These Ten-Year-Series (TYS) worked solutions with video explanations for 2015 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at gs.ude.htamnagmitobfsctd-95bbc1@troppus.
(i)
Sketch the curve with equation $y=\left| \frac{x+1}{1-x} \right|$, stating the equations of the asymptotes. On the same diagram, sketch the line with equation $y=x+2$.
[3]
(i) Sketch the curve with equation $y=\left| \frac{x+1}{1-x} \right|$, stating the equations of the asymptotes. On the same diagram, sketch the line with equation $y=x+2$.
[3]
(ii)
Solve the inequality $\left| \frac{x+1}{1-x} \right|<x+2$.
[3]
(ii) Solve the inequality $\left| \frac{x+1}{1-x} \right|<x+2$.
[3]
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(i)
Given that $\text{f}$ is a continuous function, explain, with the aid of a sketch, why the value of
$\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{n}\left[ \text{f}\left( \frac{1}{n} \right)+\text{f}\left( \frac{2}{n} \right)+…+\text{f}\left( \frac{n}{n} \right) \right]$
is $\int_{0}^{1}{\text{f}(x)\text{d}x}$.
[2]
(i) Given that $\text{f}$ is a continuous function, explain, with the aid of a sketch, why the value of
$\underset{n\,\to \,\infty }{\mathop{\lim }}\,\,\frac{1}{n}\left\{ \text{f}\left( \frac{1}{n} \right)+\text{f}\left( \frac{2}{n} \right)+…+\text{f}\left( \frac{n}{n} \right) \right\}$
is $\int_{0}^{1}{\text{f}(x)\text{d}x}$.
[2]
(ii)
Hence evaluate $\underset{n\,\to \,\infty }{\mathop{\lim }}\,\frac{1}{n}\left( \frac{\sqrt[3]{1}+\sqrt[3]{2}+…+\sqrt[3]{n}}{\sqrt[3]{n}} \right)$.
[3]
(ii) Hence evaluate $\underset{n\,\to \,\infty }{\mathop{\lim }}\,\frac{1}{n}\left( \frac{\sqrt[3]{1}+\sqrt[3]{2}+…+\sqrt[3]{n}}{\sqrt[3]{n}} \right)$.
[3]
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(i)
State a sequence of transformations that will transform the curve with equation $y={{x}^{2}}$on to the curve with equation $y=\frac{1}{4}{{\left( x-3 \right)}^{2}}$.
[2]
(i) State a sequence of transformations that will transform the curve with equation $y={{x}^{2}}$on to the curve with equation $y=\frac{1}{4}{{\left( x-3 \right)}^{2}}$.
[2]
A curve has equation $y=\text{f}\left( x \right)$, where
$\text{f}\left( x \right)=\left\{ \begin{matrix}
1\text{ for }0\le x\le 1, \\
\frac{1}{4}{{\left( x-3 \right)}^{2}}\text{ for }1<x\le 3, \\
0\text{ otherwise}\text{.} \\
\end{matrix} \right.$
(ii)
Sketch the curve for $-1\le x\le 4$.
[3]
(ii) Sketch the curve for $-1\le x\le 4$.
[3]
(iii)
On a separate diagram, sketch the curve with equation $y=1+\text{f}\left( \frac{1}{2}x \right)$, for $-1\le x\le 4$.
[2]
(iii) On a separate diagram, sketch the curve with equation $y=1+\text{f}\left( \frac{1}{2}x \right)$, for $-1\le x\le 4$.
[2]
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(a)
The complex number $w$ is such that $w=a+\text{i}b$, where $a$ and $b$ are non-zero real numbers. The complex conjugate of $w$ is denoted by $w*$. Given that $\frac{{{w}^{2}}}{w*}$ is purely imaginary, find the possible values of $w$ in terms of $a$.
[5]
(a) The complex number $w$ is such that $w=a+\text{i}b$, where $a$ and $b$ are non-zero real numbers. The complex conjugate of $w$ is denoted by $w*$. Given that $\frac{{{w}^{2}}}{w*}$ is purely imaginary, find the possible values of $w$ in terms of $a$.
[5]
Method I
Method II
Method I
Method II
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A curve $C$ has parametric equations
$x={{\sin }^{3}}\theta $, $y=3{{\sin }^{2}}\theta \cos \theta $, $0\le \theta \le \frac{\pi }{2}$.
(i)
Show that $\frac{\text{d}y}{\text{d}x}=2\cot \theta -\tan \theta $.
[3]
(i) Show that $\frac{\text{d}y}{\text{d}x}=2\cot \theta -\tan \theta $.
[3]
(ii)
Show that $C$ has a turning point when $\tan \theta =\sqrt{k}$, where $k$ is an integer to be determined.Â
Find, in non-trigonometric form, the exact coordinates of the turning point and explain why it is a maximum.
[6]
(ii) Show that $C$ has a turning point when $\tan \theta =\sqrt{k}$, where $k$ is an integer to be determined.Â
Find, in non-trigonometric form, the exact coordinates of the turning point and explain why it is a maximum.
[6]
(iii)
Show that the area of the region bounded by $C$ and the $x$-axis is given by
$\int_{0}^{\frac{1}{2}\pi }{9{{\sin }^{4}}\theta {{\cos }^{2}}\theta \text{ d}\theta }$.
Use your calculator to find the area, giving your answer correct to $3$ decimal places.
[3]
(iii) Show that the area of the region bounded by $C$ and the $x$-axis is given by
$\int_{0}^{\frac{1}{2}\pi }{9{{\sin }^{4}}\theta {{\cos }^{2}}\theta \text{ d}\theta }$.
Use your calculator to find the area, giving your answer correct to $3$ decimal places.
[3]
The line with equation $y=ax$, where $a$ is a positive constant, meets $C$ at the origin and at the point $P$.
(iv)
Show that $\tan \theta =\frac{3}{a}$ at $P$. Find the exact value of $a$ such that the line passes through the maximum point of $C$.
[3]
Show that $\tan \theta =\frac{3}{a}$ at $P$. Find the exact value of $a$ such that the line passes through the maximum point of $C$.
[3]
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(a)
The function $\text{f}$ is defined by $\text{f}:x\mapsto \frac{1}{1-{{x}^{2}}}$, $x\in \mathbb{R}$, $x>1$.
(a) The function $\text{f}$ is defined by $\text{f}:x\mapsto \frac{1}{1-{{x}^{2}}}$, $x\in \mathbb{R}$, $x>1$.
(i) Show that $\text{f}$ has an inverse.
[2]
(i) Show that $\text{f}$ has an inverse.
[2]
(ii) Find ${{\text{f}}^{-1}}\left( x \right)$ and state the domain of ${{\text{f}}^{-1}}$.
[3]
(ii) Find ${{\text{f}}^{-1}}\left( x \right)$ and state the domain of ${{\text{f}}^{-1}}$.
[3]
(b)
The function $\text{g}$ is defined by $\text{g}:x\mapsto \frac{2+x}{1-{{x}^{2}}}$, $x\in \mathbb{R}$, $x\ne \pm 1$. Find algebraically the range of $\text{g}$, giving your answer in terms of $\sqrt{3}$ as simply as possible.
[5]
(b) The function $\text{g}$ is defined by $\text{g}:x\mapsto \frac{2+x}{1-{{x}^{2}}}$, $x\in \mathbb{R}$, $x\ne \pm 1$. Find algebraically the range of $\text{g}$, giving your answer in terms of $\sqrt{3}$ as simply as possible.
[5]
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For events $A$, $B$ and $C$ it is given that $\text{P}\left( A \right)=0.45$, $\text{P}\left( B \right)=0.4$, $\text{P}\left( C \right)=0.3$ and $P\left( A\cap B\cap C \right)=0.1$. It is also given that events $A$ and $B$ are independent, and that events $A$ and $C$ are independent.
(i)
Find $\text{P}\left( B|A \right)$.
[1]
(i) Find $\text{P}\left( B|A \right)$.
[1]
(ii)
Given also that event $B$ and $C$ are independent, find $\text{P}\left( A’\cap B’\cap C’ \right)$.
[3]
(ii) Given also that event $B$ and $C$ are independent, find $\text{P}\left( A’\cap B’\cap C’ \right)$.
[3]
(iii)
Given instead that events $B$ and $C$ are not independent, find the greatest and least possible values of $\text{P}\left( A’\cap B’\cap C’ \right)$.
[4]
(iii) Given instead that events $B$ and $C$ are not independent, find the greatest and least possible values of $\text{P}\left( A’\cap B’\cap C’ \right)$.
[4]
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This question is about arrangements of all eight letters in the word CABBAGES.
(i)
Find the number of different arrangements of the eight letters that can be made.
[2]
(i) Find the number of different arrangements of the eight letters that can be made.
[2]
(ii)
Write down the number of these arrangements in which the letters are not in alphabetical order.
[1]
(ii) Write down the number of these arrangements in which the letters are not in alphabetical order.
[1]
(iii)
Find the number of different arrangements that can be made with both the A’s together and both the B’s together.
[2]
(iii) Find the number of different arrangements that can be made with both the A’s together and both the B’s together.
[2]
(iv)
Find the number of different arrangements that can be made with no two adjacent letters the same.
[4]
(iv) Find the number of different arrangements that can be made with no two adjacent letters the same.
[4]
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