Mixing Cleaning Solutions. King’s Service Station uses two kinds of cleaning solution containing acid and water. “Attack” is 2% acid and “Blast” is 6% acid. They want to mix the two in order to get 60 qt of a solution that is 5% acid. How many quarts of each should they use? Do the Familiarize and Translate steps by completing the following table. Let a = the number of quarts of Attack and b = the number of quarts of Blast. AMOUNT OF SOLUTION ATTACK BLAST MIXTURE a b PERCENT OF ACID 2 % 5% AMOUNT OF ACID IN SOLUTION 0.06 b
Mixing Cleaning Solutions. King’s Service Station uses two kinds of cleaning solution containing acid and water. “Attack” is 2% acid and “Blast” is 6% acid. They want to mix the two in order to get 60 qt of a solution that is 5% acid. How many quarts of each should they use? Do the Familiarize and Translate steps by completing the following table. Let a = the number of quarts of Attack and b = the number of quarts of Blast. AMOUNT OF SOLUTION ATTACK BLAST MIXTURE a b PERCENT OF ACID 2 % 5% AMOUNT OF ACID IN SOLUTION 0.06 b
Solution Summary: The author calculates the amount of Attack and Blast in quarts to get 60 qt. of solution which contain 5% acid.
Mixing Cleaning Solutions. King’s Service Station uses two kinds of cleaning solution containing acid and water. “Attack” is 2% acid and “Blast” is 6% acid. They want to mix the two in order to get 60 qt of a solution that is 5% acid. How many quarts of each should they use?
Do the Familiarize and Translate steps by completing the following table. Let
a
=
the number of quarts of Attack and
b
=
the number of quarts of Blast.
1.2.19. Let and s be natural numbers. Let G be the simple graph with vertex set
Vo... V„−1 such that v; ↔ v; if and only if |ji| Є (r,s). Prove that S has exactly k
components, where k is the greatest common divisor of {n, r,s}.
Question 3
over a field K.
In this question, MË(K) denotes the set of n × n matrices
(a) Suppose that A Є Mn(K) is an invertible matrix. Is it always true that A is
equivalent to A-¹? Justify your answer.
(b) Let B be given by
8
B = 0 7 7
0 -7 7
Working over the field F2 with 2 elements, compute the rank of B as an element
of M2(F2).
(c) Let
1
C
-1 1
[4]
[6]
and consider C as an element of M3(Q). Determine the minimal polynomial
mc(x) and hence, or otherwise, show that C can not be diagonalised.
[7]
(d) Show that C in (c) considered as an element of M3(R) can be diagonalised. Write
down all the eigenvalues. Show your working.
[8]
R denotes the field of real numbers, Q denotes the field of rationals, and
Fp denotes the field of p elements given by integers modulo p. You may refer to general
results from lectures.
Question 1
For each non-negative integer m, let R[x]m denote the
vector space consisting of the polynomials in x with coefficients in R and of degree ≤ m.
x²+2, V3 = 5. Prove that (V1, V2, V3) is a linearly independent
(a) Let vi = x, V2 =
list in R[x] 3.
(b) Let V1, V2, V3 be as defined in (a). Find a vector v € R[×]3 such that (V1, V2, V3, V4)
is a basis of R[x] 3.
[8]
[6]
(c) Prove that the map ƒ from R[x] 2 to R[x]3 given by f(p(x)) = xp(x) — xp(0) is a
linear map.
[6]
(d) Write down the matrix for the map ƒ defined in (c) with respect to the basis
(2,2x + 1, x²) of R[x] 2 and the basis (1, x, x², x³) of R[x] 3.
[5]
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