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SECTION A

1. Let z = 5− 2i. Find the real and imaginary parts of 1 (z − 4)2

. [4 marks]

2. Let z = − √ 3 − i. Express z in the form reiθ. (As usual, r > 0 and θ is

real.) Indicate the position of z on a diagram. Use de Moivre’s theorem to find the real and imaginary parts of z9. [6 marks]

3. Verify that (2 + i)2 = 3 + 4i. By means of the quadratic formula, or completing the square, solve the quadratic equation

z2 + (7i− 2)z − 8i− 12 = 0. [4 marks]

4. Let A,B,C be three points with position vectors a,b, c respectively. Write down the position vectors m of M which is the mid-point of AB; p of P which is on CM , one-quater of the distance from C to M .

Show that → PA +

→ PB +6

→ PC is the zero vector. [4 marks]

5. Let A = (−1, 3, 0), B = (−1, 1, 2) and C = (1, 2, 5).

(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 B to 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+rx2 passes through the points (1, 2), (−1,−12) and (2, 3). [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 = (−3, 9, 6), v = (−2, 6,−4),

(b) u = (−4, 9, 3), v = (−1, 3, 4), w = (2,−3, 5).

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 Jan-06 Page 2 of 4 CONTINUED

8. Find the determinants of the matrices A and B:

A =

−3 2 0−7 5 1 1 −4 −8

, B = 3 −7 80 4 −5

0 0 −2

. Use the rules for determinants, which should be clearly stated, to write down

the determinants of A−3B and B − 4I, where I is the 3× 3 identity matrix. [6 marks]

9. (i) Find the eigenvalues of the matrix A =

( 6 −2

−2 9

) . [2 marks]

(ii) For each eigenvalue, find an eigenvector of length 1. (You need not evaluate any square roots which arise.) [5 marks]

(iii) Write down an orthogonal matrix P and a diagonal matrix D such that P>AP = D. [2 marks]

SECTION B

10. Express the complex number a = 8i in the form |a|eiα. Find all the solutions of the equation z6 = a in the form z = reiθ and indicate their positions on a diagram. Express also two of the solutions in cartesian form z = x+ iy with no trigonometric functions involved.

[15 marks]

11. Let

A =

1 −3 10 1 α+ 2 3 α− 7 4

. (i) Show that A is invertible if and only if α 6= −1 and α 6= −3. [5 marks] (ii) Find the inverse of A when α = 0. [6 marks]

(iii) Find a condition which a, b and c must satisfy for the system of equa- tions

x − 3y + z = a y + z = b

3x − 8y + 4z = c

to be consistent. [4 marks]

Paper Code MATH103 Jan-06 Page 3 of 4 CONTINUED

12. Let L denote the line of intersection of the planes in R3 with equations

x− 2y + 2z = −1 and 3x− 7y + 4z = 5.

Let L′ denote the line joining the points A = (2,−1,−1) and B = (4, 1, 3). (i) Find in parametric form an expression for the general point of L.

[4 marks]

(ii) Write down the vector → AB and an expression for the general point of

L′. [3 marks]

(iii) Determine the point at which L′ meets the plane

x− y + z = 10.

[3 marks]

(iv) Show that L meets L′ and find the point of intersection. [5 marks]

13. Vectors v1,v2,v3,v4 in R 4 are defined by

v1 = (1,−2, 0, 8), v2 = (2, 1, 15, 5), v3 = (−2, 3,−3,−6), v4 = (1, 0, 6, 2).

(i) Show that v1,v2,v3,v4 are linearly dependent. [6 marks]

(ii) Let S be the span of v1,v2,v3,v4. Find linearly independent vectors with the same span S. Extend these linearly independent vectors to a basis of R4. [5 marks]

(iii) Decide whether the vector (−4, 0,−24, 3) lies in S. [4 marks]

14. Find the eigenvalues and eigenvectors of the matrix

A =

−2 −5 31 5 −4 1 7 −6

Hence write down a matrix C and a diagonal matrixD such that C−1AC = D.

[15 marks]

Paper Code MATH103 Jan-06 Page 4 of 4 END