Name:
Student number:
Friday, 15 Jan.
1. Let√u = h1, −1,√0i and v = h−1,
√ 0, 1i. What
√ is |(u.v)u + 2v|?
a) 6
b) 14
c) 3 2
d) 2
2. What is the distance of the point (4, −1, −3) from the plane y = 1?
a) 5
b) 4
c) 3
d) 2
3. Let P : x + y = z and Q : −x + 2c(y + 1) + z = 5 be two planes. Find c for which
these two planes are orthogonal.
a) 0
b) 1
c) −1
d) 2
4. What
of the function
z =ln(4 − x2 − y 2 )?
is the domain
2
2
2
2 : x2 + y 2 ≥ 4
a) (x, y) ∈ R : x + y ≤ 4
b)
(x,
y)
∈
R
d) (x, y) ∈ R2
c) (x, y) ∈ R2 : 4 − x2 − y 2 ≥ 1
√
1
5. What is the domain of the function z = x + y + √
?
3
−
x−y
a) (x, y) ∈ R2 : 0 ≤ x + y ≤ 3
b) (x, y) ∈ R2 : x + y ≥ 0 ∪ (x, y) ∈ R2 : x + y < 3
c) (x, y) ∈ R2 : 0 ≤ x + y < 3
d) (x, y) ∈ R2 : x + y ≥ 0 and x + y 6= 3
6. Let P : 2x + my − 3z = 50 and Q : −nx + z = 2 − y be two parallel planes where
m and n are two real numbers. What is m + n?
b) − 13
c) − 37
d) 11
a) 13
3
7. What is the angle between vectors u = 2i − 3k and v = i + j + 2k?
a) cos−1 √478
b) cos−1 √−4
c) cos−1 √−1
d) cos−1 √−4
78
78
39
8. Which it the equation of a plane passing through the point P (2, 3, 0) and parallel
to the plane x + 3z − 4 = 2(y − 1)?
a) 2x−y = 1
b) x−2y+3z = 2
c) x+2y+3z = 10
d) −x+2y−3z = 4
9. What is the shape of the level curves of z = f (x, y) = −x2 + 2 + 5y?
a) lines
b) circles
c) ellipses
d) parabolas
10. Let Q be a plane that is through the point P (−1, −2, 3) and is perpendicular to
the line that connects the point P and the origin. Which point lies on Q?
a) (1, 0, −1)
b) (−4, −5, 0)
c) (−2, −4, 6)
d)(1, 2, −3)