These Ten-Year-Series (TYS) worked solutions with video explanations for 2004 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at gs.ude.htamnagmitobfsctd-7c9f9d@troppus.
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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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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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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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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(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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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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$A$ 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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