Further Mathematics (XFM01) 考試技巧

計算機程式

常見錯誤

1. 1high涉及分數: 4Transformations using matrices

   Reversing matrix multiplication order in equations of the type XA = B, incorrectly solving as X = A^-1 B instead of X = B A^-1.

   如何避免: Always post-multiply both sides of the equation by A^-1 to preserve order, as matrix multiplication is non-commutative.
2. 2high涉及分數: 1Proof

   Skipping crucial inductive wrapper steps ('if true for n=k then true for n=k+1...') which are strictly required for the final CSO mark.

   如何避免: Write down the full structural conclusion verbatim: 'If true for n=k, then true for n=k+1. As it is true for n=1, it is true for all positive integers n by induction.'
3. 3medium涉及分數: 2Numerical solution of equations

   Differentiating fractional negative indices incorrectly during the Newton-Raphson process (e.g., differentiating -x^-2 incorrectly or missing signs).

   如何避免: Write down fractional steps to convert roots to negative powers (e.g., 7/\sqrt{x} = 7x^{-0.5}) before applying the power rule: d/dx(x^n) = n*x^{n-1}.
4. 4high涉及分數: 2Complex numbers

   Sign errors during algebraic expansion of complex conjugates (such as (z - 2i)(z* - 2i)), particularly processing the imaginary component multiplied by itself.

   如何避免: Work slowly and expand systematically: (-2i)*(-2i) = 4i^2 = -4. Do not skip writing out the middle terms.
5. 5high涉及分數: 2Series

   Incorrect summation limits subtraction when evaluating a series, e.g., subtracting f(20) instead of f(19) when summing from r=20 to r=40.

   如何避免: Use the limit rule: \sum_{r=A}^{B} g(r) = \sum_{r=1}^{B} g(r) - \sum_{r=1}^{A-1} g(r).
6. 6medium涉及分數: 1Coordinate systems

   Failing to simplify surds in the final coordinate geometry or hyperbola question (such as leaving a distance as \sqrt{1640/9} instead of simplifying it).

   如何避免: Identify perfect square factors within your surds systematically and pull them out (e.g., \sqrt{1640}/3 = 2\sqrt{410}/3).