VERSION 1

THE UNIVERSITY OF AUCKLAND MATHS 108

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,∞)

(c) R

(d) [0,∞)

2. Where is the function g(x) =x2 − 1

x2 − 4undefined?

(a) at x = ±2 only

(b) at x = ±1 and x = ±2

(c) nowhere

(d) at x = ±1 only

CONTINUED

VERSION 1

– 2 – MATHS 108

3. Suppose h(x) = 1/x is defined 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

(c) 1/x2 for all x ∈ (−∞, 0) ∪ (0,∞)

(d) 1/x2 for all x ∈ R

4. Suppose f(x) =1

x + 1is defined on its natural domain.

Which one of the following is (f ◦ f)(x) on the domain R r {−1,−2}?

(a)x

x + 1

(b)x + 2

x + 1

(c)x

x + 2

(d)x + 1

x + 2

5. What are the solutions of sin(x) = sin(−x)?

(a) all x ∈ R

(b) x = nπ for all n ∈ Z only

(c) x = 0 only

(d) x = nπ/2 for all n ∈ Z only

6. Where are the vertical asymptotes of y = tan(x)?

(a) x = nπ/4 for all n ∈ Z

(b) x = (2n + 1)π/4 for all n ∈ Z

(c) x = nπ/2 for all n ∈ Z

(d) x = (2n + 1)π/2 for all n ∈ Z

7. What is the value of limx→1

√

2 +x2 − 1

x − 1?

(a) undefined (b) 2 (c) 0 (d)√

2

CONTINUED

VERSION 1

– 3 – MATHS 108

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

9. What values of k make the following piecewise function continuous at x = k?

f(x) =

{

x2 + kx + 2 if x ≤ kkx2 + x + 2 if x > k

(a) k = 0 only

(b) k = 1 only

(c) k = 0 and k = 1

(d) no values of k

10. Where is the function defined by f(x) =√

x2 − x4 on its natural domain [−1, 1] NOTcontinuous?

(a) at x = ±1 only

(b) it is continuous on its natural domain

(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 theangle θ between the vectors?

(a) 0 < θ < π

2

(b) π < θ < 2π

(c) π

2< θ < π

(d) θ = π

2

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)z = c + s(c1 − c) + t(c2 − c)

where s, t ∈ R

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