2010 A-level H2 Mathematics (9740) Paper 1 Question 10 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(i)

Direction vector of $l \text{~is~} \begin{pmatrix}{-3}\\6\\9\end{pmatrix} = -3 \begin{pmatrix}1\\{-2}\\{-3}\end{pmatrix}$

Normal vector of $p \text{~is~} \begin{pmatrix}1\\{-2}\\{-3}\end{pmatrix}$

Since $l$ is parallel to the normal vector of $p, l$ is perpendicular to $p$ .

(ii)
$l: r = \begin{pmatrix}10\\{-1}\\{-3}\end{pmatrix} + \lambda \begin{pmatrix}1\\{-2}\\{-3}\end{pmatrix}$

$[\begin{pmatrix}10\\{-1}\\{-3}\end{pmatrix} + \lambda \begin{pmatrix}1\\{-2}\\{-3}\end{pmatrix}] \bullet \begin{pmatrix}1\\{-2}\\{-3}\end{pmatrix} = 0$

$10 + \lambda + 2 4 \lambda + 9 + 9 \lambda = 0$

$\lambda = - \frac{3}{2}$

Point of intersection $= (\frac{17}{2}, 2, \frac{3}{2})$ .

(iii)
$\begin{pmatrix}10\\{-1}\\{-3}\end{pmatrix} + \lambda \begin{pmatrix}1\\{-2}\\{-3}\end{pmatrix} = \begin{pmatrix}{-2}\\23\\33\end{pmatrix}$

$10 + \lambda = - 2$ — (1)

$-1 - 2 \lambda = 23$ — (2)

$-3 - 3 \lambda = 33$ — (3)

Since $\lambda = -12$ satisfies all 3 equations, A lies on l.

Since $= (\frac{17}{2}, 2, \frac{3}{2})$ is the midpoint of A & B, by ratio theorem,

$\begin{pmatrix}{\frac{17}{2}}\\2\\{\frac{3}{2}}\end{pmatrix} = \frac{\vec{OA} + \vec{OB}}{2}$

$\vec{OB} = 2 \begin{pmatrix}{\frac{17}{2}}\\2\\{\frac{3}{2}}\end{pmatrix} - \begin{pmatrix}{-2}\\23\\33\end{pmatrix} = \begin{pmatrix}19\\{-19}\\{-30}\end{pmatrix}$

$B(19, -19, -30)$

(iv)
Area

$= \frac{1}{2} |\vec{OA} \times \vec{OB}|$

$= \frac{1}{2} |\begin{pmatrix}{-2}\\23\\33\end{pmatrix} \times \begin{pmatrix}19\\{-19}\\{-30}\end{pmatrix}|$

$= \frac{1}{2} |\begin{pmatrix}{-63}\\{567}\\{-399}\end{pmatrix}|$

$= \frac{1}{2} \sqrt{63^2 +567^2 + 399^2}$

$= 348$ to the nearest whole number.

KS Comments:

Students must give answers in coordinates, rather than as a position vector.

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KS Teng

KS has been teaching H2/H1 Mathematics and IB mathematics for the more than 10 years. Having taught students from all various Junior Colleges, KS adapts to students' abilities and help them better understand the topics. As someone who loves to teach mathematics and sees it as a truly useful tool in life, KS seeks to enable students to appreciate math. Therefore, his tuition mission is to motivate and cultivate students to be independent and confident thinkers.

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2010 A-level H2 Mathematics (9740) Paper 1 Question 10 Suggested Solutions

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