MATH1041– ??
(9 pages—page 1)
Section A - Multiple choice questions.
Circle the letter corresponding to the correct answer.
There is only one correct answer for each question and there is no negative marking.
Question 1
Given that the vectors a, b and c are 3-dimensional, which one of the following is a real number?
(A) a × b × c
(B) a × b + c
(C) a · b × c
(D) b × c + c
(E) None of the above.
[3 marks]
Question 2
Let z = x + iy be a complex number with x, y ∈ R. Which of the following is true?
(A) Re(i z) = −Re(z).
(B) Re(i z) = −Im(z).
(C) Im(i z) = −Re(z).
(D) Im(i z) =Im(z).
(E) Im(i z) = −Im(z).
[3 marks]
MATH1041– Mathematics I (Auxiliary)
ALGEBRA
November 2018
(9 pages—page 2)
Question 3
Which one of the following operations is the real-imaginary form of the complex number
2 − 3i
?
−i
(A) 3 − 2i.
(B) 2 + 3i.
(C) 3 + 2i.
(D) 2 − 3i.
(E) −2 + 3i.
[3 marks]
Section B - Show all necessary working.
Question 4
Let a = (2, 1, 3) and b = (−1, 1, 2).
(a) Find c = a × b
(4)
(b) Is c orthogonal to a? Give a reason for your answer.
(2)
[6 marks]
MATH1041– Mathematics I (Auxiliary)
ALGEBRA
November 2018
(9 pages—page 3)
Question 5
Let P1 be the plane with equation 2x − y + 3z = 2.
(a) Write down the normal of the plane P1 .
(1)
(b) Find the equation of the plane through point (4, 7, 1) that is parallel to the plane P1 .
(3)
(c) Is the line with direction vector d = (2, 1, −1) parallel to the plane P1 ? Give a reason for
your answer.
(4)
(d) Does the line r = (2, 3, 1) + λ(2, 1, −1) lie on the plane P1 ? Give a reason for your answer.
(2)
[10 marks]
MATH1041– Mathematics I (Auxiliary)
ALGEBRA
November 2018
(9 pages—page 4)
Question 6
Given the points A(0, −1, 5), B(1, 0, 0) and C(2, 1, 1).
(a) Write down the vectors BA and BC.
(2)
(b) Find |BC|.
(2)
(c) Find the component of BA in the direction of BC.
(4)
π
(d) State one reason why the angle between vectors AC and BC is .
2
(1)
[9 marks]
MATH1041– Mathematics I (Auxiliary)
ALGEBRA
November 2018
(9 pages—page 5)
Question 7
Consider the sphere r − (1, 2, 5) = 5.
(a) Find the centre and radius of the sphere.
(2)
(b) Write the Cartesian equation of the given sphere.
(2)
(c) Show that the point (4, 2, 1) is on the sphere?
(2)
(d) Which of the following 2 planes is tangent to the given sphere at the point (4, 2, 1)?
P1 : (3, 0, −4) · (x, y, z) = (3, 0, −4) · (4, 2, 1)
P2 : (3, 0, −4) · (x, y, z) = (3, 0, −4) · (1, 2, 5)
(1)
[7 marks]
MATH1041– Mathematics I (Auxiliary)
ALGEBRA
November 2018
(9 pages—page 6)
Question 8
Consider the following equation:
2z − z = −1 + i
By first conjugating the given equation solve for z using Cramer’s rule.
[6 marks]
MATH1041– Mathematics I (Auxiliary)
ALGEBRA
November 2018
(9 pages—page 7)
Question 9
Find the complex number z = x + iy such that 5z2 + 8z + 5 = 0.
[4 marks]
MATH1041– Mathematics I (Auxiliary)
ALGEBRA
November 2018
(9 pages—page 8)
Question 10
√
Let z = 2 3 − 2i.
(a) In which quadrant is z?
(1)
π
π (b) Show that the mod-arg form of z is z = 4 cos
− i sin
.
6
6
(3)
(c) Find the fourth roots of z using the Euler formula eiθ = cos θ + i sin θ.
(6)
MATH1041– Mathematics I (Auxiliary)
ALGEBRA
(d) Plot the 4 fourth roots of z on the complex plane.
November 2018
(9 pages—page 9)
(4)
[14 marks]
[Total:60 (+5 bonus) marks]