Innova Junior College H2 Math papers, with video solutions

In established forest fires, the proportion of the total area of the forest which has been destroyed is denoted by \(x\), and the rate of change of \(x\) with respect to time, \(t\) hours, is called the destruction rate. Investigations show that the destruction rate is directly proportional to the product of the proportion destroyed and the proportion not destroyed. A particular fire is initially noticed when one half of the forest is destroyed, and it is found that the destruction rate at this time is such that, if it remained constant thereafter, the forest would be destroyed completely in a further 24 hours.

Show that \(12\frac{\mathrm{d}x}{\mathrm{d}t}=x(1-x)\) .[3]

Deduce that approximately 73% of the forest is destroyed 12 hours after it is first noticed.[6]

2. In general, the marks obtained by students in their Mathematics examination may be assumed to be a random variable with mean \(65\) and variance \(32\). After the introduction of a new program of revision classes, it is found that there are improvements to the marks of the some of the students.
   A random sample of the marks of \(50\) students is taken and the mean is found to be \(66.4\). A test is conducted and it shows that there is insufficient evidence that the new program of revision classes is effective. Find an inequality satisfied by the significance level of the test.A random sample of the marks of \(50\) students is taken and the mean is found to be \(66.4\). A test is conducted and it shows that there is insufficient evidence that the new program of revision classes is effective. Find an inequality satisfied by the significance level of the test.[3]
3. A sample of \(n\) observations is taken form a population with mean \(5\). The sample mean is found to be \(5.1\) and sample standard deviation \(0.7\). A 2-tailed normal test (\(z\)-test) is conducted and the null hypothesis is rejected at the \(5\%\) significance level. Find the least value of \(n\).