Saturday, 11th December, 2010
1.00p.m. to 3.00p.m.
EXAMINATION FOR THE DEGREES OF
M.A. AND B.Sc.
MATHEMATICS 1X
An electronic calculator may be used provided that it does not have
a facility for either textual storage or display or for graphical display.
Thursday, 13th December, 2012
1 p.m. to 3 p.m.
Solutions
A2)
(i) Write m n = ka and m + n = a for some k, Z. Then
m2 n2 = (m n)(m + n) = (ka)( a) = (k )a2 ,
where k Z, hence a2 | (m2 n2 ).
3
(ii) We have
6 10
2( 3 5)
2
2 5
10
= =
=
.
=
5
5
15 5
5( 3
Wednesday, 7th December, 2011
9.30a.m. to 11.30 a.m.
EXAMINATION FOR THE DEGREES OF
M.A. AND B.Sc.
MATHEMATICS 1X
An electronic calculator may be used provided that it does not have
a facility for either textual storage or display or for graphical display
A Table of Greek Letters
Upper case
Lower case
In English
A
B
E
Z
H
I
K
M
N
O
P
T
X
alpha
beta
gamma
delta
epsilon
zeta
eta
theta
iota
kappa
lambda
mu
nu
csi
omicron
pi
rho
sigma
tau
upsilon
phi
psi
chi
omega
o
Notes. In pronouncing the names of the lette
Thursday, 10th December, 2009
9.30a.m. to 11.30a.m.
EXAMINATION FOR THE DEGREES OF
M.A. AND B.Sc.
MATHEMATICS 1X
An electronic calculator may be used provided that it does not have
a facility for either textual storage or display or for graphical display.
Thursday, 13th December, 2012
1 p.m. to 3 p.m.
EXAMINATION FOR THE DEGREES OF
M.A. AND B.Sc.
Mathematics 1X
An electronic calculator may be used provided that it does not have
a facility for either textual storage or display, or for graphical display.
Can
27.
(a)
Let P(n) denote the statement that 4 - 62 + 3 - 23" is divisible by 7.
Whenn: 1, 4-62"+3-23" 24-62+3-23 = 144+24= 168 =7-24. So P(n) is truewhen
n = 1.
Suppose now that P(n) is true when n = k, i.e. suppose that 4 e 62" + 3 - 23 is divisible by
Thursday, 18th December, 2014
9:30 a.m. to 11:30 a.m.
EXAMINATION FOR THE DEGREES OF
M.A. AND B.Sc.
Mathematics 1X
An electronic calculator may be used provided that it does not have
a facility for either textual storage or display, or for graphical displ
MATHEMATICS 1X
Solution to Algebra Exercises
[
31
]
3
68. Let () denote the statement that = 2
.
2 3
[
]
[
]
5
3
5
3
When = 1, LHS= 1 = =
and the RHS= 20
=LHS.
3 1
3 1
1
3 + 2
3
So () is true when = 1. Suppose now that () is true when = , i.e. suppose tha
46.
47.
48.
By the Binomial theorem
(2:2 if = (1'2 + <_§>)3n : :io ($29.1. (_§>r : :0 (3:1)176W2r. (if? 2
= we)
0
The term independent of a: is the term in x0 which occurs when 671 3r 2 0, i.e. r = 271. So the
term independent of a: is
MATHEMATICS 1X
Algebra Exercises
1
1. Show that if and then ( ).
2. Find the greatest common divisor, gcd(323, 133). Represent it in the form
gcd(323, 133) = 323 + 133 where , .
3. Simplify
3 12
(a) ,
27
6+3 5
,
(c)
5 + 20
1
,
(b)
3 5
7+4 3
.
(d)
2+ 3
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Statistics 1Z Paired Data (2014-2015)
5
50
Assessing the Linear Regression Model
5.1
Informal Checks of Assumptions
Certainly always plot the tted line
y = + (x x)
on the data plot. Look for any consistent marked deviations of the data from the tted
lin
Statistics 1Z Paired Data (2014-2015)
4
41
Interval Estimation In Regression
4.1
The Point Estimates
In the previous chapter we have considered the linear regression model
yi = + xi + i
where the i are independent N (0, 2 ) random variables. The least-squ
Statistics 1Z Paired Data (2014-2015)
3
30
Simple Linear Regression
3.1
The Question of Interest
When we measure two variables of interest on the same experimental unit (e.g. patient) then we are often interested in modelling the dependence of one variabl