horizontal stretch with scale
1
factor .
2
(6)
(Total 14 marks)
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IB Math – Standard Level Year 2: May ‘ 10, Paper 1, TZ 1
9.
Let f(x) =
(a)
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cos x
, for sin x ≠ 0.
sin x
Use the quotient rule to show that f′(x) =
−1
sin 2 x
.
(5)
(b)
Find f′′(x).
(3)
π
π
In the following table, f′   = p and f′′   = q. The table also gives approximate values of f′(x)
2
2
π
and f′′(x) near x = .
2
(c)
x
π
− 0.1
2
π
2
π
+ 0.1
2
f′(x)
–1.01
p
–1.01
f″(x)
0.203
q
–0.203
Find the value of p and of q.
(3)
(d)
Use information from the table to explain why there is a point of inflexion on the graph of f
π
where x = .
2
(2)
(Total 13 marks)
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IB Math – Standard Level Year 2: May ‘ 10, Paper 1, TZ 1
10.
 −3 


The line L1 is represented by the vector equation r =  − 1  +
 − 25 


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 2 
 
p 1  .
 − 8
 
A second line L2 is parallel to L1 and passes through the point B(–8, –5, 25).
(a)
Write down a vector equation for L2 in the form r = a + tb.
(2)
5  − 7
   
A third line L3 is perpendicular to L1 and is represented by r =  0  + q − 2  .
 3  k 
   
(b)
Show that k = –2.
(5)
The lines L1 and L3 intersect at the point A.
(c)
Find the coordinates of A.
(6)
 6 


The lines L2 and L3 intersect at point C where BC =  3  .
 − 24 


(d)
(i)
Find AB .
(ii)
Hence, find | AC |.
(5)
(Total 18 marks)
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IB Math – Standard Level Year 2: May ’10 Paper 1, TZ 1: MarkScheme
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IB Practice Exam: 10 Paper 1 Zone 1 – MarkScheme
1.
(a)
evidence of setting function to zero
2
e.g. f(x) = 0, 8x = 2x
evidence of correct working
(M1)
A1
− 8 ± 64
e.g. 0 = 2x(4 – x),
−4
(b)
x-intercepts are at 4 and 0 (accept (4, 0) and (0, 0), or x = 4, x = 0)
(i)
x = 2 (must be equation)
(ii)
substituting x = 2 into f(x)
y=8
A1A1
A1
(M1)
A1
N1N1
N1
N2
[7]
2.
(a)
13 
 
WP =  5 
6
 
A1A1A1 N3
Note: Award A1 for each correct element.
Note: The first two steps may be done in any order.
(b)
subtracting
(A1)
 26 
 
e.g.  12  – 2WP
 10 
 
multiplying WP by 2
(A1)
 26 
 
e.g.  10 
 12 
 
 0 
 
S=  2 
 − 2
 
A1
N2
[6]
3.
(a)
(b)
evidence of expanding
4
3
2 2
3
4
2
2
e.g. 2 + 4(2 )x + 6(2 )x + 4(2)x + x , (4 + 4x + x )(4 + 4x + x )
4
2
3
4
(2 + x) = 16 + 32x + 24x + 8x + x
finding coefficients 24 and 1
2
term is 25x
M1
A2
(A1)(A1)
N2
A1
N3
[6]
4.
(a)
tan θ =
(b)
(i)
3
3 
 do not accept x 
4
4 
3
4
sin θ = , cos θ =
5
5
A1N1
(A1)(A1)
correct substitution
A1
 3  4 
 5  5 
e.g. sin 2θ = 2   
sin 2θ =
(ii)
24
25
A1
correct substitution
N3
A1
2
 3  4
5 5
2
3
5
2
e.g. cos 2θ = 1 – 2   ,   −  
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IB Math – Standard Level Year 2: May ’10 Paper 1, TZ 1: MarkScheme
cos 2θ =
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7
25
A1
N1
[7]
5.
(a)
(b)
(i)
(ii)
(iii)
p = 0.2
q = 0.4
r = 0.1
P(A│B′) =
2
3
Note: Award A1 for an unfinished answer such as
(c)
valid reason
e.g.
A1N1
A1
A1
N1
N1
A2
N2
0.2
.
0.3
R1
2
≠ 0.5, 0.35 ≠ 0.3
3
thus, A and B are not independent
AG
N0
[6]
6.
attempt to set up integral expression
e.g. π
∫
2
16 − 4 x 2 dx, 2 π
∫ 16dx = 16 x, ∫ 4 x
2
dx =
∫
2
0
4x
3
(16 − 4 x 2 ),
M1
∫
2
16 − 4 x 2 dx
3
(seen anywhere)
evidence of substituting limits into the integrand
32  
32 
64

e.g.  32 −
 −  − 32 + , 64 −
3 
3
3

128π
volume =
3
A1A1
(M1)
A2
N3
[6]
7.
(a)
interchanging x and y (seen anywhere)
e.g. x = log
y (accept any base)
evidence of correct manipulation
x
e.g. 3 =
(b)
(c)
(M1)
y ,3y =
1
x2 ,x
=
A1
1
log3 y, 2y = log3 x
2
–1
2x
f (x) = 3
–1
y > 0, f (x) > 0
METHOD 1
finding g(2) = log3 2 (seen anywhere)
attempt to substitute
–1
e.g. (f ° g)(2) = 3 log 3 2
evidence of using log or index rule
log 4
log 2 2
–1
e.g. (f ° g)(2) = 3 3 , 3 3
–1
(f ° g)(2) = 4
METHOD 2
attempt to form composite (in any order)
–1
e.g. (f ° g)(x) = 3 2 log 3 x
evidence of using log or index rule
log x 2
log x 2
–1
e.g.(f ° g)(x) = 3 3 , 3 3
–1
2
(f ° g)(x) = x
AG
N0
A1
N1
A1
(M1)
(A1)
A1
N1
(M1)
(A1)
A1
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IB Math – Standard Level Year 2: May ’10 Paper 1, TZ 1: MarkScheme
–1
(f ° g)(2) = 4
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A1
N1
[7]
8.
(a)
2
f′(x) = x – 2x – 3
evidence of solving f′(x) = 0
2
e.g. x – 2x – 3 = 0
evidence of correct working
e.g. (x + 1)(x – 3),
A1A1A1
(M1)
A1
2 ± 16
2
x = –1 (ignore x = 3)
evidence of substituting their negative x-value into f(x)
(A1)
(M1)
1
1
(−1) 3 − (−1) 2 − 3(−1), − − 1 + 3
3
3
5
y=
3
5

coordinates are  − 1, 
3

A1
(i)
(ii)
(iii)
(–3, –9)
(1, –4)
reflection gives (3, 9)
A1
A1A1
(A1)
N1
N2
3 
, 9
2 
A1A1
N3
e.g.
(b)
stretch gives 
N3
[14]
9.
(a)
d
d
sin x = cos x, cos x = − sin x (seen anywhere)
dx
dx
(A1)(A1)
evidence of using the quotient rule
correct substitution
M1
A1
e.g.
sin x(− sin x) − cos x(cos x) − sin 2 x − cos 2 x
,
sin 2 x
sin 2 x
− (sin 2 x + cos 2 x)
f′(x) =
f′(x) =
(b)
sin 2 x
−1
A1
AGN0
sin 2 x
METHOD 1
appropriate approach
–2
e.g. f′(x) = –(sin x)
(M1)
 2 cos x 
–3
f″(x) = 2(sin x)(cos x)  =

3

A1A1
sin x 
Note: Award A1 for 2 sin
–3
N3
x, A1 for cos x.
METHOD 2
2
derivative of sin x = 2 sin x cos x (seen anywhere)
evidence of choosing quotient rule
A1
(M1)
sin 2 x × 0 − (−1)2 sin x cos x
2
e.g. u = –1,v = sin x, f″(x) =
2
2
(sin x)
f″(x) =
2 sin x cos x  2 cos x 
=

(sin 2 x) 2  sin 3 x 
A1
N3
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IB Math – Standard Level Year 2: May ’10 Paper 1, TZ 1: MarkScheme
(c)
evidence of substituting
π
2
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M1
π
2
e.g.
,
2 π
3 π
sin
sin
2
2
−1
2 cos
(d)
p = –1, q = 0
second derivative is zero, second derivative changes sign
A1A1
R1R1
(a)
any correct equation in the form r = a + tb (accept any parameter)
A2N2
N1N1
N2
[13]
10.
 − 8  2 
   
e.g. r =  − 5  + t  1 
 25   − 8 
   
(b)
Note: Award A1 for a + tb, A1 for L = a + tb, A0 for r = b + ta.
recognizing scalar product must be zero (seen anywhere)
R1
e.g. a • b = 0
 2 
 
evidence of choosing direction vectors  1 ,
 − 8
 
(c)
correct calculation of scalar product
e.g. 2(–7) + 1(–2) – 8k
simplification that clearly leads to solution
e.g. –16 – 8k, –16 – 8k =0
k = –2
evidence of equating vectors
 −3 


e.g. L1 = L3,  − 3  +
 − 25 


(d)
− 7
 
 − 2
 k 
 
(A1)(A1)
(A1)
A1
AG
(M1)
N0
 2  5  − 7
     
p 1  =  0  + q − 2 
 − 8  3  − 2
     
any two correct equations
e.g. –3 + 2p = 5 – 7q, –1 + p = –2q, –25 – 8p = 3 –2q
attempting to solve equations
finding one correct parameter (p = –3, q = 2)
the coordinates of A are (–9, –4, –1)
(i)
evidence of appropriate approach
A1A1
(M1)
A1
A1
(M1)
N3
 − 8  − 9 
   
e.g. OA + AB = OB, AB =  − 5  −  − 4 
 25   − 1 
   
(ii)
1
 
AB =  − 1
 26 
 
A1
7
 
finding AC =  2 
 2
 
A1
evidence of finding magnitude
e.g. AC =
AC = 57
2
2
7 +2 +2
N2
(M1)
2
A1
N3
[18]
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