Victoria Junior College H2 Math papers, with video solutions

1. The graph of \(y=\mathrm{f}(x)\) is shown in the diagram. It passes through the origin and has a minimum point at \(\left( 2,\mathrm{ }0 \right)\). The lines \(x=1\) and \(y=2\) are the asymptotes of the graph.
   
   Sketch the graph of \(y=2-\mathrm{f}(x)\), stating the coordinates of the point where the graph crosses the \(y\)-axis, the coordinates of any stationary point(s) and the equations of any asymptotes.[2]
2. The curve \(C\) has equation \(y=\frac{2{{x}^{2}}+kx-2}{x+2}\), where \(k\) is a constant.
   1. Find the set of values of \(k\) for which \(C\) has \(2\) stationary points.[3]
      For the rest of the question, take \(k=-3\).
   2. By expressing the equation \(y=\frac{2{{x}^{2}}-3x-2}{x+2}\) in the form \(y=ax+b+\frac{c}{x+2}\), where \(a\), \(b\) and \(c\) are constants, write down the equations of the asymptotes of \(C\).[3]
   3. Hence sketch \(C\), giving the equations of any asymptotes, the coordinates of any stationary points and of the points where \(C\) crosses the axes.[2]
   4. Describe a pair of transformations which transform the graph of \(y=x+\frac{6}{x+2}\) to \(C\).[2]
   For the rest of the question, take \(k=-3\).

Exposure to Volatile Organic Compounds (VOCs), which have been identified in indoor air, is suspected as a cause of headaches and respiratory systems.Indoor plants have not only a positive psychological effect on humans, but may also improve the air quality.Certain species of indoor plants were found to be effective removers of VOCs.

A commonly known VOC is Benzene. The following data gives the benzene levels, $x$(in ppm) in $40$ test chambers containing the indoor plant Epipremnum aureum.

$n=40$, $\sum{\left( x-26.0 \right)=-30.1}$, ${{\sum{\left( x-26.0 \right)}}^{2}}=214.61$.

The initial mean Benzene level (in ppm) without Epipremnum aureum was found to be $26.0$.

1. Test, at the $5$% level of significance, the claim that the mean Benzene level, $\mu $(in ppm), has decreased as a result of the indoor plant Epipremnum aureum. You should state your hypotheses clearly.

   [5]

2. State, giving a reason, whether there is a need to make any assumptions about the population distribution of the Benzene level in order for the test to be valid.

   [2]

The Benzene levels of another $50$ test chambers containing the indoor plant Epipremnum aureum were recorded. The sample mean is $\bar{x}$ppm and the sample variance is $8.33$ ppm$^{2}$.

3. The acceptance region of a test of the null hypothesis $\mu =26.0$ is $\bar{x}>25.1$. State the alternative hypothesis and find the level of significance of the test.

   [4]

4. If the null hypothesis is $\mu ={{\mu }_{0}}$, where ${{\mu }_{0}}>26.0$, would the significance level of a test with the same acceptance region in part (iii) be larger or smaller than that found in (iii)? Give a reason for your answer.

   [2]

It is given that $\sum\limits_{r=1}^{n}{\frac{{{r}^{2}}}{{{3}^{r}}}}=\frac{3}{2}-\frac{{{n}^{2}}+3n+3}{2\left( {{3}^{n}} \right)}$ .

1. Find $\sum\limits_{r=1}^{\infty }{\frac{{{r}^{2}}+{{\left( -1 \right)}^{r}}}{{{3}^{r}}}}$.

   [3]

2. Show that $\sum\limits_{r=4}^{n}{\frac{{{\left( r-2 \right)}^{2}}}{{{3}^{r-2}}}}=\frac{p}{q}-\frac{a{{n}^{2}}-an+a}{2\left( {{3}^{n-2}} \right)}$, where $a$, $p$ and $q$ are integers to be determined.

   [5]

A trial for a new treatment for a particular disease is to be carried out. A sample of $n$ patients suffering from this disease is taken and these patients are invited to take part in the trial. The number of patients in the sample who accept the invitation is denoted by $X$.

1. State, in context, two assumptions needed for $X$ to be well modelled by a binomial distribution.

   [2]

Assume now that $X$ has the distribution $\mathrm{B}(10,\,0.25)$.

2. Find the most probable number of patients who accept the invitation.

   [2]

3. Find the probability that exactly $6$ patients accept the invitation given that at least $1$ patient accepts the invitation.

   [2]

A supermarket sells \(2\) different types of apples – Granny Smith and Fuji. The masses, in grams, of each type of apples follow normal distributions. The means and standard deviations of these distributions are shown in the following table.

1. Two Fuji apples are chosen. Find the probability that their masses differs by at least \(20\) g.
2. Granny Smith apples are sold at \(\$3.50\) per kg and Fuji apples at \(\$5\) per kg.

Find the probability that six Granny Smith apples and four Fuji apples cost more than \(\$6.50\).

3. State an assumption needed for your calculations in parts (i) and (ii).

1. The diagram shows the graph of \(y=\mathrm{f}(x)\) which has a minimum point at \((0, 2)\) and asymptotes \(x=2\) and \(y=5.\)
   On separate diagrams, sketch the graphs of \(y\ =\ \ \frac{1}{\mathrm{f}(x)}\).