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

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: 2x

^{2}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) **

View Answer

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

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 (-∞, +∞).

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

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

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.

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

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-a_{1})^{n1} / (x-a_{2})^{n2} / (x-a_{3})^{n3} …………… / (x-a_{k})^{nk} * (x-b_{1})^{m1} / (x-b_{2})^{m2} / (x-b_{3})^{m3} …….. /(x-b_{p})^{mp}

b) (x-a_{1})^{n1} + (x-a_{2})^{n2} +(x-a_{3})^{n3} …………… + (x-a_{k})^{nk} / (x-b_{1})^{m1} + (x-b_{2})^{m2} + (x-b_{3})^{m3} …….. + (x-b_{p})^{mp}

c) (x-a_{1})^{n1} (x-a_{2})^{n2} (x-a_{3})^{n3} …………… (x-a_{k})^{nk} / (x-b_{1})^{m1} (x-b_{2})^{m2} (x-b_{3})^{m3} …….. (x-b_{p})^{mp}

d) (x-a_{1})^{n1} – (x-a_{2})^{n2} – (x-a_{3})^{n3} …………… – (x-a_{k})^{nk} / (x-b_{1})^{m1} – (x-b_{2})^{m2} – (x-b_{3})^{m3} ……..- (x-b_{p})^{mp}

View Answer

Explanation: The method of intervals {or wavy curve} is used for solving inequalities of the form

f(x) = (x-a

_{1})

^{n1}(x-a

_{2})

^{n2}(x-a

_{3})

^{n3}…………… (x-a

_{k})

^{nk}/ (x-b

_{1})

^{m1}(x-b

_{2})

^{m2}(x-b

_{3})

^{m3}…….. (x-b

_{p})

^{mp}> 0 (< 0, ≤ 0, or ≥ 0)

where, n1, n2, ,n3, …….. nk and m1, m2, m3, …….. , mp are natural numbers .

a

_{1}, a

_{2}, a

_{3}, ……..a

_{k}and b

_{1}, b

_{2}, b

_{3}, …….. b

_{p}are any real numbers such that a

_{i}≠ b

_{j}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}

View Answer

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)/(x^{3} + 6x^{2} + 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

Explanation: f(x) = (x – 1)(x – 2)(x – 3)/(x

^{3}+ 6x

^{2}+ 11x+ 6)

After solving the cubic equation (x

^{3}+ 6x

^{2}+ 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)/(x^{3} + 6x^{2} + 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

Explanation: f(x) = (x – 1)(x – 2)(x – 3)/(x

^{3}+ 6x

^{2}+ 11x+ 6)

After solving the cubic equation (x

^{3}+ 6x

^{2}+ 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

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 = 2^{n} where n ∈ N}

c) {x: x = 4n where n ∈ N}

d) {x: x = 2n+4 where n ∈ N}

View Answer

Explanation: The sequence is a geometric progression with base 2 hence 2

^{n}is the correct answer.

13. A set can be a collection but a collection cannot be a set.

a) True

b) False

View Answer

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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