SINGAPORE INSTITUTE OF MANAGEMENT
UNIVERSITY OF LONDON
PRELIMINARY EXAM 2022
MODULE CODE
:
MT105A
MODULE TITLE :
Mathematics 1
DATE OF EXAM :
3 March 2022
TOTAL NUMBER : 3
OF PAGES
(INCLUDING
THIS PAGE)
------------------------------------------------------------------------------------------------------INSTRUCTIONS TO CANDIDATES :Candidates should answer all EIGHT questions: all SIX questions of Section A (60 marks in
total) and BOTH questions from Section B (20 marks each).
Candidates are strongly advised to divide their time accordingly.
MT105A Mathematics 1
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Section A (Answer all 6 questions in this section. 60 marks in total)
Question 1
Suppose that the function f is given by
f(x) = x6 – 6x5 + 6x4 + 8.
Show that f has three critical points. Determine whether each critical point is a local maximum,
local minimum or inflexion point.
Question 2
The function f is defined by
f(x, y) = x2 + 7x + 8xy + xy2 + x2y.
Find the critical points of f and determine, for each, whether it is a local maximum, a local
minimum or a saddle point.
Question 3
Express the following system of equations in matrix form, and solve it using row operations.
4x – y – 3z = 7
2x – 3y – z = 7
3x – 5y + z = 16.
Question 4
Determine the following integrals:
6𝑥+4
∫ 2𝑥 2+5𝑥−3 𝑑𝑥 , ∫(𝑥 + 1)[ln⁡(𝑥 2 + 2𝑥 + 8)]2 𝑑𝑥
Question 5
An arithmetic progression has the following properties:
*
the sum of the first twelve terms is –840;
*
the twelfth term is five times the second term.
Determine the first term and the common difference.
Question 6
The demand equation for a good is q(3p + 2) = 34 and the supply equation is q – 2p + 8 = 0
where p is the price and q is the quantity. Determine the equilibrium price and quantity.
Sketch the supply and demand functions for p  0.
MT105A Mathematics 1
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Section B (Answer both questions in this section. 20 marks each)
Question 7
(a)
The market value of an asset, if it is sold at time t  0, is assumed to be V(t) = (t + 2)2.
The present value P(t) of money raised if it is sold at time t is V(t)e–0.1t.
Show that there is one critical point for P(t) for t  0.
Use the second derivative test to show that P(t) is maximised at the critical value of t.
Determine the maximum value of P(t), expressing your answer in terms of e.
(b)
A company’s production function is given by
Q = 100[0.4K5/2 + 0.6L5/2]2/5
and input limitations mean that 16K + 3L = 134. Use the Lagrange multiplier method
to find the values of K and L that maximise the production function subject to the given
constraint.
Question 8
(a)
A firm is a monopoly for the good it produces.
Its marginal cost function is MC = 2q2 + 3, where q is the quantity it produces, and it
has fixed costs of 15. The demand equation for its good is given by 2p + q = 48, where
p is the price. Find expressions, in terms of q, for the total cost, total revenue and profit.
Determine the production level q that gives maximum profit.
(b)
The population of a village is 6000 at the start of 2022. Each year, 2% of the population
leave the village, and 300 new residents move in. Find an expression, in as simple a
form as possible, for the population of the village N years after the start of 2022. What
happens to the population of the village in the long run?
End Of Paper
MT105A Mathematics 1
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