St. Andrew's Junior College H2 Math papers, with video solutions

A factory manufactures a large number of erasers in a variety of colours. Each box of erasers contains $36$ randomly chosen erasers. On average, $20$% of erasers in the box are blue.

1. State, in context, two assumptions needed for the number of blue erasers in a box to be well modelled by a binomial distribution.

   [2]

2. Find the probability that a randomly chosen box of erasers contain at most six blue erasers.

   [1]

$200$ randomly chosen boxes are packed into a carton. A carton is considered acceptable if at least $40$% of the boxes contain at most six blue erasers each.

3. Find the probability that a randomly chosen carton is acceptable.

   [3]

The cartons are exported by sea. Over a one-year period, there are $30$ shipments of $150$ cartons each.

4. Using a suitable approximation, find the probability that the mean number of acceptable cartons per shipment for the year is less than $80$.

The owner decided to change the proportion of blue erasers to $p$. A box of erasers is chosen.

5. Write down in terms of $p$, the probability that the box contains exactly one blue eraser.

   [1]

6. The probability that a box contains exactly one blue eraser is twice the probability that the box contains exactly two blue erasers. Write an equation in terms of $p$, and hence find the value of $p$.

   [2]

1. Given that the two events \(A\) and \(B\) are such that \(\mathrm{P}(A|B)=\frac{2}{3},\,\,\mathrm{P}(A\cap B'\,)=\frac{1}{4}\) and \(\mathrm{P}(A\cap B)=\frac{5}{12}\),
   1. determine if \(A\) and \(B\) are independent,[1]
   2. find \(\mathrm{P}(A\cup B)\).[2]
2. Ali and Bob take turns to shoot an arrow in an Archery training session. The probability that Ali hits the bull’s eye is \(0.3\) and for Bob, it is \(0.4\). Ali shoots first.
   Find the probability that Ali hits the bull’s eye first.[3]
3. Calvin, his parents and \(7\) other people are seated in a round table of \(10\). Find the probability that his parents are seated together but Calvin is not seated beside either of his parents.[2]

The curve $C$ has equation $y=\frac{a{{x}^{2}}+2}{x-1}$ where $x\ne 1$ and $a$ is a non-zero constant.

1. Show that if the curve $C$ has no stationary points, then $-2 < a < 0$.
2. Sketch the curve for $a=1$, showing clearly the asymptotes and coordinates of any intersections with the coordinate axes.
3. Verify that $y=k(x-1)+2$ passes through $(1,2)$ for all real values of $k$.
4. By considering the equation of an appropriate line drawn on the same diagram with the curve $C$, find the range of values of $k$ for which the equation ${{x}^{2}}+2=k{{(x-1)}^{2}}+2(x-1)$ has no real roots.