CAIE IGCSE Additional Mathematics Combinatorics, Series and Vectors Question Bank

Cambridge IGCSE Additional Mathematics 0606 syllabus for exams in 2028, 2029 and 2030Exams 2028-2030

Practise counting arrangements and selections, binomial expansions, arithmetic and geometric series, and vector methods for geometry, motion and collision problems.

Covered syllabus points

11. Permutations and combinations 12. Series 13. Vectors in two dimensions

(a)

3 men and 3 women are standing in a line.
The 3 men are standing next to each other.
Find how many different arrangements are possible.

[ 2 ]

(b)

In the 13-letter word MULTIBRANCHED, there are 4 vowels, U, I, A and E.

7 different letters are selected from these 13 letters.
Find how many different selections are possible if the selection includes at least 2 vowels.

[ 3 ]

(a)

Figure for Question 3(a) — CAIE IGCSE Additional Math

The diagram shows a triangle OAB. The point P lies on AB. The ratio A P: P B is 1: 3.
Given that $\overrightarrow{O A}=\mathbf{a}$ and $\overrightarrow{O B}=\mathbf{b}$ , find an expression for $\overrightarrow{O P}$ in terms of a and b. Simplify your answer.

[ 2 ]

(b)

Vector q has magnitude $12 \sqrt{5}$ and direction $\binom{6}{-3}$ .

Vector r has magnitude $15 \sqrt{2}$ and direction $\binom{-5}{5}$ .
Find the unit vector in the direction of q+r.

[Maximum number: 6]

Find the exact value of the term independent of x in the expansion of $\left(2+\frac{3}{x^{2}}\right)^{10}\left(1-4 x^{2}\right)^{2}$ .

[Maximum number: 7]

The point O is the origin.
Two points P and Q are such that $\overrightarrow{PQ}$ is in the same direction as $-\mathbf{i}+5\mathbf{j}$ .

(a)

The point R is such that $\overrightarrow{OR}$ is in the same direction as $\overrightarrow{PQ}$ and the magnitude of $\overrightarrow{OR}$ is $3\sqrt{26}$ .
Find $\overrightarrow{OR}$ .

[ 3 ]

(b)

$\overrightarrow{OP}$ is in the same direction as $2\mathbf{i}-3\mathbf{j}$ and $\overrightarrow{OQ}=10\mathbf{i}+6\mathbf{j}$ .
Find $\overrightarrow{OP}$ .

[ 4 ]

(a)

In an arithmetic progression:
- the first term is 3
- the sum of the first 10 terms is 4 times the sum of the first 5 terms.

Find the common difference.

[ 3 ]

(b)

The 1st, 2nd and 5th terms of another arithmetic progression are the 1st, 2nd and 3rd terms of a geometric progression.

It is given that the 1st terms of the progressions are not 0.
Find the common ratio, r, where $r \neq 1$ , of the geometric progression.

[ 4 ]

[Maximum number: 8]

In this question lengths are in centimetres and time, t, is in seconds.
A particle P is moving in a straight line with a speed of 26 in the direction of the vector $\binom{5}{-12}$ .

(a)

Find the velocity vector of P.

When t=0, P passes through a point A which has position vector $\binom{3}{6}$ .

[ 2 ]

(b)

Write down the position vector of P at time t.

At the same time that P passes through A, a particle Q passes through a point B.
The position vector of Q at time t is given by $\binom{8 t-5}{2-25 t}$ .
The distance between P and Q at time t is d.

[ 2 ]

(c)

Show that $d^{2}=m t^{2}+n t+r$ , where m, n and r are integers to be found.

[ 3 ]

(d)

Hence show that P and Q do not collide.

[ 1 ]

(a)

An arithmetic progression has first term a and common difference d.
Given that $S_{20}=3\times S_{10}$ , find a in terms of d.

[ 3 ]

(b)

A geometric progression, A, has common ratio r, where |r|<1.
The terms of this progression are $a_1, a_2, a_3, \ldots$ .
Another geometric progression, B, has terms $b_1, b_2, b_3, \ldots$ , where
$b_1=a_2,\quad b_2=a_4,\quad b_3=a_6,\ldots$
The sum to infinity of A is $S_A$ and the sum to infinity of B is $S_B$ .
Find $\frac{S_B}{S_A}$ in terms of r.
Give your answer in its simplest form.

[ 5 ]

[Maximum number: 4]

The number of permutations of n items taken 4 at a time is equal to $\frac{1}{6} \times$ the number of permutations of 2 n items taken 3 at a time.

Show that n satisfies the equation $3 n^{2}-19 n+20=0$ .

[Maximum number: 5]

In this question a, b and n are constants.
When $5(2+a x)^{n}$ is written in ascending powers of x, the first three terms are $640+b^{2} x+30240 x^{2}$ . Find the value of a and the possible values of b.

Covered syllabus points

Question 3

Question 3(a)

Question 3(b)

Question 3

Question 3(a)

Question 3(b)

Question 3

Question 7

Question 7(a)

Question 7(b)

Question 5

Question 5(a)

Question 5(b)

Question 6

Question 6(a)

Question 6(b)

Question 6(c)

Question 6(d)

Question 10

Question 10(a)

Question 10(b)

Question 11(a)

Question 10