IB Analysis and approaches HL, summative assessment October 2019

IB: Analysis and approaches Higher level                 TIME: 2 hours

Instructions to candidates  

• You are permitted to access graphic display calculator for this paper.
• Section A and B:  Answer all questions.  Answers must be written within the answer boxes provided.  
• 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 HL and further mathematics HL formula booklet is required for this paper.
• The maximum mark for this examination paper is [110 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.

Section A

1. [Maximum marks: 6] 

2. [Maximum marks: 6] 

Solve the following equation, expressing x in terms of natural logarithms

3. [Maximum marks:7]

Solve the simultaneous equations below

Express the solution in terms of a and b, where a = log 3 and b= log 5

3^5x-y=10

5^x+y=100

4. [Maximum marks:6]
Solve for x

5. [Maximum marks: 4]

Consider the expansion of: (x²- 5)⁷

• Write down the number of terms in the expansion [1 mark]
• Write down the first three terms of the expansion. [1 mark]
• The fourth term is Ax⁸ [ 2 marks]

6. [Maximum marks: 4]

Let a not equal to 0 represent the first term and r represent the common ratio of a geometric series. The sum to infinity of this series equals 4 times its second term.

Find the value of r

7.  [Maximum marks:6]

Find the coefficient of x³yz² in the expansion of (x + 2y + 3z)⁶

^8. [Maximum marks:7]

Consider the expansion of ( square root of x - 4x)⁸

• Write down the number of terms in this expansion [ 1 mark]
• Write down the term in which x has the small exponent [ 1 mark]
• Write down the term in which x has the largest exponent [1 mark]
• Find the coefficient of x⁶  [4 marks]

9. [Maximum marks: 9]

Christopher wons $1000 in a competition. He decides to invest his money. His financial adviser gives him three investment options:

Option 1: An interest payment of $15 every year.

Option 2: An interest payment of 2% per annum compounded annually.

Option 3: An interest payment of 1.6% per annum, compounded monthly.

• Calculate the value of Christopher’s investment after 5 years if he invests in:

(a) Option 1

(b) Option 2

(c) Option 3

Option 2 is a high-risk investment compared to Option 3. Christopher decides to invest $400 in Option 2 and the rest in Option 3.

• Find the number of complete years Christopher will have to wait in order to earn at least $250 

Section B

10. [Maximum marks: 12]

The first three terms in an arithmetic sequence are as follows:

U₁ = 300 , u₂ = 321, u₃ = 342

• Find the value of u₁₅  [2 marks]
• Find the sum of the first 20 terms of the sequence [2 marks]

In a geometric sequence, the first three terms are:

W₁ = 1 , w₂ = 2 , w₃ = 4

• Find the value of w₁₅ [2 marks]
• Find the value of the sum of the first 20 terms of the sequence [ 2 marks]
• Find the smallest value of n for which w_n > u_n [ 4 marks]

11.  [Maximum marks: 16]

The diagram below shows the first three images in a sequence of figures.

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 according to the pattern above. ( splitting the bottom left square into fours squares and shading one of these.)

(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: 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).

•  How much is this first investment worth on John’s 18^th birthday? [2 marks]

The family also decides to put $1000 in the account on John’s every birthday.

•  How much is the second investment worth on John’s 18^th birthday? [1 mark]

• How much money is there in the account on John’s 18^th birthday (including the $1000 the family puts in the account on this day)?  [4 marks]

• 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 18^th birthday? [3 marks]

13. [Maximum marks:17]

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 x 1.12ⁿ

a.     Find the number of taxis in the city at the end of 2005    [2 marks]

b.     Find the year in which the number of taxis is double the number of taxis that were at the end of 2000.                         [3 mark]

At the end of the year 2000, there were 25600 people in the city who used taxis. After n years the number of people, P, in the city who used taxis is given by 

c.     Find the value of P at the end of 2005, giving your answer in the nearest whole number.                [3 mark]

d.     After seven complete years will the value of P be double its value at the end of 2000? Justify your answer       [4 mark]

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

e.     Find the value of r at the end of the year 2000   [2 mark]

f.     After how many complete years will be city first reduce the number of taxis? [3 marks]

Resource: IB Past Papers, HAESE, www.kognity.com