Download Math 136 - Mathematics IB 2004e1 Tutorial Problems—Week 10

Math 136 - Mathematics IB
2004e1
Tutorial Problems—Week 10
Calculus
Solve the following differential equations. Remember that you can always check your
answer by differentiating and seeing whether the equation holds.
1. y = tan y cos x, y(0) = π/6.
dy
x2 + y 2 + y
2.
=
, x > 0, y(4) = 3.
dx
x
3.
dy
3x + 4y − 5
=
dx
6x − 7y + 8
4. yexy + 1 + (xexy + 1)y = 0.
Algebra


3 4 4

1. Let A = 4 3 4 . Determine whether the following vectors are eigenvectors
4 4 3
of A. If 
it is an
of A, then write
the corresponding
eigenvalue.
 eigenvector
 
 down



−1
1
2
1
(i)  0 
(ii)  1 
(iii)  −3 
(iv)  −2 
1
1
1
1
2. For each of the following matrices find
(a) the eigenvalues, and
(b) a basis for each of the corresponding eigenspaces.
3
−2
(i) A =

1 −1

(iv) A = 1
1
2
0
4
−3

1
2
3
(ii) B =

−5

(v) B = −5
9
6 −3
−4
2
0
−1
0

−4
−4 
7
(iii) C =

2 −4
1
6
2 −1

(vi) C = −1
2
−1 −1
Note: Test 2 will be held in the tutorials in week 11.

−1
−1 
2