Question 1a

Marks: 4

q2a-set-c-ai-sl-paper-2-ib-maths

Question 1b

Marks: 2

q2b-set-c-ai-sl-paper-2-ib-maths

Question 1c

Marks: 4

q2c-set-c-ai-sl-paper-2-ib-maths

Question 1d

Marks: 6

q2d-set-c-ai-sl-paper-2-ib-maths

Question 2a

Marks: 2

Question 2b

Marks: 1

Question 2c

Marks: 2

Question 2d

Marks: 2

Question 2e

Marks: 4

Question 2f

Marks: 3

Question 3a

Marks: 1

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Question 3b

Marks: 3

q6b-6-1-extended-questions-hard-ib-ai-sl-maths

Question 3c

Marks: 2

q6c-6-1-extended-questions-hard-ib-ai-sl-maths

Question 3d

Marks: 2

q6d-6-1-extended-questions-hard-ib-ai-sl-maths

Question 3e

Marks: 2

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Question 3f

Marks: 4

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Question 3g

Marks: 3

q6g-6-1-extended-questions-hard-ib-ai-sl-maths

Question 4a

Marks: 2

In the town of Manh, all the residents belong to either one or the other of the town’s two fitness clubs – Giang’s House of Fitness (G) or Thu’s Wonder Gym (T). Each year 30% of the members of $G$ switch to $T$ and 25% of the members of $T$ switch to $G$ . Any other losses or gains of members by the two fitness clubs may be ignored.

a)

Write down a transition matrix

T

representing the movement of members between the two clubs in a particular year.

Key Concepts

Markov Chains
Transition Matrices

Question 4b

Marks: 4

(b)

Find the eigenvalues and corresponding eigenvectors of

T

.

Key Concepts

Eigenvalues & Eigenvectors

Question 4c

Marks: 2

(c)

Hence write down matrices

P

and

D

such that

T = P D P^{- 1}

.

Key Concepts

Diagonalisation

Question 4d

Marks: 6

Initially there are 2500 members of $G$ and 800 members of $T$ .

(d)

Using the matrix power formula, show that the numbers of members of

G

and

T

after

n

years will be

(1500 + 1000 ({0.45}^{n}))

and

(1800 - 1000 ({0.45}^{n}))

, respectively.

Key Concepts

Powers of Transition Matrices
Inverse Matrices

Question 4e

Marks: 2

(e)

Hence write down the number of customers that each of the fitness clubs can expect to have in the long term.

Key Concepts

Steady State & Long-term Probabilities

Question 5a

Marks: 1

In a game, enemies appear independently and randomly at an average rate of 2.5 enemies every minute.

a)

Find the probability that exactly 3 enemies will appear during one particular minute.

Question 5b

Marks: 2

b)

Find the probability that exactly 10 enemies will appear in a five-minute period.

Question 5c

Marks: 2

c)

Find the probability that at least 3 enemies will appear in a 90-second period.

Question 5d

Marks: 2

d)

The probability that at least one enemy appears in

k

seconds is 0.999. Find the value of

k

correct to 3 significant figures.

Question 5e

Marks: 4

e)

A 10-minute interval is divided into ten 1-minute periods (first minute, second minute, third minute, etc.). Find the probability that there will be exactly two of those 1-minute periods in which no enemies appear.

Question 5f

Marks: 4

On the next level of the game, there is a boss enemy and a number of additional henchmen to fight against.

The number of times that the boss enemy appears in a one-minute period can be modelled by a Poisson distribution with a mean of 1.1.

The number of times that an individual henchman appears in a one-minute period can be modelled by a Poisson distribution with a mean of 0.6.

It may be assumed that the boss enemy and the henchmen each appear randomly and independently of one another.

Each time that the boss enemy or any particular henchman appears, it is counted as one ‘enemy appearance’.

f)

Determine the least number of henchmen required in order that the probability of 40 or more ‘enemy appearances’ occurring in a 3-minute period is greater than 0.38. You may assume that neither the boss enemy nor any of the henchmen are able to be totally eliminated from the game during this 3-minute period.

Question 6a

Marks: 1

James throws a throws ball to his friend Mia. The height, $h$ , in metres, of the ball above the ground is modelled by the function

$h (t) = - 1.05 t^{2} + 3.84 t + 1.97, t \geq 0$

where $t$ is the time, in seconds, from the moment that James releases the ball.

a)

Write down the height of the ball when James releases it.

Question 6b

Marks: 2

After 4 seconds the ball is at a height of metres above the ground.

b)

Find the value of

q

.

Question 6c

Marks: 2

c)

Find

h' (t)

Question 6d

Marks: 3

d)

Find the maximum height reached by the ball and write down the corresponding time

t

.

Question 6e

Marks: 4

James then drives a remote-controlled car in a straight horizontal line from a starting position right in front of his feet. The velocity of the remote-controlled car in ${ms}^{- 1}$ is given by the equation

$v (t) = \frac{5}{4} t^{3} - \frac{19}{2} t^{2} + 18 t - 2$

e)

Find an expression for the horizontal displacement of the remote-controlled car from its starting position at time

t

seconds.

Question 6f

Marks: 3

f)

Find the total horizontal distance that the remote-controlled car has travelled in the first 5 seconds.

Question 7a

Marks: 6

Consider the following system of differential equations:

$\frac{d x}{d t} = x + 2 y$

$\frac{d y}{d t} = - 3 x - 4 y$

(a)

Find the eigenvalues and corresponding eigenvectors of the matrix

(\begin{matrix} 1 & 2 \\ - 3 & - 4 \end{matrix})

.

Question 7b

Marks: 2

(b)

Hence write down the general solution of the system.

Question 7c

Marks: 3

When $t = 0$ , $x = 2$ and $y = 4$ .

(c)

Use the given initial condition to determine the exact solution of the system.

Key Concepts

Solving Coupled Differential Equations

Question 7d

Marks: 3

(d)

(i)

Find the value of

\frac{d y}{d x}

when

t = 0 .

(ii) Find the values of $x, y$ and $\frac{d y}{d x}$ when $t = \ln \frac{9}{7}$ .

Question 7e

Marks: 3

e)

Hence sketch the solution trajectory of the system for

t \geq 0

.

DP IB Maths: AI HL

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