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(1)

(MC examination 2010, Morning) 1

1 With respect to a complex number a, we consider the matrix

A =

4 2 2

2a 1 +a 1 2a 1 +a 1

.

Answer the following questions:

(1) Find all the values of a such that rankA= 1.

(2) When rankA= 1, find a regular matrix P such that P1AP is diagonal.

(3) Find all the values of a such that A cannot be diagonalized.

(July 25th, 2009) (Continue to Next Page)

(2)

(MC examination 2010, Morning) 2

2 Let V be the linear space of polynomials of order 3 or less overR:

V ={p+qx+rx2+sx3 | p, q, r, sR}. For any f(x)V, we define

T(f(x)) = 1 x1

x

1

(t1)f(t)dt.

Answer the following questions:

(1) Show that the mapping T, which maps f(x) V to T(f(x)), is a linear map from V toV.

(2) Find the representation matrix of T, using 1, x, x2, x3 as a basis for V.

(3) Assuming g(x) = p+ qx +rx2 +sx3 V (p, q, r, s R), find a necessary and sufficient condition on p, q, r, s so that there exists an f(x) V such that T(f(x)) =g(x).

(4) For some constant k R, we fix g(x) = 1x+kx23x3 V. Check whether T(f(x)) =g(x) admits a solution f(x)V. If there are solutions, find all such f(x)V.

(July 25th, 2009) (Continue to Next Page)

(3)

(MC examination 2010, Morning) 3

3 Answer the following questions:

(1) For the 2-variable functionf(x, y) = e(x+y) cos(xy), find the 2nd order polynomial p(x, y) such that

lim

(x,y)(0,0)

f(x, y)p(x, y) x2+y2 = 0.

(2) Let g(x, y) be a function of class C1 around (x, y) = (a, b), we define

c=g(a, b), ξ= ∂g

∂x(a, b), η= ∂g

∂y(a, b).

Let F(u, v, w) be a function of class C1 around (u, v, w) = (a, b, c), we define

α= ∂F

∂u(a, b, c), β= ∂F

∂v(a, b, c), γ = ∂F

∂w(a, b, c).

Let G(x, y) = (x, y, g(x, y)) and H(x, y) = (F G)(x, y). Represent ∂H

∂x(a, b) and ∂H

∂y(a, b) usingα, β, γ, ξ, η. Here, F Gis the composition of G and F.

(3) Evaluate the following integral:

2 0

dy

2

y

exp (y

x )

dx.

(July 25th, 2009) (Continue to Next Page)

(4)

(MC examination 2010, Morning) 4

4 Answer the following questions:

(1) For α >0 andn = 1,2, . . ., show that 1 (n+ 1)α

n+1

n

dx xα.

(2) Show that, when α >1, the series

n=1

1

(n+ 1)α converges.

(3) Show that, when α → ∞,

n=1

1

(n+ 1)α 0.

(4) We assume thatα >1, the sequence{an}n=1 is such thatan>0 (n= 1,2, . . .), and the series

n=1

an converges. Under those assumptions, show that both

n=1

ann and

n=1

(

an+ 1 (n+ 1)α

)n

do converge.

(5) Under the assumptions of (4), show that, when α→ ∞,

n=1

{(

an+ 1 (n+ 1)α

)n

ann }

−→0.

(July 25th, 2009) (The End)

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