CBSE Class 12 - Model Question Paper

SECTION A — Multiple Choice Questions [1 Mark Each]

1 Mark Q1. Let R be a relation on the set L of all lines in a plane defined by R = {(L₁, L₂) : L₁ is perpendicular to L₂}. Then R is

A) Reflexive and Symmetric

B) Symmetric but neither Reflexive nor Transitive

C) Equivalence Relation

D) Transitive but not Symmetric
1 Mark Q2. The number of all equivalence relations on the set {1, 2, 3} containing (1, 2) and (2, 1) is

A) 1

B) 2

C) 3

D) 4
1 Mark Q3. Let f: R → R be defined as f(x) = x⁴. Choose the correct answer

A) f is one-one onto

B) f is many-one onto

C) f is one-one but not onto

D) f is neither one-one nor onto
1 Mark Q4. If A = {1, 2, 3}, then the number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is

A) 1

B) 2

C) 3

D) 4
1 Mark Q5. Let f: A → B and g: B → C be two functions such that gof is injective. Then

A) f must be injective

B) g must be injective

C) both f and g must be injective

D) g must be surjective
1 Mark Q6. The function f: N → N defined by f(n) = n+1 if n is odd and f(n) = n-1 if n is even, is

A) One-one but not onto

B) Onto but not one-one

C) Bijective

D) Neither one-one nor onto
1 Mark Q7. If f(x) = (3 - x³)^1/3, then f(f(x)) is

A) x^1/3

B) "x³"

C) "x"

D) 3 - x³
1 Mark Q8. Let A = {1, 2, 3}. The number of binary operations on A is

A) 3³

B) 3⁶

C) 3⁹

D) 2³
1 Mark Q9. A function f: A → B is onto if

A) Range of f ⊂ B

B) Range of f = B

C) Range of f ⊃ B

D) None of these
1 Mark Q10. If f(x) = 8x³ and g(x) = x^1/3, then fog(x) is

A) 8x

B) 2x

C) 8x³

D) x

SECTION B — Very Short Answer Type Questions [2 Marks Each]

2 Marks Q1. If $A = \{1, 2, 3\}$, check if the relation $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)\}$ is transitive.
2 Marks Q2. Let $R$ be a relation on the set of natural numbers $\mathbb{N}$ defined by $xRy$ if $x + 2y = 8$. Write $R$ as a set of ordered pairs.
2 Marks Q3. State the reason why the relation $R = \{(a, b) : a \le b^2\}$ on the set of real numbers $\mathbb{R}$ is not reflexive.
2 Marks Q4. Show that the relation $R$ in the set $A = \{1, 2, 3\}$ given by $R = \{(1, 2), (2, 1)\}$ is symmetric but neither reflexive nor transitive.
2 Marks Q5. A relation $R$ on the set of real numbers $\mathbb{R}$ is defined as $R = \{(a, b) : a - b + \sqrt{3} \text{ is an irrational number}\}$. Check if $R$ is reflexive.
2 Marks Q6. If $R = \{(x, y) : x^2 - y^2 < 1\}$ is a relation on the set $\{1, 2, 3, 4\}$, write its domain.
2 Marks Q7. Let $A = \{a, b, c\}$. Find the total number of distinct equivalence relations that can be defined on $A$ containing the element $(a, b)$.
2 Marks Q8. Check whether the relation $R$ defined on the set $A = \{1, 2, 3, 4, 5, 6\}$ as $R = \{(x, y) : y = x + 1\}$ is symmetric.
2 Marks Q9. Determine whether the relation $R$ on the set of integers $\mathbb{Z}$ defined as $R = \{(x, y) : x - y \text{ is divisible by 5}\}$ is symmetric.
2 Marks Q10. Show that the relation $R$ on the set of all straight lines in a plane defined by $L_1 R L_2 \iff L_1 \perp L_2$ is symmetric but not transitive.
2 Marks Q11. Test the transitivity of the relation $R$ on $\mathbb{R}$ defined by $aRb \iff 1 + ab > 0$.
2 Marks Q12. Let $R$ be a relation on the set $A = \{1, 2, 3, 4\}$ given by $R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (3, 1)\}$. Is $R$ an equivalence relation? Justify.
2 Marks Q13. If $R = \{(a, b) : a, b \in \mathbb{N} \text{ and } a = b^2\}$, check if $R$ is transitive.
2 Marks Q14. Let $R$ be an equivalence relation on $\mathbb{Z}$ defined by $aRb$ if $a - b$ is an even integer. Find the equivalence class $[0]$.
2 Marks Q15. For the relation $R = \{(x, y) : x, y \in \mathbb{Z}, x \equiv y \pmod 3\}$, find the equivalence class $[2]$.
2 Marks Q16. If $A = \{1, 2, 3\}$, what is the smallest equivalence relation containing $(1, 2)$?
2 Marks Q17. If $A = \{1, 2, 3\}$, what is the largest equivalence relation that can be formed on $A$?
2 Marks Q18. Let $R$ be a relation on $\mathbb{N} \times \mathbb{N}$ defined by $(a, b) R (c, d) \iff a+d = b+c$. Show that $(1, 2) R (3, 4)$.
2 Marks Q19. Prove that the empty relation on a non-empty set $A$ is symmetric and transitive but not reflexive.
2 Marks Q20. If $R_1$ and $R_2$ are two equivalence relations on a set $A$, prove that $R_1 \cap R_2$ is also an equivalence relation.
2 Marks Q21. Give an example of a relation which is reflexive and transitive but not symmetric
2 Marks Q22. Give an example of a relation which is symmetric and transitive but not reflexive.
2 Marks Q23. Let $R = \{(a, b) : a \le b\}$ be a relation on $\mathbb{R}$. Prove that $R$ is transitive.
2 Marks Q24. Find the number of all possible relations on the set $A = \{1, 2, 3\}$.
2 Marks Q25. If $R$ is a relation on the set $A = \{1, 2, 3, 4\}$ defined by $R = \{(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)\}$, find its range.
2 Marks Q26. Check the injectivity of the function $f: \mathbb{N} \to \mathbb{N}$ given by $f(x) = x^2$.
2 Marks Q27. Check the injectivity of the function $f: \mathbb{Z} \to \mathbb{Z}$ given by $f(x) = x^2$.
2 Marks Q28. Show that the constant function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = c$ (where $c$ is a constant) is neither one-one nor onto if $\mathbb{R}$ has more than one element.
2 Marks Q29. Check if the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = |x|$ is injective.
2 Marks Q30. Show that the Modulus function $f: \mathbb{R} \to \mathbb{R}$, given by $f(x) = |x|$, is not surjective.
2 Marks Q31. Check whether the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 3 - 4x$ is surjective.
2 Marks Q32. Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = x^3$. Prove that $f$ is injective.
2 Marks Q33. Examine if the function $f: \mathbb{N} \to \mathbb{N}$ given by $f(x) = x + 1$ is onto.
2 Marks Q34. Show that the Signum function $f: \mathbb{R} \to \mathbb{R}$ is neither one-one nor onto.
2 Marks Q35. Check the bijectivity of the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 2x + 5$.
2 Marks Q36. Let $A = \{1, 2, 3\}$ and $B = \{4, 5, 6, 7\}$. Let $f = \{(1, 4), (2, 5), (3, 6)\}$ be a function from $A$ to $B$. Show that $f$ is one-one but not onto.
2 Marks Q37. If $f: \mathbb{R} \to \mathbb{R}$ is defined by $f(x) = \frac{x}{x^2+1}$, find whether the function is one-one.
2 Marks Q38. Prove that the Greatest Integer Function $f: \mathbb{R} \to \mathbb{R}$, given by $f(x) = [x]$, is neither one-one nor onto.
2 Marks Q39. Let $f: \mathbb{R} - \{3\} \to \mathbb{R} - \{1\}$ be defined by $f(x) = \frac{x-2}{x-3}$. Show that $f$ is one-one.
2 Marks Q40. Using the function in Q39, check if $f$ is onto.
2 Marks Q41. If a set $A$ contains 3 elements and set $B$ contains 4 elements, find the number of one-one functions from $A$ to $B$.
2 Marks Q42. If a set $A$ contains 4 elements and set $B$ contains 3 elements, find the number of one-one functions from $A$ to $B$.
2 Marks Q43. Find the total number of onto functions from a set $A = \{1, 2, 3\}$ to itself.
2 Marks Q44. If $f: A \to B$ is a bijection where $n(A) = 5$, what must be the value of $n(B)$?
2 Marks Q45. Let $f: [0, \infty) \to [0, \infty)$ be defined by $f(x) = x^2$. Show that $f$ is a bijection.
2 Marks Q46. State the condition under which a linear function $f(x) = mx + c$ (where $m \neq 0$) from $\mathbb{R} \to \mathbb{R}$ is a bijective function.
2 Marks Q47. Let $f: \mathbb{N} \to \mathbb{Z}$ be defined by: $$f(n) = \begin{cases} \frac{n}{2}, & \text{if } n \text{ is even} \\ -\frac{n-1}{2}, & \text{if } n \text{ is odd} \end{cases}$$ Find the value of $f(4) + f(5)$.
2 Marks Q48. If $f: \mathbb{R} \to \mathbb{R}$ is defined by $f(x) = \sin x$, find the range of $f$. Is it onto?
2 Marks Q49. If $A = \{-1, 1\}$, find the number of all onto functions from $A$ to $A$.
2 Marks Q50. Give an example of a function $f: \mathbb{N} \to \mathbb{N}$ which is onto but not one-one.

SECTION C — Short Answer Type Questions [3 Marks Each]

3 Marks Q1. Show that the relation $R$ in the set $A = \{1, 2, 3, 4, 5\}$ given by $R = \{(a, b) : |a - b| \text{ is even}\}$ is an equivalence relation.
3 Marks Q2. Prove that the relation $R$ on the set $\mathbb{Z}$ of all integers defined by $(x, y) \in R \iff (x - y)$ is divisible by 5 is an equivalence relation.
3 Marks Q3. Let $A = \{x \in \mathbb{Z} : 0 \le x \le 12\}$. Show that the relation $R = \{(a, b) : |a - b| \text{ is a multiple of } 4\}$ is an equivalence relation. Find the equivalence class [1].
3 Marks Q4. Show that the relation $R$ defined on the set $A$ of all polygons as $R = \{(P_1, P_2) : P_1 \text{ and } P_2 \text{ have same number of sides}\}$ is an equivalence relation.
3 Marks Q5. Let $L$ be the set of all lines in a XY-plane and $R$ be the relation in $L$ defined as $R = \{(L_1, L_2) : L_1 \text{ is parallel to } L_2\}$. Show that $R$ is an equivalence relation.
3 Marks Q6. Prove that the relation $R$ on the set $N \times N$ defined by $(a, b) R (c, d) \iff a + d = b + c$ is an equivalence relation.
3 Marks Q7. Let $R$ be a relation on the set $N \times N$ defined by $(a, b) R (c, d) \iff ad = bc$. Show that $R$ is an equivalence relation.
3 Marks Q8. Show that the relation $R$ on the set $A = \mathbb{Z}$ defined by $R = \{(a, b) : 2 \text{ divides } (a - b)\}$ is an equivalence relation.
3 Marks Q9. Let $A$ be the set of all books in a library of a college. $R$ is a relation on $A$ given by $R = \{(x, y) : x \text{ and } y \text{ have the same number of pages}\}$. Check if $R$ is an equivalence relation.
3 Marks Q10. Show that the relation $R$ in the set $A = \{1, 2, 3\}$ given by $R = \{(1,1), (2,2), (3,3), (1,2), (2,1)\}$ is an equivalence relation.
3 Marks Q11. Show that the relation $R$ in the set $\mathbb{R}$ of real numbers, defined as $R = \{(a, b) : a \le b^2\}$, is neither reflexive nor symmetric nor transitive.
3 Marks Q12. Check whether the relation $R$ defined in the set $\{1, 2, 3, 4, 5, 6\}$ as $R = \{(a, b) : b = a + 1\}$ is reflexive, symmetric, or transitive.
3 Marks Q13. Show that the relation $R$ in the set $\mathbb{R}$ of real numbers, defined as $R = \{(a, b) : a \le b^3\}$, is neither reflexive nor symmetric nor transitive.
3 Marks Q14. Examine if the relation $R$ on the set $\mathbb{R}$ defined by $R = \{(a, b) : 1 + ab > 0\}$ is reflexive, symmetric, and transitive.
3 Marks Q15. Let $R$ be a relation on the set $A = \{1, 2, 3, 4\}$ given by $R = \{(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)\}$. Is $R$ an equivalence relation? Justify.
3 Marks Q16. Determine whether the relation $R$ in the set $A$ of human beings in a town at a particular time given by $R = \{(x, y) : x \text{ is exactly 7 cm taller than } y\}$ is reflexive, symmetric, or transitive.
3 Marks Q17. Show that the relation $R$ on the set $\mathbb{R}$ of real numbers defined as $R = \{(a, b) : a \le b\}$ is reflexive and transitive but not symmetric.
3 Marks Q18. Let $R$ be a relation on the set of natural numbers $\mathbb{N}$ defined by $R = \{(x, y) : x + 4y = 10\}$. Find the domain and range of $R$. Is it reflexive?
3 Marks Q19. Prove that the perpendicular relation $R = \{(L_1, L_2) : L_1 \perp L_2\}$ on the set of all lines in a plane is symmetric but neither reflexive nor transitive.
3 Marks Q20. Let $A = \{1, 2, 3\}$. Write the smallest equivalence relation containing $(1, 2)$ and the largest equivalence relation on $A$.
3 Marks Q21. If $R = \{(a, b) : a^2 - b^2 \text{ is divisible by 3}\}$ is an equivalence relation on integers $\mathbb{Z}$, find all the distinct equivalence classes.
3 Marks Q22. Let $R$ be a relation on the set $A = \{x \in \mathbb{Z} : 0 \le x \le 10\}$ given by $R = \{(a, b) : |a - b| \text{ is a multiple of 3}\}$. Find the equivalence class of $[2]$.
3 Marks Q23. Show that the relation $R = \{(a, b) : a \text{ divides } b\}$ on the set $\mathbb{N}$ of natural numbers is reflexive and transitive but not symmetric.
3 Marks Q24. Prove that the intersection of two equivalence relations on a set $A$ is also an equivalence relation on $A$.
3 Marks Q25. Is the union of two equivalence relations on a set $A$ necessarily an equivalence relation? Give an example to justify your answer.
3 Marks Q26. Show that the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 3 - 4x$ is bijective.
3 Marks Q27. Check the injectivity and surjectivity of the function $f: \mathbb{N} \to \mathbb{N}$ given by $f(x) = x^2$.
3 Marks Q28. Check the injectivity and surjectivity of the function $f: \mathbb{Z} \to \mathbb{Z}$ given by $f(x) = x^2$.
3 Marks Q29. Show that the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^3$ is a bijection.
3 Marks Q30. Let $A = \mathbb{R} - \{3\}$ and $B = \mathbb{R} - \{1\}$. Consider the function $f: A \to B$ defined by $f(x) = \frac{x - 2}{x - 3}$. Is $f$ one-one and onto? Justify.
3 Marks Q31. Show that the Modulus Function $f: \mathbb{R} \to \mathbb{R}$, given by $f(x) = |x|$, is neither one-one nor onto.
3 Marks Q32. Show that the Greatest Integer Function $f: \mathbb{R} \to \mathbb{R}$, given by $f(x) = [x]$, is neither one-one nor onto.
3 Marks Q33. Show that the Signum Function $f: \mathbb{R} \to \mathbb{R}$, given by: $$f(x) = \begin{cases} 1 & \text{if } x > 0 \\ 0 & \text{if } x = 0 \\ -1 & \text{if } x < 0 \end{cases}$$ is neither one-one nor onto.
3 Marks Q34. Show that the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \frac{x}{x^2 + 1}$ is neither one-one nor onto.
3 Marks Q35. Let $f: \mathbb{N} \to \mathbb{N}$ be defined by: $$f(n) = \begin{cases} \frac{n+1}{2} & \text{if } n \text{ is odd} \\ \frac{n}{2} & \text{if } n \text{ is even} \end{cases}$$ State whether the function $f$ is bijective. Justify your answer.
3 Marks Q36. Let $A = \{-1, 0, 1, 2\}$, $B = \{-4, -2, 0, 2\}$ and $f, g: A \to B$ be functions defined by $f(x) = x^2 - x$ and $g(x) = 2|x - \frac{1}{2}| - 1$. Are $f$ and $g$ equal? Justify.
3 Marks Q37. Show that an injective function $f: \{1, 2, 3\} \to \{1, 2, 3\}$ must be onto.
3 Marks Q38. Show that a surjective function $f: \{1, 2, 3\} \to \{1, 2, 3\}$ must be one-one.
3 Marks Q39. Examine whether the function $f: [0, \infty) \to \mathbb{R}$ defined by $f(x) = \sqrt{x}$ is one-one and onto.
3 Marks Q40. Consider a function $f: \mathbb{R} \to [-1, 1]$ defined by $f(x) = \sin x$. Is this function bijective? If not, how can you restrict the domain to make it bijective?
3 Marks Q41. Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = 2x^3 - 5$. Prove that $f$ is a bijective function.
3 Marks Q42. Let $A = \mathbb{R} - \{-4/3\}$. Show that the function $f: A \to \mathbb{R}$ defined as $f(x) = \frac{4x}{3x + 4}$ is one-one. Find its range to make it an onto function.
3 Marks Q43. Check the injectivity and surjectivity of the function $f: \mathbb{Q} \to \mathbb{Q}$ defined by $f(x) = 3x + 5$.
3 Marks Q44. Show that the function $f: \mathbb{N} \to \mathbb{N}$ defined by $f(x) = x + 1$ if $x$ is odd, and $f(x) = x - 1$ if $x$ is even, is a bijection.
3 Marks Q45. Prove that the function $f: \mathbb{R} \to \mathbb{R}$ given by $f(x) = a x + b$ where $a, b \in \mathbb{R}$ and $a \neq 0$ is a bijective function.
3 Marks Q46. Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = x^4$. Check if the function is injective, surjective, or both.
3 Marks Q47. Show that the function $f: (-1, 1) \to \mathbb{R}$ defined by $f(x) = \frac{x}{1 - |x|}$ is one-one and onto.
3 Marks Q48. If $f: \mathbb{R} \to \mathbb{R}$ is defined by $f(x) = \frac{3x^2 + x - 1}{x^2 + 1}$, find whether the function is many-one or one-one.
3 Marks Q49. Let $A = \{1, 2, 3, 4\}$ and $B = \{a, b, c\}$. Find the total number of functions from $A$ to $B$. How many of them are one-one?
3 Marks Q50. Prove that the function $f: \mathbb{R} \to \mathbb{R}$ given by $f(x) = x^3 + x$ is an injective function

SECTION D — Case-Based/Source-Based Integrated Questions [4 Marks Each]

4 Marks Q1. Case Study: Amusement Park Rides
An amusement park designer is mapping out the relationship between different rides using coordinates. Let $A = \{1, 2, 3\}$ be the set representing three major rides (Roller Coaster, Giant Wheel, and Bumper Cars). A relation $R$ is defined on set $A$ as: $$R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)\}$$

a) Is the relation $R$ reflexive? Justify. (1 Marks)

b) Is the relation $R$ symmetric? Justify. (1 Marks)

c) Check whether $R$ is transitive. (1 Marks)

d) Is $R$ an equivalence relation? If not, what minimum ordered pair should be added to make it one? (1 Marks)
4 Marks Q2. Case Study: Cellular Network Connections
A telecom company analyzes signal interference between four towers represented by the set $A = \{a, b, c, d\}$. The relation $R$ on $A$ is defined as: $$R = \{(a, a), (b, b), (c, c), (d, d), (a, b), (b, a), (b, c), (c, b)\}$$

a) Find whether the relation is symmetric. (1 Marks)

b) Show that $R$ is not transitive by giving a counterexample. (1 Marks)

c) If we remove $(b, c)$ and $(c, b)$ from $R$, does the new relation become an equivalence relation? (1 Marks)

d) Find the number of elements in the power set of $R$. (1 Marks)
4 Marks Q3. Case Study: Blood Group Compatibility
In a medical study, researchers define a relation $R$ on the set of all human beings such that $xRy$ if and only if $x$ and $y$ have the exact same blood group.

a) Prove that $R$ is a reflexive relation. (1 Marks)

b) Prove that $R$ is a symmetric relation. (1 Marks)

c) Prove that $R$ is a transitive relation. (1 Marks)

d) Based on your answers, what special type of relation is $R$? (1 Marks)
4 Marks Q4. Case Study: E-Commerce Delivery Logistics
An e-commerce company routes packages using a function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = 3x + 5$, where $x$ represents the weight of the package in kg and $f(x)$ represents the shipping cost in dollars.

a) Show that the function $f(x)$ is injective (one-one). (1.5 Marks)

b) Show that the function $f(x)$ is surjective (onto). (1.5 Marks)

c) Is this function bijective? What does this mean for the company regarding tracking costs back to weights? (1 Marks)
4 Marks Q5. Case Study: The Architectural Arch
An architect designs a parabolic arch for a monument. The height of the arch is modeled by a function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = x^2$. f(x) | . | / -----+---/----+--> x | / | .

a) Check if the function is one-one. Provide a counterexample if it is not. (1 Marks)

b) Check if the function is onto. Provide a counterexample if it is not. (1 Marks)

c) If the domain and co-domain are restricted to the set of natural numbers $\mathbb{N}$, check the injectivity of $f(x) = x^2$. (2 Marks)
4 Marks Q6. Case Study: School Student Database
A school database stores student records. Let $A$ be the set of all students in Class XII. A relation $R$ in $A$ is given by $R = \{(a, b) : \text{height of } a \text{ is exactly } 5 \text{ cm more than height of } b\}$.

a) Is $R$ reflexive? Why or why not? (1 Marks)

b) Is $R$ symmetric? Why or why not? (1 Marks)

c) If $(a, b) \in R$ and $(b, c) \in R$, find the relation between the heights of $a$ and $c$. Is $R$ transitive? (2 Marks)
4 Marks Q7. Case Study: Coding and Cryptography
A software engineer creates an encryption algorithm using a function $f: \mathbb{N} \rightarrow \mathbb{N}$ defined by: $$f(n) = \begin{cases} \frac{n+1}{2} & \text{if } n \text{ is odd} \\ \frac{n}{2} & \text{if } n \text{ is even} \end{cases}$$

a) Find $f(1)$ and $f(2)$. (1 Marks)

b) Based on Q1, is the function one-one? Explain. (1 Marks)

c) Is the function onto? Justify your answer. (2 Marks)
4 Marks Q8. Case Study: Digital Image Processing
In digital image processing, a pixel scaling function $f: \mathbb{R} \rightarrow \mathbb{R}$ is defined as $f(x) = [x]$, where $[x]$ denotes the greatest integer less than or equal to $x$ (Floor function).

a) Find the values of $f(2.3)$ and $f(2.9)$. (1 Marks)

b) Is this function one-one? Explain using the values from Q1. (1 Marks)

c) What is the range of this function? (1 Marks)

d) Is this function onto? Justify. (1 Marks)
4 Marks Q9. Case Study: Aviation Altitude Control
An aircraft's autopilot system utilizes a cubic function to adjust altitude smoothing. The function $f: \mathbb{R} \rightarrow \mathbb{R}$ is given by $f(x) = x^3$.

a) Prove mathematically that $f(x) = x^3$ is a one-one function. (2 Marks)

b) Prove that $f(x)$ is an onto function. (1.5 Marks)

c) What is the term used for functions that are both one-one and onto? (0.5 Marks)
4 Marks Q10. Case Study: Library Catalog System
A university library defines a relation $R$ on the set $B$ of all books in the library given by $R = \{(x, y) : x \text{ and } y \text{ have the same number of pages}\}$.

a) Check if $R$ is reflexive. (1 Marks)

b) Check if $R$ is symmetric. (1 Marks)

c) Check if $R$ is transitive. (1 Marks)

d) If Book $X$ has 350 pages, what does the equivalence class $[X]$ represent? (1 Marks)
4 Marks Q11. Case Study: Absolute Value Sensors
A thermal sensor records temperature fluctuations using an absolute value function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = |x|$.

a) Evaluate $f(-5)$ and $f(5)$. (1 Marks)

b) Is the sensor function one-one? Justify based on Q1. (1 Marks)

c) Can the output of this sensor ever be negative? What is the range of $f(x)$? (1 Marks)

d) Is the function onto? Explain (1 Marks)
4 Marks Q12. Case Study: Real Estate Plot Subdivisions
A developer divides a plot of land. Let $L$ be the set of all straight lines in a XY-plane. Let $R$ be a relation in $L$ defined as $R = \{(L_1, L_2) : L_1 \text{ is parallel to } L_2\}$.

a) Prove that $R$ is reflexive. (1 Marks)

b) Prove that $R$ is symmetric. (1 Marks)

c) Prove that $R$ is transitive. (1 Marks)

d) Find the set of all lines related to the line $y = 2x + 4$. (1 Marks)
4 Marks Q13. Case Study: Perpendicular Structural Beams
In a construction blueprint, $L$ is the set of all lines in a 2D plane. A relation $R$ is defined as $R = \{(L_1, L_2) : L_1 \text{ is perpendicular to } L_2\}$.

a) Can a line be perpendicular to itself? Is $R$ reflexive? (1 Marks)

b) If $L_1 \perp L_2$, is $L_2 \perp L_1$? Is $R$ symmetric? (1 Marks)

c) If $L_1 \perp L_2$ and $L_2 \perp L_3$, what is the geometric relationship between $L_1$ and $L_3$? Is $R$ transitive? (2 Marks)
4 Marks Q14. Case Study: Social Media Networks
On a social media application, a relation $R$ is defined on the set of all active users. User $A$ is related to User $B$ ($(A, B) \in R$) if $A$ follows $B$.

a) If user $A$ follows themselves, the relation is reflexive. Is it universally true that everyone follows themselves on social media? (1 Marks)

b) If $A$ follows $B$, does it imply $B$ must follow $A$? What does this say about symmetry? (1 Marks)

c) If $A$ follows $B$ and $B$ follows $C$, does it mean $A$ follows $C$? What does this say about transitivity? (1 Marks)

d) Is this relation an equivalence relation? (1 Marks)
4 Marks Q15. Case Study: Set Theory Operations
Let $A = \{1, 2, 3\}$. A relation $R$ on set $A$ is given by $R = \{(1, 2), (2, 3)\}$.

a) Write the minimum number of ordered pairs to be added to $R$ so that it becomes reflexive. (1 Marks)

b) Write the ordered pairs to be added to the original $R$ to make it symmetric. (1 Marks)

c) What ordered pair must be added to $R$ to satisfy the transitivity condition for the existing elements? (1 Marks)

d) Find the total number of possible relations on set $A$. (1 Marks)
4 Marks Q16. Case Study: Mathematical Mapping
Consider a function $f: \mathbb{R} - \{3\} \rightarrow \mathbb{R} - \{1\}$ defined by: $$f(x) = \frac{x-2}{x-3}$$

a) Show that $f(x)$ is a one-one function. (2 Marks)

b) Show that $f(x)$ is an onto function. (2 Marks)
4 Marks Q17. Case Study: Divisibility in Inventory Systems
An inventory tracking system assigns unique identification numbers to items. Let $\mathbb{Z}$ be the set of integers. A relation $R$ on $\mathbb{Z}$ is defined by $R = \{(a, b) : 2 \text{ divides } (a - b)\}$.

a) Show that $R$ is reflexive. (1 Marks)

b) Show that $R$ is symmetric. (1 Marks)

c) Show that $R$ is transitive. (1.5 Marks)

d) How many distinct equivalence classes are formed by this relation? (0.5 Marks)
4 Marks Q18. Case Study: Linear Economic Projections
An economic forecast tool uses a linear function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = 4x - 7$.

a) Check if the function is injective. (1.5 Marks)

b) Check if the function is surjective. (1.5 Marks)

c) Find the value of $x$ if $f(x) = 5$. (1 Marks)
4 Marks Q19. Case Study: Human Genetics Mapping
Let $X$ be the set of all human beings alive in the world at a given time. A relation $R$ on $X$ is defined as $R = \{(a, b) : a \text{ is the brother of } b\}$.

a) Is $R$ reflexive? Explain with a reason involving female individuals. (1 Marks)

b) Is $R$ symmetric? (Hint: Consider if $a$ is a brother of a female $b$). (1.5 Marks)

c) Is $R$ transitive? Explain carefully. (1.5 Marks)
4 Marks Q20. Case Study: The Signum Signal
An electrical engineer uses a signum function $f: \mathbb{R} \rightarrow \mathbb{R}$ to switch control states in a circuit. The function is defined as: $$f(x) = \begin{cases} 1 & \text{if } x > 0 \\ 0 & \text{if } x = 0 \\ -1 & \text{if } x < 0 \end{cases}$$

a) Find $f(5)$ and $f(10)$. Is the function one-one? (1.5 Marks)

b) What is the range of this function? (1 Marks)

b) Is this function onto? Explain based on its co-domain. (1.5 Marks)

SECTION E — Long Answer Type Questions [5 Marks Each]

5 Marks Q1. Show that the relation $R$ in the set $A = \{x \in \mathbb{Z} : 0 \le x \le 12\}$, given by $R = \{(a, b) : |a - b| \text{ is a multiple of } 4\}$, is an equivalence relation. Find the set of all elements related to 1.
5 Marks Q2. Let $\mathbb{N}$ be the set of all natural numbers and $R$ be the relation on $\mathbb{N} \times \mathbb{N}$ defined by $(a, b) R (c, d) \iff ad(b + c) = bc(a + d)$. Show that $R$ is an equivalence relation.
5 Marks Q3. Let $A = \mathbb{R} - \{3\}$ and $B = \mathbb{R} - \{1\}$. Consider the function $f: A \to B$ defined by $f(x) = \frac{x - 2}{x - 3}$. Is $f$ one-to-one and onto? Justify your answer.
5 Marks Q4. Show that the relation $R$ defined on the set $A$ of all polygons as $R = \{(P_1, P_2) : P_1 \text{ and } P_2 \text{ have same number of sides}\}$, is an equivalence relation. What is the set of all elements in $A$ related to the right-angled triangle $T$ with sides 3, 4, and 5?
5 Marks Q5. Prove that the relation $R$ on the set $\mathbb{Z}$ of all integers defined by $(x, y) \in R \iff (x - y)$ is divisible by $n$ (where $n$ is a fixed positive integer) is an equivalence relation.
5 Marks Q6. Let $L$ be the set of all lines in $XY$-plane and $R$ be the relation in $L$ defined as $R = \{(L_1, L_2) : L_1 \text{ is parallel to } L_2\}$. Show that $R$ is an equivalence relation. Find the set of all lines related to the line $y = 2x + 4$.
5 Marks Q7. Let $A = \{1, 2, 3, \dots, 9\}$ and $R$ be the relation in $A \times A$ defined by $(a, b) R (c, d)$ if $a + d = b + c$ for $(a, b), (c, d)$ in $A \times A$. Prove that $R$ is an equivalence relation. Also, obtain the equivalence class $[(2, 5)]$.
5 Marks Q8. Show that the relation $R$ on the set $A = \mathbb{Z}$ of integers given by $R = \{(a, b) : 2 \text{ divides } a - b\}$ is an equivalence relation. Write all its equivalence classes.
5 Marks Q9. Let $R$ be a relation on the set $A$ of ordered pairs of positive integers defined by $(x, y) R (u, v) \iff xv = yu$. Show that $R$ is an equivalence relation.
5 Marks Q10. Let $f: X \to Y$ be a function. Define a relation $R$ in $X$ given by $R = \{(a, b) : f(a) = f(b)\}$. Examine if $R$ is an equivalence relation.
5 Marks Q11. Show that the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 4x + 3$ is a bijective function.
5 Marks Q12. Consider $f: \mathbb{R}_+ \to [4, \infty)$ given by $f(x) = x^2 + 4$. Show that $f$ is bijective, where $\mathbb{R}_+$ is the set of all non-negative real numbers.
5 Marks Q13. Let $f: \mathbb{N} \to \mathbb{N}$ be defined by: $$f(n) = \begin{cases} \frac{n + 1}{2}, & \text{if } n \text{ is odd} \\ \frac{n}{2}, & \text{if } n \text{ is even} \end{cases}$$ State whether the function $f$ is bijective. Justify your answer.
5 Marks Q14. Show that the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^3$ is a bijection.
5 Marks Q15. Consider $f: \mathbb{R} - \left\{-\frac{4}{3}\right\} \to \mathbb{R}$ data as $f(x) = \frac{4x}{3x + 4}$. Show that $f$ is a one-to-one function. Is it onto if the codomain is changed to the range of $f$?
5 Marks Q16. Show that the Signum function $f: \mathbb{R} \to \mathbb{R}$, given by: $$f(x) = \begin{cases} 1, & \text{if } x > 0 \\ 0, & \text{if } x = 0 \\ -1, & \text{if } x < 0 \end{cases}$$ is neither one-to-one nor onto.
5 Marks Q17. Check the injectivity (one-to-one) and surjectivity (onto) of the function $f: \mathbb{Z} \to \mathbb{Z}$ given by $f(x) = x^2$. What happens if the domain and codomain are changed to $\mathbb{N}$?
5 Marks Q18. Let $A = \mathbb{R} - \{2\}$ and $B = \mathbb{R} - \{1\}$. If $f: A \to B$ is a function defined by $f(x) = \frac{x - 1}{x - 2}$, show that $f$ is one-to-one and onto.
5 Marks Q19. Show that a function $f: \mathbb{R} \to \mathbb{R}$ defined as $f(x) = a x + b$, where $a, b \in \mathbb{R}$ and $a \neq 0$, is a bijective function.
5 Marks Q20. Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = x^4$. Choose the correct answer with full mathematical justification: (A) $f$ is one-to-one onto (B) $f$ is many-one onto (C) $f$ is one-to-one but not onto (D) $f$ is neither one-to-one nor onto

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Important Announcement for Students & Teachers!

CBSE Class 12 Maths Chapter 1 Model Questions

CBSE CLASS XII – MODEL QUESTION PAPER

SECTION A — Multiple Choice Questions [1 Mark Each]

SECTION B — Very Short Answer Type Questions [2 Marks Each]

SECTION C — Short Answer Type Questions [3 Marks Each]

SECTION D — Case-Based/Source-Based Integrated Questions [4 Marks Each]

SECTION E — Long Answer Type Questions [5 Marks Each]

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