These Ten-Year-Series (TYS) worked solutions with video explanations for 1993 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at support@timganmath.edu.sg.
(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.
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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.
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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.
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