Download MATH1231 Mathematics 1B Exam - Semester 2, 2014 - University of New South Wales and more Summaries Physics in PDF only on Docsity!
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
Semester 2 2014
MATH
MATHEMATICS 1B
(1) TIME ALLOWED – TWO (2) HOURS
(2) TOTAL NUMBER OF QUESTIONS – 4
(3) ANSWER ALL QUESTIONS
(4) THE QUESTIONS ARE OF EQUAL VALUE
(5) ANSWER EACH QUESTION IN A SEPARATE BOOK
(6) THIS PAPER MAY BE RETAINED BY THE CANDIDATE
(7) ONLY CALCULATORS WITH AN AFFIXED “UNSW APPROVED” STICKER
MAY BE USED
(8) A SHORT TABLE OF INTEGRALS and A STANDARD NORMAL TABLE
ARE APPENDED ON THE LAST PAGES
All answers must be written in ink. Except where they are expressly required pencils
may only be used for drawing, sketching or graphical work.
Use a SEPARATE book clearly marked Question 1
- i) By expanding sin(A+B) + sin(A−B), or otherwise, find I 1 =
sin(5x) cos(x) dx.
ii) Evaluate the integral I 2 =
0
x
2
√ 4 − x
2
dx.
iii) Use appropriate tests to determine whether each of the following series
converges or diverges
a)
∑^ ∞
n=
n(ln n)^3
b)
∑^ ∞
n=
2 n^ + 3n^
iv) Let S =
x
y
z
(^) ∈ R^3 : z^2 = x^2 + y^2
a) Prove that S is closed under scalar multiplication.
b) Prove that S is not a subspace of R
3 .
v) Let A =
a) Find the eigenvalues and eigenvectors for the matrix A.
b) Write down an invertible matrix M and a diagonal matrix D such
that
D = M
− 1 AM.
vi) Let
A =
Using the MAPLE output below, find a basis for ker(A).
with(LinearAlgebra):
A := <<1,2,3,1>|<2,4,-3,2>|<2,10,-3,1>|<-1,-44,24,6>>;
ReducedRowEchelonForm(A);
Please see over...
Use a SEPARATE book clearly marked Question 2
- i) Consider the initial value problem
dy
dx
x
)y =
x
, with y(1) = 0,
defined for x > 0.
a) Show that an integrating factor for this equation is xe
2 x .
b) Hence solve the initial value problem.
ii) Find the general solution to
d
2 y
dx^2
dy
dx
2 x .
iii) Consider the MAPLE session:
a:=n->n^n/n!*(x-1)^n;
a := n →
n
n (x − 1)
n
n!
a(n+1);
(n + 1)
(n+1) (x − 1)
(n+1)
(n + 1)!
limit(a(n+1)/a(n),n=infinity);
ex − e
Using MAPLE session above, or otherwise, find the open interval of con-
vergence I = (a, b) for the power series
∑^ ∞
n=
n
n (x − 1)
n
n!
iv) Consider the set S consisting of the vectors v 1 =
(^) , v 2 =
from R
3 and let u =
a) Find scalars λ and μ such u = λv 1 + μv 2.
b) A linear transformation T : R
3 → R
3 has v 1 , v 2 as eigenvectors with
eigenvalues 2 and −1, respectively.
α) Find T (u) as a linear combination of v 1 , v 2.
β) Denote T (T (u)) by T
2 (u), T (T (T (u))) by T
3 (u), and so on.
Express T
n (u) as a linear combination of v 1 , v 2 , where n is a
positive integer.
Please see over...
v) The two most popular soft drinks in Old South Wales are AppleAde and
BananAde. Assume that no-one in Old South Wales likes both of these
drinks equally (that is, everyone has a preference for one or the other).
Past statistics show that 50 % of the population prefer AppleAde.
Last month the manufacturer advertised AppleAde on television for a
week. After that, a survey was conducted by taking a random sample of
100 people. Of the 100 people sampled, 60 preferred AppleAde and 40
preferred BananAde.
a) Assuming that the advertising had no effect on people’s preferences,
write down an expression for the tail probability that 60 or more
people preferred AppleAde in a sample of 100.
b) Use the normal approximation to the binomial to calculate the tail
probability in (a), giving your answer to 3 decimal places.
c) Giving reasons, is there evidence that the advertising campaign in-
creased the percentage of the population that prefer AppleAde?
Please see over...
a) Find a basis for span(S).
b) Write down the dimension of span(S).
iii) Employment data at a large company reveal that 40 % of the employees
are thirty years of age or younger; 60 % are older than thirty. All em-
ployees belong to exactly one of the types: full-time, part-time or casual.
Among those who are thirty years of age or younger, 25 % are full-time;
15 % are part-time; the others are casual. Among those who are older
than thirty, 45 % are full-time; 20 % are part-time; the others are casual.
a) What proportion of the employees are full-time?
b) What is the probability that a randomly chosen employee is older
than thirty given that the employment type of the employee is full-
time?
iv) The discrete random variable X can only take the values − 2 , − 1 , 1 , 2 , 5.
The probability distribution for X is given in the following table:
x − 2 − 1 1 2 5
P (X = x) 0. 3 4 c 0. 35 6 c 0. 05
Another random variable Y is defined by Y = X
2 .
a) Find the value of c.
b) Calculate P (Y = 4).
v) The density function f for a random variable X is defined by
f (x) =
3 x
2 for 0 < x ≤ 1
0 otherwise.
a) Find the cumulative probability density function F corresponding to
f.
b) Calculate E(X) for the probability density function f.
c) Calculate Var(X).
vi) Let B = {v 1 , v 2 , v 3 } be a set of three non-zero vectors in R
3 .
a) State the definition for the set B to be a linearly independent set.
b) Prove that if B is an orthogonal set then B is linearly independent.
c) Hence explain why any orthogonal set of 3 non-zero vectors in R
3
forms a basis for R
3 .
Please see over...
Use a SEPARATE book clearly marked Question 4
- i) A right circular cone with base radius r cm and a perpendicular height
h cm is formed by rotating the line segment given by
y =
r
h
x, 0 ≤ x ≤ h,
about the x-axis.
a) Use calculus to show that the surface area S of the curved surface of
the cone is given by
S = πr
r
2
2 .
b) Find the partial derivatives
∂S
∂r
and
∂S
∂h
c) If the values of r and h are measured to be 3.0 and 4.0 cms respec-
tively and each of these measurements is made with an error whose
absolute value is at most 0.05 cm, then use the total differential ap-
proximation of S to estimate the maximum absolute error in the
measured value of the total surface area S.
ii) During the winter the daytime temperature in the Physics Theatre is
maintained at 20
◦ C. The heating is turned off at 10pm and turned on
again at 6am. On a certain day, the temperature inside the Theatre at
11pm was found to be 18
◦ C. The outside temperature was found to be
constant throughout the night at 10
◦ C. Let P (t) be the temperature (in ◦ C) in the Physics Theatre at time t (in hours), from 10pm. The rate of
cooling of the air inside the Theatre can be modelled by the differential
equation
dP
dt
= −k(P − 10),
where k is a positive constant. (Do not prove this.)
a) By first solving the differential equation, show that k = ln(5/4).
b) Find, to two decimal places, the temperature inside the Physics The-
atre when the heating was turned on at 6am.
Please see over...
BASIC INTEGRALS
x
dx = ln |x| + C = ln |kx|, C = ln k
∫
e
ax dx =
a
e
ax
∫
a
x dx =
ln a
a
x
∫
sin ax dx = −
a
cos ax + C
∫
cos ax dx =
a
sin ax + C
∫
sec
2 ax dx =
a
tan ax + C
∫
cosec
2 ax dx = −
a
cot ax + C
∫
tan ax dx =
a
ln | sec ax| + C
∫
cot ax dx =
a
ln | sin ax| + C
∫
sec ax dx =
a
ln | sec ax + tan ax| + C
∫
sinh ax dx =
a
cosh ax + C
∫
cosh ax dx =
a
sinh ax + C
∫
sech
2 ax dx =
a
tanh ax + C
∫
cosech
2 ax dx = −
a
coth ax + C
∫ dx
a^2 + x^2
a
tan
− 1 x
a
+ C
dx
a
2 − x
2
a
tanh
− 1 x
a
a
coth
− 1 x
a
2 a
ln
a + x
a − x
2 6 = a
2
dx √ a
2 − x
2
= sin
− 1 x
a
+ C
dx √ x^2 + a^2
= sinh
− 1 x
a
+ C
dx √ x
2 − a
2
= cosh
− 1 x
a
Standard normal probabilities P (Z ≤ z)
- z .00 .01 .02 .03 .04 .05 .06 .07 .08.
- − 2 9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.
- − 2 8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.
- − 2 7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.
- − 2 6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.
- − 2 5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.
- − 2 4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.
- − 2 3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.
- − 2 2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.
- − 2 1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.
- − 2 0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.
- − 1 9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.
- − 1 8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.
- − 1 7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.
- − 1 6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.
- − 1 5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.
- − 1 4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.
- − 1 3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.
- − 1 2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.
- − 1 1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.
- − 1 0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.
- − 0 9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.
- − 0 8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.
- − 0 7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.
- − 0 6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.
- − 0 5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.
- − 0 4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.
- − 0 3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.
- − 0 2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.
- − 0 1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.
- − 0 0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.
- 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.
- 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.
- 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.
- 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.
- 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.
- 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.
- 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.
- 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.
- 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.
- 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.
- 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.
- 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.
- 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.
- 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.
- 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.
- 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.
- 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.
- 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.
- 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.
- 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.
- 2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.
- 2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.
- 2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.
- 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.
- 2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.
- 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.
- 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.
- 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.
- 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.
- 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.
