Practice Paper 1 (DP IB Maths: AI SL)

Practice Paper Questions

1a
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1 mark

In a trial for a new drug, scientists found that the amount of the drug in the bloodstream decreased over time, according to the model

D left parenthesis t right parenthesis space equals space 1.4 space X space 0.77 to the power of t comma space space space space space space space t greater or equal than 0

where D is the amount of the drug in the bloodstream in mg per litre (mg L-1) and t is the time in hours.

Write down the amount of the drug in the bloodstream at t space equals space 0.

1b
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2 marks

Calculate the amount of the drug in the bloodstream after four hours.

1c
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3 marks

Calculate the time, in hours, for the amount of the drug in the bloodstream to decrease to 0.22 mg L-1.

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2a
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2 marks

Two points, straight A (2, 1, 3) and space straight B space(5, 2, 6), are located on an x y z coordinate grid as shown in the diagram below.

q2-practice-paper1-setc-ib-dp-ai-sl

Find the length of AB.

2b
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2 marks

Find the coordinates of the midpoint of [AB].

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3a
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3 marks

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 x cm.

q3-practice-paper1-setc-ib-dp-ai-sl

Calculate the surface area of the first container.

3b
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4 marks

Both containers have the same surface area.

Find the value of x.

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4a
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3 marks

Adah would like to estimate the height of a tree located at point straight P on the edge of a riverbank, with the top of the tree at point straight Q. 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 straight N (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 straight M and straight N she measures a horizontal distance of 12 m. Points M, straight N and straight P all lie on a single horizontal line, and point straight Q is vertically above point straight P. The diagram below shows this information.

q4-practice-paper1-setc-ib-dp-ai-sl

Calculate the length of NQ.

4b
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2 marks

Calculate the height of the tree.

4c
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3 marks

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 straight N to point P, find out how long it takes her to cross the river.

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5
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4 marks

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

q5-practice-paper1-setc-ib-dp-ai-sl

Using the box plot above:

i)
Write down the median weight of the otters.

ii)
Write down the lower quartile.

iii)
Find the interquartile range.

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6a
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2 marks

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

Find the equation of l subscript 1. Give your answer in the form of space y space equals space m x space plus space c.

6b
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2 marks

A new line, l subscript 2, is perpendicular to l subscript 1 and passes through the point (4, 8).

Find the equation of l subscript 2. Give your answer in the form of y space equals space m x space plus space c.

6c
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2 marks

The point Z is the intersection of l subscript 1and l subscript 2.

Find the coordinates of Z.

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7a
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3 marks

A function is defined by f space left parenthesis x right parenthesis space equals 4 minus fraction numerator 12 over denominator 5 x plus 9 end fraction, for negative 8 space less or equal than space space x space less or equal than space 8 comma space x equals negative 9 over 5.

Find the range of f.

7b
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2 marks

Find the value of space f to the power of negative 1 end exponent space left parenthesis 2 right parenthesis. Give your answer as a fraction.

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8a
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2 marks

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.

Japan India
173.0 155.2
158.2 157.8
148.5 156.0
150.6 142.7
168.7 149.6
149.8 150.1
158.8 152.6
155.3 148.2
159.2 151.3
158.9 147.6
166.0 168.0

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

State the null and alternative hypotheses.

8b
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2 marks

Find the p-value for this test.

8c
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2 marks

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

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9a
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4 marks

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

q9-practice-paper1-setc-ib-dp-ai-sl

Calculate the value of x.

9b
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3 marks

Calculate the length of the outer arc.

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10a
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3 marks

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.

10b
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3 marks

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 r.

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11a
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1 mark

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.

Month January February March April May June
Number of sales 16 12 14 20 15 19

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.

11b
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1 mark

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

11c
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4 marks

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

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12a
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4 marks

The function g left parenthesis x right parenthesis space equals space a x squared space plus space b x space plus space c intercepts the y-axis at —16, has an x-intercept when x = —4 and can be obtained by an appropriate translation of the graph y space equals space 2 x squared .

i)
Find the values of a comma space b and c.

ii)
Write down g space left parenthesis x right parenthesis.
12b
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1 mark

Find the other x-intercept of g space left parenthesis x right parenthesis.

12c
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2 marks

Write down the coordinates of the vertex of g space left parenthesis x right parenthesis.

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13a
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3 marks

A rice farm sells x kg of rice every week.

It is known that fraction numerator d P over denominator d x end fraction equals negative 0.02 x plus 6 comma space x greater or equal than 0 where P is the weekly profit, in dollars ($), from the sale of x kg of rice.

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

13b
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5 marks

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

Find P left parenthesis x right parenthesis.

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