IB Math – Standard Level Year 1 – Final Exam Practice
Alei - Desert Academy 2011-12
Final Exam Prep – Practice Problems
1.
Consider the arithmetic sequence 2, 5, 8, 11, ....
(a)
Find u101.
(b) Find the value of n so that un = 152.
(3)
(3)
(Total 6 marks)
2.
Consider the infinite geometric sequence 3000, – 1800, 1080, – 648, … .
(a) Find the common ratio.
(2)
th
(b)
Find the 10 term.
(c)
Find the exact sum of the infinite sequence.
(2)
(2)
(Total 6 marks)
8
3.
Find the term in x3 in the expansion of  2 x − 3  .
3

(Total 5 marks)
4.
2
3
Consider the infinite geometric sequence 3, 3(0.9), 3(0.9) , 3(0.9) , … .
(a) Write down the 10th term of the sequence. Do not simplify your answer.
(1)
(b)
Find the sum of the infinite sequence.
(4)
(Total 5 marks)
5.
(a)
Expand (x – 2)4 and simplify your result.
(b)
Find the term in x3 in (3x + 4)(x – 2)4.
(3)
(3)
(Total 6 marks)
6.
Let f (x) = ln (x + 5) + ln 2, for x > –5.
(a) Find f −1(x).
(4)
Let g (x) = ex.
(b) Find (g ◦ f) (x), giving your answer in the form ax + b, where a, b,∈
.
(3)
(Total 7 marks)
7.
Let f (x) = 3(x + 1)2 – 12.
(a) Show that f (x) = 3x2 + 6x – 9.
(2)
(b)
For the graph of f
(i)
write down the coordinates of the vertex;
(ii) write down the equation of the axis of symmetry;
(iii) write down the y-intercept;
(iv) find both x-intercepts.
(c)
Hence sketch the graph of f.
(8)
(2)
(d)
2
Let g (x) = x . The graph of f may be obtained from the graph of g by the two transformations:
a stretch of scale factor t in the y-direction
followed by a translation of  p .
q
 
p
Find   and the value of t.
q
(3)
(Total 15 marks)
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IB Math – Standard Level Year 1 – Final Exam Practice
8.
Alei - Desert Academy 2011-12
Consider f(x) = x − 5 .
(a) Find
(i)
f(11);
(ii) f(86);
(iii) f(5).
(3)
(b)
Find the values of x for which f is undefined.
(c)
Let g(x) = x2. Find (g ° f)(x).
(2)
(2)
(Total 7 marks)
9.
The quadratic function f is defined by f(x) = 3x2 – 12x + 11.
(a) Write f in the form f(x) = 3(x – h)2 – k.
(3)
(b)
The graph of f is translated 3 units in the positive x-direction and 5 units in the positive
y-direction. Find the function g for the translated graph, giving your answer in the form
g(x) = 3(x – p)2 + q.
(3)
(Total 6 marks)
10.
 2 − 1
 , and O =
Let M = 
−3 4 
 0 0

 . Given that M2 – 6M + kI = O, find k.
0
0


(Total 6 marks)
11.
Let f (x) = 2x2 – 12x + 5.
(a) Express f(x) in the form f(x) = 2(x – h)2 – k.
(3)
(b)
Write down the vertex of the graph of f.
(c)
Write down the equation of the axis of symmetry of the graph of f.
(d)
Find the y-intercept of the graph of f.
(2)
(1)
(2)
(e)
The x-intercepts of f can be written as
p± q
, where p, q, r ∈
.
r
Find the value of p, of q, and of r.
(7)
(Total 15 marks)
12.
1
, x ≠ 0.
x
Sketch the graph of f.
Let f(x) =
(a)
(2)
2
The graph of f is transformed to the graph of g by a translation of   .
3
(b) Find an expression for g(x).
(2)
(c)
(i)
(ii)
(iii)
Find the intercepts of g.
Write down the equations of the asymptotes of g.
Sketch the graph of g.
(10)
(Total 14 marks)
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IB Math – Standard Level Year 1 – Final Exam Practice
13.
The function f is defined by f(x) =
Alei - Desert Academy 2011-12
3
(a)
, for –3 < x < 3.n (Calculator OK)
9 − x2
On the grid below, sketch the graph of f.
(b)
Write down the equation of each vertical asymptote.
(c)
Write down the range of the function f.
(2)
(2)
(2)
(Total 6 marks)
14.
The functions f and g are defined by f : x ֏ 3x, g : x ֏ x + 2.
(a) Find an expression for (f ° g)(x).
(2)
(b)
Find f–1(18) + g–1(18).
(4)
(Total 6 marks)
15.
The following diagram shows part of the graph of f, where f (x) = x2 − x − 2.
(a)
Find both x-intercepts.
(b)
Find the x-coordinate of the vertex.
(4)
(2)
(Total 6 marks)
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IB Math – Standard Level Year 1 – Final Exam Practice
16.
Alei - Desert Academy 2011-12
In an arithmetic sequence u21 = –37 and u4 = –3.
(a) Find
(i)
the common difference;
(ii) the first term.
(4)
(b)
Find S10.
(3)
(Total 7 marks)
17.
Let un = 3 – 2n.
(a) Write down the value of u1, u2, and u3.
(3)
20
(b)
Find
∑ (3 − 2n) .
n =1
(3)
(Total 6 marks)
18.
Solve the following equations.
(a) logx 49 = 2
(3)
(b)
log2 8 = x
(c)
1
log25 x = −
2
(d)
log2 x + log2(x – 7) = 3
(2)
(3)
(5)
(Total 13 marks)
19.
A theatre has 20 rows of seats. There are 15 seats in the first row, 17 seats in the second row, and
each successive row of seats has two more seats in it than the previous row.
(a) Calculate the number of seats in the 20th row.
(4)
(b)
Calculate the total number of seats.
(2)
(Total 6 marks)
20.
A sum of $ 5000 is invested at a compound interest rate of 6.3 % per annum.
(a) Write down an expression for the value of the investment after n full years.
(1)
(b)
What will be the value of the investment at the end of five years?
(c)
The value of the investment will exceed $ 10 000 after n full years.
(i)
Write down an inequality to represent this information.
(ii) Calculate the minimum value of n.
(1)
(4)
(Total 6 marks)
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IB Math – Standard Level Year 1 – Final Exam Practice
21.
Alei - Desert Academy 2011-12
Part of the graph of a function f is shown in the diagram below.
y
4
3
2
1
–2
–1
0
1
3
2
4
x
–1
–2
–3
–4
(a)
On the same diagram sketch the graph of y = − f (x).
(b)
Let g (x) = f (x + 3).
(i)
Find g (−3).
(ii) Describe fully the transformation that maps the graph of f to the graph of g.
(2)
(4)
(Total 6 marks)
22.
Let f (x) = 3x – ex–2 – 4, for –1 ≤ x ≤ 5.
(a) Find the x-intercepts of the graph of f.
(3)
(b)
On the grid below, sketch the graph of f.
y
3
2
1
–2 –1 0
–1
1
2
3
4
5
6
x
–2
–3
–4
–5
–6
–7
–8
–9
–10
(3)
(c)
Write down the gradient of the graph of f at x = 2.
(1)
(Total 7 marks)
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IB Math – Standard Level Year 1 – Final Exam Practice
23.
Alei - Desert Academy 2011-12
A city is concerned about pollution, and decides to look at the number of people using taxis. At the
end of the year 2000, there were 280 taxis in the city. After n years the number of taxis, T, in the city
is given by
T = 280 × 1.12n.
(a) (i)
Find the number of taxis in the city at the end of 2005.
(ii) Find the year in which the number of taxis is double the number of taxis there were at
the end of 2000.
(6)
(b)
At the end of 2000 there were 25 600 people in the city who used taxis. After n years the
number of people, P, in the city who used taxis is given by
2 560 000
.
P=
10 + 90e – 0.1n
(i)
Find the value of P at the end of 2005, giving your answer to the nearest whole number.
(ii) After seven complete years, will the value of P be double its value at the end of 2000?
Justify your answer.
(c)
Let R be the ratio of the number of people using taxis in the city to the number of taxis. The
city will reduce the number of taxis if R < 70.
(i)
Find the value of R at the end of 2000.
(ii) After how many complete years will the city first reduce the number of taxis?
(6)
(5)
(Total 17 marks)
24.
Let f be the function given by f(x) = e0.5x, 0 ≤ x ≤ 3.5. The diagram shows the graph of f.
(a)
On the same diagram, sketch the graph of f–1.
(b)
Write down the range of f–1.
(c)
Find f–1(x).
(3)
(1)
(3)
(Total 7 marks)
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IB Math – Standard Level Year 1 – Final Exam Practice
25.
Alei - Desert Academy 2011-12
[Note: Trig Functions will not be included on the final exam. But transformations of functions
will be. Do this problem only if you want to explore transformations more thoroughly.]
Let f(t) = a cos b (t – c) + d, t ≥ 0. Part of the graph of y = f(t) is given below.
When t = 3, there is a maximum value of 29, at M.
When t = 9 , there is a minimum value of 15.
(a) (i)
Find the value of a.
π
(ii) Show that b = .
6
(iii) Find the value of d.
(iv) Write down a value for c.
(7)
1
The transformation P is given by a horizontal stretch of a scale factor of , followed by a translation
2
of  3  .


 −10 
(b)
Let M′ be the image of M under P. Find the coordinates of M′.
(2)
The graph of g is the image of the graph of f under P.
(c) Find g(t) in the form g(t) = 7 cos B(t – C) + D.
(4)
(d)
Give a full geometric description of the transformation that maps the graph of g to the graph of
f.
(3)
(Total 16 marks)
26.
Let f(x) = 2x2 + 4x – 6.
(a) Express f(x) in the form f(x) = 2(x – h)2 + k.
(3)
(b)
Write down the equation of the axis of symmetry of the graph of f.
(c)
Express f(x) in the form f(x) = 2(x – p)(x – q).
(1)
(2)
(Total 6 marks)
27.
Let f(x) = x cos (x – sin x), 0 ≤ x ≤ 3.
(a) Sketch the graph of f on the following set of axes.
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IB Math – Standard Level Year 1 – Final Exam Practice
Alei - Desert Academy 2011-12
(3)
(b)
The graph of f intersects the x-axis when x = a, a ≠ 0. Write down the value of a.
(1)
(Total 4 marks)
28.
Consider the points A (1, 5, 4), B (3, 1, 2) and D (3, k, 2), with (AD) perpendicular to (AB).
(a) Find
(i)
AB ;
(ii)
AD , giving your answer in terms of k.
(3)
(b)
Show that k = 7.
(3)
1
The point C is such that BC = AD.
2
(c) Find the position vector of C.
(4)
(d)
Find cosAB̂C .
(3)
(Total 13 marks)
29.
Let v = 3i + 4 j + k and w = i + 2 j – 3k. The vector v + pw is perpendicular to w.
Find the value of p.
(Total 7 marks)
30.
The point O has coordinates (0, 0, 0), point A has coordinates (1, –2, 3) and point B has coordinates
(–3, 4, 2).
 − 4
 
(a) (i)
Show that AB =  6 .
 −1
 
(ii) Find BÂO .
(8)
(b)
 x   − 3  − 4 
     
The line L1 has equation  y  =  4  + s  6  .
 z   2   −1
     
Write down the coordinates of two points on L1.
(2)
(c)
The line L2 passes through A and is parallel to OB .
(i)
Find a vector equation for L2, giving your answer in the form r = a + tb.
(ii) Point C (k, –k, 5) is on L2. Find the coordinates of C.
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IB Math – Standard Level Year 1 – Final Exam Practice
Alei - Desert Academy 2011-12
(6)
(d)
 x  3 
 1 
   
 
The line L3 has equation  y  =  − 8  + p  − 2  , and passes through the point C.
z  0 
 −1
   
 
Find the value of p at C.
(2)
(Total 18 marks)
31.
 2
1
 
 
The line L1 is represented by r1 =  5  + s  2  and the line L2 by r2 =
 3
 3
 
 
The lines L1 and L2 intersect at point T. Find the coordinates of T.
 3   −1
   
 − 3  + t  3 .
 8   − 4
   
(Total 6 marks)
32.
A particle is moving with a constant velocity along line L. Its initial position is A(6, –2, 10). After
one second the particle has moved to B(9, –6, 15).
(a) (i)
Find the velocity vector, AB .
(ii) Find the speed of the particle.
(4)
(b)
Write down an equation of the line L.
(2)
(Total 6 marks)
33.
1 2 
 and B =
Let A = 
 3 − 1
Find
(a) A + B;
 3 0

 .
− 2 1
(2)
(b)
−3A;
(c)
AB.
(2)
(3)
(Total 7 marks)
34.
2 1 
 .
Let M = 
 2 − 1
(a) Write down the determinant of M.
(1)
(b)
−1
Write down M .
(2)
(c)
 x   4
Hence solve M   =  .
 y 8
(3)
(Total 6 marks)
35.
(a)
− 2 1 

1 .
 q

2

Find AB in terms of p and q.
(b)
Matrix B is the inverse of matrix A. Find the value of p and of q.
1 − 2
 and B =
Let A = 
3 p 
(2)
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(5)
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