Further Mathematics

1. 1.

   Solve z^2 - 4z + 13 = 0 over the complex numbers. Find the modulus of each root.

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. A negative discriminant gives complex roots. Calculate each modulus as its distance from the origin in the complex plane.
   2. The discriminant is 16 - 52 = -36.
   3. z = 2 + 3i or z = 2 - 3i.
   4. Both roots have modulus sqrt(13).

   Marking points

   • The discriminant is 16 - 52 = -36.
   • z = 2 + 3i or z = 2 - 3i.
   • Both roots have modulus sqrt(13).

   Examiner tip: The modulus is sqrt(real^2 + imaginary^2), not the sum of the two parts.

2. 2.

   For A = [[2, 1], [1, 2]], find both eigenvalues and give one eigenvector for each.

   [4 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Find the values that make A - lambda I singular; then solve the homogeneous equations for each value.
   2. det(A - lambda I) = (2 - lambda)^2 - 1.
   3. The eigenvalues are 3 and 1.
   4. For 3, an eigenvector is (1, 1).
   5. For 1, an eigenvector is (1, -1).

   Marking points

   • det(A - lambda I) = (2 - lambda)^2 - 1.
   • The eigenvalues are 3 and 1.
   • For 3, an eigenvector is (1, 1).
   • For 1, an eigenvector is (1, -1).

   Examiner tip: An eigenvector must be nonzero; any nonzero scalar multiple of a valid vector is acceptable.

3. 3.

   Use the Maclaurin series for e^x to write the expansion of e^(2x) through the x^3 term.

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Substitute 2x into every power of the exponential series, not just the linear term.
   2. e^x = 1 + x + x^2/2! + x^3/3! + ...
   3. Substitute 2x: 1 + 2x + (2x)^2/2 + (2x)^3/6.
   4. The expansion is 1 + 2x + 2x^2 + (4/3)x^3 + ...

   Marking points

   • e^x = 1 + x + x^2/2! + x^3/3! + ...
   • Substitute 2x: 1 + 2x + (2x)^2/2 + (2x)^3/6.
   • The expansion is 1 + 2x + 2x^2 + (4/3)x^3 + ...

   Examiner tip: Keep the factorial denominators when substituting and simplify afterwards.

Original, unofficial practice questions from SubjectScout — not official past-paper questions.