Entrance Examination for the Ph. D. Program
Graduate School of Mathematics
Nagoya University
2012 Admission
Part 1 of 2
Tuesday, February 7, 2012, 9:00 a.m.〜12:00 noon

Note:
1. Please do not turn pages until told to do so.
2. The problem sheet consists of the cover page and 4 single-sided pages. Afterthe exam has begun, please first confirm that the number of pages and theirprinting and order are correct. Please report any problem immediately.


3. There are a total of 4 problems labeled 1 , 2 , 3 , and 4 , respectively.
Please answer all 4 problems.
4. The answering sheet consists of 4 single-sided pages. Please confirm thenumber of pages, and please do not remove the staple.






5. Pleasewrite the answers
to problems 1  , 2  ， 3  ，and 4  on pages
1  , 2  , 3  , and 4  of the answering sheet, respectively.
6. Please write name and application number in the space provided
on each of the 4 pages in the answering sheet．
7. The back side of the 4 pages in the answering sheet may also be used. Ifused, please check the box at the lower right-hand corner on the front side.
8. If the answering sheet staple is torn, or if additional paper is needed for
calculations, please notify the exam proctor.
9. After the exam has ended, please hand in the 4 page answering sheet. The
problem sheet and any additional sheets used for calculations may be taken
home.

Notation:
The symbols Z, Q, R, and C denote the sets of integers, rational numbers, real
numbers, and complex numbers, respectively.

(MC Examination 2012, Morning)1

1 Let f : R → R be the linear map defined by f (x) = Ax, where A is the matrix
A33A




01 −2


A =  1 −10 .
−204
Please answer the following questions.
(1) Find a basis for the kernel Ker fA and a basis for the image Im fA .
(2) Find a 3 × 3-matrix P which satisfies that for all x ∈ Ker fA , P x = 0 and forall x ∈ Im fA , P x = x.

(February 7th, 2012)(Continue to Next Page)

(MC Examination 2012, Morning)2

2 Decide if each of the following statements is true or false. If true, then please give aproof. If false, then please give a counterexample and prove that this is so.
(1) Let P and Q be planes in the Euclidean space R3 and suppose that an invertible
linear transformation f of R3 maps P to Q. Then f maps a normal vector tothe plane P to a normal vector to the plane Q.
(2) Suppose that all eigenvalues of a real 2 × 2-matrix A have absolute value atmost 1. Then, for all x ∈ R2 , the subset {An x | n = 1, 2, . . . } of R2 is bounded.
((3) Suppose that!
a11 a12a21 a22!)b11 b12,b21 b22

is a linearly independent subset ofthe complex vector space that consists of all complex 2 × 2-matrices. Then the(!
!)
a11b11subset,of C2 is linearly independent.
a21b21(February 7th, 2012)(Continue to Next Page)

(MC Examination 2012, Morning)3

3 Please answer the following questions.
(1) Find the value of the integral
Z ∞ log
1xx+1

1+x+1
dx.

(2) Show that if 0 < θ < π2 , then 2θ < sin θ + tan θ.
(3) Consider the following system of differential equations with f (x, y) a functionof class C 1 defined over R2 :−y∂f∂f+x= 0,∂x∂yx

∂f∂f+y= 2f.
∂x∂yExpress the system of differential equations in polar coordinates and find allsolutions.

(February 7th, 2012)(Continue to Next Page)

(MC Examination 2012, Morning)4

4 Let D be the subset of the plane defined by
D = {(x, y) ∈ R2 | 1 ≤ x + y ≤ 2, −1 ≤ xy ≤ 0}.
Please answer the following questions.
(1) Make a sketch of the subset D.

(2) Find the Jacobian∂(u, v)
associated with the change of variables u = x + y and∂(x, y)v = xy, where x, y ∈ R.
(3) Find the value of the double integral
ZZ|x3 − y 3 | dxdy.
D(February 7th, 2012)(The End)