Q: Failing to divide by $ \frac{dx}{dt} $ when calculating the second parametric derivative $ \frac{d^2y}{dx^2} $.

Consistently write the formula $ \frac{d^2y}{dx^2} = \frac{d}{dt}\left(\frac{dy}{dx}\right) / \frac{dx}{dt} $ as your first line of working for parametric second derivatives.

Q: Omitting the base case check or writing an incomplete concluding statement in mathematical induction proofs.

Explicitly show the calculation for $ n = 1 $ (or the first positive integer), write 'assume true for $ n = k $', and conclude with a formal paragraph summarizing the inductive principle.

Q: Applying boundary values or dividing by the integrating factor before adding the arbitrary constant of integration $ C $ in differential equations.

Add $ + C $ immediately at the integration step before you rearrange or attempt to solve for the constant with initial conditions.

Question 1

Forgetting the factor of 1/2 in the polar curve area integral formula $ \frac{1}{2} \int r^2 d\theta $.

Accepted Answer

Write the general formula down with the 1/2 factor explicitly written outside the integral sign before substituting your r-expression.

Question 2

Failing to divide by $ \frac{dx}{dt} $ when calculating the second parametric derivative $ \frac{d^2y}{dx^2} $.

Accepted Answer

Consistently write the formula $ \frac{d^2y}{dx^2} = \frac{d}{dt}\left(\frac{dy}{dx}\right) / \frac{dx}{dt} $ as your first line of working for parametric second derivatives.

Question 3

Omitting the base case check or writing an incomplete concluding statement in mathematical induction proofs.

Accepted Answer

Explicitly show the calculation for $ n = 1 $ (or the first positive integer), write 'assume true for $ n = k $', and conclude with a formal paragraph summarizing the inductive principle.

Question 4

Multiplying transformation matrices in the incorrect order when finding a composite transformation matrix.

Accepted Answer

Remember that consecutive transformations are applied right-to-left. For transformation A followed by B, the composite matrix is BA.

Question 5

Applying boundary values or dividing by the integrating factor before adding the arbitrary constant of integration $ C $ in differential equations.

Accepted Answer

Add $ + C $ immediately at the integration step before you rearrange or attempt to solve for the constant with initial conditions.

Question 6

Omitting the 'r =' prefix when writing vector equations of lines.

Accepted Answer

Always write vector equations of lines in the complete format, beginning with $ \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} $. Giving only the right-hand expression loses direct marks.

試卷	時間	分數	題數	比重	題型
Paper 1: Further Pure Mathematics 1	2小時	75	7	50%	結構題
Paper 2: Further Pure Mathematics 2	2小時	75	8	50%	結構題

Mathematics - Further (9231) 考試技巧

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Table mode for roots & turning points

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Equation solver — to CHECK your working

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