Ten-Year-Series (TYS) Solutions | Past Year Exam Questions

2015 A Level H2 Math

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2015 A Level H2 Math Paper 1 Question 2

(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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2015 A Level H2 Math Paper 1 Question 3

(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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2015 A Level H2 Math Paper 1 Question 5

(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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2015 A Level H2 Math Paper 1 Question 11

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. 
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. 
Find, in non-trigonometric form, the exact coordinates of the turning point and explain why it is a maximum.

[6]

The line with equation $y=ax$, where $a$ is a positive constant, meets $C$ at the origin and at the point $P$.

(iii)

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]

(iii) 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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2015 A Level H2 Math Paper 2 Question 3

(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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2015 A Level H2 Math Paper 2 Question 11

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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