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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)