SECTION A

Answer all 24 questions from this Section on the multiple choice answer sheet.

Each question is worth 3  marks.

1.  [3 MARKS]  Consider the following sets:

A   =   [3, 6)

B   =   [6, 8]

C   =   [2, 9)

D   =   [4, 6]

Which of the following is correct?

(a) x e A or x e B ÷ x e C .

(b) x e D ÷ x e A .

(c) x e C ÷ x e B .

(d) x e A or x e B  >÷  x e C . (e)  None of the above.

2.  [3 MARKS]  Consider the following equation in x :

b = (x - a)2 + c,  x e R

where a, b and c are parameters.  Which of the following is a necessary and sufficient condition for ONE solution to exist?

(a) a = c

(b)  b = c

(c)  b ● c

(d) c < b

(e) a = b = c

3.  [3 MARKS]  Consider the following function

What is its DOMAIN?

(a)  (-o, 1)

(b)  [-1, o)

(c)  (-o, 4]

(d)  [4, o)

(e)  (-1, o)

4.  [3 MARKS]  Let y = f (x) = 2x2 . What is the inverse of f (x)?

(a) g(y) =^(^) 

(b) g(y) =^(^) 

(c) g(y) = -^4y

(d) g(y) = ^4y

(e) The inverse of f (x) is not a function as it is not a one-to-one mapping.

5.  [3 MARKS]  Consider the function f (x) =  exp(x6 + x-1 ).  Which of the following is correct?

(a)  f\ (x) = 0.5(3x5 + 0.5x2 ) exp(x6 + x-1 )

(b)  f\ (x) = (3x5 ) exp(x6 + x-1 )

(c)  f\ (x) =  exp(3x5 - 0.5x-2 )

(d)  f\ (x) = (3x5 - 0.5x-2 ) exp(x6 + x-1 )

(e)  None of the above.

6.  [3 MARKS]  Consider the function f (z) =  for z < 0.  Which of the following is correct?

(a)  f\  = -3 z-4

(b)  f\  = 2 z-3

(c)  f\\ (z) = 20 z-6

(d)  f\\ (z) = 6 z-4

(e)  None of the above.

7.  [3 MARKS]  Consider the function f (x)  =  exp(x ln(x)).   Which of the following  is correct?

(a)  f\ (x) = exp(x ln(x))

(b)  f\ (x) = exp(x ln(x)) ln(x)

(c)  f\ (x) = exp(x ln(x))(ln(x) + 1)

(d)  f\ (x) = exp(ln(x))(ln(x) + 1) (e)  None of the above.

8.  [3 MARKS]  Let y = y(x) be a differentiable function of x, satisfying 2y+5-x2 -y3  = 0. What is y\ (x) at the point (x, y) = (2, -1)?

(a) 4

(b)  -2

(c)  2

(d) 0

(e)  -4

This information relates to Questions 9 to 11.

Consider the function f (x) = 7 ln(x - 2) - x2 + x.

9.  [3 MARKS] The function f (x) is concave because:   (a)  f\ (x) = 北7-2  - 2x, which is positive for all x > 2.

(b)  f\ (x) = 北7-2  - 2x + 1, which is increasing for all x > -2.            (c)  f\\ (x) = (北7-2)2    - 6 > 0, which could be negative or positive.       (d)  f\\ (x) = -  (北7-2)2    - 6 < 0, which is decreasing for all x > -2.      (e)  f\\ (x) = -  (北7-2)2    - 6 < 0, which is negative in the entire domain.

10.  [3 MARKS] The global maximum of f (x) occurs at:

(a) x = 0.25 * (5 +^65)

(b) x = 0.25 * (5 - ^65)

(c)  f (x) has no global maximum.

(d) x = 0.25 * (5 +^65) and x = 0.25 * (5 - ^65) are the global maximum. (e)  None of the above

11.  [3 MARKS] Assume that the constraint x ● 3, is imposed.  Which of the following is correct?

(a)  f (x) has a constrained maximum value at x = 0.25 * (5 - ^65). (b)  f (x) = f (3) for all (2, 3].

(c)  f (x) has no constrained maximum value on (2, 3].

(d)  f (x) < f (3) for all x e (2, 3]. (e)  None of the above.

This information relates to questions 12 to 16.

Consider the matrices A, B, C and vectors x and b:

A =                 B =                 C =                       x =              b =

and the matrix

12.  [3 MARKS] The matrix D = CT + B equals:

┌         ┐

(b) D =  ┌ 9(5)   5(1) ┐

┌         ┐

(d) D =  ┌ 2(4)   4(8) ┐      (e)  None of the above.

13.  [3 MARKS] The rank of the matrix E is: (a) 0

(b)  1

(c)  2

(d) 3

(e) 4

14.  [3 MARKS] Which of the following statements is correct? (a) The determinant of C is positive and C has rank of 2.

(b) The determinant of C is positive and C has a reduced rank of 1.

(c) The determinant of C is 0 and C has rank of 0.

(d) The determinant of C is 0 and C has rank of 2. (e)  None of the above

15.  [3 MARKS] The inverse of the matrix F = AB equals: (a) F-1  does not exist.

(b) F-1  =   ┐

(c) F-1  = -104 ┌ 1(2)3(5)

(d) F-1  =    (e) F-1  = -104 ┌ 1(2)3(5)

1(1)3(7) ┐

 ┐

┐

16.  [3 MARKS] The problem Cx = b has:

(a)  Infinitely many solutions.

(b)  No solution.

(c)  One solution.

(d) Two solutions.

(e)  Four solutions.

The following information relates to questions 17 and 18

Consider the function f (x1 , x2 ) = x 1(-)2 x2(-)3 + ln(x2(2)) + exp(x1 ).

Calculate the first and second order partial derivatives and then answer the following questions.

17.  [3 MARKS] The first partial derivatives are:

(a)  f1 (x1 , x2 ) = -2x1(-)2 x2(-)3 + exp(x1 ) and f2 (x1 , x2 ) = -3x1(-)2 x2(-)3 +北2(2) . (b)  f1 (x1 , x2 ) = -2x1(-)3 x2(-)3 + exp(x1 ) and f2 (x1 , x2 ) = -3x1(-)2 x2(-)4 +北2(2) .

(c)  f1 (x1 , x2 ) = -2x1(-)3 x2(-)3 + exp(x1 ) and f2 (x1 , x2 ) = -3x1(-)2 x2(-)4 +北2(1) .

(d)  f1 (x1 , x2 ) = -3x1(-)2 x2(-)4 +北2(2)    and f2 (x1 , x2 ) = -2x1(-)3 x2(-)3 + exp(x1 ). (e)  None of the above.

18.  [3 MARKS] Which of the following results involving second order partial derivatives of f (x1 , x2 ) is correct?

(a)  f11  = 6x1(-)4  x2(-)3  and f22  = 12x1(-)2  2x

(b)  f11  = 6x1(-)4  x2(-)3 + exp(x1 ) and f22  = 12x1(-)2 x2(-)5

(c)  f22  = 12x1(-)2  2x2(-)2  and f12  = 6x1(-)3 x

(d)  f22  = 12x1(-)2 x2(-)5 + 2x2(-)2  and f12  = 6x1(-)3

(e)  None of the above.

19.  [3 MARKS] You have the following multivariate function: f (x1 , x2 , x3 ) = x +ln1(2)  ╱ 、. Which of the following is the gradient vector of f (x1 , x2 , x3 )?

(a)  (2x1 , -1/x2 , -1/x3 )T

(b)  (2x1 , x2 /x3 )T

(c)  (2x1 , 1/x2 , 1/x3 )T

(d)  (2x1 , ln(x2 /x3 ))T

(e)  None of the above.

20.  [3 MARKS]  In the process of maximising an objective function f (x1 , x2 ), which is de- fined for x1  < 0 and x2  < 0, you obtain the following Hessian matrix:

H =  ┌ -10(/x)1(2)     -5/x(0)2(2)┐

Which of the following statements is correct?

(a) The determinant of H is negative and hence any stationary point is a maximum. (b)  H is negative semi-definite, but not negative definite.

(c)  For some values of x1  and x2  in the domain H will be positive definite, for others negative definite.

(d)  H is negative definite. (e)  None of the above.

21.  [3 MARKS]  Consider a function g(x1 , x2 ), which is of the form g(x1 , x2 ) = α + βx1 + γx2 . Which of the following is the Hessian matrix of g(x1 , x2 )?

(a)  H =  ┌ -10(/x)1(2)     -5/x(0)2(2)┐

(b)  H =  ┌0(β)   γ(0)┐

(c)  H =  ┌0(0)   0(0)┐

(d)  H =  ┌0(1)   1(0)┐         (e)  None of the above.

22.  [3 MARKS]  Consider a function g(x1 , x2 ), which is of the form g(x1 , x2 ) = α + βx1 + γx2 . Which of the following is correct?

(a) The function g(x1 , x2 ) is concave and convex.

(b) The function g(x1 , x2 ) is strictly concave and strictly convex.

(c) The function g(x1 , x2 ) is strictly concave.

(d) The function g(x1 , x2 ) is neither concave nor convex. (e)  None of the above.

23.  [3 MARKS]  In a bi-variate optimisation problem you you want to impose the following constraint:  3^x1 x2   <  6.   Which of the following is the correct specification for the constraint function g(x1 , x2 ) to be used in a Lagrangian maximisation problem?

(a) g(x1 , x2 ) = 3^x1 x2

(b) g(x1 , x2 ) = 3^x1 x2 - 6

(c) g(x1 , x2 ) = 3^x1 x2 + 6

(d) g(x1 , x2 ) = 6 - 3^x1 x2

(e)  None of the above.

24.  [3 MARKS]  Consider the bi-variate function f (x1 , x2 ) = (x1  - 2)2 + (x2  - 6)2 + x1 x2 which we want to minimise subject to x1 + x2  < 10.

Which of the following is the Lagrangian function which requires ma↓imisation?

(a)  L(x1 , x2 ) = -(x1 - 2)2 - (x2 - 6)2 - x1 x2 + λ(10 - x1 - x2 )

(b)  L(x1 , x2 ) = -(x1 - 2)2 - (x2 - 6)2 - x1 x2 + λ(x1 + x2 - 10)

(c)  L(x1 , x2 ) = (x1 - 2)2 + (x2 - 6)2 + x1 x2 + λ(x1 + x2 - 10)

(d)  L(x1 , x2 ) = (x1 - 2)2 + (x2 - 6)2 + x1 x2 + λ(10 - x1 - x2 )

(e)  None of the above.

SECTION B

Answer this question in the answer booklet provided.This question is worth 20

marks.

25.  [20 MARKS]  Maximise f (x1 , x2 ) = 2 ln x1 +  ln x2 , x1  > 0, x2  > 0, subject to 2x1 + 3x2  ● m, m > 0.

Ensure that you clearly describe your steps, interpret any result and show how you come to any conclusions.