Vectors, matrices and complex numbers

Written by Hassan Raza (Mathematics)

NET · Mathematics · 8 questions

Name: ____________________ Date: ____________

What is the eccentricity of the ellipse x² / 25 + y² / 16 = 1?
- A 4/5
- B 9/25
- C 3/5
- D 5/4
What is the angle between the vectors (1, 0) and (1, 1)?
- A 90 degrees
- B 60 degrees
- C 45 degrees
- D 30 degrees
What is the magnitude of the cross product of a = (2, 0, 0) and b = (0, 3, 0)?
- A 1
- B 5
- C 0
- D 6
What are the eigenvalues of the matrix [[2, 0], [0, 3]]?
- A 2 and -3
- B 2 and 3
- C 1 and 6
- D 0 and 5
If A is a 2 x 2 matrix with det A = 3, what is det(2A)?
- A 9
- B 12
- C 6
- D 24
What is (1 + i) / (1 - i)?
- A i
- B -i
- C -1
- D 1
What is the argument of the complex number -1?
- A pi
- B -pi / 2
- C 0
- D pi / 2
What is the sum of the three cube roots of unity?
- A -1
- B 1
- C 0
- D 3

Answer key

1. C, 2. C, 3. D, 4. B, 5. B, 6. A, 7. A, 8. C

Here a² = 25 and b² = 16. e = sqrt(1 - b² / a²) = sqrt(9/25) = 3/5.
cos(theta) = (1 x 1 + 0 x 1) / (1 x sqrt(2)) = 1 / sqrt(2), so theta = 45 degrees.
The vectors are perpendicular, so |a x b| = |a||b| = 2 x 3 = 6.
The eigenvalues of a diagonal matrix are its diagonal entries.
For a 2 x 2 matrix, det(kA) = k² det(A) = 4 x 3 = 12.
Multiply top and bottom by (1 + i): (1 + i)² / 2 = 2i / 2 = i.
The number -1 lies on the negative real axis, at an angle of pi from the positive real axis.
The roots of z³ = 1 sum to the negative of the coefficient of z², which is 0.