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The math103 january 2010 exam, which covers various topics in mathematics including complex numbers, vectors, matrices, and systems of linear equations. The exam is divided into two sections, with section a carrying 55% of the available marks and requiring the student to answer all questions, while section b requires the student to answer three questions out of six. The exam includes a mix of theoretical and computational problems.
Typology: Exams
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Examiner: Prof. V.V.Goryunov, Extension 44041.
Time allowed: Two and a half hours
Answer all of Section A and THREE questions from Section B. The marks shown against questions, or parts of questions, indicate their relative weight. Section A carries 55% of the available marks.
2 z + 5i 3 i − z
. [4 marks]
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 z^9. [6 marks]
m of M which is the mid-point of AC; p of P which is on BM, twelve-seventeenth of the distance from B to M.
Show that 6
→ AP + 5
→ BP + 6
→ CP is the zero vector. [4 marks]
→ KL,
→ KM and
→ KL ×
→ KM.
Verify that your vector
→ KL ×
→ KM is perpendicular to the vectors
→ KL and → KM , stating your method for doing this. [4 marks]
(ii) Write down the area of the triangle KLM and find the length of the perpendicular from K to the side LM. (You need not evaluate any square roots occurring.) [3 marks]
(iii) Find an equation for the plane containing the triangle KLM. [3 marks]
[8 marks]
A =
1 − α 3 − 2 1 − 2 α − 2 2 − 1 0
.
(i) Show that A is invertible if and only if α 6 = 4 and α 6 = 5. [5 marks] (ii) Find the inverse of A when α = 3. [6 marks] (iii) Find a condition which a, b and c must satisfy for the system of equa- tions − 4 x + 3 y − 2 z = a x − 2 y + 3 z = b 2 x − y = c
to be consistent. [4 marks]
− 4 x + y + 2z = 5 and 7 x − 2 y − 3 z = − 14.
Let L′^ denote the line joining the points A = (1, 4 , −5) and B = (− 1 , − 1 , −7).
(i) Find in parametric form an expression for the general point of L. [3 marks]
(ii) Write down the vector
→ AB and an expression for the general point of L′. [2 marks]
(iii) Determine the point P at which L′^ meets the plane
3 x − y + 4z = 6.
[3 marks] (iv) Find the distance from the point P to the line L. [3 marks] (v) Find the distance between the lines L and L′. [4 marks]
v 1 = (− 3 , 10 , 6 , −14), v 2 = (4, 8 , − 8 , 8), v 3 = (− 1 , 4 , 2 , −5), v 4 = (2, 0 , − 4 , 6).
(i) Show that v 1 , v 2 , v 3 , v 4 are linearly dependent. [4 marks] (ii) Let S be the span of v 1 , v 2 , v 3 , v 4. Find linearly independent vectors with the same span S. Extend these linearly independent vectors to a basis of R^4. [4 marks]
(iii) Decide whether the vector u 1 = (2, 1 , − 4 , −5) is in S. [2 marks] (iv) Let T be the span of u 1 and u 2 = (0, 2 , 1 , 0). What is the sum S + T? What is the intersection S ∩ T? [4 marks]
.
Hint: one of the eigenvalues is 5. [5 marks]
(ii) For each eigenvalue, find an eigenvector of length 1. [8 marks] (iii) Hence write down an orthogonal matrix P and a diagonal matrix D such that P ⊤A P = D. [2 marks]
Paper Code MATH103 January 2010 Page 5 of 5 END
