Dunman High School H2 Math papers, with video solutions

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by

$\mathrm{f}:x\mapsto {{x}^{2}}+cx+d,x\in \mathbb{R},$ where $c$ and $d$ are constants,

$\mathrm{g}:x\mapsto \ln x,x\in \mathbb{R},x>0$.

1. Given that the composite function $\mathrm{gf}$ exists, find an inequality involving $c$ and $d$.

   [2]

For the rest of the question, take $c=2,d=4$.

2. Write down an expression for $\mathrm{gf}\left( x \right)$ and find exactly the range of $\mathrm{gf}$.

   [3]

3. The graph of a function $\mathrm{h}$ is symmetrical about the line $x=k$ if $\mathrm{h}\left( x+k \right)=\mathrm{h}\left( -x+k \right)$ for all valid values of $x$. Using this definition, show that the graph of $\mathrm{gf}$ is symmetrical about the line $x=-1$.

   [2]

Using the result $\sum\limits_{r=1}^{n}{\frac{r}{{{2}^{r}}}}=2-\frac{n+2}{{{2}^{n}}}$, show that $\sum\limits_{r=1}^{n}{\left( r-n \right)\left( {{2}^{-r}}+1 \right)}$ can be expressed in the form $C\left( 1-\frac{1}{{{2}^{n}}} \right)+Dn\left( n+1 \right)$, where $C$ and $D$ are constants to be determined.

[4]

1. Find $\displaystyle\int{\frac{x}{\sqrt{2x-1}}}\,\mathrm{d}x$.

   [3]

2. Using the substitution $t=\tan x$, find $\displaystyle\int{\frac{1}{4{{\cos }^{2}}x+9{{\sin }^{2}}x}\,\mathrm{d}x}$.

   [4]

   
3. The diagram above shows part of the graph of $y={{x}^{2}}+3x$ , with rectangles approximating the area under the curve from $x=0$ to $x=1$. The area under the curve may be approximated by the total area, $A$, of $\left( n-1 \right)$ rectangles each of width $\frac{1}{n}$. Given that $\underset{r=1}{\overset{n}{\mathop \sum }}\,{{r}^{2}}=\frac{1}{6}n\left( n+1 \right)\left( 2n+1) \right)$, show that $A=\frac{\left( n-1 \right)\left( 11n-1 \right)}{6{{n}^{2}}}$.
   Explain briefly how the value of $\mathop{\displaystyle\int }_{0}^{1}{{x}^{2}}+3x\,\mathrm{d}x$ can be deduced from this expression, and hence find this value exactly without integration.

   [6]

The discrete random variable $X$ has probability mass function given by

$\mathrm{P}(X=x)=\left\{ \begin{matrix} & \frac{ax}{n(n+1)}\,\,\,\,\mathrm{for}\,\,x=1,2,...,n, \\ & 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{otherwise,} \\ \end{matrix} \right.$

where $a$ is a constant.

1. Show that $a=2$.

   [2]

2. Find $\mathrm{E}(X)$ in terms of $n$. You may use the result that $\sum\limits_{r=1}^{n}{{{r}^{2}}=\frac{n}{6}(n+1)(2n+1)}$.

   [3]

Two players ${{P}_{1}}$ and ${{P}_{2}}$ play a game with $n+1$ tokens labelled $1$ to $n+1$. Each player randomly picks one token without replacement and the player who picks the token with the smaller number loses. The amount of money lost by the losing player, in dollars, will be the number on the winning token. For example, if ${{P}_{1}}$ and ${{P}_{2}}$ pick the tokens labelled $5$ and $3$ respectively, ${{P}_{2}}$ loses $\$\,5$ to ${{P}_{1}}$.

3. Explain why the probability of ${{P}_{2}}$ losing a game is $0.5$.

   [1]

4. Given that ${{P}_{2}}$ loses, find the probability ${{P}_{2}}$ loses $\$\,(m+1)$ in terms of $m$ and $n$, where $m$ is such that $1\le m\le n$.

   [3]

5. Using the result in part (i), when ${{P}_{2}}$ loses a game, find the amount that he is expected to lose in terms of $n$.

   [3]

Data is transmitted in bytes, where each byte consists of $8$ bits. The probability of a bit being corrupted during its transmission is $0.03$. A byte is considered ‘corrupted’ if it contains at least $2$ corrupted bits.

Assume that all bits are not corrupted prior to their transmission.

1. Find the probability that a randomly chosen byte is corrupted during its transmission.
2. Given that a randomly chosen byte is not corrupted during its transmission, find the probability that it contains no corrupted bits.
3. Find the probability that between $5$ and $10$ bytes are corrupted during the transmission of $100$ bytes.

In another transmission system, the probability of having at least $3$ corrupted bytes during the transmission of $20$ bytes is $0.85$. Find the probability that a bit is corrupted during its transmission.