Draw a Venn diagram to show this information. Use T to represent the set of students who have a television, C the set of students who have a computer and I the set of students who have an iPhone.

Write down the number of students that

(i) have a computer only;

(ii) have an iPhone and a computer but no television.

Write down \(n[T \cap (C \cup I)']\).

Calculate the number of students who have none of the three.

Two students are chosen at random from the 450 students. Calculate the probability that

(i) neither student has an iPhone;

(ii) only one of the students has an iPhone.

The students are asked to collect money for charity. In the first month, the students collect x dollars and the students collect y dollars in each subsequent month. In the first 6 months, they collect 7650 dollars. This can be represented by the equation x + 5y = 7650.

In the first 10 months they collect 13 050 dollars.

(i) Write down a second equation in x and y to represent this information.

(ii) Write down the value of x and of y .

The students are asked to collect money for charity. In the first month, the students collect x dollars and the students collect y dollars in each subsequent month. In the first 6 months, they collect 7650 dollars. This can be represented by the equation x + 5y = 7650.

In the first 10 months they collect 13 050 dollars.

Calculate the number of months that it will take the students to collect 49 500 dollars.

     (A1)(A1)(A1)(A1)

Notes: Award (A1) for labelled sets T, C, and I included inside an enclosed universal set. (Label U is not essential.) Award (A1) for central entry 40. (A1) for 20, 30 and 35 in the other intersecting regions. (A1) for 60, 110 and 115 or T(150), C(205), I(220).

[4 marks]

In parts (b), (c) and (d) follow through from their diagram.

(i) 110     (A1)(ft)

(ii) 35     (A1)(ft)

[2 marks]

In parts (b), (c) and (d) follow through from their diagram.

60     (A1)(ft)

[2 marks]

In parts (b), (c) and (d) follow through from their diagram.

450 − (60 + 20 + 40 + 30 + 115 + 35 + 110)     (M1)

Note: Award (M1) for subtracting all their values from 450.

= 40     (A1)(ft)(G2)

[2 marks]

(i) \(\frac{{230}}{{450}} \times \frac{{229}}{{449}}\)     (A1)(M1)

Note: Award (A1) for correct fractions, (M1) for multiplying their fractions.

\(\frac{{52670}}{{202050}}\left( {\frac{{5267}}{{20205}},{\text{ 0}}{\text{.261, 26}}{\text{.1% }}} \right)(0.26067...)\)     (A1)(G2)

Note: Follow through from their Venn diagram in part (a).

(ii) \(\frac{{220}}{{450}} \times \frac{{230}}{{449}} + \frac{{230}}{{450}} \times \frac{{220}}{{449}}\)     (A1)(A1)

Note: Award (A1) for addition of their products, (A1) for two correct products.

OR

\(\frac{{230}}{{450}} \times \frac{{220}}{{449}} \times 2\)     (A1)(A1)

Notes: Award (A1) for their product of two fractions multiplied by 2, (A1) for correct product of two fractions multiplied by 2. Award (A0)(A0) if correct product is seen not multiplied by 2.

\(\frac{{2024}}{{4041}}(0.501,{\text{ 50}}{\text{.1% )(0}}{\text{.50086}}...{\text{)}}\)     (A1)(G2)

Note: Follow through from their Venn diagram in part (a) and/or their 230 used in part (e)(i).

Note: For consistent use of replacement in parts (i) and (ii) award at most (A0)(M1)(A0) in part (i) and (A1)(ft)(A1)(A1)(ft) in part (ii).

[6 marks]

(i) x + 9y = 13050     (A1)

(ii) x = 900     (A1)(ft)

y = 1350     (A1)(ft)

Notes: Follow through from their equation in (f)(i). Do not award (A1)(ft) if answer is negative. Award (M1)(A0) for an attempt at solving simultaneous equations algebraically but incorrect answer obtained.

[3 marks]

49500 = 900 + 1350n     (A1)(ft)

Notes: Award (A1)(ft) for setting up correct equation. Follow through from candidate’s part (f).

n = 36     (A1)(ft)

The total number of months is 37.     (A1)(ft)(G2)

Note: Award (G1) for 36 seen as final answer with no working. The value of n must be a positive integer for the last two (A1)(ft) to be awarded.

OR

49500 = 900 + 1350(n − 1)     (A2)(ft)

Notes: Award (A2)(ft) for setting up correct equation. Follow through from candidate’s part (f).

n = 37     (A1)(ft)(G2)

Note: The value of n must be a positive integer for the last (A1)(ft) to be awarded.

[3 marks]

The question was moderately well answered. The majority of candidates answered part (a) and at least parts of (b), and (d).

The question was moderately well answered. The majority of candidates answered part (a) and at least parts of (b), and (d).

The question was moderately well answered. Part (c) proved to be difficult, as it required understanding and interpreting set notation.

The question was moderately well answered. The majority of candidates answered part (a) and at least parts of (b), and (d).

The question was moderately well answered. Part (e) was rarely answered in its entirety.

The question was moderately well answered. Part (f) was answered by many candidates, but most of them offered a partial answer to part (g); a typical response was 36 instead of 37.

The question was moderately well answered. Part (f) was answered by many candidates, but most of them offered a partial answer to part (g); a typical response was 36 instead of 37.