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

2005 A Level H2 Math

These Ten-Year-Series (TYS) worked solutions with video explanations for 2005 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at gs.ude.htamnagmitobfsctd-8c02c7@troppus.

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2005 A Level H2 Math Paper 1 Question 14 [Either]

The indefinite integral $\int{\frac{\text{P}(x)}{{{x}^{3}}+1}\,\text{d}x}$, where $\text{P}\left( x \right)$ is a polynomial in $x$, is denoted by $\text{I}$.

(i)

Find $\text{I}$ when $\text{P}(x)={{x}^{2}}$.

[2]

(i) Find $\text{I}$ when $\text{P}(x)={{x}^{2}}$.

[2]

(ii)

By writing ${{x}^{3}}+1=\left( x+1 \right)\left( {{x}^{2}}+Ax+B \right)$, where $A$ and $B$ are constants, find $\text{I}$ when

(ii) By writing ${{x}^{3}}+1=\left( x+1 \right)\left( {{x}^{2}}+Ax+B \right)$, where $A$ and $B$ are constants, find $\text{I}$ when

(a) $\text{P}\left( x \right)={{x}^{2}}-x+1$,

[3]

(a) $\text{P}\left( x \right)={{x}^{2}}-x+1$,

[3]

(b) $\text{P}\left( x \right)=x+1$.

[3]

(b) $\text{P}\left( x \right)=x+1$.

[3]

(iii)

Using the results of parts (i) and (ii), or otherwise, find $\text{I}$ when $\text{P}\left( x \right)=1$.

[4]

(iii) Using the results of parts (i) and (ii), or otherwise, find $\text{I}$ when $\text{P}\left( x \right)=1$.

[4]

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2005 A Level H2 Math Paper 2 Question 1

Verify that $z=\text{i}$ is a root of the equation

${{z}^{4}}-2{{z}^{3}}+6{{z}^{2}}-2z+5=0$.

[1]

Hence determine the other roots.

[4]

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2005 A Level H2 Math Paper 2 Question 4

It is given that $a$, $b$, $c$ are the first three terms of a geometric progression. It is also given that $a$, $c$, $b$ are the first three terms of an arithmetic progression.

(i)

Show that ${{b}^{2}}=ac$ and $c=\frac{a+b}{2}$.

[2]

(i) Show that ${{b}^{2}}=ac$ and $c=\frac{a+b}{2}$.

[2]

(ii)

Hence show that $2{{\left( \frac{b}{a} \right)}^{2}}-\left( \frac{b}{a} \right)-1=0$.

[2]

(ii) Hence show that $2{{\left( \frac{b}{a} \right)}^{2}}-\left( \frac{b}{a} \right)-1=0$.

[2]

(iii)

Given that the sum to infinity of the geometric progression is $S$, find $S$ in terms of $a$.

[4]

(iii) Given that the sum to infinity of the geometric progression is $S$, find $S$ in terms of $a$.

[4]

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2005 TYS 2005


2005 TYS 2005


2005 TYS 2005


2005 TYS 2005

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