Further Mathematics

1. 1.

   Express (3 + 4i)/(1 - 2i) as a + bi and find its modulus.

   [3 marks] · no calculator

   Answer explanation

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

   1. The real denominator is 1^2 + 2^2 = 5; the numerator expands to -5 + 10i.
   2. The distance of (-1, 2) from the origin is sqrt(1 + 4).

   Marking points

   • Multiply numerator and denominator by 1 + 2i.
   • The quotient is -1 + 2i.
   • Its modulus is sqrt(5).

   Examiner tip: The conjugate of the denominator has +2i, not -2i.

2. 2.

   For z = x + iy, identify and sketch the locus |z - (2 + i)| = 3, labelling its real-axis intersections.

   [3 marks] · no calculator

   Answer explanation

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

   1. Subtracting 2 + i measures distance from a fixed point, not from the origin.
   2. Put y = 0 for the real axis; (x - 2)^2 = 8 gives both crossings.

   Marking points

   • A circle with centre (2, 1) and radius 3.
   • Equation: (x - 2)^2 + (y - 1)^2 = 9.
   • Real-axis intersections are (2 - sqrt(8), 0) and (2 + sqrt(8), 0), labelled on a circle sketch.

   Examiner tip: The centre is above the real axis, so the intercepts are not 2 +/- 3.

3. 3.

   Find all cube roots of -8i in modulus-argument form with arguments in (-pi, pi].

   [3 marks] · no calculator

   Answer explanation

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

   1. Take k = 0, 1, 2 before reducing arguments; this prevents loss of roots.
   2. The last angle 7pi/6 is equivalent to -5pi/6; cubing each result gives -8i.

   Marking points

   • Use modulus 8 and argument -pi/2 + 2k*pi.
   • Root modulus is 2 and arguments are -pi/6 + 2k*pi/3.
   • Roots: 2 cis(-pi/6), 2 cis(pi/2), 2 cis(-5pi/6), where cis(theta) = cos(theta) + i sin(theta).

   Examiner tip: Divide the complete argument, including 2k*pi, by three.

4. 4.

   A real-coefficient cubic x^3 + ax^2 + bx - 10 has root 1 + 2i. Find a, b and the remaining roots.

   [3 marks] · no calculator

   Answer explanation

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

   1. Real coefficients force non-real roots to occur in conjugate pairs.
   2. Their product is 5. The constant term is minus the product of all three roots, so 5r = 10.

   Marking points

   • The conjugate root is 1 - 2i; their factor is x^2 - 2x + 5.
   • The third root is 2, since the product of roots is 10.
   • Expansion (x - 2)(x^2 - 2x + 5) gives a = -4, b = 9.

   Examiner tip: For a monic cubic, the root product is the negative of its constant term.

5. 5.

   Find the points satisfying both |z - 1| = |z + i| and |z| = sqrt(2). Give exact answers and explain the geometry.

   [3 marks] · no calculator

   Answer explanation

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

   1. Squaring is safe because both moduli are non-negative; cancellation leaves a straight line.
   2. Intersect that line with the origin-centred circle. Both intersections are equally distant from the two fixed points.

   Marking points

   • Equality of squared distances gives (x - 1)^2 + y^2 = x^2 + (y + 1)^2.
   • This reduces to y = -x, the perpendicular bisector of (1, 0) and (0, -1).
   • With x^2 + y^2 = 2, x = +/-1; z = 1 - i or -1 + i.

   Examiner tip: z + i measures distance from -i, not from i.

6. 6.

   Use De Moivre's theorem to derive cos(3theta) = 4cos^3(theta) - 3cos(theta). Then solve 4u^3 - 3u = 0 for real u.

   [3 marks] · no calculator

   Answer explanation

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

   1. In the expansion the terms containing i and i^3 are imaginary; the i^2 term is negative and real.
   2. Once the identity is established, factor the cubic rather than divide by u: dividing would lose u = 0.

   Marking points

   • Take real parts of (cos(theta) + i sin(theta))^3 to get cos^3(theta) - 3cos(theta)sin^2(theta).
   • Substitute sin^2(theta) = 1 - cos^2(theta) to obtain the identity.
   • u(4u^2 - 3) = 0 gives u = 0, sqrt(3)/2, -sqrt(3)/2.

   Examiner tip: An identity derivation must start from De Moivre, not assume the triple-angle formula.

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