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Quiz 1: Numbers and sets

Last unanswered question  Question  Next unanswered question
 

Question 1

 
 
Which of the following are correct ways of writing the set
A = {x ∈ ℝ | - 3 < x ≤ - 1 or x ≥ 0} ?
a) (-3,∞)   b) [-3,∞)
c) [-3,-1] ∩ [0,∞)   d) (-3,-1) ∪ [0,∞)
e) (-3,-1] ∪ [0,∞)

 

Not correct. Choice (a) is false.
(-3,∞) includes the real numbers between -1 and 0,
which do not belong to A.
Not correct. Choice (b) is false.
[-3,∞) includes -3, as well as the real numbers between -1 and 0, none of which belong to A.
Not correct. Choice (c) is false.
[-3,-1] ∩ [0,∞) is the empty set ∅. As A is not empty (for example, A includes -1), this option can’t be correct.
Not correct. Choice (d) is false.
-1 is not in (-3,-1) ∪ [0,∞), but -1 is in A.
Your answer is correct.
 

Question 2

 
 
What is another way of writing the set
B = {x ∈ ℝ | |x - 3| < 2} ?
a) (2,3]   b) [2,4]
c) (1,5)   d) [1,5]
e) [2,3)

 

Not correct. Choice (a) is false.
For example, 4 belongs to B but is not in (2,3].
Not correct. Choice (b) is false.
For example, 1.5 belongs to B but is not in [2,4].
Your answer is correct.
B is the set of all points whose distance from 3 on the number line is less than 2.
The solution to |x - 3| < 2 is 1 < x < 5.
Not correct. Choice (d) is false.
Neither 1 nor 5 belong to B, but both 1 and 5 belong to [1,5].
Not correct. Choice (e) is false.
For example, 4 belongs to B but is not in [2,3).
 

Question 3

 
 
If A = {7,8,9,10} and B = {5,6,7,8} then (A\B) ∪ (B\A) is
a) {5,6,7,8,9,10}   b) {5,6,9,10}
c) ∅, the empty set.   d) {7,8}
e) None of the above.

 

Not correct. Choice (a) is false.
Your answer is correct.
A\B = {9,10} and B\A = {5,6} so (A\B) ∪ (B\A) = {5,6,9,10}.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
 

Question 4

 
 
The set {0,1,±√--- - 1,π,12} is a subset of
a) ℕ   b) ℤ
c) ℚ   d) ℝ
e) ℂ

 

Not correct. Choice (a) is false.
The number π is not a natural number.
Not correct. Choice (b) is false.
The number π is not an integer.
Not correct. Choice (c) is false.
The number π is not a rational number.
Not correct. Choice (d) is false.
√ --- - 1 is not real.
Your answer is correct.
Since ±√--1 denotes the two imaginary numbers i and -i, the given set cannot be in any of the sets ℕ, ℤ, ℚ or ℝ.
Hence the right answer must be ℂ which contains all imaginary numbers.
 

Question 5

 
 
Which of the following alternatives is the best response to ‘Solve x2 - 3x + 4 = 0 over ℂ’.
a) There are no real solutions.   b) x = 1,4
c) √ - x = 3±---7 2   d) 3± i√7- x = ------- 2
e) None of the above is correct.

 

Not correct. Choice (a) is false.
As the question asks us to solve the equation over ℂ (that is, to find all solutions belonging to the set of complex numbers), this is not the best response.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
Using the quadratic formula, 3± √9---16- 3± √ - 7 x = ----2------= ---2---- .
Not correct. Choice (e) is false.
 

Question 6

 
 
If z = 9 + 3i and w = 2 - i then z + w equals
a) 9 - i   b) 11 + 2i
c) 6 + 3i   d) 8
e) None of the above

 

Not correct. Choice (a) is false.
Your answer is correct.
z + w = (9 + 3i) + (2 - i) = (9 + 2) + (3 - 1)i = 11 + 2i.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
 

Question 7

 
 
If w = 2 - i then w equals
a) 2 - i   b) 2
c) 2 + i   d) -2 + i
e) None of the above

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
w = 2 - i = 2 + i.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
 

Question 8

 
 
If p = 9 + 3i and q = 2 - i then pq equals
a) 15 + 15i   b) 21 + 15i
c) 18 + 3i   d) 1 - i
e) None of the above

 

Your answer is correct.
pq = (9 + 3i)(2 - i)
= (9 + 3i)(2 + i) = (18 - 3) + (6 + 9)i
= 15 + 15i.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
 

Question 9

 
 
If z = 9 + 3i and w = 2 - i then z- w equals
a) 15 + 15i   b) 6 + 3i
c) 12 + 15i   d) 3 - 3i
e) None of the above

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Your answer is correct.
z 9+ 3i 9+ 3i 2 + i 15 + 15i w- = 2---i = 2---i × 2-+-i =--5--- = 3+ 3i .
 

Question 10

 
 
The shaded region in the graph
iy 3i x 3
corresponds to which set of complex numbers?
a) {z ∈ ℂ : |z - (i+ 1)| < 2}   b) {z ∈ ℂ : |z|- |1 + i| < 2}
c) {z ∈ ℂ : Re (z + (i +1)) < 2}   d) {z ∈ ℂ : |z - 2| < |i+ 1- 2|}
e) None of the above.

 

Your answer is correct.
Not correct. Choice (b) is false.
This set corresponds to the interior of a circle, centre the origin, radius 2 + √2-.
Not correct. Choice (c) is false.
This set corresponds to the open half plane containing all complex numbers z = x + iy with x < 1.
Not correct. Choice (d) is false.
This set corresponds to the interior of a circle, centre 2, radius √ - 2.
Not correct. Choice (e) is false.