Case Study / Scenario-Based MCQs for Sub-Topics of Topic 9: Sets, Relations & Functions

Question 1. A teacher is making a list of students in Class XI of a school who are good at sports. She lists students Rahul, Priya, Amit, and Sneha, noting their preferred sport: Rahul (Cricket), Priya (Badminton), Amit (Cricket), Sneha (Tennis). She also notes that students are considered 'good' based on their performance in the previous year's inter-school competition. Which of the following statements correctly describes the nature of the collection of 'good' sports students she is trying to form?

(A) It is not a set because 'good' is a subjective term.

(B) It is a set because 'good' is defined based on a specific criterion (inter-school performance).

(C) It is a set, but the definition of 'good' is not well-defined enough.

(D) It is a relation, not a set.

Question 2. Referring to Question 1, let the set of students be $S = \{\text{Rahul, Priya, Amit, Sneha}\}$ and the set of sports be $P = \{\text{Cricket, Badminton, Tennis}\}$. The teacher decides to represent the information as a set of ordered pairs (Student, Sport). Which of the following is the correct representation in roster form?

(A) $\{(\text{Rahul, Cricket}), (\text{Priya, Badminton}), (\text{Amit, Cricket}), (\text{Sneha, Tennis})\}$

(B) $\{\text{Rahul, Priya, Amit, Sneha, Cricket, Badminton, Tennis}\}$

(C) $\{(\text{Cricket, Rahul}), (\text{Badminton, Priya}), (\text{Cricket, Amit}), (\text{Tennis, Sneha})\}$

(D) $\{(\text{Rahul, Priya, Amit, Sneha}), (\text{Cricket, Badminton, Tennis})\}$

Question 3. A shopkeeper is creating a list of items available in his shop, categorized by type. He lists Mobiles (M), Laptops (L), and Tablets (T). For Mobiles, he has models A and B. For Laptops, models X and Y. For Tablets, models P, Q, and R. He wants to represent all possible combinations of (Item Type, Model) as a set. Which of the following describes this set using set-builder form?

(A) $\{(t, m) : t \in \{\text{M, L, T}\}, m \in \{\text{A, B, X, Y, P, Q, R}\}\}$

(B) $\{(t, m) : t \in \{\text{M, L, T}\} \text{ and } m \text{ is a model of type } t\}$

(C) $\{\text{M, L, T}\} \times \{\text{A, B, X, Y, P, Q, R}\}$

(D) $\{(\text{M, A}), (\text{M, B}), (\text{L, X}), (\text{L, Y}), (\text{T, P}), (\text{T, Q}), (\text{T, R})\}$

Question 4. An economist is analyzing populations of states in India. She considers the set of natural numbers $\mathbb{N}$ for population counts (in lakhs). She wants to represent the population of states A, B, and C as a set. State A has a population of 120 lakhs, State B has 85 lakhs, and State C has 250 lakhs. Which of the following correctly describes the set of populations $\{120, 85, 250\}$ using standard set notations and representation?

(A) $\{p \in \mathbb{N} : p \text{ is the population of state A, B, or C}\}$

(B) $\{120, 85, 250\} \subset \mathbb{N}$

(C) $\{x \in \mathbb{Z} : x \text{ is the population of state A, B, or C}\}$

(D) $\{120, 85, 250\}$ is a subset of $\mathbb{R}$.

Question 5. A restaurant offers two types of starters (Veg, Non-Veg) and three types of main courses (Indian, Chinese, Continental). A customer chooses one starter and one main course. We can represent the possible choices as ordered pairs (Starter Type, Main Course Type). Let $S = \{\text{Veg, Non-Veg}\}$ and $M = \{\text{Indian, Chinese, Continental}\}$. Which of the following is the Cartesian product $S \times M$ in roster form?

(A) $\{(\text{Veg, Indian}), (\text{Veg, Chinese}), (\text{Veg, Continental}), (\text{Non-Veg, Indian}), (\text{Non-Veg, Chinese}), (\text{Non-Veg, Continental})\}$

(B) $\{(\text{Veg, Non-Veg}), (\text{Indian, Chinese, Continental})\}$

(C) $\{(\text{Veg, Indian}), (\text{Non-Veg, Chinese}), (\text{Veg, Continental})\}$

(D) $\{(\text{Indian, Veg}), (\text{Chinese, Veg}), (\text{Continental, Veg}), (\text{Indian, Non-Veg}), (\text{Chinese, Non-Veg}), (\text{Continental, Non-Veg})\}$

Question 1. Consider the following collections: Collection 1: The set of all Indian citizens living in Mumbai. Collection 2: The set of all numbers $x$ such that $x \in \mathbb{Q}$ and $x^2 = 7$. Collection 3: The set of all grains of sand on all beaches in Goa. Collection 4: The set of all odd integers. Which of these collections represent empty sets?

(A) Collection 1

(B) Collection 2

(C) Collection 3

(D) Collection 4

Question 2. Referring to Question 1, which of the collections represent infinite sets?

(A) Collection 1

(B) Collection 2

(C) Collection 3

(D) Collection 4

Question 3. A survey is conducted in a class of 40 students to find out how many have a driving license. Let A be the set of students with a driving license. If 15 students have a license, what is the cardinal number of set A?

(A) $n(A) = 40$

(B) $n(A) = 15$

(C) $n(A) = 40 - 15$

(D) The cardinal number cannot be determined from the given information.

Question 4. Consider the set $S = \{\text{Red, Blue}\}$. Which of the following is the power set $P(S)$?

(A) $\{\text{Red, Blue}\}$

(B) $\{\phi, \{\text{Red}\}, \{\text{Blue}\}, \{\text{Red, Blue}\}\}

(C) $\{\text{Red}\}, \{\text{Blue}\}, \{\text{Red, Blue}\}$

(D) $\phi, \{\text{Red}\}, \{\text{Blue}\}, \{\text{Red, Blue}\}$

Question 5. In a study of different types of vehicles on the road in a city, the set of all vehicles on the road is considered as the universal set U. Let A be the set of all cars and B be the set of all two-wheelers. Which statement is true regarding A and B in this context?

(A) A is a subset of B.

(B) B is a subset of A.

(C) A and B are subsets of U.

(D) U is a subset of A.

Question 1. A group of students {Rahul, Simran, Rohit, Kavita} are in Class 11. Their ages are Rahul (16), Simran (17), Rohit (16), Kavita (17). We define a relation R on the set of students by $x R y$ if "x and y are of the same age". Which of the following ordered pairs is in the relation R?

(A) (Rahul, Rohit)

(B) (Rahul, Simran)

(C) (Simran, Rohit)

(D) (Rohit, Kavita)

Question 2. Referring to Question 1, let the set of students be $S = \{\text{Rahul, Simran, Rohit, Kavita}\}$. What is the set representation of the relation R?

(A) $\{(\text{Rahul, Rahul}), (\text{Simran, Simran}), (\text{Rohit, Rohit}), (\text{Kavita, Kavita})\}$

(B) $\{(\text{Rahul, Rohit}), (\text{Rohit, Rahul}), (\text{Simran, Kavita}), (\text{Kavita, Simran})\}$

(C) $\{(\text{Rahul, Rahul}), (\text{Simran, Simran}), (\text{Rohit, Rohit}), (\text{Kavita, Kavita}), (\text{Rahul, Rohit}), (\text{Rohit, Rahul}), (\text{Simran, Kavita}), (\text{Kavita, Simran})\}$

(D) $\{(\text{Rahul, Simran}), (\text{Rohit, Kavita})\}$

Question 3. Consider two sets: Set A = {Cities in India with population > 1 Crore}, Set B = {States in India}. We define a relation R from A to B by $c R s$ if "city c is the capital of state s". Which of the following statements is true?

(A) The ordered pair (Delhi, Delhi) might be in R.

(B) The ordered pair (Mumbai, Maharashtra) is in R.

(C) The ordered pair (Kolkata, West Bengal) is in R.

(D) The domain of R is the set B.

Question 4. Let $A=\{1, 2, 3, 4\}$. A relation R on A is defined by $R = \{(x, y) : x \text{ is a factor of } y\}$. What is the domain of this relation R?

(A) $\{1, 2, 3, 4\}$

(B) $\{1, 2, 3\}$

(C) $\{1, 2\}$

(D) $\{4\}$

Question 5. Referring to Question 4, what is the range of the relation R?

(A) $\{1, 2, 3, 4\}$

(B) $\{1, 2, 3\}$

(C) $\{1, 2\}$

(D) $\{4\}$

Question 1. Consider the set of all integers $\mathbb{Z}$. Let R be the relation defined by $a R b$ if $a+b$ is an even integer.

Check if R is reflexive. Is $a+a$ always even for any integer $a$?

(A) Yes, R is reflexive because $a+a = 2a$ which is always even.

(B) No, R is not reflexive because $a+a$ is not always even.

(C) R is reflexive only if 'a' is an even integer.

(D) R is reflexive only if 'a' is an odd integer.

Question 2. Referring to Question 1, check if R is symmetric. If $a+b$ is even, is $b+a$ also even?

(A) Yes, R is symmetric because $b+a = a+b$, and if $a+b$ is even, $b+a$ is also even.

(B) No, R is not symmetric in general.

(C) R is symmetric only if $a=b$.

(D) R is symmetric only if $a+b=0$.

Question 3. Referring to Question 1, check if R is transitive. If $a+b$ is even and $b+c$ is even, is $a+c$ always even?

Hint: If $a+b = 2k_1$ and $b+c = 2k_2$, then $(a+b)+(b+c) = 2k_1 + 2k_2 = 2(k_1+k_2)$. This implies $a+2b+c = 2(k_1+k_2)$. Since $2b$ is even, for $a+2b+c$ to be even, $a+c$ must be even.

(A) Yes, R is transitive because if $a+b$ and $b+c$ are even, $a+c$ must be even.

(B) No, R is not transitive in general.

(C) R is transitive only if $b=0$.

(D) R is transitive only if $a=c$.

Question 4. Based on your findings in Questions 1, 2, and 3, what type of relation is R (defined by $a R b$ if $a+b$ is even) on the set of integers $\mathbb{Z}$?

(A) Reflexive only

(B) Symmetric and Transitive only

(C) Reflexive, Symmetric, and Transitive (Equivalence Relation)

(D) None of the above properties hold for all integers.

Question 5. Referring to the equivalence relation R from Question 4, what is the equivalence class of the integer 0?

The equivalence class of 0 is the set of all integers $x$ such that $x R 0$, i.e., $x+0$ is even.

(A) The set of all odd integers

(B) The set of all even integers

(C) $\{0\}$

(D) $\mathbb{Z}$

Question 1. In a colony, there are 100 families. 60 families own a car (C), and 40 families own a scooter (S). 20 families own both a car and a scooter. We can use set operations to analyze this data. What is the set representing families who own a car OR a scooter OR both?

(A) $C \cap S$

(B) $C \cup S$

(C) $C - S$

(D) $(C \cup S)'$

Question 2. Referring to Question 1, how many families own only a car?

(A) $n(C \cup S) - n(S)$

(B) $n(C) - n(C \cap S)$

(C) $n(C)$

(D) $n(C \cap S)$

Question 3. Referring to Question 1, how many families own at least one of the two vehicles (car or scooter)?

(A) $n(C) + n(S)$

(B) $n(C) + n(S) + n(C \cap S)$

(C) $n(C) + n(S) - n(C \cap S)$

(D) $100 - n((C \cup S)')$

Question 4. Referring to Question 1, how many families own neither a car nor a scooter? (The universal set U is the set of 100 families).

(A) $n(U) - n(C \cup S)$

(B) $n(U) - n(C \cap S)$

(C) $n(C') + n(S')$

(D) $n(C' \cap S')$

Question 5. A Venn diagram is used to illustrate the relationships between sets C (families with car) and S (families with scooter) within the universal set U (all families). Which area in the Venn diagram would represent the families who own a car but not a scooter?

(Assume the regions are labeled: I - C only, II - C and S, III - S only, IV - Neither)

(A) Region I

(B) Region II

(C) Region III

(D) Region IV

Question 1. In a market survey of 120 customers, 80 liked product A (set A) and 60 liked product B (set B). Every customer liked at least one product. Using the principle of inclusion-exclusion, find the number of customers who liked both products A and B.

(A) $n(A \cap B) = n(A) + n(B) + n(A \cup B)$

(B) $n(A \cap B) = n(A) + n(B) - n(A \cup B)$

(C) $n(A \cap B) = 80 + 60 + 120$

(D) $n(A \cap B) = 80 + 60 - 120$

Question 2. Referring to Question 1, calculate the numerical value for the number of customers who liked both products A and B.

(A) 20

(B) 40

(C) 140

(D) 260

Question 3. Consider a group of 90 people. 40 read newspaper A, 30 read newspaper B, and 20 read newspaper C. 10 read A and B, 8 read B and C, 5 read A and C. 3 people read all three newspapers. Let A, B, and C represent the sets of people reading newspapers A, B, and C respectively. Which formula would you use to find the number of people who read at least one newspaper?

(A) $n(A \cup B \cup C) = n(A)+n(B)+n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C)$

(B) $n(A \cup B \cup C) = n(A)+n(B)+n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$

(C) $n(A \cup B \cup C) = n(A)+n(B)+n(C) + n(A \cap B \cap C)$

(D) $n(A \cup B \cup C) = 90 - n(\text{none})$

Question 4. Referring to Question 3, calculate the number of people who read at least one newspaper.

(A) 70

(B) 72

(C) 80

(D) 90

Question 5. Referring to Question 3, how many people read exactly one newspaper?

Hint: Use the formula $n(\text{only A}) = n(A) - n(A \cap B) - n(A \cap C) + n(A \cap B \cap C)$, and similarly for B and C, then sum them up.

(A) 40

(B) 30

(C) 50

(D) 60

Question 1. A taxi service charges a fare based on the distance traveled. The fare F (in $\textsf{₹}$) is calculated as $F(d) = 10d + 50$, where $d$ is the distance in kilometers. Here, the distance $d$ must be a non-negative real number. What is the domain of this function in the context of real-world distance?

(A) $\mathbb{R}$ (All real numbers)

(B) $[0, \infty)$ (Non-negative real numbers)

(C) $\mathbb{N}$ (Natural numbers)

(D) $(0, \infty)$ (Positive real numbers)

Question 2. Referring to Question 1, what is the range of the function $F(d) = 10d + 50$, given the domain $[0, \infty)$?

(A) $[50, \infty)$

(B) $[0, \infty)$

(C) $\mathbb{R}$

(D) $[60, \infty)$

Question 3. Consider the mapping that assigns each student in a class to their roll number. Let the set of students be A and the set of roll numbers be B. For this mapping to be a function, which condition must hold?

(A) Every roll number in B must be assigned to a student in A.

(B) Every student in A must be assigned to exactly one roll number in B.

(C) Each roll number must be unique.

(D) The number of students must be equal to the number of roll numbers.

Question 4. A function is defined as $f(x) = \frac{1}{\sqrt{x-3}}$. This function is used to calculate a value based on an input $x$. What is the domain of this real function?

(A) $x > 3$

(B) $x \geq 3$

(C) $x < 3$

(D) $\mathbb{R} - \{3\}$

Question 5. The area A of a circle is a function of its radius r, given by $A(r) = \pi r^2$. In the context of a real-world circle, the radius must be a positive value. Given the domain $(0, \infty)$ for $r$, what is the range of this function $A(r)$?

(A) $(0, \infty)$

(B) $[0, \infty)$

(C) $\mathbb{R}$

(D) Positive integers

Question 1. A school assigns a unique roll number to each student. Let A be the set of students and B be the set of assigned roll numbers. The function maps each student to their roll number. What type of function is this mapping?

(A) Many-to-one

(B) Onto

(C) One-to-one

(D) Constant function

Question 2. Consider a function $f: \mathbb{Z} \to \mathbb{Z}$ defined by $f(x) = x + 2$. We are mapping integers to integers by adding 2. Is this function one-to-one? If $f(x_1) = f(x_2)$, does $x_1 = x_2$?

(A) Yes, if $x_1+2 = x_2+2$, then $x_1 = x_2$, so it's one-to-one.

(B) No, different integers can map to the same integer.

(C) It's one-to-one only for positive integers.

(D) It's one-to-one only for negative integers.

Question 3. Referring to Question 2, is the function $f: \mathbb{Z} \to \mathbb{Z}$ defined by $f(x) = x + 2$ onto? Is every integer $y$ in the codomain $\mathbb{Z}$ the image of some integer $x$ in the domain $\mathbb{Z}$?

(A) Yes, for any integer $y$, we can find an integer $x = y-2$ such that $f(x) = y$. So, it's onto.

(B) No, only even integers are in the range.

(C) No, only integers greater than 2 are in the range.

(D) It's onto only for positive integers.

Question 4. Based on your findings in Questions 2 and 3, what type of function is $f: \mathbb{Z} \to \mathbb{Z}$ defined by $f(x) = x+2$?

(A) One-to-one but not onto

(B) Onto but not one-to-one

(C) Bijective (Both one-to-one and onto)

(D) Neither one-to-one nor onto

Question 5. A company offers a discount of $\textsf{₹}10$ on all products priced above $\textsf{₹}100$. For products priced $\textsf{₹}100$ or below, there is no discount. Let $P$ be the original price and $D(P)$ be the discounted price. This can be modeled by a function. If the domain is the set of positive real numbers representing prices, which of the following is true about the function's type?

(A) It is a one-to-one function.

(B) It is an onto function (assuming the range is the set of possible discounted prices).

(C) It is a many-to-one function (e.g., original prices $\textsf{₹}101, \textsf{₹}102$ map to discounted prices $\textsf{₹}91, \textsf{₹}92$, while prices $\textsf{₹}50, \textsf{₹}60$ map to $\textsf{₹}50, \textsf{₹}60$).

(D) It is a constant function.

Question 1. The cost C (in $\textsf{₹}$) of producing x units of a product is given by the function $C(x) = 10x + 500$, where x is a non-negative integer representing the number of units produced. Is this a real function?

(A) Yes, because the input x is from a subset of $\mathbb{R}$ (non-negative integers) and the output C(x) is also a real number.

(B) No, because the domain is not $\mathbb{R}$ or an interval of $\mathbb{R}$.

(C) Yes, because the formula involves arithmetic operations on real numbers.

(D) No, because the domain is restricted to integers.

Question 2. A stone is dropped from a height. The distance fallen, $s$ (in metres), after time $t$ (in seconds) is given by $s(t) = 5t^2$ (approximately, ignoring air resistance). In this scenario, $t$ must be non-negative. What is the domain of this real function in the context of this physical scenario?

(A) $\mathbb{R}$

(B) $[0, \infty)$

(C) $\mathbb{N}$

(D) $(-\infty, \infty)$

Question 3. Referring to Question 2, what is the range of the function $s(t) = 5t^2$ given the domain $[0, \infty)$?

(A) $(0, \infty)$

(B) $[0, \infty)$

(C) $(0, \infty)$

(D) Integers $\geq 0$ starting from 0

Question 4. The graph of the function $f(x) = \lfloor x \rfloor$ (the greatest integer function) is a series of horizontal line segments. For example, $\lfloor 1.5 \rfloor = 1$, $\lfloor 2.9 \rfloor = 2$. What is the range of this function?

(A) $\mathbb{R}$

(B) $\mathbb{Z}$

(C) $\mathbb{N}$

(D) $(-\infty, \infty)$

Question 5. A company's daily profit P (in $\textsf{₹}$ lakhs) is modeled by the function $P(x) = -x^2 + 10x - 20$, where $x$ is the number of units produced (in thousands). Assume the domain is $[0, 10]$ (thousand units). What is the maximum profit the company can make based on this model?

Hint: Find the vertex of the parabola $y = -x^2 + 10x - 20$. The x-coordinate of the vertex is $-b/(2a)$. The y-coordinate is the maximum value.

(A) $\textsf{₹}5$ lakhs

(B) $\textsf{₹}30$ lakhs

(C) $\textsf{₹}0$ lakhs

(D) $\textsf{₹}25$ lakhs

Question 1. A manufacturer's cost C (in $\textsf{₹}$) for producing $x$ items is $C(x) = 50x + 1000$. The revenue R (in $\textsf{₹}$) from selling $x$ items is $R(x) = 150x$. The profit P is given by $P(x) = R(x) - C(x)$. What is the profit function $P(x)$?

(A) $P(x) = (150x)(50x + 1000)$

(B) $P(x) = 150x + 50x + 1000$

(C) $P(x) = 150x - (50x + 1000)$

(D) $P(x) = (50x + 1000) - 150x$

Question 2. Referring to Question 1, calculate the numerical value of the profit function $P(x)$.

(A) $100x - 1000$

(B) $100x + 1000$

(C) $200x + 1000$

(D) $100x$

Question 3. Suppose the cost of raw material increases, changing the cost function to $C_2(x) = 60x + 1200$. The revenue function remains $R(x) = 150x$. The new profit function is $P_2(x) = R(x) - C_2(x)$. What is $P_2(x)$?

(A) $90x - 1200$

(B) $90x + 1200$

(C) $210x + 1200$

(D) $150x - 60x$

Question 4. A tailor first cuts a length of cloth $L$ by a function $f(L) = L - 0.5$ (losing 0.5m due to error/waste). Then, he stitches the remaining cloth, and the time taken to stitch a length $l$ is given by the function $g(l) = 2l + 10$ minutes. The total time taken for stitching a length $L$ of initial cloth is given by the composite function $(g \circ f)(L)$. What is the expression for $(g \circ f)(L)$?

(A) $g(f(L)) = g(L - 0.5) = 2(L - 0.5) + 10$

(B) $f(g(L)) = f(2L + 10) = (2L + 10) - 0.5$

(C) $g(L) + f(L) = (2L + 10) + (L - 0.5)$

(D) $g(L) \times f(L) = (2L + 10)(L - 0.5)$

Question 5. Referring to Question 4, simplify the expression for $(g \circ f)(L)$.

(A) $2L + 9.5$

(B) $3L + 9.5$

(C) $2L + 10$

(D) $2L + 0.5$

Question 1. A currency exchange office converts US Dollars (USD) to Indian Rupees (INR). The function for conversion is approximately $R(D) = 75D$, where D is the amount in USD and R is the amount in INR. If someone wants to convert INR back to USD, they would need the inverse function. Assuming this function is invertible for positive real numbers, what is the inverse function $D(R)$ that converts INR (R) back to USD (D)?

(A) $D(R) = R/75$

(B) $D(R) = 75R$

(C) $D(R) = R - 75$

(D) $D(R) = R + 75$

Question 2. In a code, a message represented by a number $x$ is encrypted using the function $E(x) = 2x + 5$. To decrypt the message, the inverse function $D(y)$ is needed. If the encryption is done over the set of real numbers $\mathbb{R}$, what is the decryption function $D(y)$?

(A) $D(y) = 2y - 5$

(B) $D(y) = (y - 5)/2$

(C) $D(y) = (y + 5)/2$

(D) $D(y) = y/2 - 5$

Question 3. Consider the set of even integers $E = \{..., -4, -2, 0, 2, 4, ...\}$. Is addition (+) a binary operation on E?

(A) Yes, because the sum of any two even integers is always an even integer, which is in E.

(B) No, because the sum of two even integers can be odd.

(C) Yes, but only for positive even integers.

(D) No, the set E is not closed under addition.

Question 4. Consider the set of rational numbers $\mathbb{Q}$. We define a binary operation $*$ by $a * b = a + b - ab$. Is this operation commutative?

(A) Yes, because $a+b-ab = b+a-ba$ for all $a, b \in \mathbb{Q}$.

(B) No, because $a+b-ab$ is not always equal to $b+a-ba$.

(C) It is commutative only if $a=b$.

(D) It is commutative only if $ab=0$.

Question 5. Consider the set of positive integers $\mathbb{N} = \{1, 2, 3, ...\}$. Is there an identity element for the operation of multiplication ($\times$) on $\mathbb{N}$?

(A) Yes, 0 is the identity element because $a \times 0 = a$ for all $a \in \mathbb{N}$.

(B) Yes, 1 is the identity element because $a \times 1 = a$ and $1 \times a = a$ for all $a \in \mathbb{N}$.

(C) No, there is no element $e \in \mathbb{N}$ such that $a \times e = a$ for all $a \in \mathbb{N}$.

(D) The identity element exists, but it is not unique.