# Mathematics Questions and Answers – Sets and their Representations – 2

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This set of Mathematics MCQs for Class 11 focuses on “Sets and their Representations – 2”.

1. How to define a set?
a) A collection of well-defined objects or element
b) A collection of unordered objects or element
c) Any random elements
d) A collection of special characters

Explanation: Generally, a set is defined as a collection of well defined objects or elements.
Each element in a set is unique.
Say for example, if S a set it is represented as,
S = {x: 2x2 ᵾ x< 5 and x € N}
Then the elements present in the set will be
S = {2, 8, 18, 32}.

2. How is a set denoted?
a) ()
b) {}
c) []
d) **

Explanation: A set is represented by {}.
Usually, but not necessarily a set is denoted by a capital letter e.g. A, B……. V, W, X, Y, Z.
The elements are enclosed between { } denoted by small letters a, b, ……., y, z.

3. How will you define a set of all real numbers?
a) {x: -1 < x < 1}
b) [x: -∞ < x < ∞]
c) {x: -∞ < x < ∞}
d) {x: -Z < x < +Z}

Explanation: All the numbers whether it is an integer or rational number or irrational number is defined as Real Number. The range of the real number lies between in the range (-∞, +∞).
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4. How will you define Union of two sets A and B?
a) {x: x € A or x € B}
b) {x: x € A or x € B (or both)}
c) {x: x € A and B}
d) {x: x € A – B}

Explanation: Union of two or more sets is the set of all elements that belongs to any of these sets.
The symbol used for this union of sets is ‘∪‘.
If A = {1, 2, 3, 4} and B = {2, 4, 5, 6} and C = {1, 2, 6, 8}
Then, A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}.

5. How will you define the difference of two sets B-A?
a) {x: x € A and x Ɇ B}
b) {x: x Ɇ A and x € B}
c) {x: x € A and x € B}
d) {x: x Ɇ A and x Ɇ B}

Explanation: The difference of a set A and B is denoted as A-B. A-B is a set of those elements that are in the set A but not in the set B. Similarly, the difference of a set B and A is denoted as B-A. It is a set of those elements that are in the set B but not in the set A.
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6. What will be the set of the interval (a, b]?
a) {x: a < x < b}
b) {x: a ≤ x ≤ b}
c) {x: a < x ≤ b}
d) {x: a ≤ x < b}

Explanation: The symbol ( ) implies that the value will always be less than or greater than the x value i.e. end points are not included.
{ } implies that all the values that does not satisfy a given interval are included inside {}.
[ ] implies that the value will always be less than equal to or greater than equal to the x value i.e. end points are included. This is possible only when both a and b are finite.

7. How to define Wavy Curve Method f(x)?
a) (x-a1)n1 / (x-a2)n2 / (x-a3)n3 …………… / (x-ak)nk * (x-b1)m1 / (x-b2)m2 / (x-b3)m3 …….. /(x-bp)mp
b) (x-a1)n1 + (x-a2)n2 +(x-a3)n3 …………… + (x-ak)nk / (x-b1)m1 + (x-b2)m2 + (x-b3)m3 …….. + (x-bp)mp
c) (x-a1)n1 (x-a2)n2 (x-a3)n3 …………… (x-ak)nk / (x-b1)m1 (x-b2)m2 (x-b3)m3 …….. (x-bp)mp
d) (x-a1)n1 – (x-a2)n2 – (x-a3)n3 …………… – (x-ak)nk / (x-b1)m1 – (x-b2)m2 – (x-b3)m3 ……..- (x-bp)mp

Explanation: The method of intervals {or wavy curve} is used for solving inequalities of the form
f(x) = (x-a1)n1 (x-a2)n2 (x-a3)n3 …………… (x-ak)nk / (x-b1)m1 (x-b2)m2 (x-b3)m3 …….. (x-bp)mp > 0 (< 0, ≤ 0, or ≥ 0)
where, n1, n2, ,n3, …….. nk and m1, m2, m3, …….. , mp are natural numbers .
a1, a2, a3, ……..ak and b1, b2, b3, …….. bp are any real numbers such that ai ≠ bj where i = 1, 2, 3, ……. , k and j = 1, 2, 3, ….. , p.

8. How to solve for x, if |x-1| ≥ 3?
a) (-∞, -2) ∪ (4, ∞)
b) (-∞, -2] ∪ [4, ∞)
c) (0, -2] ∪ (4, 0)
d) (-∞, ∞) – {-2, 4}

Explanation: Given, |x-1| ≥ 3
= x-1 < -3 or x – 1 ≥ 3
= x ≤ -2 or x ≥ 4
Hence, x c (-∞, -2] ∪ [4, ∞).

9. What is the interval of f(x) = (x – 1)(x – 2)(x – 3)/(x3 + 6x2 + 11x + 6) where f(x) is positive?
a) (-∞, -3) ∪ (3, ∞)
b) (3, -2) ∪ (1, 1) ∪ (2, 3)
c) (-∞, -3) ∪ (2, -1) ∪ (1, 2) ∪ (3, ∞)
d) (-∞, ∞)

Explanation: f(x) = (x – 1)(x – 2)(x – 3)/(x3 + 6x2 + 11x+ 6)
After solving the cubic equation (x3 + 6x2 + 11x+ 6) we get (x+1)(x+2)(x+3)
Now, we can see that this implies f(x) = (x – 1)(x – 2)(x – 3)/(x + 1)(x + 2)(x + 3)
So, the critical points of x are, x = 1, 2, 3, -1, -2, -3
So, for f(x) > 0 ᵾ x € (-∞, -3) ∪ (2, -1) ∪ (1, 2) ∪ (3, ∞).

10. What is the interval of f(x) = (x – 1)(x – 2)(x – 3)/(x3 + 6x2 + 11x+ 6) where f(x) is negative?
a) (-∞, -3) ∪ (3, ∞)
b) (3, -2) ∪ (1, 1) ∪ (2, 3)
c) (-∞, -3) ∪ (2, -1) ∪ (1, 2) ∪ (3, ∞)
d) (-∞, ∞)

Explanation: f(x) = (x – 1)(x – 2)(x – 3)/(x3 + 6x2 + 11x+ 6)
After solving the cubic equation (x3 + 6x2 + 11x+ 6) we get (x+1)(x+2)(x+3)
Now, we can see that this implies f(x) = (x – 1)(x – 2)(x – 3)/(x + 1)(x + 2)(x + 3)
So, the critical points of x are, x = 1, 2, 3, -1, -2, -3
So, for f(x) < 0 ᵾ x € (3, -2) ∪ (1, 1) ∪ (2, 3).

11. What is the set of all x for which 1/(x – 1)(3 – x) ≤ 1?
a) (-∞, 1) ∪ (3, ∞)
b) (-∞, 1) ∪ (3, ∞) ∪ {2}
c) (-∞, 1) ∪ {2}
d) (3, ∞) ∪ {2}

Explanation: 1/(x – 1)(3 – x) ≤ 1
Now, on solving the equation further we get,
1/(x – 1)(3 – x) -1 ≤ 0
This also implies,
1- 1/(x – 1)(3 – x) ≥ 0
(x – 1)(3 – x) – 1/(x – 1)(3 – x) ≥ 0
So, (x – 2)2/(x – 1)(3 – x) ≥ 0.
This implies, (-∞, 1) ∪ (3, ∞) ∪ {2}.

12. Which one of the following is the correct representation of set A = {2,4,8,16….} in set builder form?
a) {x: x = 2n where n ∈ N}
b) {x: x = 2n where n ∈ N}
c) {x: x = 4n where n ∈ N}
d) {x: x = 2n+4 where n ∈ N}

Explanation: The sequence is a geometric progression with base 2 hence 2n is the correct answer.

13. A set can be a collection but a collection cannot be a set.
a) True
b) False

Explanation: A collection becomes a set when it is well defined for example a collection of good football players is not a set since the phrase “good football players” is vague and not defined.

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