AQA Maths: What "Fully Justify Your Answer" Actually Means

If you're sitting AQA A Level Maths, you'll see the words "fully justify your answer" many times. The phrase tells you that some of the marks are for explaining rather than calculating, and in my experience it's a common way for students to drop marks on a question they've otherwise got right.

It's AQA's own phrase. The other boards have their own versions, so if a friend on a different board hasn't come across it, that's why.

What the phrase is telling you

AQA publishes a list explaining its command words. For this one it says:

Any reasoning to reach the required conclusion should be explicitly stated in the solution.

The two words that matter there are "explicitly" and "reasoning".

Explicitly means written down. It isn't enough for it to be implied by your working or obvious from the numbers, or for the examiner to be able to assume you knew it.

Reasoning means the why, not the what. You've usually done the calculation already. This asks you to say what the calculation proves.

What it looks like in a real question

Here's a typical example. A question gives you a model for the velocity of a runner:

\[ v = 11.71 - 11.68\mathrm{e}^{-0.9t} - 0.03\mathrm{e}^{0.3t} \]

and asks you to find the maximum velocity, fully justifying your answer. It's worth 8 marks.

You'd differentiate, set the derivative equal to zero, solve for \( t \) and substitute back to get the maximum velocity. That's most of the question.

But two of the eight marks aren't for any of that. They're for stating two things:

• That a maximum occurs where \( \dfrac{\mathrm{d}v}{\mathrm{d}t} = 0 \). You need to state the principle, not just use it.
• That the value you found is a maximum, rather than a minimum or a point of inflection.

So a student who differentiates, solves correctly and gets the right velocity to one decimal place, but doesn't state either of those, loses a quarter of the marks on the question.

Maximum and minimum questions

This is where the phrase comes up most often, so it's worth knowing what AQA will accept.

To show that your stationary point is a maximum, any one of these four is enough:

• Use the second derivative. Work out \( \dfrac{\mathrm{d}^2v}{\mathrm{d}t^2} \) at your value and show it's negative. This is the one most students are taught.
• Check the gradient either side. Show the derivative is positive just before your point and negative just after.
• Sketch the curve with the maximum clearly marked.
• Argue it's the only one. If there's only one stationary point and the context makes it clear it's a maximum, say so.

I'd suggest using whichever is quickest for the function in front of you. If the second derivative is awkward to work out, testing either side is often faster, and it scores the same.

Three other AQA words worth knowing

Hence

AQA says that hence means use the result from the previous part, and that using another method "will typically lead to fewer or no marks being given".

So if part (a) asks you to show that \( 3x^2 - 12x + 9 = 3(x-1)(x-3) \) and part (b) starts with "hence", part (b) wants you to use those factors. Starting again with the quadratic formula may get you the same answer but few of the marks.

Show that

You're given the answer, so the marks are for how you get there. AQA asks for "every step of a process that will lead to the required outcome", plus a closing statement saying what you've shown.

A common mistake is working backwards from the printed answer. Start from what you were given and work towards it.

Exact

Leave it as a fraction, a surd, or in terms of \( \ln \) or \( \mathrm{e} \). A rounded decimal won't get full marks, however many figures you give. If a question asks for the exact solution of \( \ln x = 2 \), the answer is \( \mathrm{e}^2 \), not \( 7.389 \).

A checklist for the exam

When you see "fully justify your answer", it's worth checking these before you move on:

• Have I said why my method finds what the question asked for? For a maximum, that means stating that it occurs where the derivative is zero.
• Have I shown it's the right kind of point? Second derivative, gradient either side, or a sketch. Any one is enough.
• Have I written a sentence, not just a number? The mark is for the conclusion in words.
• Have I answered in context? If the question is about a runner, the maximum velocity is a speed, not just a value of \( v \).

None of this needs extra revision. It usually takes about thirty seconds once you've done the hard part of the question.