Download Tutorial 2

3COM0091
Quantum Computing
Quantum Computing
Tutorial 2 - Algebra 1 Answers
The following questions were set in Tutorial 1 to assess the abilities of the group
as a whole with algebra. If you experienced difficulties with any of these then do
please let me know. The aim is to enable you to work with these and other
algebraic concepts and for me to assess the requirements of the group
Addition, Subtraction, Multiplication and Division
Complete the following:
1
2
3
4
a + a + a + a + a + a + a + a + a + a = 10a
2b + 5b = 7b
7h – 4m – 6h = h – 4m
6x – 3y + 12x2 – 4x + 5y = 12x2 + 2x + 2y
Directed Numbers
State the value of the following:
5
6
7
8
9
10
7 + (- 4) = 7 – 4 = 3
(17) - (- 12) = 17 + 12 = 29
(- 8)2 + (- 4)3 = 64 + (- 64) = 64 – 64 = 0
(- 2) * 3 / (- 12) = (- 6) / (- 12) = 6/12 = 1/2
(- 3g2) * 4g5 = (-3)*4*g2*g5 = (-12)*g2+5 = - 12g7
4g2 * (-4g5) / (- g3) = 4*(-4)*g2*g5/(-1)*g3 = -16*g2+5/(-1)*g3 = 16g2+5-3 = 16g4
Evaluating Expressions
Let a = 6, b = - 2 and c = 1/3
Calculate the following:
11
12
13
14
15
16
17
18
19
20
ab = 6*(-2) = - 12
ac = 6*(1/3) = (6*1)/3 = 2
a – b = 6 – (-2) = 6 + 2 = 8
b – c = (-2) – 1/3 = - 21/3
a2 – b2 = 6*6 – (-2)*(-2) = 36 – 4 = 32
(a - b)2 = (6 – (-2))2 = (8)2 = 82 = 64
a/b = 6/(-2) = - 6/2 = - 3
abc = 6*(-2)*(1/3) = (-12)*(1/3) = -12/3 = - 4
(a + b) = (6 + (-2)) = (6 - 2) = 4 = 2
(a + b)/c = (6 + (-2))/(1/3) = 4/(1/3) = 4*3 = 12
QC Tutorial 2
1/2
06/10/05
3COM0091
Quantum Computing
Expanding Brackets
Expand the following:
21
22
23
24
25
26
27
28
29
30
5(x + 7) = 5*x + 5*7 = 5x + 35
3(4b –3c) = 3*4b – 3*3c = 12b – 9c
–p2(2p3 – p + 3) = (–p2)*2p3 –(–p2)*p + (–p2)*3) = -2p5 + p3 – 3p2
n(4 – 3 n) – (n – 6) = n*4 – n*3n – n + 6 = -3n2 + 3n + 6
rt( r + t + q) = rt*r + rt*t + rt*q = r2t + rt2 + rtq
(a + b)(c + d) = a*(c + d) + b*(c + d) = ac + ad + bc + bd
(5b – 3)(2b – 1) = 5b*(2b – 1) – 3*(2b – 1) = 10b2 – 5b – 6b +3 = 10b2 – 11b +3
(a + b)2 = (a + b)*(a + b) = a*(a + b) + b*(a + b) = a2 +ab + ba + b2 = a2 + 2ab + b2
(a - b)2 = (a – b)*(a – b) = a*(a – b) – b*(a – b) = a2 – ab – ba +b2 = a2 - 2ab + b2
(a + b)(a – b) = a*(a – b) + b(a – b) = a2 – ab + ba – b2 = a2 – ab + ab + b2 = a2 – b2
Factorising Brackets
31
32
33
34
35
36
37
38
39
40
3x2- 12 = 3*x2 – 3*4 = 3(x2 – 4) = 3(x2 – 22) = 3(x + 2)(x – 2)
12b – 9c = 3*4b – 3*3c = 3(4b – 3c)
pt + t2 – tx = t(p + t – x)
2p5 – p3 + 3p2 = p2(2p3 – p + 3)
r2t + rt2 + qrt = rt(r + t + q)
6lx – 15mx = 3x(2l – 5m)
x2 – 9 = x2 – 32 = (x + 3)(x – 3)
a2 - 1 = a2 – 12 = (a + 1)(a - 1)
1 - b2 = 12 - b2 = (1 + b)(1 – b)
7y2 - 63 = 7y2 – 7*9 = 7(y2 – 9) = 7(y2 – 32) = 7(y + 3)(y – 3)
Algebraic Fractions
41
42
43
44
45
46
47
48
49
50
1/a = ?/ab
5/c = ?/cd
x/5 = ?/15
m/n = m3/?
d/e = 5df/?
R/t = ?/t2
3 = ?/5
c = ?/3
1 = ?/f
y = ?/y
QC Tutorial 2










?=b
? = 5d
? = 3x
? = m2n
? = 5ef
? = Rt
? = 15
? = 3c
?=f
? = y2
2/2
06/10/05