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Math 115 Section 03 (Barsamian) Midterm 1 Solutions
( x + 2 ) ( x 0 ) 0 , makes it clear that the important x-values
1 Rewriting one of the factors and rearranging,
( x 3)
are -2, 0, and 3
M2AA3, Introduction to Numerical Analysis
Dr Robert N
urnberg
Solutions to Problem Sheet 4
1. (i) Normal equations
T
A A=
1 + k2
1
1
1 + k2
" 1+k #
2+k2
1+k
2+k2
AT A x = AT b x =
T
A b=
and
1+k
1+k
.
.4
Question 4. Given events E, F, G :
(i) Prove that (E C )C = E.
(ii) If F C and G are disjoint, prove that (E \ F )C \ G = E C \ G.
(iii) If E and F are elements of a -field, F, prove that E \ F 2 F
M1P2 Algebra test I solution
Let U and W be the following subspaces of
4
:
U = Spancfw_(1, 1, 1, 2), (2, 1, 0, 5),
W = Spancfw_(0, 1, 1, 1), (3, 2, 4, 1)
1. Find a basis B for U \ W .
2. Find a basis
M3S11 / M4S11 GAMES, RISKS AND DECISIONS
EXERCISES (non-assessed)
1. Show that for any real numbers a, b, c, d, e, the two-person zero-sum game
with pay-off matrix
a
c
c
a b
d c
e c
b
d
e
always has a
M1M2 - Mathematical Methods II
Test 2 Monday, 2 February 2015
1. Using the method of variation of parameters (or reduction of
order), derive the general solution of the following dierential
equation:
Galois Theory
Lectured in Autumn 2016 at Imperial College by Prof. A. Corti.
Most humbly typed by Karim Bacchus.
Caveat Lector: unofficial notes. Comments and corrections should be sent to
[email protected]
Protactinium234
Uranium-238
Question 11 (2 points)
Which statement about freshwater use is FALSE?
Question 11 options:
As the population increases, water scarcity and sanitation issues
will also incre
Question 5 options:
A smooth, clear surface such as a parking lot or other paved
area
A forested area with lots of undergrowth and a variety of trees,
shrubs, and forest floor plants
A pretty flower b
The pH of surface waters is a problem in many systems, with some
surface waters showing too high a pH and some too low. Which of
the following is a key contributor to low pH surface waters?
Question 3
Suggestions for managing Chesapeake Bay pollution include all of
the following EXCEPT _.
Question 17 options:
using green roofs to reduce stormwater runof
protect native plant species
utilizing ripari
.
Question 1. The functions f (x) and g(x) are defined for all x by
f (x) =
x2
x+2
,
+x+1
g(x) = f (x2 ).
Find the even part of f (x) and the even part of g(x), which we write as fe (x) and ge (x).
Ex
M2AA3, Introduction to Numerical Analysis
Dr Robert N
urnberg
Solutions to Problem Sheet 1
1. Note that ka bk2 = ha b, a bi = kak2 + kbk2 2 ha, bi. Adding yields the desired result.
2.
q1 =
a1
= e1 .
M2AA3, Introduction to Numerical Analysis
Dr Robert N
urnberg
Solutions to Problem Sheet 0
1. Let A = (a1 an ) Rmn .
(i) (AT )ij = Aji = (ai )j . ith row of AT is aTi .
m
m
X
X
(ii) (B A)ij =
Bik Akj
M2AA3, Introduction to Numerical Analysis
Dr Robert N
urnberg
Solutions to Problem Sheet 7
1. The following MATLAB function can be used to plot all the required graphs. Note that
for [a, b] = [5, 5] t
M2AA3, Introduction to Numerical Analysis
Dr Robert N
urnberg
Solutions to Problem Sheet 2
1. If g(x) = a + bT x + 21 xT A x, then
[ g(x)]i =
g
(x) = bi + 12 [A x]i + 12 [AT x]i = bi + [A x]i
xi
as AT
M2AA3, Introduction to Numerical Analysis
Dr Robert N
urnberg
Solutions to Problem Sheet 5
1. f (x) = x2 is an even function, and d = . Therefore bm = 0, m = 1 ,
Z
Z
2
2
2 2
2 sin(m x)
x cos (m x) dx
M1P1 Analysis test question
Week 17
(a) Define what it means for a sequence (an )n1 of real numbers to converge to a R.
(b) Prove carefully from first principles that the sequence 1, 2, 3, . . . (i.e.
.(-"
Question.
(a) Vectors x, y, z 2 R2 are such that |x + y + z| = |x| + |y| + |z|. Prove that
x, y and z are multiples of the same vector. (You need to give complete details of
your argument, but yo
M2AA3, Introduction to Numerical Analysis
Dr Robert N
urnberg
Solutions to Problem Sheet 3
1. Assume that C is an eigenvalue of A Rnn . Then there exists a z Cn \ cfw_0 such
that A z = z. Denoting the
M2AA3, Introduction to Numerical Analysis
Dr Robert N
urnberg
Solutions to Problem Sheet 6
(x5)
(x5)
x (x5)
x (x2)
1. (a) p3 (x) = 3 x (x2)
1 (x+1) (x2)
+ 4 (x+1)18
+ 1 (x+1)90
.
18
10
3
(b) p3 (x) =
M1A1 Mechanics: Test 1
26 January 2015
1. Consider the motion of a point on the surface of a rolling wheel of radius R
r(t) = rc (t) + R cos[(t)]
+ R sin[(t)]
|,
where the position of the wheel centr
It takes approximately _ gallons of water to produce 1-pound of
beef and _ gallons of water to produce 1-pound of chicken and
_ gallons of water to produce 1-pound of corn.
Question 15 options:
100 ;
Acid deposition results from burning fossil fuels that release
_ that react in the atmosphere to form acids that fall back
to Earth as acid rain, snow, and fog.
Question 21 options:
sulfur oxides and
Question 1 (2 points)
If too much water is removed from an aquifer in coastal areas,
_.
Question 1 options:
desalinization will decrease.
the water table will rise.
saltwater intrusion will occur.
the
q
The image shows a device that generates a form of renewable
energy that we have discussed in class. (Hint: this device converts
mechanical energy to electrical energy). What is this device called?
Q
The process of capturing CO2 from a large point source and storing it
in a way so that it does not enter the atmosphere is called:
Question 36 options:
burning fossil fuels
air pollution
the greenhous
Earth's atmospheric circulation and prvailing winds are caused by
_ and _.
Question 25 options:
infiltration; hypoxia
industrial emissions; coal-fired power plants
the Coriolis Effect; Solar energy
el