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IB Mathematics AI HL - Revision Ladder

Level 2

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Paper 1

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Question 1

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easy

[Maximum mark: 4]

Cylindrical logs of length 2.22.2 m and radius 7.67.6 cm are used to construct a fence line. Part of the logs are removed, as illustrated in the diagram below.

7ab1702add176182b785d942e7d7230ac6752635.svg

Find the volume of each log.

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Question 2

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easy

[Maximum mark: 6]

Mike wants to deposit part of his savings in a bank account that pays an annual interest rate of 4.1%compounded annually4.1\,\%\, \textbf{compounded annually}. The annual inflation rate is expected to be 3%3\% per year throughout the following 88 years. Mike wants his initial deposit to have a real value of $5000\$5\,000 after 88 years, compared to current values.

The bank gives Mike two proposals:

Proposal 1:A one-time investment at the start of the 8-year period.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\textbf{Proposal 1:} \,\,\,\text{A one-time investment at the start of the 8-year period.}

Proposal 2:Invest $2000 at the start of the 8-year period and make payments of $ x at the end of each year.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\textbf{Proposal 2:} \,\,\,\text{Invest \$2\,000 at the start of the 8-year period and make payments of \$ $x$ at the end of each year.}

  1. Find the minimum amount Mike should deposit if he accepts proposal 1. Round your answer to the nearest dollar. [3]

  2. Find the minimum value of the annual payments, xx, if Mike accepts proposal 2. Round your answer to the nearest dollar. [3]

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Question 3

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easy

[Maximum mark: 4]

The magnitude of an earthquake, MM, on the Richter scale, can be modelled by the function

M(E)=23log10(E)3.2,M(E) = \dfrac{2}{3}\log_{10}(E) - 3.2, \\[3pt]

where EE is the amount of energy, in joules, released by the earthquake.

  1. Find the magnitude of an earthquake which releases 6.3×10136.3\times10^{13} joules of energy\text{energy}. [2]

The Great Chilean Earthquake of 19601960, the most powerful earthquake recorded in the 2020th century, had a magnitude of 9.59.5 on the Richter scale.

  1. Find the amount of energy, in joules, released by the Great Chilean Earthquake\text{Earthquake}. [2]

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Question 4

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easy

[Maximum mark: 6]

A survey was conducted at a New York high school on the preferred sports of a cohort of students from the options of of football, basketball, athletics or swimming. The results are shown in the table below, with results categorised by gender.

f927fd4558a0d486c785e348d70360dcb0a58221.svg

A χ2\chi^2-test was performed at the 5%5\% significance level to determine if sport preference is independent of gender.

  1. State the null hypothesis for this test, H0H_0. [1]

  2. Calculate the expected number of boys who preferred basketball. [2]

  3. Use your graphic display calculator to find the pp-value for this test. [2]

  4. State whether a students preferred sport is independent of [1]
    their gender. Give a reason for your answer.

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Question 5

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easy

[Maximum mark: 6]

A 33D printer builds a set of 4949 Eif\text{f}fel Tower Replicas in different sizes. The height of the largest tower in this set is 6464 cm. The heights of successive smaller towers are 9595 % of the preceding larger tower, as shown in the diagram below.

AI110

  1. Find the height of the smallest tower in this set. [3]

  2. Find the total height if all 4949 towers were placed one on top of another. [3]

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Question 6

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easy

[Maximum mark: 6]

The cumulative frequency curve below shows the number of hours spent by 6060 students on their IB homework during one week.

e4f96999c2aac8ece4e4ff865f8aa1d973517b8b.svg

  1. Write down the median value of this data. Give your answer correct to the nearest hour. [1]

  2. The upper quartile of this data is 10.110.1. Find the interquartile range, giving your answer correct to one decimal place. [2]

  3. The minimum number of hours spent by students on their homework is zero. The maximum number hours spent by students is 1414. Draw a box-and-whisker diagram to represent this data. [2]

  4. The 6060 students sampled were chosen by randomly sampling 3030 students from Diploma Program Year 11 and 3030 students from Diploma Program Year 22. Both the Year 11 and Year 22 cohorts had the same number of total students. Write down the sampling method used. [1]

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Question 7

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easy

[Maximum mark: 6]

Peter is a salesman who works for a skin care products company. He travels to a nearby city to find new customers. He researches six beauty salons on Yelp and pins their locations on Google Map.

The numbers on the graph below indicate the travel times in minutes between Peter's hotel (at A) and the beauty salons (B-G).

Peter is currently located at his hotel and wants to visit each beauty salon once, before returning to the hotel.

03eccc51b0bedb43cf6c315c4959bdd71b39011e.svg

  1. Write down the term which is used to describe this type of route. [1]

Peter decides to visit beauty salon E first.

  1. Find the route that he can take. [3]

  2. Calculate the total travel time, in hours and minutes, for his route. [2]

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Question 8

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easy

[Maximum mark: 6]

The equation of a curve is y=x2+8x12y = -x^2 + 8x - 12. A section of the curve is shown on the diagram below, with the two xx-intercepts labelled.

c2fbee5ada527ee7ac66bd331318622398a5d354.svg

  1. Find dydx\dfrac{\mathrm{d}y}{\mathrm{d}x}. [2]

  2. Write down the coordinates of the local maximum. [1]

  3. Write down an integral representing the area of the shaded region. [1]

  4. Find the area of the shaded region. [2]

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Question 9

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easy

[Maximum mark: 6]

The runtime, RR, of lithium batteries made by a battery manufacturing company follows a normal distribution. RR is measured in hours of continuous use in a camping head lamp.

The normal distribution curve of RR is shown below.

ea19a30bee9041db718ef8c59822d9d75879bbb1.svg

  1. Write down the mean runtime of the batteries. [1]

The standard deviation of the runtime of the batteries is 1212 hours.

  1. Find the probability that 85R10085 \leq R \leq 100, for a randomly chosen battery. [2]

The company states that 95%95\hspace{0.1em}\% of the batteries have a runtime of at least nn hours.

  1. Find the value of nn. Give your answer correct to the nearest integer. [3]

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Question 10

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easy

[Maximum mark: 6]

An object is moving along a straight line such that its velocity, vv, ms1^{-1}, is given by v(t)=5te1.2tv(t) = 5 t e^{-1.2t}, for t0t \geq 0.

  1. On the grid below, sketch the graph of vv, for 0t30 \leq t \leq 3. [2]

    9e220f7900c7ae0744aee17bcbc0834aa0ea52fb.svg

  2. Find the distance travelled by the object in the first 33 seconds.[2]

  3. Find the velocity of the object when its acceleration is zero. [2]

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