These Ten-Year-Series (TYS) worked solutions with video explanations for 2000 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at gs.ude.htamnagmitobfsctd-969f03@troppus.
A ten-digit number is formed by writing down the digits $0,1,2,3,4,5,6,7,8,9$ in some order. No number is allowed to start with $0$. Find how many such numbers are odd.Â
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It is given that $x$ and $y$ satisfy the equation ${{\tan }^{-1}}x+{{\tan }^{-1}}y+{{\tan }^{-1}}\left( xy \right)=\frac{7}{12}\pi $.
(i)
Find the value of $y$ when $x=1$.
[2]
(i) Find the value of $y$ when $x=1$.
[2]
(ii)
Express $\frac{\text{d}}{\text{d}x}\left( {{\tan }^{-1}}\left( xy \right) \right)$ in terms of $x$, $y$ and $\frac{\text{d}y}{\text{d}x}$.
[2]
(ii) Express $\frac{\text{d}}{\text{d}x}\left( {{\tan }^{-1}}\left( xy \right) \right)$ in terms of $x$, $y$ and $\frac{\text{d}y}{\text{d}x}$.
[2]
(iii)
Show that, when $x=1$, $\frac{\text{d}y}{\text{d}x}=-\frac{1}{3}-\frac{1}{2\sqrt{3}}$.
[3]
(iii) Show that, when $x=1$, $\frac{\text{d}y}{\text{d}x}=-\frac{1}{3}-\frac{1}{2\sqrt{3}}$.
[3]
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Relative to the origin $O$, the points $A$, $B$, $C$ have position vectors $5\mathbf{i}+4\mathbf{j}+10\mathbf{k}$, $-4\mathbf{i}+4\mathbf{j}-2\mathbf{k}$ and $-5\mathbf{i}+9\mathbf{j}+5\mathbf{k}$ respectively.
(i)
Find the cartesian equation of the line $AB$.
[2]
(i) Find the cartesian equation of the line $AB$.
[2]
(ii)
Find the length of projection of the vector $\overrightarrow{AC}$ on line $AB$.
[2]
(ii) Find the length of projection of the vector $\overrightarrow{AC}$ on line $AB$.
[2]
(iii)
Find the position vector of the foot of the perpendicular,$N$, from $C$ to line $AB$.
[3]
(iii) Find the position vector of the foot of the perpendicular,$N$, from $C$ to line $AB$.
[3]
(iv)
The point $D$ lies on the line $CN$ produced and is such that $N$ is the mid-point of $CD$.
Find the position vector of $D$.
[2]
(iv) The point $D$ lies on the line $CN$ produced and is such that $N$ is the mid-point of $CD$.
Find the position vector of $D$.
[2]
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