Practice Paper 2 (DP IB Maths: AA HL)

An arithmetic sequence with a common difference -3.5 has first term 77.

Given that the th term of the sequence is zero, find the value of .

Find the maximum value of the sum of the first  terms of the sequence.

and  are independent events, such that = 0.25 and  = 0.52.  is another event, such that  and  are mutually exclusive and  = 0.09.

Given that = 0.95, find

Let  and ,  where each function has the largest possible valid domain.

Write down the range of .

Write down the domain and range of .

Find

.

The number of bacteria, , in a dish, after  minutes is given by.

Find the initial amount of bacteria.

Find the amount of bacteria after 12 minutes. Give your answer in the form , where 

Find the value of  when .

A UK energy company charges £0.22 per kilowatt hour (kWh) of electricity used.
The amount of energy used per day by the company’s customers,  kWh, follows the following probability density function

Show that .

A customer’s total daily charge consists of a fixed (standing) charge of £0.38 per day plus the charge for the electricity used.

Consider the nine letters in the word MAGNITUDE.

Find the number of ways that the nine letters may be arranged if

Consider  where .

Show that .

The binomial series expansion for  is given by

Using the above result and the Maclaurin series for , show that the Maclaurin series for  is

By using the result from part (a) and the Maclaurin series for , find the value of the limit

The Strike A Light! matchstick company produces matchsticks with a length,  mm, that is normally distributed with mean 45 and variance .

The probability that  is greater than 45.37 is 0.1714.

Find .

Andrew has a box of Strike A Light! matches with fifteen matchsticks remaining in it.Those matchsticks may be assumed to be a random sample. Let represent the number of matchsticks in Andrew's box with lengths less than 44.5 mm.

Find

Find the probability that exactly one of the matchsticks in Andrew's box has a length less than 44.5 mm.

A Strike A Light! matchstick is selected at random and is found to have a length greater than 44.5 mm.

Find the probability that the length of the matchstick is between 44.63 mm and 45.37 mm.

Paola is modelling a small vase from her house for her maths project. To model the edge of the vase in cross-section, she decides to use a function  of the form

where  and . 

The function and the vase are represented in the diagrams below.

The vertical height of the vase, OB, is measured along the -axis. The radius of the vase’s opening is OA, and its base radius is BC. 

To model the vase, she will rotate by  radians about the -axis the region enclosed by the graph of  ,  the -axis, the -axis, and the line  . 

Show that the volume of the solid of revolution thus formed is .

The volume of the actual vase is 100

Use this information to find the value of .

Find the cross-sectional radius of the vase

Paola wants to investigate how the cross-sectional radius of the vase changes.

Sketch a graph of the derivative of , and use it to find the value of  at which the cross-sectional radius of the vase is decreasing most rapidly.

A function is defined by  

Show that  is an even function.

By considering the limit of as tends to infinity, show that the graph of   has a horizontal asymptote and state its equation.

A new function, , is created by restricting the domain of , such that ,  ,  .

Find an expression for , carefully considering the range of  in determining your final answer.

State the domain of .