Title: InkMaths / Further Maths: definitions and slips
On every note of this deck, unless a note says otherwise:
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Front: CHECK YOURSELF
Evaluate (1 + i)⁴ without expanding four brackets.
Back: (1 + i)² = 2i, and (2i)² = −4. Two squarings beat a binomial expansion; De Moivre gives the same via modulus (√2)⁴ = 4 and argument 4 × π/4 = π.

Front: SPOT THE ERROR
Say what is wrong with this claim:
Differentiating cosh introduces a minus sign, just as cos does.
Back: cosh' = sinh and sinh' = cosh, both signs positive: the minus in sinh's definition supplies internally what trig needs a sign flip for. No minus appears anywhere in the pair.

Front: SPOT THE ERROR
Say what is wrong with this claim:
Once the inductive step is proved, the statement holds for all n.
Back: The step only passes truth along; without a verified base case there is no truth to pass. False formulae can survive the step and die only at n = 1, which is why both halves earn marks.

Front: DEFINE
Define: improper integral.
Back: An integral with an infinite limit or an integrand unbounded at a limit, defined as the limit of an ordinary integral, and said to converge when that limit is finite.

Front: DEFINE
Define: invariant line.
Back: A line whose every point maps to a point of the same line under the transformation. Points may move along it; none leaves it.

Front: CHECK YOURSELF
Find the mean value of f(x) = x³ on [0, 2].
Back: ∫x³ dx from 0 to 2 is 4; dividing by the width 2 gives mean 2. Integral over width, always.

Front: CHECK YOURSELF
Find the distance from the origin to the plane 2x + 2y + z = 9.
Back: |0 + 0 + 0 − 9|/√(4 + 4 + 1) = 9/3 = 3: substitute the point, subtract d, divide by the normal's length.

Front: DEFINE
Define: principal argument.
Back: The angle a complex number's position vector makes with the positive real axis, measured anticlockwise, in the interval (−π, π].

Front: SPOT THE ERROR
Say what is wrong with this claim:
Polar area is the integral of r dθ, by analogy with area under a graph.
Back: Polar regions are swept by sectors, not strips, and a sector's area is ½r² dθ. The r comes in squared, with a half in front.