Additional Mathematics 4037/12 — May/June 2013

Cambridge O-Level · Calculator · worked solutions for every part, with the mark scheme

questions

marks

120

minutes

Topics Calculus · Trigonometry · Quadratic functions · Straight-line graphs · [Legacy] Set language and notation · Permutations and combinations · +4 more

Q1[Legacy] Set language and notationFree sample

Fig. 1

The Venn diagram shows the universal set $\mathscr{E}$ , the set $A$ and the set $B$ . Given that $\mathrm{n}(B) = 5$ , $\mathrm{n}(A') = 10$ and $\mathrm{n}(\mathscr{E}) = 26$ , find

(i)

$\mathrm{n}(A \cap B)$ ,

DifficultyEasy

Worked solution

Approach

From the Venn diagram, $B$ lies entirely inside $A$ , so $B \subset A$ . Therefore every element of $B$ is also in $A$ , and $A \cap B = B$ .

Working

\mathrm{n}(A \cap B) = \mathrm{n}(B) = 5

Answer

\mathrm{n}(A \cap B) = 5

Final answer

Detailed explanation

Walkthrough

The diagram shows the circle for $B$ drawn completely inside the oval for $A$ , which means $B$ is a subset of $A$ . The intersection of a set with its own subset is just the subset itself, so $A \cap B = B$ . Since $\mathrm{n}(B) = 5$ , we get $\mathrm{n}(A \cap B) = 5$ .

Key Takeaways

If $B \subset A$ , then $A \cap B = B$ and $A \cup B = A$ .
Reading the containment relationship from the picture is the whole task here.

Common Mistakes

Treating $A$ and $B$ as overlapping sets and trying to compute the intersection from totals.
Confusing $A \cap B$ with $A \cup B$ .

Things to Be Careful About

The answer is exact (a B1 mark, cao): it must be $5$ , not any other value.
Note the diagram shows $B$ strictly inside $A$ — there is no region of $B$ outside $A$ .

Techniques used

read the subset relationship from the Venn diagramidentify the intersection of a set with its subset

(ii)

$\mathrm{n}(A)$ ,

DifficultyEasy

Worked solution

Approach

The complement $A'$ contains all elements of the universal set $\mathscr{E}$ that are not in $A$ , so $\mathrm{n}(A) = \mathrm{n}(\mathscr{E}) - \mathrm{n}(A')$ .

Working

\mathrm{n}(A) = \mathrm{n}(\mathscr{E}) - \mathrm{n}(A') = 26 - 10 = 16

Answer

\mathrm{n}(A) = 16

Final answer

Detailed explanation

Walkthrough

The universal set has $26$ elements and $10$ of them lie outside $A$ (in $A'$ ). The elements of $A$ are therefore the remaining $26 - 10 = 16$ .

Key Takeaways

$\mathrm{n}(A) + \mathrm{n}(A') = \mathrm{n}(\mathscr{E})$ for any set $A$ .

Common Mistakes

Subtracting the wrong way round, or using $\mathrm{n}(B) = 5$ unnecessarily.

Things to Be Careful About

The value $\mathrm{n}(B) = 5$ is not needed in this part — do not mix it in.

Techniques used

use the complement rule n(A) = n(E) - n(A')

(iii)

$\mathrm{n}(B' \cap A)$ .

DifficultyEasy

Worked solution

Approach

The region $B' \cap A$ is the part of $A$ that lies outside $B$ — the crescent between the oval $A$ and the inner circle $B$ . Its count is $\mathrm{n}(A) - \mathrm{n}(B)$ .

Working

\mathrm{n}(B' \cap A) = \mathrm{n}(A) - \mathrm{n}(B) = 16 - 5 = 11

Answer

\mathrm{n}(B' \cap A) = 11

Final answer

Detailed explanation

Walkthrough

Since $B \subset A$ , the set $A$ splits into two disjoint pieces: the part inside $B$ (with $5$ elements) and the part of $A$ outside $B$ , which is exactly $B' \cap A$ . So $\mathrm{n}(B' \cap A) = 16 - 5 = 11$ , using $\mathrm{n}(A) = 16$ from part (ii).

Key Takeaways

$B' \cap A$ means 'in $A$ but not in $B$ '; when $B \subset A$ its count is $\mathrm{n}(A) - \mathrm{n}(B)$ .
Parts of a multi-part Venn question build on each other — the value $\mathrm{n}(A) = 16$ is carried forward.

Common Mistakes

Using $\mathrm{n}(\mathscr{E}) = 26$ instead of $\mathrm{n}(A) = 16$ .
Misreading $B' \cap A$ as the whole of $A'$ .

Things to Be Careful About

The answer must be exactly $11$ (B1, cao). Keep the intermediate value $\mathrm{n}(A) = 16$ exact — no rounding issues arise here.

Techniques used

identify the region B' intersect A on the Venn diagramsubtract the subset count from the containing set count

The rest of this paper

11 more questions

Q2Permutations and combinations5M
Q3Trigonometry3M
Q4Quadratic functions4M
Q5Simultaneous equations · Straight-line graphs6M
Q6Calculus · Straight-line graphs6M
Q7Factors of polynomials · Quadratic functions8M
Q8[Legacy] Matrices9M
Q9Series7M
Q10Calculus9M
Q11Trigonometry9M
Q12Calculus11M

Loading the full paper…