IB DP Mathematical Studies 1.1 Natural numbers, N ; integers, Z ; rational numbers, Q ; and real numbers, R Paper 1

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Question

Five pipes labelled, “6 metres in length”, were delivered to a building site. The contractor measured each pipe to check its length (in metres) and recorded the following;

5.96, 5.95, 6.02, 5.95, 5.99.

(i) Find the mean of the contractor’s measurements.

(ii) Calculate the percentage error between the mean and the stated, approximate length of 6 metres.

[3]

Calculate \(\sqrt {{{3.87}^5} – {{8.73}^{ – 0.5}}} \), giving your answer

(i) correct to the nearest integer,

(ii) in the form \(a \times 10^k\), where 1 ≤ a < 10, \(k \in {\mathbb{Z}}\) .

[3]

Answer/Explanation

Markscheme

(i) Mean = (5.96 + 5.95 + 6.02 + 5.95 + 5.99) / 5 = 5.974 (5.97) (A1)

(ii) \({\text{% error}} = \frac{{error}}{{actualvalue}} \times 100\% \)

\( = \frac{{6 – 5.974}}{{5.974}} \times 100\% = 0.435\% \) (M1)(A1)(ft)

(M1) for correctly substituted formula.

Allow 0.503% as follow through from 5.97

Note: An answer of 0.433% is incorrect. (C3)

[3 marks]

number is 29.45728613

(i) Nearest integer = 29 (A1)

(ii) Standard form = 2.95 × 10¹ (accept 2.9 × 10¹) (A1)(ft)(A1)

Award (A1) for each correct term

Award (A1)(A0) for 2.95 × 10 (C3)

[3 marks]

Question

Write down the following numbers in increasing order.

\(3.5\), \(1.6 \times 10^{−19}\), \(60730\), \(6.073 \times 10^{5}\), \(0.006073 \times 10^6\), \(\pi\), \(9.8 \times 10^{−18}\).

[3]

Write down the median of the numbers in part (a).

[1]

State which of the numbers in part (a) is irrational.

[1]

Answer/Explanation

Markscheme

\(1.6 \times 10^{−19}\), \(9.8 \times 10^{−18}\), \(\pi\), \(3.5\), \(0.006073 \times 10^6\), \(60730\), \(6.073 \times 10^{5}\) (A4)

Award (A1) for \(\pi\) before 3.5

Award (A1) for \(1.6 \times 10^{−19}\) before \(9.8 \times 10^{−18}\)

Award (A1) for the three numbers containing 6073 in the correct order.

Award (A1) for the pair with negative indices placed before 3.5 and \(\pi\) and the remaining three numbers placed after (independently of the other three marks).

Award (A3) for numbers given in correct decreasing order.

Award (A2) for decreasing order with at most 1 error (C4)

[3 marks]

The median is 3.5. (A1)(ft)

Follow through from candidate’s list. (C1)

[1 mark]

\(\pi\) is irrational. (A1) (C1)

[1 mark]

Question

Consider the numbers \(2\), \(\sqrt 3 \), \( – \frac{2}{3}\) and the sets \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{Q}\) and \(\mathbb{R}\).

Complete the table below by placing a tick in the appropriate box if the number is an element of the set, and a cross if it is not.

[3]

A function \(f\) is given by \(f(x) = 2{x^2} – 3x{\text{, }}x \in \{ – 2{\text{, }}2{\text{, }}3\} \).

Write down the range of function \(f\).

[1]

Answer/Explanation

Markscheme

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

Note: Accept any symbol for ticks. Do not penalise if the other boxes are left blank.

[3 marks]

\({\text{Range}} = \{ 2{\text{, }}9{\text{, }}14\} \) (A1)(ft) (C1)

Note: Brackets not required.

[1 mark]

Question

Consider the numbers \(\sqrt 3 \), \(6\), \(2\frac{1}{2}\), \(\pi \), \( – 5\), and the sets \(\mathbb{N}\), \(\mathbb{Z}\), and \(\mathbb{Q}\). Complete the following table by placing a tick in the appropriate box if the number is an element of the set.

Answer/Explanation

Markscheme

Note: Accept any symbol for ticks. Do not penalize if candidate had also indicated, by a different symbol, that the number is not an element of the set.

Row \(\mathbb{N}\) correct, no extra entries. (A1) (C1)

Row \(\mathbb{Z}\) (A1)(A1) (C2)

Note: Award (A1) for each correct tick and no extra entries. Award (A1) only for both ticks correct and 1 extra entry, (A0) otherwise.

Row \(\mathbb{Q}\) (A1)(A1)(A1) (C3)

Note: Award (A1) for each correct tick and no extra entries. Award (A2) only for all 3 correct and one extra entry. Award (A1) only for 2 correct and one extra entry. (A0) otherwise.

[6 marks]

Question

The Venn diagram shows the number sets \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{Q}\) and \(\mathbb{R}\). Place each of the following numbers in the appropriate region of the Venn diagram.

\(\frac{{1}}{{4}}\), −3, π, cos 120°, 2.7 × 10³, 3.4 × 10^–2

Answer/Explanation

Markscheme

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

Note: Award (A1) for each number placed once in the correct section. Accept equivalent forms for numbers.

[6 marks]

Question

The sets \(P\), \(Q\) and \(U\) are defined as

U = {Real Numbers} , P = {Positive Numbers} and Q = {Rational Numbers}.

Write down in the correct region on the Venn diagram the numbers

\(\frac{{22}}{7}\) , \(5 \times {10^{ – 2}}\) , \(\sin (60^\circ )\) , \(0\) , \(\sqrt[3]{{ – 8}}\) , \( – \pi \).

Answer/Explanation

Markscheme

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

Note: Award (A1) for each number placed once in the correct region. Accept equivalent forms for numbers.

[6 marks]

Question

\(U\) is the set of all the positive integers less than or equal to \(12\).
\(A\) , \(B\) and \(C\) are subsets of \(U\) .
\[A = \{ 1{\text{, }}2{\text{, }}3{\text{, }}4{\text{, }}6{\text{, }}12\} \]\[B = \{ {\text{odd integers}}\} \]\[C = \{ 5{\text{, }}6{\text{, }}8\} \]

Write down the number of elements in \(A \cap C\) .

[1]

List the elements of \(B\) .

[1]

Complete the following Venn diagram with all the elements of \(U\) .

[4]

Answer/Explanation

Markscheme

\(1\) (one) (A1) (C1)

Note: \(6\), \(\{6\} \) or \(\{1\} \) earns no marks.

[1 mark]

\(1\), \(3\), \(5\), \(7\), \(9\), \(11\) (A1) (C1)

Note: Do not penalise if braces, parentheses or brackets are seen.

[1 mark]

(A1)(A1)(ft)(A1)(ft)(A1)(ft) (C4)

Notes: Award (A1) for the empty set \(A \cap B \cap C\) .

Award (A1)(ft) for the correct placement of \(6\), \(5\), \(1\) and \(3\).

Award (A1)(ft) for the correct placement of \(2\), \(4\), \(12\), \(7\), \(9\), \(11\), \(8\).

Award (A1)(ft) for the correct placement of \(10\).

Follow through from part (b).

[4 marks]

Question

Consider c = 5200 and d = 0.0000037.

Write down the value of r = c × d.

[1]

Write down your value of r in the form a × 10^k, where 1 ≤ a < 10 and \(k \in \mathbb{Z}\).

[2]

Consider the following statements about c, d and r. Only three of these statements are true.

Circle the true statements.

[3]

Answer/Explanation

Markscheme

r = 0.01924 (A1) (C1)

Note: Accept 0.0192 and 0.019

[1 mark]

r = 1.924 × 10⁻² (A1)(ft)(A1)(ft) (C2)

Notes: Award (A1) for 1.924, (A1) for 10⁻². Accept 1.92 and 1.9. Follow through from their part (a).

[2 marks]

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

Notes: Award (A1) for each true statement circled. Do not follow through from part (a). Award (A1)(A1)(A0) if 1 extra term seen. Award (A1)(A0)(A0) if 2 extra terms seen. Award (A0)(A0)(A0) if all terms circled. Accept other indications of the correct statements i.e. highlighted or ticks.

[3 marks]

Question

\(U = \{ x|x{\text{ is an integer, }}2 < x < 10\}\)

A and B are subsets of U such that A = {multiples of 3}, B = {factors of 24}.

List the elements of

(i) U ;

(ii) B .

[2]

Write down the elements of U on the Venn diagram.

[3]

Write down \(n (A \cap B)\).

[1]

Answer/Explanation

Markscheme

(i) 3, 4, 5, 6, 7, 8, 9 (A1)

(ii) 3, 4, 6, 8 (A1)(ft) (C2)

Notes: Follow through from part (a)(i).

[2 marks]

(A1)(ft) for their 3, 6
(A1)(ft) for their 4, 8, 9
(A1)(ft) for their 5, 7 (A1)(ft)(A1)(ft)(A1)(ft) (C3)

Note: Follow through from their universal set and set B in part (a).

[3 marks]

2 (A1)(ft) (C1)

Note: Follow through from their Venn diagram.

[1 mark]

Question

Consider the numbers 3, −5 , \(\sqrt{7}\), \(2^{−3}\) and 1.75.

Complete the table below, placing a tick () to show which of the number sets, \(\mathbb{N}, \mathbb{Q} {\text{ and }} \mathbb{R}\) these numbers belong to. The first row has been completed as an example.

Answer/Explanation

Markscheme

(A1) for \(\mathbb{N}\) column correct.

(A2) for \(\mathbb{R}\) column correct, award (A1) if one is missing, award (A0) if two or more missing.

(A3) for \(\mathbb{Q}\) column correct, award (A2) for two correctly placed and no extra entries, award (A1) for one correctly placed and no extra entries or placed in all entries. (A6) (C6)

Examiners report

[N/A]

Question

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

\[\mathbb{N},{\text{ }}\mathbb{Z}{\text{, }}\mathbb{Q}{\text{ and }}\mathbb{R}{\text{.}}\]

The number –3 belongs to the set of \(\mathbb{Z}{\text{, }}\mathbb{Q}\) and \(\mathbb{R}\), but not \(\mathbb{N}\), 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.

a. 4 [1]

b. \(\frac{1}{3}\)[1]

c. \(\pi \)[1]

d. \(0.38\)[1]

e. \(\sqrt 5 \)[1]

f. \(-0.25\)[1]

Answer/Explanation