These Ten-Year-Series (TYS) worked solutions with video explanations for 2017 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at support@timganmath.edu.sg.
(i)
On the same axes, sketch the graphs of $y=\frac{1}{x-a}$ and $y=b\left| x-a \right|$, where $a$ and $b$ are positive constants.
[2]
(i) On the same axes, sketch the graphs of $y=\frac{1}{x-a}$ and $y=b\left| x-a \right|$, where $a$ and $b$ are positive constants.
[2]
(ii)
Hence, or otherwise, solve the inequality $\frac{1}{x-a}<b\left| x-a \right|$.
[4]
(ii) Hence, or otherwise, solve the inequality $\frac{1}{x-a}<b\left| x-a \right|$.
[4]
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When the polynomial ${{x}^{3}}+a{{x}^{2}}+bx+c$ is divided by $\left( x-1 \right)$, $\left( x-2 \right)$ and $\left( x-3 \right)$, the remainders are $8$, $12$ and $25$ respectively.
(i)
Find the values of $a$, $b$ and $c$.
[4]
(i) Find the values of $a$, $b$ and $c$.Â
[4]
A curve has equation $y=\text{f}\left( x \right)$, where $\text{f}\left( x \right)={{x}^{3}}+a{{x}^{2}}+bx+c$, with the values of $a$, $b$ and $c$ found in part (i).
(ii)
Show that the gradient of the curve is always positive. Hence explain why the equation $\text{f}\left( x \right)=0$ has only one real root and find this root.
[3]
(ii) Show that the gradient of the curve is always positive. Hence explain why the equation $\text{f}\left( x \right)=0$ has only one real root and find this root.
[3]
(iii)
Find the $x$- coordinates of the points where the tangent to the curve is parallel to the line $y=2x-3$.
[3]
(iii) Find the $x$- coordinates of the points where the tangent to the curve is parallel to the line $y=2x-3$.
[3]
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(i)
Interpret geometrically the vector equation $\mathbf{r}=\mathbf{a}+t\mathbf{b}$, where $\mathbf{a}$ and $\mathbf{b}$ are constant vectors and $t$ is a parameter.
[2]
(i) Interpret geometrically the vector equation $\mathbf{r}=\mathbf{a}+t\mathbf{b}$, where $\mathbf{a}$ and $\mathbf{b}$ are constant vectors and $t$ is a parameter.
[2]
(ii)
Interpret geometrically the vector equation $\mathbf{r}\cdot \mathbf{n}=d$, where $\mathbf{n}$ is a constant unit vector and $d$ is a constant scalar, stating what $d$ represents.
[3]
(ii) Interpret geometrically the vector equation $\mathbf{r}\cdot \mathbf{n}=d$, where $\mathbf{n}$ is a constant unit vector and $d$ is a constant scalar, stating what $d$ represents.
[3]
(iii)
Given that $\mathbf{b}\cdot \mathbf{n}\ne 0$, solve the equations $\mathbf{r}=\mathbf{a}+t\mathbf{b}$ and $\mathbf{r}\cdot \mathbf{n}=d$ to find $\mathbf{r}$ in terms of $\mathbf{a}$, $\mathbf{b}$, $\mathbf{n}$ and $d$. Interpret the solution geometrically.
[3]
(iii) Given that $\mathbf{b}\cdot \mathbf{n}\ne 0$, solve the equations $\mathbf{r}=\mathbf{a}+t\mathbf{b}$ and $\mathbf{r}\cdot \mathbf{n}=d$ to find $\mathbf{r}$ in terms of $\mathbf{a}$, $\mathbf{b}$, $\mathbf{n}$ and $d$. Interpret the solution geometrically.
[3]
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It is given that $\text{f}\left( x \right)=\sin 2mx+\sin 2nx$, where $m$ and $n$ are positive integers and $m\ne n$.
(i)
Find $\int{\sin 2mx\sin 2nx\text{ d}x}$.
[3]
(i) Find $\int{\sin 2mx\sin 2nx\text{ d}x}$.
[3]
(ii)
Find $\int_{0}^{\pi }{{{\left( \text{f}\left( x \right) \right)}^{2}}\text{d}x}$ .
[5]
(ii) Find $\int_{0}^{\pi }{{{\left( \text{f}\left( x \right) \right)}^{2}}\text{d}x}$ .
[5]
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(a)
A sequence of numbers ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … has a sum ${{S}_{n}}$ where ${{S}_{n}}=\sum\limits_{r=1}^{n}{{{u}_{r}}}$. It is given that ${{S}_{n}}=A{{n}^{2}}+Bn$, where $A$ and $B$ are non-zero constants.
(a) A sequence of numbers ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … has a sum ${{S}_{n}}$ where ${{S}_{n}}=\sum\limits_{r=1}^{n}{{{u}_{r}}}$. It is given that ${{S}_{n}}=A{{n}^{2}}+Bn$, where $A$ and $B$ are non-zero constants.
(i) Find an expression for ${{u}_{n}}$ in terms of $A$, $B$ and $n$. Simplify your answer.
[3]
(i) Find an expression for ${{u}_{n}}$ in terms of $A$, $B$ and $n$. Simplify your answer.
[3]
(ii) It is also given that the tenth term is $48$ and the seventeenth term is $90$. Find $A$ and $B$.
[2]
(ii) It is also given that the tenth term is $48$ and the seventeenth term is $90$. Find $A$ and $B$.
[2]
(b)
Show that ${{r}^{2}}{{(r+1)}^{2}}-{{(r-1)}^{2}}{{r}^{2}}=k{{r}^{3}}$, where $k$ is a constant to be determined. Use this result to find a simplified expression for $\sum\limits_{r=1}^{n}{{{r}^{3}}}$.
[4]
(b) Show that ${{r}^{2}}{{(r+1)}^{2}}-{{(r-1)}^{2}}{{r}^{2}}=k{{r}^{3}}$, where $k$ is a constant to be determined. Use this result to find a simplified expression for $\sum\limits_{r=1}^{n}{{{r}^{3}}}$.
[4]
(c)
D’Alembert’s ratio test states that a series of the form $\sum\limits_{r=0}^{\infty }{{{a}_{r}}}$ converges when $\underset{n\to \infty }{\mathop{\lim }}\,\left| \frac{{{a}_{n+1}}}{{{a}_{n}}} \right|<1$, and diverges when $\underset{n\to \infty }{\mathop{\lim }}\,\left| \frac{{{a}_{n+1}}}{{{a}_{n}}} \right|>1$. When $\underset{n\to \infty }{\mathop{\lim }}\,\left| \frac{{{a}_{n+1}}}{{{a}_{n}}} \right|=1$, test is inconclusive. Using the test, explain why the series $\sum\limits_{r=0}^{\infty }{\frac{{{x}^{r}}}{r!}}$converges for all real values of $x$ and state the sum to infinity of this series, in terms of $x$.
[4]
(c) D’Alembert’s ratio test states that a series of the form $\sum\limits_{r=0}^{\infty }{{{a}_{r}}}$ converges when $\underset{n\to \infty }{\mathop{\lim }}\,\left| \frac{{{a}_{n+1}}}{{{a}_{n}}} \right|<1$, and diverges when $\underset{n\to \infty }{\mathop{\lim }}\,\left| \frac{{{a}_{n+1}}}{{{a}_{n}}} \right|>1$. When $\underset{n\to \infty }{\mathop{\lim }}\,\left| \frac{{{a}_{n+1}}}{{{a}_{n}}} \right|=1$, test is inconclusive. Using the test, explain why the series $\sum\limits_{r=0}^{\infty }{\frac{{{x}^{r}}}{r!}}$converges for all real values of $x$ and state the sum to infinity of this series, in terms of $x$.
[4]
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Electrical engineers are installing electricity cables on a building site. Points $(x,y,z)$ are defined relative to a main switching site at $(0,0,0)$, where units are metres. Cables are laid in straight lines and the widths of cables can be neglected.
An existing cable $C$ starts at the main switching site and goes in the direction $\left( \begin{matrix}3 \\1 \\-2 \\\end{matrix} \right)$. A new cable is installed which passes through points $P\,(1,2-1)$ and $Q\,(5,7,a)$.
(i)
Find the value of $a$ for which $C$ and the new cable will meet.
[4]
(i) Find the value of $a$ for which $C$ and the new cable will meet.
[4]
To ensure that the cables do not meet, the engineers use $a=-3$. The engineers wish to connect each of the points $P$ and $Q$ to a point $R$ on $C$.
(ii)
The engineers wish to reduce the length of cable required and believe in order to do this that angle $PRQ$ should be $90{}^\circ $. Show that this is not possible.
[4]
(ii) The engineers wish to reduce the length of cable required and believe in order to do this that angle $PRQ$ should be $90{}^\circ $. Show that this is not possible.
[4]
(iii)
The engineers discover that the ground between $P$ and $R$ is difficult to drill through and now decide to make the length of $PR$ as small as possible. Find the coordinates of $R$ in this case and the exact minimum length.
[5]
(iii) The engineers discover that the ground between $P$ and $R$ is difficult to drill through and now decide to make the length of $PR$ as small as possible. Find the coordinates of $R$ in this case and the exact minimum length.
[5]
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(a)
The curve $y=\text{f}(x)$ cuts the axes at $(a,0)$ and $(0,b)$. It is given that ${{\text{f}}^{-1}}(x)$ exists. State, if it is possible to do so, the coordinates of the points where the following curves cut the axes.
(a) The curve $y=\text{f}(x)$ cuts the axes at $(a,0)$ and $(0,b)$. It is given that ${{\text{f}}^{-1}}(x)$ exists. State, if it is possible to do so, the coordinates of the points where the following curves cut the axes.
(i) $y=\text{f}(2x)$
(ii) $y=\text{f}(x-1)$
(iii) $y=\text{f}(2x-1)$
(iv) $y={{\text{f}}^{-1}}(x)$
[4]
(b)
The function $\text{g}$ is defined by
$\text{g}:x\mapsto 1-\frac{1}{1-x}$, where $x\in \mathbb{R}$, $x\ne a$.
(b) The function $\text{g}$ is defined by
$\text{g}:x\mapsto 1-\frac{1}{1-x}$, where $x\in \mathbb{R}$, $x\ne a$.
(i) State the value of $a$ and explain why this value has to be excluded from the domain of $\text{g}$.
[2]
(ii) Find ${{\text{g}}^{2}}(x)$ and ${{\text{g}}^{-1}}(x)$, giving your answers in simplified form.
[4]
(iii) Find the values of $b$ such that ${{\text{g}}^{2}}(b)={{\text{g}}^{-1}}(b)$.
[2]
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(a)
A flat novelty plate for serving food on is made in the shape of the region enclosed by the curve $y={{x}^{2}}-6x+5$ and the line $2y=x-1$. Find the area of the plate.
[4]
(a) A flat novelty plate for serving food on is made in the shape of the region enclosed by the curve $y={{x}^{2}}-6x+5$ and the line $2y=x-1$. Find the area of the plate.
[4]
(b)
A curved container has a flat circular top. The shape of the container is formed by rotating the part of the curve $x=\frac{\sqrt{y}}{a-{{y}^{2}}}$, where $a$ is a constant greater than $1$, between the points $(0,0)$ and $\left( \frac{1}{a-1},1 \right)$ through $2\pi $ radians about the $y$-axis.
(b) A curved container has a flat circular top. The shape of the container is formed by rotating the part of the curve $x=\frac{\sqrt{y}}{a-{{y}^{2}}}$, where $a$ is a constant greater than $1$, between the points $(0,0)$ and $\left( \frac{1}{a-1},1 \right)$ through $2\pi $ radians about the $y$-axis.
(i) Find the volume of the container, giving your answer as a single fraction in terms of $a$ and $\pi $.
[4]
(ii) Another curved container with a flat circular top is formed in the same way from the curve $x=\frac{\sqrt{y}}{b-{{y}^{2}}}$ and the points $(0,0)$ and $\left( \frac{1}{b-1},1 \right)$. It has a volume that is four times as great as the container in part (i). Find an expression for $b$ in terms of $a$.
[3]
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A children’s game is played with $20$ cards, consisting of $5$ sets of $4$ cards. Each set consists of a father, mother, daughter and son from the same family. The family names are Red, Blue, Green, Yellow and Orange. So, for example, the Red family cards are father Red, mother Red, daughter Red and son Red.
The $20$ cards are arranged in a row.
(i)
In how many different ways can the $20$ cards be arranged so that the $4$ cards in each family set are next to each other?
[2]
(i) In how many different ways can the $20$ cards be arranged so that the $4$ cards in each family set are next to each other?
[2]
(ii)
In how many different ways can the cards be arranged so that all five father cards are next to each other, all four Red family cards are next to each other and all four Blue family cards are next to each other?
[3]
(ii) In how many different ways can the cards be arranged so that all five father cards are next to each other, all four Red family cards are next to each other and all four Blue family cards are next to each other?
[3]
The cards are now arranged at random in a circle.
(iii)
Find the probability that no two father cards are next to each other.
[4]
(iii) Find the probability that no two father cards are next to each other.
[4]
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