UNIVERSITY OF TECHNOLOGY, JAMAICA
FACULTY OF SCIENCE AND SPORT
SCHOOL OF MATHEMATICS AND STATISTICS
Final Examination, Summer
Module Name: Engineering Mathematics 1
Module Code: MAT 1032
Date: July 2012
Theory / Practical: Theory
Groups: CM1, SE1, Eng 1(Elec/Mech)
Duration: Two (2) hours
Instructions:
1. This question paper consists of five (5) printed pages, which includes a cover page,
six(5) questions and a formulae sheet.
2. You are required to ANSWER ANY FOUR (4) questions in the answer booklet provided.
3. Full marks will be awarded for full workings / explanations.
4. You are allowed to use silent electronic calculators.
5. Begin the answer to each question on a fresh page and number your solutions carefully.
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
1
QUESTION# 1
a) Rationalise the denominator of
5
[3 marks]
2− 5
b) Determine the set of values of x that satisfy
x 2 + x − 12 0
Express your answer in interval notation. [4 marks]
c) A polynomial is defined by
P( x) = 2 x 3 + kx 2 − 11x + 60 .
Given that ( x − 3) is a factor of P(x):
(i) Determine the value of k
(ii) Factorize 2 x3 + kx 2 − 11x + 60 completely.
(iii) Hence solve 2 x3 + kx 2 − 11x + 60 = 0 [2+4+2 marks ]
QUESTION# 2
a) A and B are real intervals defined by (−, −2] and [−5, 0] , respectively. Using interval notation,
find:
(i) A'
(ii) ( A B) ' [2+3 marks]
b) Find the sum to infinity of the following series:
3 + 1.2 + 0.48 + ..... [4 marks]
c) Solve the following system of linear equations:
xy = 14
[6 marks]
4 x + 3 y = 29
2
QUESTION# 3
a) The standard form of a straight line is 5 x + 3 y − 4 = 0 .
Determine
(i) The gradient of the straight line.
(ii) The equation of a line perpendicular to the line 5 x + 3 y − 4 = 0 and passing through
the point (−2,5) .
[2+3 marks]
b) The table given below shows experimental values of two quantities x and y which are known to be
1
connected by an equation of the form = a x +b.
y
x 0.50 1.00 1.50 2.00 2.50 3.00
y 1.61 0.83 0.61 0.50 0.42 0.38
1
Plot against x and use the graph to estimate the values of ‘a’ and ‘b’.
y
[10 marks]
QUESTION# 4
1
a) Compute (−1) ( n − 2 ) .
n =−1
n +1
[3 marks]
b) Solve the following equations, for the unknown variable:
(i) 52 x−9 + 2 = 627
(ii) 7 2 x − 8(7 x ) + 7 = 0
2
(iii) log 2 = 3 + log 2 x
x
[3+4+5marks]
3
QUESTION# 5
a) Two functions, f and g, are defined as follows:
x−2
f ( x) = 3x + 4 g ( x) =
3
Find an expression for:
(i) fg (x)
(ii) g −1 ( x) [2 +3 marks]
4
5
b) Expand and simplify 2 + 3x using the binomial theorem. [5 marks]
x
c) Seats in an Auditorium were arranged as follows: first row, 12 seats; second row, 15 seats;
third raw ,18 seats; etc.
(i) Find the number of seats in the auditorium if there are 30 rows?
(ii) Find the number of seats in the 20th row?
[3 +2 marks]
***END OF PAPER***
4
USEFUL FORMULAE
−b b 2 − 4ac
Quadratic Formula , x =
2a
x m x n = x m+n
x m x n = x m−n
(x )
m n
= x mn
log( xy ) = log x + log y
x
log = log x − log y
y
log( x) p = p log x
Arithmetic series
an = a + (n − 1)d
n
sn = [2a + (n − 1)d ]
2
Geometric series
an = ar n −1
1− rn
sn = a
1− r
a
S =
1− r
5
ANSWER KEY
MAT 1032 summer 2012
Question#1
(x+4)(x+3) 1
x = -4,3
Test intervals 1
Answer (-4,3) 2
b.
k=-9 2
division 2
(x-4)((2x+5) 2
X=3,4,-5/2 2
c.
Question #2
a.
b.
Answer =5 3
c.
4x2-29x+42=0 3
x= 21/4,y=8/3
x=2,y=7 3
Question #3
a.
(i) m=-5/3 2
(ii)5y-3x-31=0 3
b.
Root(x) 0.71 1 1.22 1.41 1.58 1.73
1/y 0.62 1.2 1.64 2.00 2.38 2.63
Graph 4
6
c=-0.75 1
m= 2.03 2
Question #4
a.
2 +-1+0 2
answer =1 1
b.
x=6.5 3
(ii)
y=7,1 3
x=0,1 1
(iii)
(2/x2)=8 3
x=1/2,-1/2 2
Question #5
fg=x+2 2
g-1(x) =3x+2 3
b.
(625/x2)+(1500/x5)+(1350/x2)+540x3+81x4 5
c.
30=12+(n-1)3 1
n=7 2
a20= 30 2
