Question 1. Binomial
Everyone was basically good on this. A lot of students lost a mark unnecessarily in part (c) by giving a correct approximation to √23/5, rather than an approximation to √23, which the question asked for.
Question 2. Binomial
Most students got this completely right. A significant minority messed up the first step, of writing (9+4x2)^(−1/2) as 9^(−1/2).(1+4x2/9)^(−1/2).
Question 3. Rates of flow and chain rule
Most of you did this efficiently and accurately. But almost everyone who calculated dV/dh by using the product rule, rather than multiplying out the formula for V as
V = (1/4)πh2 − (1/3)πh3
got themselves entangled in their working and made a mistake. The simpler, quicker way is also the more accurate.
Question 4. Differentiating with parameters
Almost everyone was good with finding dy/dx = (dy/dt)/(dx/dt), and then finding the tangent.
More than half the class, though, failed on finding the cartesian equation (the equation with just x and y, no t).
Most of you seemed to have the correct idea that you must use cos2 + sin2 = 1 to get rid of t, but failed to carry it through.
But x = tan2 t
so x = (sin2 t)/(cos2 t)
and so x = y2/ (1−y2)
Then just rearrange to make this a y2 =…. equation.
Question 5. Implicit differentiation
Almost everyone was good on the basic implicit differentiation (part a). Quite a few had difficulties with part b, finding the point where the tangent is parallel to the y-axis.
Some wrote that at that point dy/dx = 0. But that would be parallel to the x-axis. Parallel to the y-axis means dx/dy = 0, or in this case y=−2x
Oddly, many then substituted −y/2 for x in the starting equation, rather than the simpler and more direct substitution of −2x for y.
Question 6. Vectors
Most people were good with most of this. Only a few, however, used the easy method for part (f), namely drawing the right-angled triangle AXY and using your knowledge of AX and cos(angle XAY) to find AY = AX / cos(angle).
Of those who tried more complicated methods, two especially able students were able to get the right answer (with ten times as much work as they needed to do), but everyone else got lost.
Extra questions on integration.
The first question was deliberately chosen to be as hard as they come, so the fact that on the whole most people did as well on the integration questions as on the “official” part of the paper should give us some confidence.
Most students have a lot more practice to do with integration – it takes a lot of practice – but most have the right basic ideas.
There were still a few attempts to invent a “product rule” for integration. Remember, for integration there is no chain rule, no product rule, no quotient rule. Think of a rule, and almost certainly it doesn’t work for integration. It is the Wild West of maths.
In the partial-fractions bit of question 2, several students chose to go with comparing coefficients rather than substituting values of x to find A, B, C. In this case comparing coefficients is much more complicated, and almost everyone who went that way ended up making slips.