Practice Paper 2 (DP IB Maths: AA SL)

Jennifer sells cups of tea at her shop and has noticed that she sells more tea on cooler days. On five different days, she records the maximum daily temperature, , measured in degrees Celsius, and the number of cups of teas sold, . The results are shown in the following table.      

Maximum Daily Temparature , .	3	5	8	9	12
Cups of tea sold, .	37	34	33	26	21

The relationship between  and  can be modelled by the regression line of  on  with equation 

Find the value of and the value of .

Write down the value of Pearson's product-moment correlation coefficient, .

Use your regression equation from part (a) (i) to estimate the number of teas that Jennifer will sell on a day when the maximum temperature is 11°C.

A scientist is studying the movement of snails and has observed that the distribution of their speeds, , follows a normal distribution with a mean of 48 m/h and a standard deviation of 1.5 m/h.

Sketch a diagram to represent this information.

Find the probability that a randomly selected snail has a speed of less than 46.5 m/h.

From a sample of 80 snails, calculate the expected number of snails that would have a speed of less than 46.5 m/h. Give your answer to the nearest integer.

A hamster runs in its exercise wheel, rotating the wheel at a constant speed. The wheel has a diameter of 14 centimetres and the top of the wheel is positioned at a height of  centimetres above the floor of the cage.

A point at the top of the wheel is marked before the hamster starts to run, turning the wheel clockwise. The hamster takes 4 seconds to turn the wheel one complete revolution.

After  seconds, the height of the mark on the wheel above the floor of the cage is given by

       for 

After 26 seconds, the mark is 3 cm above the cage floor. Find .

Find the value of .

A particle moves along a straight line with a velocity,  ms^-1, given by  where  is measured in seconds such that 

Find the acceleration of the particle at time 

State the time when the particle comes to rest.

Find the total distance travelled by the particle.

In the expansion of , the coefficient of the  term is , where .

Find .

Three triplets, Aneirin, Bran and Culhwch, are each given a gift of £12000 by their grandmother on their twenty-first birthday.

Aneirin invests his £12000 in a bank account that offers an interest rate of 5.84% per annum compounded annually.

Bran invests his £12 000 in a bank account that offers an interest rate of % per annum, compounded monthly, where  is set to two decimal places.

Find the minimum value of  needed if Bran is to have at least as much money in his account after 9 years as Aneirin has in his.

Culhwch doesn't trust banks, and so he puts all of his £12000 in a metal box buried in his garden where he does not earn any interest. His savings plan is simply to add more money to the box each year, such that the money added each year will be a fixed multiple of the money added the previous year.

James has an awkwardly-shaped corner of his garden that is hard to mow. The corner is bordered by two straight fences that meet at point , with the garden lying between them. James' solution to his mowing problem is to buy a goat and tie it to one of the fences at point , so that the goat can eat all the grass it can reach. The rope is of length  metres. Points  and  (each on a different fence) are such that and  radians.  This is shown in the following diagram.

The length of arc  in the diagram is 16 m.

Write down an expression for in terms of .

Show that the area of James' garden that the goat can reach is m².
You may treat the goat as a point particle located at the end of the rope.

The area of James' garden that the goat can reach is equal to 240 m².

Find the value of .

Hence find the size of 

After the goat chews through the rope and eats all the flowers in a flowerbed at the far end of the garden, James decides to build a new fence between points  and  to enclose the goat's portion of the garden. This is shown in the following diagram.

Point  is due north of point  and  = 40 m. The bearing of  from  is 45°.

Find the size of .

The diagram below shows a part of the graph of the function

Point $\mathrm{A}$ is the point of intersection between the graph and the $y$-axis. Write down the coordinates of point $A$.

Find 

Using the graph, explain why the equation  must have exactly two distinct real solutions.

Point  is the point on the graph with -coordinate  .

Find the gradient of the tangent line to the graph at point .

Points  and  are the points on the graph at which the tangent lines are perpendicular to the tangent line at point .

By first determining the gradient of the tangents at points  and , find the -coordinates of points  and .

Given that point  lies between points  and  on the graph, find the equation of the tangent line to the graph at point .