These Ten-Year-Series (TYS) worked solutions with video explanations for 2016 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at support@timganmath.edu.sg.
The curve $y={{x}^{4}}$ is transformed onto the curves with equation $y=\text{f}\left( x \right)$. The turning point on $y={{x}^{4}}$ corresponds to the point with coordinates $\left( a,b \right)$ on $y=\text{f}\left( x \right)$. The curve $y=\text{f}\left( x \right)$ also passes through the point with coordinate $\left( 0,c \right)$. Given that $\text{f}\left( x \right)$ has the form $k{{\left( x-l \right)}^{4}}+m$ and that $a$, $b$ and $c$ are positive constants with $c>b$, express $k$, $l$ and $m$ in terms of $a$, $b$ and $c$.
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By sketching the curve $y=\text{f}\left( x \right)$, or otherwise, sketch the curve $y=\frac{1}{\text{f}\left( x \right)}$. State, in terms of $a$, $b$ and $c$, the coordinates of any points where $y=\frac{1}{\text{f}\left( x \right)}$ crosses the axes and of any turning points.
[4]
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(a)
Verify that $-1+5\mathbf{i}$ is a root of the equation ${{w}^{2}}+\left( -1-8\mathbf{i} \right)w+\left( -17+7\mathbf{i} \right)=0$. Hence, or otherwise, find the second root of the equation in Cartesian form, $p+\mathbf{i}q$, showing your working.
(a) Verify that $-1+5\mathbf{i}$ is a root of the equation ${{w}^{2}}+\left( -1-8\mathbf{i} \right)w+\left( -17+7\mathbf{i} \right)=0$. Hence, or otherwise, find the second root of the equation in Cartesian form, $p+\mathbf{i}q$, showing your working.Â
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It is given that $y=\text{f}\left( x \right)$, where $\text{f}\left( x \right)=\tan \left( ax+b \right)$ for constants $a$ and $b$.
(i)
Show that $\text{{f}’}\left( x \right)=a+a{{y}^{2}}$. Use this result to find $\text{{f}”}\left( x \right)$ and $\text{{f}”’}\left( x \right)$ in terms of $a$ and $y$.
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(i) Show that $\text{{f}’}\left( x \right)=a+a{{y}^{2}}$. Use this result to find $\text{{f}”}\left( x \right)$ and $\text{{f}”’}\left( x \right)$ in terms of $a$ and $y$.
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(ii)
In the case where $b=\frac{1}{4}\pi $, use your results from part (i) to find the Maclaurin series for $\text{f}\left( x \right)$ in terms of $a$, up to and including the term in ${{x}^{3}}$.
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(ii) In the case where $b=\frac{1}{4}\pi $, use your results from part (i) to find the Maclaurin series for $\text{f}\left( x \right)$ in terms of $a$, up to and including the term in ${{x}^{3}}$.
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(iii)
Find the first two non-zero terms in the Maclaurin series for $\tan 2x$.
[3]
(iii) Find the first two non-zero terms in the Maclaurin series for $\tan 2x$.
[3]
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(a)
The function $\text{f}$ is given by $\text{f}:x\mapsto 1+\sqrt{x}$, for $x\in \mathbb{R}$, $x\ge 0$.
(a) The function $\text{f}$ is given by $\text{f}:x\mapsto 1+\sqrt{x}$, for $x\in \mathbb{R}$, $x\ge 0$.
(i) Find ${{\text{f}}^{-1}}\left( x \right)$ and state the domain of ${{\text{f}}^{-1}}$.
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(i) Find ${{\text{f}}^{-1}}\left( x \right)$ and state the domain of ${{\text{f}}^{-1}}$.
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(ii) Show that if $\text{ff}\left( x \right)=x$ then ${{x}^{3}}-4{{x}^{2}}+4x-1=0$. Hence find the value of $x$ for which $\text{ff}\left( x \right)=x$. Explain why this value of $x$ satisfies the equation $\text{f}\left( x \right)={{\text{f}}^{-1}}\left( x \right)$.
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(ii) Show that if $\text{ff}\left( x \right)=x$ then ${{x}^{3}}-4{{x}^{2}}+4x-1=0$. Hence find the value of $x$ for which $\text{ff}\left( x \right)=x$. Explain why this value of $x$ satisfies the equation $\text{f}\left( x \right)={{\text{f}}^{-1}}\left( x \right)$.
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(b)
The function $\text{g}$, with domain the set of non-negative integers, is given by
$g\left( n \right)=\left\{ \begin{matrix}
1\text{for}\,n=0, \\
2+\text{g}\left( \frac{1}{2}n \right)\text{for }n\text{ even,} \\
1+\text{g}\left( n-1 \right)\text{for }n\text{ odd}. \\
\end{matrix} \right.$
(b) The function $\text{g}$, with domain the set of non-negative integers, is given by
$g\left( n \right)=\left\{ \begin{matrix}
1\text{for}\,n=0, \\
2+\text{g}\left( \frac{1}{2}n \right)\text{for }n\text{ even,} \\
1+\text{g}\left( n-1 \right)\text{for }n\text{ odd}. \\
\end{matrix} \right.$
(i) Find $\text{g}\left( 4 \right)$, $\text{g}\left( 7 \right)$and $\text{g}\left( 12 \right)$.
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(i) Find $\text{g}\left( 4 \right)$, $\text{g}\left( 7 \right)$and $\text{g}\left( 12 \right)$.
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(ii) Does $\text{g}$ have an inverse? Justify your answer.
[2]
(ii) Does $\text{g}$ have an inverse? Justify your answer.
[2]
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The plane $p$ has equation $\mathbf{r}=\left( \begin{matrix}
1 \\
-3 \\
2 \\
\end{matrix} \right)+\lambda \left( \begin{matrix}
1 \\
2 \\
0 \\
\end{matrix} \right)+\mu \left( \begin{matrix}
a \\
4 \\
-2 \\
\end{matrix} \right)$ , and the line $l$ has equation $\mathbf{r}=\left( \begin{matrix}
a-1 \\
a \\
a+1 \\
\end{matrix} \right)+t\left( \begin{matrix}
-2 \\
1 \\
2 \\
\end{matrix} \right)$, where $a$ is a constant and $\lambda $, $\mu $ and $t$ are parameters.
(i)
In the case where $a=0$,
(i) In the case where $a=0$,
(a) show that $l$ is perpendicular to $p$ and find the values of $\lambda $, $\mu $ and $t$ which give the coordinates of the point at which $l$ and $p$ intersect,
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(a) show that $l$ is perpendicular to $p$ and find the values of $\lambda $, $\mu $ and $t$ which give the coordinates of the point at which $l$ and $p$ intersect,
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(b) find the Cartesian equations of the planes such that the perpendicular distance from each plane to $p$ is $12$.
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(b) find the Cartesian equations of the planes such that the perpendicular distance from each plane to $p$ is $12$.
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(ii)
Find the value of $a$ such that $l$ and $p$ do not meet in a unique point.
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(ii) Find the value of $a$ such that $l$ and $p$ do not meet in a unique point.
[3]
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