MALAWI UNIVERSITY OF
BUSINESS AND APPLIED SCIENCES
Mathematical Sciences Department
BUSINESS MATHEMATICS I : BUM 111
EBS 1.
Date: 6th March, 2023
Tutorial 1
Instruction: Attempt all questions
Question 1
(a) Use De Morgans laws to simplify (X ∩ Y 0 )0 ∪ Y 0 and (X 0 ∪ Y 0 )0 ∩ X 0 .
(b) Show using Venn diagrams that:
i. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
ii. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C). Use these identities to simplify:
iii. A ∩ (B ∪ A0 ).
iv. A0 ∪ (B ∩ A).
v. (A ∩ B) ∪ (A0 ∪ B).
(c) Suppose S and T are sets. Under what conditions is S ∪ T = T ?.
(d) Suppose X and Y are sets and X ∪ Y = X. What can you deduce about X and Y ?.
(e) If A and B are sets and A ∪ B = A ∩ B, what can be deduced?
(f) Suppose A, B, C are sets. Illustrate the relation
A ∪ (B ∩ C) ⊇ (A ∪ B) ∩ C
using a Venn diagram. From your diagram (or otherwise) show that if
A ∪ (B ∩ C) = (A ∪ B) ∩ C then C ⊇ A.
Give a specific example of sets A, B, C where A ∪ (B ∩ C) 6= (A ∪ B) ∩ C.
(g) Show using a Venn diagram that A − B = A ∩ B 0 . Use this identity and other well known
identities, but not Venn diagrams, giving your reasons, to simplify the following:
i. A − (B − A)
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ii. (A − B) − A.
iii. (A − B) ∩ (A − C).
(h) If A, B, C are sets that A − B = C − B, does it follow that A = C? Give reasons.
(i) Consider the sets U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, A = {1, 2, 3, 7, 9, 10}, B = {3, 4, 7, 8, 10}
and C = {5, 6, 7, 8, 9, 10} where U is the universal set. Find:
i. (A ∩ B)0 ∩ C.
ii. (B∆C) ∩ A0 .
iii. A × B × C.
iv. A0 ∪ B 0 .
Question 2
(a) Let C = {4n + 1 : n ∈ Z}. Give three different elements of C.
i. Show that if x, y ∈ X then xy ∈ C.
ii. Give an example to show that the statement”if x, y are integers and xy ∈ C, then
x, y ∈ C”, is false.
(b) Suppose n is an integer. Give a precise condition for n to be even. Show that
i. (−1)n = 1 if n is even.
ii. (−1)n = −1 if n is odd.
n
Hence simplify (−1)(−1) , where n is any integer.
(c) Write down two irrational numbers. Show that the sum and difference of any two rational
numbers is rational, and give examples to show that the sum and product of two irrational
numbers may be rational or irrational.
(d) If x, y ∈ Q and y 6= 0, prove the following statements:
i. x + y ∈ Q.
ii. xy ∈ Q.
x
∈ Q.
y
(e) Prove that the following are irrational:
√
i. 3.
iii.
1
ii. 2 3 .
iii.
√
√
2 + 3.
Question 2 continues on the next page. . .
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√
4
iv. 5 2 − .
17
Question 3
(a) Solve the following inequalities:
i. −2x + 4 ≤ −3x − 4.
ii. x2 ≤ −2x.
iii. x2 < −4.
iv. x ≤ x3 .
v. x3 ≥ 8. x + 4 < 2x − 1 < 5 − 3x.
vi. 32x − 4.3x + 3 < 0.
vii.
2
3x − 2
viii.
1
1
<
.
x−6
≤ 3.
2x − 1
(b) Suppose a < b < 0. Show that a2 > b2 .
(c) Find the domain and range of the following functions:
√
1
− x + 3.
i. f (x) =
1 − 2x
1
ii. g(x) = √
.
2
x + 3x − 10
iii. h(x) =
√
3
+ 2x + 5.
2
1−x
(d) Find the inverse of the following functions f :
√
i. f (x) = 3 2x3 − 1.
1
ii. f (x) = 1 + 2x 7 .
(e) Let f (x) = 2x − 3 and g(x) =
2x
. Find:
4+x
i. (f og)(x).
ii. f −1 (g(x)).
iii. (f og)−1 (x).
Question 3 continues on the next page. . .
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(f) Factorise the following:
i. 3x2 + 2x − 8.
ii. x4 − 81.
iii. 7(x2 − 6x + 9) + 5(x − 3).
iv. −8p2q − 12pq 2 + 16p2 q 2 .
v. 3 − 27x2 .
(g) Find the middle term of the expansion for the following:
1 10
i. 3x2 +
.
2x
1 7
ii. 2x +
.
x
a b 11
−
iii.
.
2 3
Question 4
(a) Find the constant term in the expansion of :
√
1 10
x+ 2 .
i.
3x
1 9
ii. x2 −
.
x
(b)
1 10
i. Find the coefficient of x in the expansion of 2x −
.
3x
10
a2
+ 2b2 .
ii. Find the coefficient of b6 in the expansion of
2
5
2
iii. Find to four decimal places the value of (1.02)4 using binomial theorem.
(c)
i. Simplify and write (−3 + i) − (−8 + 2i) in the form of z = a + ib.
ii. Simplify (5 + 7i)(5 − 7i).
iii. Simplify
4+i
.
3 + 5i
iv. Solve the quadratic equation 2z 2 + z + 3 = 0 in the form z = a + ib.
v. Show that z1 .z2 = z1 + z2 .
√
vi. Show that if x = 1 + i 3 then x2 − 2x + 4 = 0.
END OF TUTORIAL QUESTIONS
End of Tutorial 1