Catholic Junior College H2 Math papers, with video solutions

A student reaches the bus-stop outside his house at $7.20$ am every morning to take a bus to school and has to reach his school by $7.40$ am. Assume that the waiting time for his bus is normally distributed with mean $8$ minutes and variance $5$ minutes$^{2}$, and the journey time is normally distributed with mean $11$ minutes and variance $4$ minutes$^{2}$.

1. Find the probability that he will take more than $20$ minutes to reach school (i.e. late for school) on a randomly chosen day.

   [2]

2. In a month of $30$ days, what is the expected number of days he will be late for school?

   [1]

3. Find the latest time he would have to reach the bus-stop outside his house so that the probability of him being late for school is less than $5$%.

   [4]

4. Find the probability that the average time taken to travel from the bus stop outside his house (including waiting for the bus) to school in $30$ days is between $15$ to $20$ minutes.

   [2]

Without using a calculator, solve

\(x<\frac{3}{x-2}\).[3]

Hence solve

\({{\mathrm{e}}^{-x}}<\frac{3}{{{\mathrm{e}}^{-x}}-2}\).[2]

The complex number $z$ has modulus $2$ and argument $\frac{\pi }{8}$ . It is also given that $w=1+\mathrm{i}$.

1. Given that $n$ is an integer, find $\frac{{{z}^{n}}}{w*}$ in terms of $n$, giving your answer in the form $r{{e}^{i\theta }}$ .

   [3]

2. Hence, find the smallest two positive integers $n$ such that $\frac{{{z}^{n}}}{w*}$ is real and negative.

   [3]

The security guard of a particular school claims that the average speed of the cars in the school compound is greater than the speed limit of $25$ km/h. To investigate the security guard’s claim, the traffic police randomly selected $50$ cars and the speed was recorded. The total speed and the standard deviation of the $50$ cars are found to be $1325$ km/h and $7.75$ km/h respectively.

1. Find the unbiased estimates of the population mean and variance.

   [2]

2. Test at $5$% level of significance whether there is sufficient evidence to support the security guard’s claim.

   [4]

3. Without any further test, state the conclusion if instead the traffic police is testing that the average speed of the cars is not $25$ km/h at $20$% level of significance

   [2]

1. The complex numbers \(z\) and \(w\) satisfy the simultaneous equations
   \(z+{{w}^{*}}+5\mathbf{i}=10\) and \({{\left| w \right|}^{2}}=z+18+\mathbf{i}\).
   Find \(z\) and \(w\).[4]
2. 1. It is given that \(2+\mathbf{i}\) is a root of the equation \({{z}^{2}}-5z+7+\mathbf{i}=0\). Find the second root of the equation in Cartesian form, showing your working clearly.[2]
   2. Hence find the roots of the equation \(-\mathbf{i}{{w}^{2}}+5w+7\mathbf{i}-1=0\).[2]
3. The complex number \(z\) is given by \(z=-a+a\,\mathbf{i}\), where \(a\) is a positive real number.
   1. It is given that \(w=-{{\frac{\sqrt{2}z}{{{z}^{4}}}}^{*}}\). Express \(w\) in the form \(r{{\mathrm{e}}^{i\theta }}\), in terms of \(a\), where \(r>0\) and \(-\pi <\theta \le \pi \).[4]
   2. Find the two smallest positive whole number values of \(n\) such that \(\operatorname{Re}\,({{w}^{n}})=0\).[3]

1. 1. Without using a calculator, solve the inequality $\frac{x+6}{{{x}^{2}}-3x-4}\le \frac{1}{4-x}$.

      [3]

   2. Hence, deduce the range of values of $x$ that satisfies
      $\frac{\left| x \right|+6}{{{x}^{2}}-3\left| x \right|-4}\le \frac{1}{4-\left| x \right|}$.

      [2]

2. Solve the inequality $\ln \left( x+6 \right)\le -\frac{x}{3}$.

   [3]