IB Math SL 1 Review Homework
Please complete on a separate sheet.
Topic 1:
1.
In an arithmetic sequence, the first term is –2, the fourth term is 16, and the nth term is 11,998.
(a) Find the common difference d.
(b) Find the value of n.
(Total 6 marks)
2.
Consider the infinite geometric series 405 + 270 + 180 +....
(a) For this series, find the common ratio, giving your answer as a fraction in its simplest form.
(b) Find the fifteenth term of this series.
(c) Find the exact value of the sum of the infinite series.
(Total 6 marks)
1
+ log9 3 = log9 x.
9
3.
Solve the equation log9 81 + log9
4.
Solve the equation log27 x = 1 – log27 (x – 0.4).
(Total 4 marks)
(Total 6 marks)
5.
Find the coefficient of x5 in the expansion of (3x – 2)8.
(Total 4 marks)
Topic 2:
1.
The functions f (x) and g (x) are defined by f (x) = ex and g (x) = ln (1+ 2x).
(a) Write down f −1(x).
(b) (i)
Find ( f ◦ g) (x).
(ii)
Find ( f ◦ g)−1 (x).
(Total 6 marks)
2.
2
Let f (x) = 3(x + 1) – 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
 p
 p
of scale factor t in the y-direction, followed by a translation of  . Find   and the value of t.
q
q
(3)
(Total 15 marks)
3.
The quadratic equation 4x2 + 4kx + 9 = 0, k > 0 has exactly one solution for x.
Find the value of k.
4.
Consider the function f (x) = 2x2 – 8x + 5.
(a) Express f (x) in the form a (x – p)2 + q, where a, p, q 
(b) Find the minimum value of f (x).
(Total 4 marks)
.
(Total 6 marks)
5.
$1000 is invested at 15% per annum interest, compounded monthly. Calculate the minimum number of
months required for the value of the investment to exceed $3000.
Topic 3:
1.
A formula for the depth d metres of water in a harbour at a time t hours after midnight is
 
d  P  Q cos  t , 0  t  24,
6 
where P and Q are positive constants. In the following graph the point (6, 8.2) is a minimum point and the
point (12, 14.6) is a maximum point.
d
15
(12, 14.6)
10.
(6, 8.2)
5
0
6
(i)
Q;
12
(ii)
24 t
18
(a)
Find the value of
P.
(b)
Find the first time in the 24-hour period when the depth of the water is 10 metres.
(c)
(i)
(ii)
(3)
(3)
Use the symmetry of the graph to find the next time when the depth of the water is 10 metres.
Hence find the time intervals in the 24-hour period during which the water is less than 10 metres
deep.
(4)
2.
The diagram below shows a sector AOB of a circle of radius 15 cm and centre O. The
angle  at the centre of the circle is 2 radians. Diagram not to scale
(a)
(b)
Calculate the area of the sector AOB.
Calculate the area of the shaded region.
B
A
(Total 4 marks)
3.
(a)
(b)
Express 2 cos2 x + sin x in terms of sin x only.
Solve the equation 2 cos2 x + sin x = 2 for x in the interval 0  x  , giving your answers
exactly.
O
(Total 4 marks)
4.
Solve the equation 2 cos2 x = sin 2x for 0  x  π, giving your answers in terms of π.
(Total 6 marks)
5.
If A is an obtuse angle in a triangle and sin A =
5
, calculate the exact value of sin 2A.
13
(Total 4 marks)
6.
The following diagram shows a triangle with sides 5 cm, 7 cm, 8 cm.
Diagram not to scale
Find
(a) the size of the smallest angle, in degrees;
(b) the area of the triangle.
5
7
8
(Total 4 marks)
Topic 4 + limits: 

1.
The vectors i , j are unit vectors along the x-axis and y-axis respectively.






The vectors u = – i + 2 j and v = 3 i + 5 j are given.




(a) Find u + 2 v in terms of i and j .



A vector w has the same direction as u + 2 v , and has a magnitude of 26.



(b) Find w in terms of i and j .
(Total 4 marks)
2.
The quadrilateral OABC has vertices with coordinates O(0, 0), A(5, 1), B(10, 5) and C(2, 7).
(a)
(b)
Find the vectors OB and AC .
Find the angle between the diagonals of the quadrilateral OABC.
(Total 4 marks)
3.
Consider the vectors u = 2i + 3 j − k and v = 4i + j − pk.
(a) Given that u is perpendicular to v find the value of p.
(b) Given that q u =14, find the value of q.
(Total 6 marks)
4.
 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)
5. Multiple choice: Which of the following statements is true
about the graph?
(a) g(x) is continuous at x = 5
(b)
lim g ( x)  DNE
x 3
lim g ( x)  3

x4
(c)
(d) g(x) is continuous at x = 6
(e)
lim g ( x)  2.5
x 6
6. Evaluate each of the following:
lim ( x  4)
3
a) x 1
b)
 3x  2 
lim 

x2 
x 2 
 x 2  2 x  1, x  2
f
(
x
)


lim f ( x )
k  x, x  2
7. For what value of k does x 2
exist for
c)
1
1

lim x  2 2
x 0
x
Topic 5:
1. The histogram below shows the time T seconds taken by 93
children to solve a puzzle.
The following is the frequency distribution for T .
a. (i) Write down the value of p and of q .
(ii) Write down the median class.
b. A child is selected at random. Find the probability that the child takes less than 95 seconds to solve the puzzle.
c. Consider the class interval
.
(i) Write down the interval width.
(ii) Write down the mid-interval value.
d. Hence find an estimate for the
(i) mean;
(ii) standard deviation.
e. John assumes that T is normally distributed and uses this to estimate the probability that a child takes less than 95
seconds to solve the puzzle. Find John’s estimate.
2. Bag A contains three white balls and four red balls. Two balls are chosen at random without replacement.
a. (i) Complete the following tree diagram.
(ii) Find the probability that two white balls are chosen.
Bag A contains three white balls and four red balls. Two balls are chosen at random without replacement. Bag B contains
four white balls and three red balls. When two balls are chosen at random without replacement from bag B, the probability
that they are both white is . A standard die is rolled. If 1 or 2 is obtained, two balls are chosen without replacement
from bag A, otherwise they are chosen from bag B.
b. Find the probability that the two balls are white.
c. Given that both balls are white, find the probability that they were chosen from bag A.
3. The cumulative frequency curve to the right represents the
marks obtained by 100 students.
a. Find the median mark.
b. Find the interquartile range.
4. The weights of chickens for sale in a shop are normally
distributed with mean 2.5 kg and standard deviation 0.3 kg.
(a)A chicken is chosen at random.
(i)Find the probability that it weighs less than 2kg.
(ii) more than 2.8 kg.
(iii)Copy the diagram below. Shade the areas that
represent the probabilities from parts (i) and (ii).
(iv)Hence show that the probability that it weighs
between 2 kg and 2.8 kg is 0.7936 (to four significant
figures).
(b)A customer buys 10 chickens.
(i)Find the probability that all 10 chickens weigh
between 2 kg and 2.8 kg.
(ii)Find the probability that at least 7 of the chickens weigh between 2 kg and 2.8 kg. (Total 13 marks)
5. The following table shows the average weights ( y kg) for given heights (x cm) in a population of men.
Heights (x cm)
165
170
175
180
185
Weights (y kg)
67.8
70.0
72.7
75.5
77.2
a. The relationship between the variables is modelled by the regression equation
of
.Write down the value
and of .
b. Hence, estimate the weight of a man whose height is 172 cm.
c. Write down the correlation coefficient.
d. State which two of the following describe the correlation between the variables: strong, zero, positive, negative,
no correlation, weak