mathematics studies mock 16-17 paper...

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Mathematics Studies Mock 16-17 Paper 1

Mark Scheme

1a. [2 marks]

.

Find the value of z when x = 12.5, a = 0.572 and b = 0.447. Write down your full calculator display.

Markscheme

(M1)

Note: Award (M1) for correct substitution into formula.

= 21250 (A1) (C2)

[2 marks]

1b. [2 marks]

Write down your answer to part (a)

(i) correct to the nearest 1000 ;

(ii) correct to three significant figures.

Markscheme

(i) 21000 (A1)(ft)

(ii) 21300 (A1)(ft) (C2)

Note: Follow through from part (a).

[2 marks]

1c. [2 marks]

Write your answer to part (b)(ii) in the form a × 10, where 1 ≤ a < 10, .

Markscheme

(A1)(ft)(A1)(ft) (C2)

2

Notes: Award (A1)(ft) for 2.13, (A1)(ft) for . Follow through from part (b)(ii).

[ 2 marks]

2a. [1 mark]

The diagram shows a wheelchair ramp, , designed to descend from a height of .

Use the diagram above to calculate the gradient of the ramp.

Markscheme

(A1) (C1)

[1 mark]

2b. [1 mark]

The gradient for a safe descending wheelchair ramp is .

Using your answer to part (a), comment on why wheelchair ramp is not safe.

Markscheme

(A1)(ft) (C1)

Notes: Accept “less than” in place of inequality.

Award (A0) if incorrect inequality seen.

Follow through from part (a).

[1 mark]

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2c. [4 marks]

The equation of a second wheelchair ramp, B, is .

(i) Determine whether wheelchair ramp is safe or not. Justify your answer.

(ii) Find the horizontal distance of wheelchair ramp .

Markscheme

(i) ramp is safe (A1)

the gradient of ramp is (R1)

Notes: Award (R1) for “the gradient of ramp is ” seen.

Do not award (A1)(R0).

(ii) (M1)

Note: Accept alternative methods.

(A1) (C4)

[4 marks]

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3a. [2 marks]

A group of 100 students gave the following responses to the question of how they get to school.

A test for independence was conducted at the significance level. The null hypothesis was

defined as

: Method of getting to school is independent of gender.

Find the expected frequency for the females who use public transport to get to school.

Markscheme

OR (M1)

Note: Award (M1) for correct substitution into correct formula.

(A1) (C2)

[2 marks]

3b. [2 marks]

Find the statistic.

Markscheme

(A2) (C2)

Note: Award (A1)(A0) for .

[2 marks]

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3c. [2 marks]

The critical value is at the significance level.

State whether or not the null hypothesis is accepted. Give a reason for your answer.

Markscheme

the null hypothesis is not accepted (A1)(ft)

OR (R1)

OR

the null hypothesis is not accepted (A1)(ft)

p-‐value (R1) (C2)

Notes: Follow through from their answer to part (b).

Do not award (A1)(ft)(R0).

[2 marks]

4a. [2 marks]

In this question give all answers correct to the nearest whole number.

Fumie is going for a holiday to Great Britain. She changes Japanese Yen (JPY) into British

Pounds (GBP) with no commission charged.

The exchange rate between GBP and JPY is

1 GBP = 129 JPY.

Calculate the value of JPY in GBP.

Markscheme

(M1)

(A1) (C2)

[2 marks]

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4b. [4 marks]

At the end of Fumie’s holiday in Great Britain she has 239 GBP. She converts this back to JPY at a bank,

which does not charge commission, and receives 30 200 JPY

(i) Find the exchange rate of this second transaction.

(ii) Determine, when changing GBP back to JPY, whether the exchange rate found in part (b)(i) is better

value for Fumie than the exchange rate in part (a). Justify your answer.

Markscheme

(i) (M1)

(A1)

Note: Accept ( ).

Award (M1) for .

Award (A0) for

(ii) No, the part (b)(i) rate is not better value than the part (a) rate. (A1)(ft)

(R1)

OR

No, the part (b)(i) rate is not better value than the part (a) rate. (A1)(ft)

(R1) (C4)

Note: Accept “part (a) rate is better” for the (A1)(ft).

Follow through from part (b)(i).

A numerical comparison must be seen to award (R1).

[4 marks]

5a. [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

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where is the amount of the drug in the bloodstream in mg per litre and is the time in

hours.

Write down the amount of the drug in the bloodstream at .

Markscheme

(A1) (C1)

[1 mark]

5b. [2 marks]

Calculate the amount of the drug in the bloodstream after 3 hours.

Markscheme

(M1)

Note: Award (M1) for correct substitution into given formula.

(A1) (C2)

[2 marks]

5c. [3 marks]

Use your graphic display calculator to determine the time it takes for the amount of the drug in the

bloodstream to decrease to .

Markscheme

(M1)

Note: Award (M1) for setting up the equation.

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(M1)

Notes: Some indication of scale is to be shown, for example the window used on the calculator.

Accept alternative methods.

(hours) ( , 9 hours 12 minutes, 9:12) (A1) (C3)

[3 marks]

6a. [1 mark]

The heights of apple trees in an orchard are normally distributed with a mean of and a

standard deviation of .

Write down the probability that a randomly chosen tree has a height greater than .

Markscheme

(A1) (C1)

[1 mark]

6b. [1 mark]

Write down the probability that a randomly chosen tree will be within 2 standard deviations of the

mean of .

Markscheme

(A1) (C1)

Note: Accept or .

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[1 mark]

6c. [2 marks]

Use your graphic display calculator to calculate the probability that a randomly chosen tree will have a

height greater than .

Markscheme

(M1)


(A1) (C2)

[2 marks]

6d. [2 marks]

The probability that a particular tree is less than metres high is . Find the value of .

Markscheme

(M1)


(A1) (C2)

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[2 marks]

7a. [1 mark]

A class of 13 Mathematics students received the following grades in their final IB examination.

3 5 3 4 7 3 2 7 5 6 5 3 4

For these grades, find the mode;

Markscheme

3 (A1) (C1)

[1 mark]

7b. [2 marks]

For these grades, find the median;

Markscheme

4 (M1)(A1) (C2)

Note: Award (M1) for ordered list of numbers seen.

[2 marks]

7c. [1 mark]

For these grades, find the upper quartile;

Markscheme

5.5 (A1) (C1)

[1 mark]

7d. [2 marks]

For these grades, find the interquartile range.

Markscheme

5.5 – 3 (M1)

Note: Award (M1) for 3 and their 5.5 seen.

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= 2.5 (A1)(ft) (C2)

Note: Follow through from their answer to part (c).

[2 marks]

8a. [3 marks]

Consider the three propositions p, q and r.

p: The food is well cooked

q: The drinks are chilled

r: Dinner is spoilt

Write the following compound proposition in words.

Markscheme

If the food is well cooked and the drinks are chilled then dinner is not spoilt. (A1)(A1)(A1) (C3)

Note: Award (A1) for “If…then” (then must be seen), (A1) for the two correct propositions connected

with “and”, (A1) for “not spoilt”.

Only award the final (A1) if correct statements are given in the correct order.

[3 marks]

8b. [3 marks] Complete the following truth table.

(A1)(A1)(A1)(ft) (C3)

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Notes: Award (A1) for each correct column.

The final column must follow through from the previous two columns.

[3 marks]

9a. [3 marks]

The probability that it snows today is 0.2. If it does snow today, the probability that it will snow

tomorrow is 0.6. If it does not snow today, the probability that it will not snow tomorrow is 0.9.

Using the information given, complete the following tree diagram.

Markscheme

(A1)(A1)(A1) (C3)

Note: Award (A1) for each correct pair of probabilities.

[3 marks]

9b. [3 marks]

Calculate the probability that it will snow tomorrow.

Markscheme

(A1)(ft)(M1)

Note: Award (A1)(ft) for two correct products of probabilities taken from their diagram, (M1) for the

addition of their products.

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(A1)(ft) (C3)

Note: Accept any equivalent correct fraction.

Follow through from their tree diagram.

[3 marks]

10a. [1 mark]

The diagram shows the points M(a, 18) and B(24, 10) . The straight line BM intersects the y-‐axis at A(0,

26). M is the midpoint of the line segment AB.

Write down the value of .

Markscheme

12 (A1) (C1)

Note: Award (A1) for .

[1 mark]

10b. [2 marks]

Find the gradient of the line AB.

Markscheme

(M1)

Note: Accept or (or equivalent).

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(A1) (C2)

Note: If either of the alternative fractions is used, follow through from their answer to part (a).

The answer is now (A1)(ft).

[2 marks]

10c. [3 marks]

Decide whether triangle OAM is a right-‐angled triangle. Justify your answer.

Markscheme

gradient of (A1)(ft)

Note: Follow through from their answer to part (b).

(M1)

Note: Award (M1) for multiplying their gradients.

Since the product is , OAM is a right-‐angled triangle (R1)(ft)

Notes: Award the final (R1) only if their conclusion is consistent with their answer for the product of

the gradients.

The statement that OAM is a right-‐angled triangle without justification is awarded no marks.

OR

and (A1)(ft)

(M1)

Note: This method can also be applied to triangle OMB.

Follow through from (a).

Hence a right angled triangle (R1)(ft)

Note: Award the final (R1) only if their conclusion is consistent with their (M1) mark.

OR

(cm) an isosceles triangle (A1)

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Note: Award (A1) for (cm) and (cm).

Line drawn from vertex to midpoint of base is perpendicular to the base (M1)

Conclusion (R1) (C3)

Note: Award, at most (A1)(M0)(R0) for stating that OAB is an isosceles triangle without any

calculations.

[3 marks]

11. [1 mark]

The following Venn diagram shows the relationship between the sets of numbers

The number –3 belongs to the set of and , but not , and is placed in the appropriate position on

the Venn diagram as an example.

Write down the following numbers in the appropriate place in the Venn diagram.

4, 1/3, pi, 0.38, sq rt 5, -‐0.25

Markscheme

(A1)(A1)(A1)(A1)(A1)(A1) (C6)

Note: Award (A1) for each number correctly placed.

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Award (A0) for any entry in more than one region.

[6 mark]

12a. [2 marks]

Chocolates in the shape of spheres are sold in boxes of 20.

Each chocolate has a radius of 1 cm.

Find the volume of 1 chocolate.

Markscheme

The first time a correct answer has incorrect or missing units, the final (A1) is not awarded.

(M1)

Notes: Award (M1) for correct substitution into correct formula.

(A1) (C2)

[2 marks]

12b. [1 mark]

Write down the volume of 20 chocolates.

Markscheme


(A1)(ft) (C1)

Note: Follow through from their answer to part (a).

[1 mark]

12c. [2 marks]

The diagram shows the chocolate box from above. The 20 chocolates fit perfectly in the box with each

chocolate touching the ones around it or the sides of the box.

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Calculate the volume of the box.

Markscheme


(M1)

Note: Award (M1) for correct substitution into correct formula.

(A1) (C2)

[2 marks]

12d. [1 mark]

The diagram shows the chocolate box from above. The 20 chocolates fit perfectly in the box with each

chocolate touching the ones around it or the sides of the box.

Calculate the volume of empty space in the box.

Markscheme


(A1)(ft) (C1)

Note: Follow through from their part (b) and their part (c).

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[1 mark]

13a. [1 mark]

Ludmila takes a loan of 320 000 Brazilian Real (BRL) from a bank for two years at a nominal annual

interest rate of 10%, compounded half yearly.

Write down the number of times interest is added to the loan in the two years.

Markscheme

4 (A1) (C1)

[1 mark]

13b. [3 marks]

Calculate the exact amount of money that Ludmila must repay at the end of the two years.

Markscheme

(M1)(A1)

Note: Award (M1) for substituted compound interest formula, (A1) for correct substitutions.

OR

(A1)(M1)

Note: Award (A1) for seen, (M1) for correctly substituted values from the question into the

finance application.

OR

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(A1)(M1)

Note: Award (A1) for seen, (M1) for correctly substituted values from the question into the

finance application.

amount to repay (A1) (C3)

Note: Award (C2) for final answer if not seen previously.

[3 marks]

13c. [2 marks]

Ludmila estimates that she will have to repay BRL at the end of the two years.

Calculate the percentage error in her estimate.

Markscheme

(M1)

Note: Award (M1) for correctly substituted percentage error formula.

(A1)(ft) (C2)

Notes: Follow through from their answer to part (b).

[2 marks]

14a. [1 mark]

The graph of the quadratic function intersects the y-‐axis at point A (0, 5) and

has its vertex at point B (4, 13).

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Write down the value of .

Markscheme

5 (A1) (C1)

[1 mark]

14b. [3 marks]

By using the coordinates of the vertex, B, or otherwise, write down two equations in and .

Markscheme

at least one of the following equations required

(A2)(A1) (C3)

Note: Award (A2)(A0) for one correct equation, or its equivalent, and (C3) for any two correct

equations.

Follow through from part (a).

The equation earns no marks.

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[3 marks]

14c. [2 marks]

Find the value of and of .

Markscheme

(A1)(ft)(A1)(ft) (C2)

Note: Follow through from their equations in part (b), but only if their equations lead to unique

solutions for and .

[2 marks]

15a. [3 marks]

competitors enter round 1 of a tennis tournament, in which each competitor plays a match against

one other competitor.

The winning competitor progresses to the next round (round 2); the losing competitor leaves the

tournament.

The tournament continues in this manner until there is a winner.

Find the number of competitors who play in round 6 of the tournament.

Markscheme

(M1)(A1)

Note: Award (M1) for substituted geometric progression formula, (A1) for correct substitution.

If a list is used, award (M1) for a list of at least six terms, beginning with and (A1) for first six

terms correct.

(A1) (C3)

[3 marks]

15b. [3 marks]

Find the total number of matches played in the tournament.

Markscheme

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OR (M1)(A1)

Note: Award (M1) for substituted sum of a GP formula, (A1) for correct substitution.

If a list is used, award (A1) for at least 9 correct terms, including , and (M1) for their 9 terms,

including , added together.

(A1) (C3)

[3 marks]

mathematics studies mock 16-17 paper...

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