# 1993 A Level H2 Math

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Paper 1
Paper 2
##### 1993 A Level H2 Math Paper 1 Question 10Â

(a)

A tennis club has $n$ male players and $n$ female players. For a tournament the players are to be arranged in $n$ pairs, with each pairing consisting of one male and one female. Find the number of possible pairings.

(a) A tennis club has $n$ male players and $n$ female players. For a tournament the players are to be arranged in $n$ pairs, with each pairing consisting of one male and one female. Find the number of possible pairings.

(b)

In the state of Utopia, the alphabet contains $25$ letters. A car registration number consists of two different letters of the alphabet followed by an integer $n$ such that $100\le n\le 999$.

(b) In the state of Utopia, the alphabet contains $25$ letters. A car registration number consists of two different letters of the alphabet followed by an integer $n$ such that $100\le n\le 999$.

Find the number of possible car registration numbers.

Find the number of possible car registration numbers.

##### 1993 A Level H2 Math Paper 1 Question 14

The equation of a straight line $\ell$ is $r\,=\,\left( \begin{matrix} 1 \\ 2 \\ 3 \\ \end{matrix} \right)\,+\,t\left( \begin{matrix} -1 \\ 1 \\ 1 \\ \end{matrix} \right)\,,\,\,\,t\,\in \mathbb{R}$.

The point $A\,$ on $\ell$ is given by $t\,=\,0\,$, and the origin of position vectors is $O$.

(i)

Calculate the acute angle between $OA$ and $\ell$, giving your answer correct to the nearest degree.

Calculate the acute angle between $OA$ and $\ell$, giving your answer correct to the nearest degree.

(ii)

Find the position vector of the point $P$ on $\ell$ such that $OP$ is perpendicular to $\ell$.

Find the position vector of the point $P$ on $\ell$ such that $OP$ is perpendicular to $\ell$.

(iii)

A point $Q$ on $\ell$ is such that the length of $OQ$ is $5$ units. Find the two possible position vectors of $Q$.

A point $Q$ on $\ell$ is such that the length of $OQ$ is $5$ units. Find the two possible position vectors of $Q$.

(iv)

The points $R$ and $S$ on $\ell$ are given by $t\,=\,\lambda$ and $t\,=\,2\lambda$ respectively. Show that there is no value of $\lambda$ for which $OR$ and $OS$ are perpendicular.

The points $R$ and $S$ on $\ell$ are given by $t\,=\,\lambda$ and $t\,=\,2\lambda$ respectively. Show that there is no value of $\lambda$ for which $OR$ and $OS$ are perpendicular.

##### 1993 A Level H2 Math Paper 2 Question 12 (a)

Write down the modulus and argument of the complex number $w$, where $w=1+\mathbf{i}\sqrt{3}$. Hence find the set of values of the positive integer $n$ for which ${{w}^{n}}$is real, and show that, for these values of $n$, the value of ${{w}^{n}}$ is either ${{2}^{n}}$or $-{{2}^{n}}$.

Express ${{w}^{100}}-{{\left( w* \right)}^{100}}$ in the form $k\mathbf{i}$, where $k$ is real, giving the exact value of $k$ in non-trigonometrical form.