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
View Answer

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

Answer: b
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}
View Answer

Answer: c
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}
View Answer

Answer: 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}
View Answer

Answer: 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}
View Answer

Answer: c
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
View Answer

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

Answer: b
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) (-∞, ∞)
View Answer

Answer: c
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, ∞).
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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) (-∞, ∞)
View Answer

Answer: b
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}
View Answer

Answer: b
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}
View Answer

Answer: b
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
View Answer

Answer: a
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.

Sanfoundry Global Education & Learning Series – Mathematics – Class 11.

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn | Youtube | Instagram | Facebook | Twitter