VERSION 1
THE UNIVERSITY OF AUCKLAND
SUMMER 2011 TEST MATHEMATICS General Mathematics 1 (Time allowed: 60 MINUTES)
NOTE: Write all answers on the coloured sheet. Use dark ink or pencil. 1. What is the natural domain of f (x) = eln x ? (a) (, 0) (0, ) (b) (0, ) 2. Where is the function g (x) = (a) at x = 2 only (b) at x = 1 and x = 2 (c) R (d) [0, ) x2 1 undened? x2 4
MATHS 108
(c) nowhere
(d) at x = 1 only
CONTINUED
VERSION 1 2 MATHS 108
3. Suppose h(x) = 1/x is dened on its natural domain. Which one of the following is (h h)(x)? (a) x for all x (, 0) (0, ) (b) x for all x R 4. Suppose f (x) = (c) 1/x2 for all x (, 0) (0, ) (d) 1/x2 for all x R
1 is dened on its natural domain. x+1 Which one of the following is (f f )(x) on the domain R (a) (b) x x+1 (c) (d) x x+2
{1, 2}?
x+2 x+1
x+1 x+2
5. What are the solutions of sin(x) = sin(x)? (a) all x R (b) x = n for all n Z only (c) x = 0 only (d) x = n/2 for all n Z only
6. Where are the vertical asymptotes of y = tan(x)? (a) x = n/4 for all n Z (b) x = (2n + 1)/4 for all n Z 7. What is the value of lim (a) undened
x1
(c) x = n/2 for all n Z (d) x = (2n + 1)/2 for all n Z
2+
(b) 2
x2 1 ? x1
(c) 0
(d)
CONTINUED
VERSION 1 3 8. What is the horizontal asymptote of y = ex+1 ? (a) it has no horizontal asymptote (b) y = e (c) y = 1 (d) y = 0 MATHS 108
9. What values of k make the following piecewise function continuous at x = k ? f (x) = (a) k = 0 only (b) k = 1 only 10. Where is the function dened by f (x) = continuous? (a) at x = 1 only (b) it is continuous on its natural domain x2 + kx + 2 if x k kx2 + x + 2 if x > k (c) k = 0 and k = 1 (d) no values of k x2 x4 on its natural domain [1, 1] NOT (c) at x = 1 and x = 0 (d) at x = 0 only
11. Which of the vectors u = (1, 2, 2), v = (2, 2, 1) and w = (0.5, 1, 1) are parallel? (a) v and w (b) none of them (c) u and w (d) u and v
12. Which of the vectors u = (1, 2, 2), v = (2, 2, 1) and w = (0.5, 1, 1) are orthogonal? (a) u and w (b) none of them (c) v and w (d) u and v
CONTINUED
VERSION 1 4 MATHS 108
13. Suppose two vectors have v1 v2 = 0.82. Which one of the following is TRUE about the angle between the vectors? (a) 0 < <
2
(c)
<<
2
(b) < < 2
(d) =
14. Which one of the following vectors CANNOT be written as a linear combination of (1, 2, 3) and (3, 2, 1)? (a) (0, 2, 5) (b) (1, 2, 3) (c) (2, 4, 6) (d) (1, 1, 1)
15. Which one of the following represents the line through the point (1, 5, 1) parallel to the vector (2, 3, 1), where s, t R? (a) x = (1 + 2t, 5 + 3t, 1 + t) (b) x = (1, 5, 1) + t(2, 3, 1) (c) x = s(1, 5, 1) + t(2, 3, 1) (d) x = t(1, 5, 1) + (2, 3, 1)
16. Three of the following represent the same line. Which one DOES NOT? (a) x = (0, 0, 2) + t(2, 6, 2) (b) x = (1, 2, 3) + t(2, 6, 2) (c) x = (1, 2, 3) + t(1, 3, 1) (d) x = (2, 5, 4) + t(1, 3, 1)
CONTINUED
VERSION 1 5 MATHS 108
17. A line in R3 passes through the points (a, b, c) and (a1 , b1 , c1 ). A second line passes through the point (a, b, c), and also passes through (a2 , b2 , c2 ). A student writes the following to give the equation of the plane containing both lines. x = a + s(a1 a) + t(a2 a) y = b + s(b1 b) + t(b2 b) where s, t R z = c + s(c1 c) + t(c2 c) What, if anything, is wrong with this answer? (a) The equations may give a line (b) The equations must be in vector form (c) The parameters s and t are used incorrectly (d) The answer is correct 18. Where does the line x = (0, 1, 2) + t(1, 1, 0) intersect the plane x + y + 7z = 25? (a) (3, 4, 2) (b) (1, 2, 2) (c) (5, 6, 2) (d) nowhere
CONTINUED
VERSION 1 6 MATHS 108
19. Consider the line x = u + rv and the plane x = sw1 + tsw2 where r, s, t R. Condition A: u is a linear combination of w1 and w2 Condition B: v is a linear combination of w1 and w2 When does the line intersect the plane at u ONLY? (a) When neither condition holds (b) When condition A holds and B does not (c) When both conditions hold (d) When condition B holds and A does not 20. Which one of the following planes is parallel to the plane x + y + 2z = 1? (a) 2x + 2y + z = 1 (b) x = s(1, 1, 2) + t(0, 0, 1) where s, t R (c) (1, 1, 2) (x, y, z ) = 0 (d) 2x + 2y + 2z = 0
