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

2004 A Level H2 Math

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2004 A Level H2 Math Paper 1 Question 1

The equation

$x\sin x+\cos x=1.015$

has a positive root $\alpha $ close to zero. Use small-angle approximations for $\sin x$ and $\cos x$ to obtain an approximation to $\alpha $. Give your answer correct to $3$ significant figures.

[3]

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2004 TYS 2004

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2004 A Level H2 Math Paper 1 Question 5

The numbers $x$ and $y$ satisfy the equation $4{{x}^{2}}+16xy+{{y}^{2}}+16x+14y+13=0$.
If a real value of $x$ is substituted, the equation becomes a quadratic equation in $y$.

Given that two distinct real values of $y$ may be found from this equation, show that $5{{x}^{2}}+8x+3>0$, and hence find the set of possible values of $x$.

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2004 TYS 2004

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2004 A Level H2 Math Paper 1 Question 5 [Modified]

The numbers $x$ and $y$ satisfy the equation $4{{x}^{2}}+16xy+{{y}^{2}}+16x+14y+13=0$.
If a real value of $x$ is substituted, the equation becomes a quadratic equation in $y$. Given that two distinct real values of $y$ may be found from this equation, show that

$5{{x}^{2}}+8x+3>0$,

and hence find the set of possible values of $x$.

[5]


What can you deduce about the curve with equation

$4{{x}^{2}}+16xy+{{y}^{2}}+16x+14y+13=0$?

[1]

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2004 TYS 2004
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2004 A Level H2 Math Paper 1 Question 9

Find how many positive integers, less than $1000$, are

(i)

odd numbers,

[1]

(i) odd numbers,

[1]

(ii)

odd numbers which are not divisible by $5$.

[3]

(ii) odd numbers which are not divisible by $5$.

[3]

Find the sum of the odd numbers, less than $1000$, which are not divisible by $5$.

[4]

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2004 TYS 2004

2004 TYS 2004

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2004 TYS 2004

2004 TYS 2004

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2004 TYS 2004

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2004 A Level H2 Math Paper 1 Question 12

(a)

Express ${{\left( 3-\text{i} \right)}^{2}}$ in the form of $a+\text{i}b$.

[1]

(a) Express ${{\left( 3-\text{i} \right)}^{2}}$ in the form of $a+\text{i}b$.

[1]

Hence or otherwise find the roots of the equation

${{\left( z+\text{i} \right)}^{2}}=-8+6\text{i}$.

[3]

Hence or otherwise find the roots of the equation

${{\left( z+\text{i} \right)}^{2}}=-8+6\text{i}$.

[3]

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2004 TYS 2004
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2004 A Level H2 Math Paper 2 Question 4
2004 TYS 2004

A rectangular shed, with a door at each end, contains ten fixed concret bases marked $A$, $B$, $C$, … $J$. Five on each side (see diagram). Ten canisters, each containing a different chemical, are placed with one canister on each base. In how many different ways can the canisters be placed on the bases?

[1]

Find the number of ways in which the canisters can be placed

(i)

if 2 particular canisters must not be placed on any of the 4 bases $A$, $E$, $F$ and $J$ next to a door,

[3]

(i) if 2 particular canisters must not be placed on any of the 4 bases $A$, $E$, $F$ and $J$ next to a door,

[3]

(ii)

if 2 particular canisters must not be placed next to each other on the same side of the shed.

[3]

(ii) if 2 particular canisters must not be placed next to each other on the same side of the shed.

[3]

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2004 TYS 2004


2004 TYS 2004


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2004 A Level H2 Math Paper 2 Question 23

$A$ and $B$ are events such that $\text{P}\left( A\cup B \right)=0.9$, $\text{P}\left( A\cap B \right)=0.2$ and $\text{P}\left( A|B \right)=0.8$. Find $\text{P}\left( A \right)$ and $\text{P}\left( B’ \right)$.

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