MA1S11: SOLUTIONS TO 2010 ANNUAL EXAM
1. (a) x + y = 1, x + y = 2. Graphically this system represents two
parallel lines.
(b)
(i) Augmented matrix.

20
6
21
 0 5

 0 −10 −60 −4 −34


 2 1
40 −2 15


2 −9 −20 −6 −19









(ii) Reduced row echelon form.


28
63
5 
 1 0 0 5


 0 1 0 14 29 

5
5 


 0 0 1 −2 −2 

5
5 


0 0 0 0
0
(iii) There are infinitely many solutions.
x1 =
63 28
− x4
5
5
29 14
− x4
5
5
2 2
x3 = − + x4
5 5
x2 =
x4 is a free variable.
2. (a)
→
(i) P =< 4, 0, −1 >, Q =< 4, 2, 7 >, QP =< 0, −2, −8 >.
√
P
(ii) kPk = 17. Unit vector kPk
= √117 < 4, 0, −1 >.
1
2
MA1S11: SOLUTIONS TO 2010 ANNUAL EXAM
(iii) Projection of P along Q.
projQ P =
P·Q
3
Q
=
< 4, 2, 7 >
kQk2
23
(b) Parametric equations:
x = 4 − 2t,
y = 5t − 2,
z = 3 − 2t
Cartesian equations:
4−x
y+2
3−z
=
=
2
5
2
(c) A normal vector is < −1, 6, 5 >. The point (−21, 0, 0) is on the
plane.
(d)
cos θ =
u·v
11
= √ √ ≈ 0.21
kukkvk
69 38
3. (a)


14
0
5
− 17
2
13

8

m,n
t
(b) If A = (aij )m,n
i,j=1 then the transpose of A is A = (aji )i,j=1 . (i.e.
swap rows for columns). A is symmetric if At = A.
(c) Ax = b where


1
−4 
 5 −2


 2
3
7
2 


A=

 −1 −12 −11 −16 




1
2
−1 −1
MA1S11: SOLUTIONS TO 2010 ANNUAL EXAM



3

 −3 


 18 


b=

 −37 




−3
 x1 


 x2 


x=

 x 
 3 


x4
If B is the inverse of A then x = Bb.
(d) Your picture should have 3 vertices. A 1 in the (i, j)-entry of
the matrix means there should be a edge joining vertex i to
vertex j.
(e) Take for example

A=
0 1

1 0

1 0

Note that

A2 = 
0 1

Then (trace(A))2 = (0)2 and trace(A2 ) = 2.
4. (a) (69)10 converted to base 2 is (1000101)2 = (1.000101)2 × 26 .
The exponent (6)10 converted to base 2 is 110. The number is
stored as follows: Bit 1 is the sign bit, which is 0. The next 8
bits store the exponent, which we pad out with zeros, 00000110.
The remaining bits store the mantissa, 000101000 · · · 0.
(b) ( 40
) = (101.1011011011 · · · )2 .
7 10
(c) Generate your own spreadsheet to see what happens.
(d) Check this by inputting the commands into Mathematica.