SECTION A
1. Let z= 1 + 2i. Find the real and imaginary parts of 1
(z−2i)2. [4 marks]
2. Let z= 1 −i√3. Express zin the form reiθ. (As usual, r > 0 and θis
real.) Indicate the position of zon a diagram. Use de Moivre’s theorem to find
the real and imaginary parts of z9. [6 marks]
3. Verify that (3i−2)2=−5−12i. By means of the quadratic formula, or
completing the square, solve the quadratic equation
z2+ (8 −6i)z+ 12 −12i= 0. [4 marks]
4. Let A, B, C be three points with position vectors a,b,crespectively. Write
down the position vectors
pof Pwhich is on AB, one-fifth of the distance from Ato B;
mof Mwhich is the mid-point of CP .
Show that 4
→
MA +
→
MB + 5
→
MC is the zero vector. [4 marks]
5. Let A= (−2,0,3), B = (−2,2,1) and C= (0,4,3).
(i) Find the vectors
→
AB,
→
AC and
→
AB ×
→
AC.
Verify that your vector
→
AB ×
→
AC is perpendicular to the vectors
→
AB and
→
AC, stating your method for doing this. [4 marks]
(ii) Write down the area of the triangle ABC and find the length of the
perpendicular from Bto the side AC . (You need not evaluate any square roots
occurring.) [3 marks]
(iii) Find an equation for the plane containing the triangle ABC . [3 marks]
6. Find the values of p, q, r such that the curve y=p+qx+rx2passes through
the points (1,1),(−1,11) and (2,2). [5 marks]
7. For each set of vectors (a) and (b) decide, giving reasons, whether the
vectors are linearly independent and also whether they span R3.
(a) u= (2,−8,4),v= (−3,12,−6),
(b) u= (2,−1,5),v= (−1,6,4),w= (1,2,−3).
If the vectors in (a) or (b) are linearly dependent, find a non-trivial linear combi-
nation equalling the zero vector. [7 marks]
Paper Code MATH103 Sept-06 Page 2 of 4 CONTINUED
