TIME:
1
hr 30minutes
Instructions
to candidates
- · You are permitted to access a graphic display calculator for this paper.
- · Section A and Section B: answer all questions.
- · Unless otherwise stated in the question, all numerical answers should be given exactly or correct to three significant figures.
- · A clean copy of the mathematics SL formula booklet is required for this paper.
- · The maximum mark for this examination paper is [90 marks]
Full marks are not necessarily awarded for a correct
answer with no working. Answers must be
supported by working and/or explanations.
Where an answer is incorrect, some marks may be given for a correct
method, provided this is shown by written working. You are therefore advised to show all
working.
Write the following expression in terms of p, q, and r
2. [Maximum marks: 5] :
(b) Hence or otherwise, solve the equation=\log_{3}1 "\log_{3}x-\log_{3}(x-5)=\log_{3}1")
Section A
1. [Maximum
marks: 4]
Write the following expression in terms of p, q, and r
2. [Maximum marks: 5] :
When the expression (2+ax)10 is
expanded, the coefficient of the term x3 is 414720. Find the value of
a
3. [Maximum marks: 6]:
(a).=\log_{3}A\::express\:A\:in\:terms\:of\:x "Given\:that\:\log_{3}x-\log_{3}(x-5)=\log_{3}A\::express\:A\:in\:terms\:of\:x")
(b) Hence or otherwise, solve the equation
4. [Maximum marks: 6]
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.
5. [Maximum
marks: 6]
The company Snakezen’s Ladders makes
ladders of different lengths. All the ladders that the company makes have the
same design such that:
The first rung is 30cm from the base of
the ladder,
The second rung is 57cm from the base of
the ladder,
The distance between the first and the second rung is equal to the distance between all adjacent rungs on the ladder
If the company made a ladder having
seven equally spaced rungs
(a) find the distance from the base of this ladder to the top rung
The company
also, make a ladder that is 1050cm long
(b) Find the
maximum number of rungs in this 1050cm long ladder
6. [Maximum marks: 5]
A bacterial cell is 4.6 X 10-7m
long. Behind it are flagella 2.15 X 10-6 m long which allows the
bacterium to move. Find the total length of the bacterium.
7. [Maximum marks: 5]
An arithmetic sequence has the first
term ln a and a common difference ln 3. The 13th term in the
sequence is 8 ln 9. Find the value of a.
8. [Maximum marks: 8]
The sum of the terms of a sequence
follow the pattern
S1 = 1+k, S2= 5+
3k, S3= 12+7k, S4= 22 + 15k,…. Where k is an integer.
(a) Given that u1= 1+ k, find u2,
u3 and u4
(b) Find a general expression for un
Section B
9. [ Maximum marks: 10]
John’s family set up a bank account when John was born and they put $1000
in this account. The bank pays 2% interest on the money in the account on the day before each of John’s birthdays (starting with the first birthday).
(a) How much is this first investment worth on John’s 18th
birthday? [2 marks]
The family also decides to put $1000 in the account on John’s every
birthday.
(b) How much is the second investment worth on John’s 18th
birthday? [1 mark]
(c) How much money is there in the account on John’s 18th birthday
(including the $1000 the family puts in the account on this day)? [4 marks]
(d) How much money would the family need to put in the account on every
birthday if they want John to have %25000 in the account on his 18th
birthday? [3 marks]
10. [Maximum marks: 10]
The population of the town of Pristine, at the end of 1991, was 132000.
10. [Maximum marks: 10]
The population of the town of Pristine, at the end of 1991, was 132000.
The population has since increased by 1.9% annually.
(a) What was the population
of Pristine by the end of 2010? [3 marks]
(b) In what year will the
population first exceed 230000? [3 marks]
Unfortunately, in
Pristine as well as any other town, murders are committed now and then. The murder
rate has been stable since 1991, at two murders per 10000 inhabitants.
(c) How many murders were
committed in 1991? [1 mark]
(d) In what year will the number of
murders first be at least 100? [3 marks]
11. [Maximum marks: 16]
The first figure has four square regions, one of which is shaded. The second figure has seven square regions, two of which are shaded. On all of these figures, the length of the side of the outer square is 1 cm. We continue constructing the according to the pattern above.( splitting the bottom left square into fours squares and shading one of
these.)
12. [Maximum marks: 9 ]
} "Let\:p\;=\frac{z\;sin\;x-1}{\sqrt{y}-3(z-1)}")
11. [Maximum marks: 16]
The diagram below shows the first three
images in a sequence of figures.
(a) How many square regions are there in the fourth and fifth
figures? [2 marks]
(b) What is the area of the shaded squares individually and
in total in the fifth figure?[3 marks]
(c) How many square regions are there in the nth figure? [3 marks]
(d) Which figure is the first with more than 1000 square
regions? [2 marks]
(e) The shaded region fills more than 33.33% of
the outer square in the nth figure. Find the smallest possible value of n. [4
marks]
(f) We continue the construction infinitely(we
keep adding the smaller and smaller shaded squares in the same outer square).
What is the total area of the shaded squares? [ 2 marks]
12. [Maximum marks: 9 ]
(a) Calculate the exact value of p when x
=30degree ; y = 7225 and z = 8. Express your answer as a fraction p/q where p and
q are integers
(b) Write down the answer to part (a):
- Correct to three decimal places
- Correct to three significant figures
- In the form a X 10k where
Questions are taken from Haese mathematics textbook, kognity.com, and IB past papers
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