Practice Paper 1

Two points,  (2, 1, 3) and (5, 2, 6), are located on an  coordinate grid as shown in the diagram below.

Find the length of .

Find the coordinates of the midpoint of [AB].

Vivian has two containers. The first container is in the shape of a cylinder with diameter 28 cm and height 37 cm. The second container is in the shape of a cuboid with width 28 cm, height 37 cm and length  cm.

Calculate the surface area of the first container.

Both containers have the same surface area.

Find the value of .

Adah would like to estimate the height of a tree located at point  on the edge of a riverbank, with the top of the tree at point . However, due to a raging river, she is unable to reach the base of the tree. From point M she measures an angle of elevation of 20° to the top of the tree, and then from point  (which is on the edge of Adah's bank of the river) she measures an angle of elevation of 35° to the top of the tree. Between the points and  she measures a horizontal distance of 12 m. Points , and  all lie on a single horizontal line, and point  is vertically above point . The diagram below shows this information.

Calculate the length of .

Calculate the height of the tree.

Adah borrows a boat and crosses the river at a rate of 50 metres per 15 minutes.

Assuming that she crosses in a straight line directly from point  to point , find out how long it takes her to cross the river.

Jeanette works for a conservation charity who rescue orphaned otters. Over many years she records the weight () of each otter when it first arrives. The data is illustrated in the following box and whisker diagram:

Using the box plot above:

The line  passes through the points (1, 7) and (5, 5).

Find the equation of . Give your answer in the form of .

A new line, , is perpendicular to and passes through the point (4, 8).

Find the equation of . Give your answer in the form of .

The point  is the intersection of and .

Find the coordinates of .

A function is defined by , for .

Find the range of .

Find the value of . Give your answer as a fraction.

It is claimed that women from Japan are taller on average than women from India. The heights, in cm, of 11 women from each country have been collated in the table below.

A -test is to be performed at the 5% significance level.

State the null and alternative hypotheses.

Find the -value for this test.

State whether or not the initial claim is justified. Give a reason for your answer.

A lawn sprinkler sprays water over a lawn covering an arc of 160° with a maximum spray distance of m as shown in the diagram below. The lawn sprinkler waters 20 m² of the lawn.

Calculate the value of .

Calculate the length of the outer arc.

Daniel and Jonah have each been given $5000 to save for university.

Daniel invests his money in an account that pays a nominal annual interest rate of 2.24%, compounded quarterly.

Calculate the amount Daniel will have in his account after 8 years. Give your answer to 2 decimal places.

Jonah wants to invest his money in an account such that his investment will double in 10 years. Assume the account pays a nominal annual interest of r0/o, compounded half-yearly. 

Determine the value of .

A carpet salesman in interested how his sales are distributed and records his sales results over a period of six months. The data is shown in the table.

A chi-squared goodness of fit test is to be performed on the data at the 5% significance level to find out whether the data fits a uniform distribution.

The critical value for the test is 11.070 and the hypotheses are

      H₀ : The data satisfies the model.
      H₁ : The data does not satisfy the model.

Find an estimate of how many carpets the salesman expects to sell each month.

Write down the number of degrees of freedom for this test.

State the conclusion of the test. Give a reason for your answer.

The function  intercepts the -axis at —16, has an -intercept when = —4 and can be obtained by an appropriate translation of the graph  .

Find the values of and .

Write down .

Find the other -intercept of .

Write down the coordinates of the vertex of 

A rice farm sells  kg of rice every week.

It is known that where  is the weekly profit, in dollars ($), from the sale of  kg of rice.

Find the amount of rice, in kg, that should be sold each week to maximise the profit.

The profit from selling 250 kg of rice is $480.

Find