CAIE IGCSE Additional Mathematics Combinatorics, Series and Vectors Question Bank

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

[Maximum number: 9]

(a)

Find the first three terms in the expansion of $\left(1+\frac{x}{7}\right)^5$ , in ascending powers of x. Simplify the coefficient of each term.

[ 2 ]

$1+\frac57x+\frac{10}{49}x^2$
B2
B1 for any two correct terms or for the three terms listed but not summed

(b)

The expansion of $7(1+x)^n\left(1+\frac{x}{7}\right)^5$ , where n is a positive integer, is written in ascending powers of x. The first two terms in the expansion are 7+89x. Find the value of n.

[ 2 ]

12
B2
B1 for 7n+5=89 or $7(n+\frac57)=89$ oe or ${}^nC_1=12$

(c)

In the expansion of $(k-2x)^8$ , where k is a constant, the coefficient of $x^4$ divided by the coefficient of $x^2$ is $\frac{5}{8}$ . The coefficient of x is positive. Form an equation and hence find the value of k.

[ 5 ]

${}^8C_4\times k^4\times(-2)^4[x^4]$
oe, or $1120k^4[x^4]$ soi
B1
${}^8C_2\times k^6\times(-2)^2[x^2]$
oe, or $112k^6[x^2]$ soi
B1
$\frac{1120k^4}{112k^6}=\frac58$
oe, soi
M1 FT providing at least B1 awarded and correct terms attempted
$k^2=16$
soi
A1
For coefficient of x to be positive, k<0, therefore k=-4
A1

[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 $3 n^{2}-19 n+20=0$ .

$\frac{n!}{(n-4)!}=\frac{(2 n)!}{6(2 n-3)!}$

B1

Dep on previous B1
B1 for simplification of one side

e.g. 3(n-2)(n-3)=2(2 n-1)
$3 n^{2}-19 n+20=0 \mathbf{A G}$

B1

Dep on all previous B marks

With sufficient detail shown

\begin{array}{r} n(n-1)(n-2)(n-3)= \frac{2 n(2 n-1)(2 n-2)}{6} \end{array}

[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 ]

$[\overrightarrow{OR}=]k(-\mathbf{i}+5\mathbf{j})$ or $k\binom{-1}{5}$
B1 soi
$\sqrt{1^2+5^2}=\sqrt{26}$
B1 for finding the magnitude of $\overrightarrow{PQ}$ ; allow unsimplified.
$-3\mathbf{i}+15\mathbf{j}$ or $\binom{-3}{15}$
B1
Allow $3(-\mathbf{i}+5\mathbf{j})$ or $3\binom{-1}{5}$ . Do not isw.

(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 ]

$a(2\mathbf{i}-3\mathbf{j})+b(-\mathbf{i}+5\mathbf{j})=10\mathbf{i}+6\mathbf{j}$ oe
B1
2a-b=10 oe, -3a+5b=6 oe
or $\frac{10-2a}{6+3a}=\frac{-1}{5}$ or $\frac{10+b}{6-5b}=\frac{-2}{3}$
M1 dep on previous B1 for equating like vectors.
a=8 or b=6
A1
$16\mathbf{i}-24\mathbf{j}$ oe
A1
Allow $8(2\mathbf{i}-3\mathbf{j})$ or $8\binom{2}{-3}$ .

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Covered syllabus points

Combinatorics, Series and Vectors question 1

Question (a)

Question (b)

Question (c)

Combinatorics, Series and Vectors question 2

Combinatorics, Series and Vectors question 3

Question (a)

Question (b)