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IB Mathematics AI HL - Mock Exams

Mock Exam Set 1 - Paper 1

Trial Examinations for IB Mathematics AI HL

Paper 1

17 Questions

120 mins

110 marks

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Question 1

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easy

[Maximum mark: 4]

Simon loads his truck with cartons of eggs and transports them from a farm directly to a local market. In his experience, the mean number of eggs broken while transporting a load is 3.53.5. The number of eggs broken while transporting a load is assumed to follow a Poisson distribution.

On a day in which Simon has to transport 44 separate loads to the market, find the probability that there will be less than 1010 eggs broken in total.

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Question 2

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easy

[Maximum mark: 5]

Joe creates a small baseball field in his backyard to play with his children. The field is covered by grass in the shape of a sector of a circle with radius 88 metres, as shown in the following diagram. The angle at the centre of the sector is θ\theta^{\circ}.

AI997a

A total of 4848 m2^2 of grass was used to cover the field.

  1. Find the value of θ\theta. [2]

Joe asks his son, Lancer, to run around the field twice to warm up before playing a game.

  1. Find the total distance that Lancer runs. [3]

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Question 3

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easy

[Maximum mark: 5]

The cross-section of a ship's hull below the surface of the ocean can be modelled by a parabola. The depth of the boat's hull, dd metres, is given by d(x)=0.5x26.4xd(x) = 0.5x^2-6.4x, where xx is the horizontal distance from the left-hand side of the hull at the surface, also in metres, as shown in the following diagram. The xx-axis represents the ocean surface.

AI1018a

  1. Find the equation of the axis of symmetry of the parabola that models the hull of the ship. [2]

The ship has a horizontal storage deck at a depth of 1010 metres, with a ceiling at surface level. The storage deck is used to transport containers in the shape of cuboids.

  1. Determine the maximum width of a container that the ship can transport on the storage deck, given that the container is 1010 metres high. Give your answer to the nearest metre. [3]

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Question 4

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easy

[Maximum mark: 5]

The manager of a small movie theatre recorded the number of tickets purchased each day for 1414 days. The number of tickets purchased each day were:

31, 36, 37, 17, 25, 27, 32, 54, 37, 21, 19, 26, 26, 29

For these data, the lower quartile is 2525 and the upper quartile is 3636.

  1. Show that the value 54 would be considered an outlier.[3]

The box and whisker diagram for tickets purchased is given below.

AI1014a

The manager of a second small theatre also recorded the number of tickets purchased for the same 14 days, giving the following box and whisker diagram:

AI1014b

The manager of the second theatre claims that, in general, his theatre has more customers than the first theatre.

  1. By comparing the two box and whisker diagrams, give one piece of evidence that supports this claim and one that may refute it. [2]

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Question 5

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easy

[Maximum mark: 6]

Edward wants to buy a new car, and he decides to take out a loan of 7000070\hspace{0.15em}000 Australian dollars from a bank. The loan is for 66 years, with a nominal annual interest rate of 7.2%7.2\%, compounded monthly. Edward will pay the loan in fixed monthly instalments.

  1. Determine the amount Edward should pay each month. Give your answer to the nearest dollar.[3]

  2. Calculate the amount Edward will still owe the bank at the end of the third year. [3]

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Question 6

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[Maximum mark: 6]

The Great Pyramid of Giza can be modelled by a right-pyramid with a square base. A scale model of the Great Pyramid of Giza is shown below.

4abc6a4f018c2d92a630babb30f95c228896a1b6.svg

The vertices A, B and C have coordinates A(1,5.5,0)(1,5.5,0), B(6.5,0,7)(6.5,0,7) and C(12,5.5,0)(12,5.5,0) relative to a fixed origin O near the pyramid. All distances are measured in centimetres.

    1. Find AB\vv{\text{AB}}.

    2. Find AC\vv{\text{AC}}. [2]

  1. Find AB×AC\vv{\text{AB}} \times \vv{\text{AC}}. [2]

  2. Hence, or otherwise, find the surface area of the scale model, not including the base.[2]

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Question 7

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[Maximum mark: 6]

A large flower export company in the Netherlands packs tulips in fibreboard boxes for transportation. It is claimed by the owner of the company that each box contains an average of 7878 tulips.

Yashada, a client in Japan, chooses 2525 boxes at random from a container shipped to him and records the number of tulips in each box to check the claim. Given that the number of tulips is represented by xx, he found that

x=1962 and x2=154090.\begin{aligned} \sum x &= 1962\text{ and } \sum x^2 = 154\hspace{0.15em}090. \end{aligned}
  1. Find an unbiased estimate for the mean number (μ\mu) of tulips per box. [1]

To find an unbiased estimate for the standard deviation of the number of tulips per box, Yashada uses the formula

sn1=x2(x)2nn1\begin{aligned} \\ s_{n-1} = \sqrt{\dfrac{\sum x^2 - \dfrac{(\sum x)^2}{n}}{n-1}}\end{aligned}
  1. Determine the unbiased estimate for the standard deviation found by Yashada. [2]

  2. Find a 99%99\% confidence interval for μ\mu, assuming the conditions for a normally distributed population has been met. [2]

  3. Based on the result in part (c), state a valid conclusion that Yashada could make. Justify your answer. [1]

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Question 8

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[Maximum mark: 7]

Every night a street cleaner is tasked with cleaning the streets in his designated area.

The diagram below shows his designated area, with the weights indicating the time, in minutes, it takes to clean each street, and intersections indicated by letters.

b4a84a9dbf92fa43c424a84e3e26819d15b79a75.svg

The street cleaner must clean every street and wishes to minimise the time the task will take.

  1. The street cleaner starts and finishes at the company depot, located at point K. Determine the total time it will take him to complete his task. [5]

A new road is to be constructed directly from the intersection at point B to the intersection at point G, and it is estimated that this street will take 25 minutes to clean. The street cleaner has also been told he can now start the job at any intersection, and finish at any other intersection, and another employee will pick him up and drop him off.

  1. Determine the difference in total time the job will take once the new road is completed.[2]

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Question 9

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[Maximum mark: 7]

Elon is challenged to a speed climb at a local mountain. He has to reach a height of 400400 metres above the ground within four hours.

Elon knows he can climb 150150 metres in the first hour. Due to increasing tiredness, each hour he can only climb 75%75\hspace{0.05em}\% of the height climbed in the previous hour.

  1. Verify that Elon reaches his target height of 400400 metres in four hours.[2]

The mountain has a height of 650650 metres. Elon decides to attempt to climb to the summit.

  1. Determine whether he can reach the summit of the mountain if he continues climbing, given his increasing tiredness. Justify your answer.[2]

On a different day, Elon climbs with energy snacks, which help to reduce his tiredness as he climbs. On this day, Elon again climbs 150 metres in the first hour, but then k%k\hspace{0.05em}\% of the height he climbed in the previous hour, where k>75k > 75.

  1. Calculate the minimum value of kk, given that on this day Elon is able to reach the summit. Give your answer as a percentage, to the nearest whole number.[3]

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Question 10

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[Maximum mark: 7]

Let f(x)=exf(x) = e^x, for xRx \in \mathbb{R}. The graph of ff is translated by (pq)\left(\hspace{-0.2em}\begin{matrix} -p \\[2pt] \hspace{0.2em}q \end{matrix}\right) to give the graph of a function gg that passes through the points A(0,1.5)(0,1.5) and B(ln2,2)(\ln2, 2).

Find the value of pp and the value of qq.

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Question 11

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[Maximum mark: 8]

Melody is camping with her friends in a forest. After setting up the tent, she goes for a walk to see the nearby surroundings. She walks 200200 m on a bearing of 280°\ang{280}, changes direction, then walks another 150150 m on a bearing of 035°\ang{035}.

  1. Find the distance Melody is from the tent at this point. [4]

  2. Determine the bearing Melody should walk on to return to the tent. [4]

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Question 12

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medium

[Maximum mark: 8]

BioNature Inc. is considering the monthly cost of producing straws made from paper. It is estimated that the rate of change in monthly cost with respect to the number of paper straws produced is modelled by

C(q)=0.006q20.48q12, for q>0C'(q) = 0.006q^2 -0.48q - 12, \text{ for } q > 0

Where the monthly cost, $C\,C, is in thousands and the number of straws, qq, is also in thousands.

  1. Find the number of paper straws that the company should produce in a month to minimize the cost. [4]

The company produces 110000110\hspace{0.15em}000 paper straws with a cost of $88000\$88\hspace{0.15em}000 in a certain month.

  1. Find an expression for CC in terms of qq. [4]

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Question 13

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[Maximum mark: 8]

After school, a group of six students play a soccer passing game. Alex, Bella, Cleo, Dixie, and Emmy stand in a circle and pass the ball to each other while Ben, standing in the middle, tries to intercept the passes.

The following diagram shows the possible paths that the ball can be passed between the players, in the form of a directed graph. Some of the students are more likely to pass the ball to their friends than to other students. The paths shown by dotted lines represent a pass that is twice as likely as a pass shown by a solid line. For example, Dixie can pass the ball to Alex and Cleo with probability 0.250.25 and to Emmy with probability 0.50.5. Dixie won't pass the ball to Bella.

849f3ed828f405629daaf9fc551b51a21f93e433.svg

It is assumed that each player keeps the ball for a constant time before passing it. At the start of the game, Alex has the ball.

  1. Determine the transition matrix for the graph. [3]

  2. Calculate the probability that Cleo has the ball after exactly four passes have been completed, assuming that Ben has not intercepted a pass. [3]

  3. If the players continue passing indefinitely, without an interception, determine which player will spend the least amount of time with the ball. [2]

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Question 14

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hard

[Maximum mark: 5]

The diagram below shows the slope field for the differential equation

dydx=cos(xy),6.5x4.5,0y5.5.\begin{aligned} \hspace{3.9em} \dfrac{\mathrm{d}y}{\mathrm{d}x} = \cos(x-y), \hspace{0.75em}-6.5\leq x \leq 4.5, \hspace{0.75em}0 \leq y \leq 5.5. \\ \end{aligned}

The graphs of the two solutions to the differential equation passing through points P(0,1)(0,1) and Q(0,3)(0,3) are drawn over the slope field.

90e5f091d5b901c342c544173a01c19a09907f69.svg

For the two graphs given, the local maximum points lie on the straight line L1L_1.

  1. Find the equation of L1L_1, giving your answer in the form y=mx+cy = mx+c. [3]

For the two graphs given, the local minimum points lie on the straight line L2L_2.

  1. Find the equation of L2L_2, giving your answer in the form y=mx+cy = mx+c. [2]

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Question 15

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hard

[Maximum mark: 7]

A geometric transformation T:(xy)(xy)T: \hspace{-0.15em}\bigg(\hspace{0.1em}\begin{matrix}{x} \\[2pt] {y} \end{matrix}\hspace{0.1em}\bigg) \hspace{-0.15em}\mapsto \hspace{-0.15em} \bigg(\hspace{0.1em}\begin{matrix} {x'} \\[2pt] {y'} \end{matrix}\hspace{0.1em}\bigg)\hspace{-0.1em} is defined by

(xy)=(cos(π3)sin(π3)sin(π3)cos(π3))(xy)32(12).\begin{aligned} \Bigg(\hspace{0.1em}\begin{matrix}{x'} \\[6pt] {y'} \end{matrix}\hspace{0.1em}\Bigg)\hspace{-0.05em} &= \Bigg(\hspace{0.1em}\begin{matrix} \mathrm{cos}\big(\dfrac{\pi}{3}\big) & \mathrm{sin}\big(\dfrac{\pi}{3}\big) \\[6pt] \mathrm{sin}\big(\dfrac{\pi}{3}\big) & \mathrm{cos}\big(\dfrac{\pi}{3}\big) \end{matrix}\hspace{0.1em}\Bigg)\hspace{-0.15em}\Bigg(\hspace{0.1em}\begin{matrix} {x} \\[6pt] {y} \end{matrix}\hspace{0.1em}\Bigg)\hspace{-0.05em} - \dfrac{\sqrt{3}}{2}\Bigg(\hspace{0.1em}\begin{matrix} 1 \\[6pt] 2 \end{matrix}\hspace{0.1em}\Bigg)\hspace{-0.1em}. \\ \end{aligned}
  1. Find the coordinates of the image of the point P(1,3)(1,-\hspace{0.2em}\sqrt{3}). [2]

  2. Given that T:(ab)12(ab)T: \hspace{-0.15em}\bigg(\hspace{0.1em}\begin{matrix} {a} \\[2pt] {b} \end{matrix}\hspace{0.1em}\bigg) \hspace{-0.15em}\mapsto \hspace{-0.05em} \dfrac{1}{2}\hspace{-0.05em} \bigg(\hspace{0.1em}\begin{matrix} {a} \\[2pt] {b} \end{matrix}\hspace{0.1em}\bigg)\hspace{-0.1em}, find the value of aa and the value of bb. [3]

A rhombus RR with vertices lying on the xyxy-plane is transformed by TT.

  1. Show that the area of the image is half the size of RR. [2]

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Question 16

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hard

[Maximum mark: 8]

A particle PP moves is a straight line such that its displacement xx at time t0\text{\(t \geq 0\)} is given by the diff\text{f}erential equation x˙=2x(tet2)\dot{x} = 2x\big(\hspace{-0.25em}-\hspace{-0.15em}te^{-t^2}\big). At time t=0\text{\(t = 0\)}, x=2x = 2.

  1. Use Euler's method with step length 0.10.1 to find an approximation for xx when t=0.4t = 0.4, giving your answer to 44 significant figures. [3]

  2. By solving the differential equation, find the percentage error in your approximation for xx when t=0.4t = 0.4.[5]

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Question 17

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hard

[Maximum mark: 8]

Let z=2cis(3π8)z = \sqrt{2}\mathop{\mathrm{cis}}\hspace{-0.1em}\left(\dfrac{3\pi}{8}\right) and w=2cis(nπ24)w = 2\mathop{\mathrm{cis}}\hspace{-0.1em}\bigg(\dfrac{n\pi}{24}\bigg) , where nZ+n \in \mathbb{Z}^+.

  1. Find the value of z6z^6. Give your answer in the form reiθre^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\theta}, where r0r \geq 0, π<θπ-\pi < \theta \leq \pi. [2]

  2. Find the value of (wz)4(wz)^4 for n=5n = 5. Give your answer in the form reiθre^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\theta}, where r0r \geq 0, π<θπ-\pi < \theta \leq \pi. [3]

  3. Find the smallest integer n>9n > 9 such that zwR\dfrac{z}{w} \in \mathbb{R}. [3]

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