Question 1. Trig – R cos (x+α) question
Most of you were good with this. Two students had apparently completely forgotten the method. A few lost marks through rounding errors with the answer. If your final answer has to be correct to 2 d.p., then you need to do all your calculations leading up to that to at least 3 d.p.
Question 2. Trig – identities and turning an equation in tan and sec into a quadratic in sec
Most of you were good with this. Three students had apparently completely forgotten the whole idea of converting trig equations into quadratics to solve them.
Question 3. Trig – R cos (x+α) question
Most of you were good with this. A couple of students made a sign error here which they had not made in the similar Q.1.
cos (x+α) = cos x cos α − sin x sin α (minus sign on right hand side)
Question 4. Partial fractions
Most of you were good with this. Lost marks were usually due to slips like sign errors, or in one dramatic case to reading the question wrong. One student who usually gets high grades and can’t have found this question difficult mysteriously didn’t attempt it at all.
No-one saw the easier way to do part (b)
f(x) = (1−x)/(x−3) = −1−2/(x−3), from which you can differentiate in one line without using the quotient rule to get f'(x) = 2/(x-3)2.
But then whoever wrote the mark scheme missed that easier way, too.
Question 5. Exponentials and logs
Almost everyone was good with part (a) and (b), but a sizeable minority got confused with part (c).
Because of the confusion, students wrote things like
(d/dt) exp (−kt) = −kt exp (−kt)
(d/dt) exp (−kt) = −k exp (−kt)
or replaced (ln 3)/4 midway in their working by ln (3/4), which is not the same thing
or wrote exp [−(1/4)(ln 3)t] = exp[−(1/4)].exp(ln 3).exp t
when really exp(A).exp(B) = exp(A+B)
Question 6. Trig (double angle formulas)
Some got into trouble with part (a). You had to write the double-angle formulas
1 − cos 2θ = 2 (sin θ)2
sin 2θ = 2 sin θ cos θ
Some blanked on (b) (ii)
cosec 4x − cot 4x = 1
is the same as (1/ sin 2θ) − (cos 2θ)/(sin 2θ) = 1
with 2x in place of θ.
Question 7. Partial fractions and range
Almost everyone was good on the partial-fractions calculation (part a), and the differentiation (part b).
Part (c), finding the range, was very simple. You just had to calculate when h'(x) =0 and look at the graph. Look at the graph! Look at the graph! Plug in x=0 for the bottom of the range, and the value of x which gave h'(x) =0 for the top of the range.
Some just made a slight error: < in the range where they should have had ≤. Some got it completely wrong.
I think almost everyone should work through the little workbook I did on domain and range, and we should spend some class time on this.
Question 8. Exponential function
Almost everyone was good on parts (a) and (b).
Some students who did very well on the paper overall got part (c) wrong. They wrote either 10 or 20 for the starting amount of the drug in the bloodstream, where part (a) had told them it must be 15.353. Or they got themselves into much more complicated calculations than the simple equation needed:
15.353 × exp(−T/8) = 3
One of those who got into those complicated calculations worked it through to a correct conclusion after about ten times as much working as was necessary. Others got lost and wrote things like
ln [exp(−(T+5)/8) + exp(−T/8)] = −(T+5)/8 − T/8
ln A + ln B is ln AB, not ln (A+B)!
Some of this can be put down to it being the last question (and the lesson is, train yourself to keep focused right to the end of the 90 minutes). Some of it is down to not developing the skill of looking at answers and thinking: does this look right?
For example, one student who lost no marks on any other question in the paper got the answer 9.63. But that’s not likely. You know from part (a) that the amount in the bloodstream roughly halves after 5 hours. So after 9.63 hours – about 10 hours – roughly a quarter of the first dose, and roughly half of the second dose will be left in circulation. A lot more than 3 mg.