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ECON0006: INTRODUCTION TO MATHEMATICS FOR ECONOMICS

ECON0010: MATHEMATICS FOR ECONOMICS

 

TIME ALLOWANCE: 1 HOUR

 

 

1. If and , which one of the following must be true?

a) where a is a real number

b) and p,r can take any values

c)

d)

e)

 

2. If are linearly dependent vectors, which one of the following statements is correct?

a) s can be expressed as a linear combination of

b) There exist nonzero constants such that

c) None of can be the zero vector

d) One of must be the zero vector

e) One of could be the zero vector

 

3. If A,B and C are matrices, which one of the following is false?

a)

b)

c)

d)

e)

 

4. If and then the system of equations has the solution:

a)

b)

c)

d)

e) there are no solutions

 

 

5. For a system of 6 linear equations in 7 unknowns, which one of the following

   statements must be true?

a) there is an infinite number of solutions

b) there is either a unique solution or an infinite number of solutions

c) there is either a unique solution or there are no solutions

d) there are either no solutions or an infinite number of solutions

e) either there is a unique solution or there is no solution or there are an infinite number of solutions

 

 

6. For a system of 7 linear equations in 6 unknowns, which one of the following statements must be true?

a) there is an infinite number of solutions

b) there is either a unique solution or an infinite number of solutions

c) there is either a unique solution or there are no solutions

d) there are either no solutions or an infinite number of solutions

e) either there is a unique solution or there is no solution or there are an infinite number of solutions

 

 

7. The rank of the matrix

 

 

is which one of the following?

a) 0

b) 1

c) 2

d) 3

e) None of the above

 

 

8. The matrix

 

 

has an inverse for which of the following values of k ?

a) all values of k

b) all values of k except 0

c) all values of k except 0 and -1

d) all values of k except 0, -1 and -2

e) all values of k except 0, -1 and -3

 

 

 

 

 

 

 

 

9. Let . For which one of the following matrices,, is  not equal to

a)  

b)

c)

d)

e)

 

 

10. Let . Which one of the following statements is true?

a)  is positive definite

b)  is negative definite

c)  is positive semidefinite

d)  is negative semidefinite

e)   is none of the above

 

 

11. If  which one of the following is the value of   when  and       ?

a) 

b) 

c) 

d) 

e) None of the above

 

 

12. If  which one of the following is the value of   when  and      ?

a) 0

b) 9

c) 

d) 

e) 

 

 

13. If  where  and  , the value of   at  is which one of the following?

a) 0

b) 3

c) 9

d) 6

e) None of the above

 

 

 

14. For the Cobb-Douglas production function

 

where K and L denote capital and labour respectively, which one of the following statements is true?

a) Returns to scale are constant

b) Returns to scale are decreasing

c) Returns to scale are increasing

d) Returns to each factor are diminishing

e) None of the above

 

 

15. Consider the part of the ellipse

 

satisfying . Which one of the following is an expression for  ?

a) 

b) 

c) 

d) 

e) None of the above

 

 

16. For the Cobb-Douglas production function

 

where K and L denote capital and labour respectively, which one of the following statements is not true about the isoquants?

a) They are negatively sloped

b) They are convex

c) They meet the axes

d) They exhibit the property of diminishing marginal rate of substitution

e) They have the axes as asymptotes

 

 

17. The smooth function  has a local maximum at . Which one of the following statements is true?

a) The Hessian of f is negative definite at

b) The Hessian of f is positive definite at

c) The Hessian of f is negative semidefinite at

d) The Hessian of f is positive semidefinite at

e) None of the above

 

 

18. The smooth function  is concave. Which one of the following statements is true?

a) The Hessian of f is negative semidefinite everywhere

b) The Hessian of f is positive semidefinite everywhere

c) The function f must have a global maximum

d) The Hessian of f is negative definite everywhere

e) The Hessian of f is positive definite everywhere

 

 

19. For the function

 

 

which one of the following statements is not true?

a) f is a homogeneous function

b) f has a global minimum at

c) The Hessian of f is positive definite at

d) The Hessian of f is positive semidefinite for all

e) f is a convex function

 

 

20. Which one of the following is the maximum value of  subject to

                                                  ?

a) 8

b) 5

c) 7

d) 6

e) 9

 

 

21. Consider the utility function

 

 

where  denote the consumption levels of the two goods and  are positive constants.

Which one of the following functions will not give rise to the same demand functions?

a) 

b) 

c) 

d) 

e) 

 

 

22. For the production function

 

where K and L denote capital and labour respectively, which of the following statements is not true?

a) The production function is concave

b) The isoquants are negatively sloped

c) The isoquants are convex

d) The production function is quasi-concave

e) Returns to scale are increasing

 

 

 

 

 

 

 

23. For the differential equation

 

the solution satisfying  when  is

a) 

b) 

c) 

d) 

e) none of the above

 

 

24. For the differential equation

 

 

which one of the following statements is true?

a) The general solution exhibits explosive behaviour

b) The general solution tends to 24 as

c) The general solution tends to 15 as

d) The general solution tends to 40 as

e) None of the above

 

 

25. For the difference equation

 

 

 

which one of the following statements is true?

 

a) The general solution alternates about 24

b) The general solution alternates about 15

c) The general solution does not alternate

d) The general solution alternates about 40

e) The general solution exhibits damped alternations