These Ten-Year-Series (TYS) worked solutions with video explanations for 1998 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at gs.ude.htamnagmitobfsctd-813d8b@troppus.
(a)
Given that $x$ is real, prove that ${{x}^{2}}-4x+9$ is always positive.
Solve the inequality $\frac{{{x}^{3}}+2{{x}^{2}}+x+14}{{{x}^{2}}+5}>x+1$.
(a) Given that $x$ is real, prove that ${{x}^{2}}-4x+9$ is always positive.
Solve the inequality $\frac{{{x}^{3}}+2{{x}^{2}}+x+14}{{{x}^{2}}+5}>x+1$.
Share with your friends!
A computer can give independent observations of a random variable $X$ with probability distribution given by $\text{P}\left( X=0 \right)=\frac{3}{4}$ and $\text{P}\left( X=2 \right)=\frac{1}{4}$. It is programmed to output a value for the random variable $Y$ defined by $Y={{X}_{1}}+{{X}_{2}}$, where ${{X}_{1}}$ and ${{X}_{2}}$ are independent observations of $X$. Tabulate the probability distribution of $Y$, show that $E\left( Y \right)=1$.
[3]
The random variable $T$ is defined by $T={{Y}^{2}}$. Find $\text{E}\left( T \right)$ and show that $\text{Var}\left( T \right)=\frac{63}{4}$.
[3]
The computer is programmed to produce a large number $n$ of independent values of $T$ and to calculate the mean $M$ of these values. Find the smallest value of $n$ such that $\text{P}\left( M<3 \right)>0.99$.
[4]
Share with your friends!
(b)
The complex number $q$ is given by $q=\frac{{{e}^{i\theta }}}{1-{{e}^{i\theta }}}$, where $0<\theta \le 2\pi $.
In either order,
The complex number $q$ is given by $q=\frac{{{e}^{i\theta }}}{1-{{e}^{i\theta }}}$, where $0<\theta \le 2\pi $.
In either order,
(i) find the real part of $q$,
(i) find the real part of $q$,
(ii) show that the imaginary part of $q$ is $\frac{1}{2}\cot \left( \frac{1}{2}\theta \right)$.
(ii) show that the imaginary part of $q$ is $\frac{1}{2}\cot \left( \frac{1}{2}\theta \right)$.
Share with your friends!
How can we help?
