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

2014 A Level H2 Math

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2014 A Level H2 Math Paper 1 Question 1 [Modified]

The function $\text{f}$ is defined by

$\text{f}:x\mapsto \frac{1}{1-x},x\in \mathbb{R},x\ne 1,x\ne 0$ .

(i)

Show that ${{\text{f}}^{2}}\left( x \right)={{\text{f}}^{-1}}\left( x \right)$.

[4]

(i) Show that ${{\text{f}}^{2}}\left( x \right)={{\text{f}}^{-1}}\left( x \right)$.

[4]

(ii)

Find ${{\text{f}}^{3}}\left( x \right)$ and ${{\text{f}}^{2020}}\left( x \right)$ in simplified forms.

[3]

(ii) Find ${{\text{f}}^{3}}\left( x \right)$ and ${{\text{f}}^{2020}}\left( x \right)$ in simplified forms.

[3]

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2014 A Level H2 Math Paper 1 Question 10

The mass, $x$ grams, of a certain substance present in a chemical reaction at time $t$ minutes satisfies the differential equation $\frac{\text{d}x}{\text{d}t}=k\left( 1+x-{{x}^{2}} \right)$, where $0\le x\le \frac{1}{2}$ and $k$ is a constant. It is given that $x=\frac{1}{2}$ and $\frac{\text{d}x}{\text{d}t}=-\frac{1}{4}$ when $t=0$.

(i)

Show that $k=-\frac{1}{5}$.

[1]

(i) Show that $k=-\frac{1}{5}$.

[1]

(ii)

By first expressing $1+x-{{x}^{2}}$ in completed square from, find $t$ in terms of $x$.

[5]

(ii) By first expressing $1+x-{{x}^{2}}$ in completed square from, find $t$ in terms of $x$. [5]

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2014 A Level H2 Math Paper 2 Question 3

In a training exercise, athletes run from a starting point $O$ to and from a series of points, ${{A}_{1}}$, ${{A}_{2}}$, ${{A}_{3}}$,….., increasingly far away in a straight line. In the exercise, atheletes start at $O$ and run stage 1 from $O$ to ${{A}_{1}}$ and back to $O$, then stage 2 from $O$ to ${{A}_{2}}$ and back to $O$, and so on.

(i)

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In Version 1 of the exercise, the distances between adjacent points are all $4$m (see Fig. 1).

(a) Find the distance run by an athlete who completes the first $10$ stages of Version 1 of the exercise.

[2]

In Version 1 of the exercise, the distances between adjacent points are all $4$m (see Fig. 1).

(a) Find the distance run by an athlete who completes the first $10$ stages of Version 1 of the exercise.

[2]

(b) Write down an expression for the distance run by an athlete who completes $n$ stages of Version I. Hence find the least number of stages that the athlete needs to complete to run at least $5$km.

[4]

(b) Write down an expression for the distance run by an athlete who completes $n$ stages of Version I. Hence find the least number of stages that the athlete needs to complete to run at least $5$km.

[4]

(ii)

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In Version 2 of the exercise, the distances between the points are such that $O{{A}_{1}}=4$m, ${{A}_{1}}{{A}_{2}}=4$m, ${{A}_{2}}{{A}_{3}}=8$m, ${{A}_{n}}{{A}_{n+1}}=2{{A}_{n-1}}{{A}_{n}}$ (see Fig. 2). Write down an expression for the distance run by an athlete who completes $n$ stages of Version 2.

In Version 2 of the exercise, the distances between the points are such that $O{{A}_{1}}=4$m, ${{A}_{1}}{{A}_{2}}=4$m, ${{A}_{2}}{{A}_{3}}=8$m, ${{A}_{n}}{{A}_{n+1}}=2{{A}_{n-1}}{{A}_{n}}$ (see Fig. 2). Write down an expression for the distance run by an athlete who completes $n$ stages of Version 2.

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2014 A Level H2 Math Paper 2 Question 4 (b)

It is given that $w=\sqrt{3}-\text{i}$.

(i)

Without using a calculator, find an exact expression for ${{w}^{6}}$. Give your answer in the form $r{{\text{e}}^{\text{i}\theta }}$, where $r>0$ and $0\le \theta <2\pi .$

(i) Without using a calculator, find an exact expression for ${{w}^{6}}$. Give your answer in the form $r{{\text{e}}^{\text{i}\theta }}$, where $r>0$ and $0\le \theta <2\pi .$

(ii)

Without using a calculator, find the three smallest positive whole number values for $n$ for which $\frac{{{w}^{n}}}{{{w}^{*}}}$ is a real number.

(ii) Without using a calculator, find the three smallest positive whole number values for $n$ for which $\frac{{{w}^{n}}}{{{w}^{*}}}$ is a real number.

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2014 A Level H2 Math Paper 2 Question 6

A team in a particular sport consists of 1 goalkeeper, 4 defenders, 2 midfielders and 4 attackers.
A certain club has 3 goalkeepers, 8 defenders, 5 midfielders and 6 attackers.

(i)

How many different teams can be formed by the club?

[2]

(i) How many different teams can be formed by the club?

[2]

One of the midfielders in the club is the brother of one of the attackers in the club.

(ii)

How many different teams can be formed which include exactly one of the two brothers?

[3]

(ii) How many different teams can be formed which include exactly one of the two brothers?

[3]

The two brothers leave the club. The club manager decides that one of the remaining midfielders can play as either a midfielder or as a defender.

(iii)

How many different teams can now be formed by the club?

[3]

(iii) How many different teams can now be formed by the club?

[3]

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