Grade 11 : Mathematics - Sets (Set 2) | Concept of Maths

Question 1

1. In a class of 40 students, 25 students like to play Cricket and 20 students like to play Football. If each student likes at least one game, how many students like both Cricket and Football?

A) 3

B) 5

C) 10

D) 15

Answer

Correct Answer: B

Using the formula for the union of two sets: $$n(C \cup F) = n(C) + n(F) - n(C \cap F)$$ Here, $n(C \cup F) = 40$, $n(C) = 25$, and $n(F) = 20$. $$40 = 25 + 20 - n(C \cap F)$$ $$40 = 45 - n(C \cap F)$$ $$n(C \cap F) = 45 - 40 = 5$$. So, 5 students like both.

Question 2

2. A software company labels its projects using the set of distinct letters in the word "RECURSION". Which of the following represents this set in roster form?

A) $\{R, E, C, U, R, S, I, O, N\}$

B) $\{R, E, C, U, S, I, O, N\}$

C) $\{R, E, C, U, S, I, O\}$

D) $\{R, E, C, U, R, S, I, O\}$

Answer

Correct Answer: B

In a set, elements are not repeated. The word "RECURSION" contains the letters R, E, C, U, R, S, I, O, N. Removing the duplicate 'R', we get the distinct letters: $\{R, E, C, U, S, I, O, N\}$.

Question 3

3. An ice-cream parlor offers 4 different toppings: Sprinkles, Nuts, Chocolate Chips, and Caramel. How many unique combinations of toppings (including the option of no toppings) can a customer choose?

A) 16

B) 4

C) 8

D) 15

Answer

Correct Answer: A

The number of combinations of toppings is the number of subsets of the set of toppings. If a set has $n$ elements, it has $2^n$ subsets. Here, $n = 4$, so the number of subsets is $2^4 = 16$. This includes the empty set (no toppings).

Question 4

4. Let $U$ be the set of all days in a week. If set $A = \{Saturday, Sunday\}$, what is the complement set $A'$?

A) $\{Monday, Tuesday\}$

B) $\{Saturday, Sunday\}$

C) $\{Friday, Saturday, Sunday\}$

D) $\{Monday, Tuesday, Wednesday, Thursday, Friday\}$

Answer

Correct Answer: D

The complement $A'$ contains all elements of the universal set $U$ that are not in $A$. $U = \{Mon, Tue, Wed, Thu, Fri, Sat, Sun\}$ and $A = \{Sat, Sun\}$. Therefore, $A' = \{Mon, Tue, Wed, Thu, Fri\}$.

Question 5

5. In a library, set $M$ consists of all Mathematics books and set $P$ consists of all Physics books. If there are no books that cover both subjects, the sets $M$ and $P$ are called:

A) Equal sets

B) Infinite sets

C) Disjoint sets

D) Power sets

Answer

Correct Answer: C

Two sets are disjoint if they have no common elements, i.e., $M \cap P = \emptyset$. Since no book covers both subjects, $M$ and $P$ are disjoint.

Question 6

6. Which of the following is an example of an empty set?

A) The set of months with 32 days.

B) The set of even prime numbers.

C) The set of whole numbers less than 1.

D) $\{0\}$

Answer

Correct Answer: A

An empty set (null set) has no elements. A) No month has 32 days, so this is $\emptyset$. B) Contains $\{2\}$. C) Contains $\{0\}$. D) Contains the element 0.

Question 7

7. If set $A = \{x : x \text{ is a vowel in the English alphabet}\}$ and set $B = \{a, e, i, o, u\}$, then:

A) $A \subset B$ but $A \neq B$

B) $A = B$

C) $A \cap B = \emptyset$

D) $n(A) = 6$

Answer

Correct Answer: B

Set $A$ in roster form is $\{a, e, i, o, u\}$. Set $B$ is also $\{a, e, i, o, u\}$. Since they contain exactly the same elements, $A = B$.

Question 8

8. A researcher defines a set $S = \{x : x \in \mathbb{N}, 10 < x < 15\}$. What is the cardinality $n(S)$?

A) 5

B) 3

C) 4

D) 6

Answer

Correct Answer: C

The natural numbers strictly between 10 and 15 are 11, 12, 13, and 14. $S = \{11, 12, 13, 14\}$. The number of elements is 4, so $n(S) = 4$.

Question 9

9. If a set $A$ has 5 elements, how many proper subsets does it have?

A) 32

B) 31

C) 16

D) 30

Answer

Correct Answer: B

Total number of subsets of a set with $n$ elements is $2^n$. The number of proper subsets is $2^n - 1$ (excluding the set itself). For $n=5$, proper subsets = $2^5 - 1 = 32 - 1 = 31$.

Question 10

10. Let $X = \{1, 2, 3, 4, 5\}$ and $Y = \{4, 5, 6, 7, 8\}$. Find $X - Y$.

A) $\{1, 2, 3\}$

B) $\{6, 7, 8\}$

C) $\{4, 5\}$

D) $\{1, 2, 3, 4, 5, 6, 7, 8\}$

Answer

Correct Answer: A

The difference $X - Y$ is the set of elements that are in $X$ but not in $Y$. Elements in $X$: 1, 2, 3, 4, 5. Common elements in both $X$ and $Y$: 4, 5. Removing these from $X$, we get $\{1, 2, 3\}$.

Grade 11 : Mathematics - Sets (Set 2) | Concept of Maths