MCQ Mathematics Chapter 2: RELATIONS AND FUNCTIONS, HS 1st year

 

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1. What is the Cartesian product of two sets A and B?

a) A union of sets A and B
b) A subset of elements common to A and B
c) A set of all ordered pairs where the first element is from A and the second is from B
d) A set of differences between elements of A and B

Answer: c) A set of all ordered pairs where the first element is from A and the second is from B

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2. If A = {1, 2} and B = {3, 4, 5}, how many elements are there in A × B?

a) 2
b) 5
c) 6
d) 9

Answer: c) 6

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3. If A × B = {(p, q), (p, r), (m, q), (m, r)}, what are the sets A and B?

a) A = {p, m}, B = {q, r}
b) A = {p, q}, B = {m, r}
c) A = {p, r}, B = {m, q}
d) A = {m, r}, B = {p, q}

Answer: a) A = {p, m}, B = {q, r}

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4. In a relation $R = \{(x, y) : y = x + 1 \}$, what is the domain if $x \in \{1, 2, 3, 4, 5\}$?

a) {2, 3, 4, 5, 6}
b) {1, 2, 3, 4, 5}
c) {1, 2, 3, 4}
d) {3, 4, 5, 6, 7}

Answer: b) {1, 2, 3, 4, 5}

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5. If $f(x) = 2x + 1$, what is the value of $f(3)$?

a) 5
b) 6
c) 7
d) 9

Answer: c) 7

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6. Which of the following is NOT true about the Cartesian product $A × B$?

a) $A × B$ is equal to $B × A$
b) $A × B$ can represent coordinates in a plane
c) If $A$ or $B$ is empty, $A × B$ is also empty
d) The order of pairs in $A × B$ matters

Answer: a) $A × B$ is equal to $B × A$

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7. What is the range of the function $f(x) = x^2$ for $x \in \mathbb{R}$?

a) All real numbers
b) Non-negative real numbers
c) All integers
d) Negative real numbers

Answer: b) Non-negative real numbers

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8. Which of the following represents the modulus function $f(x)$?

a) $f(x) = x^2$
b) $f(x) = x$
c) $f(x) = |x|$
d) $f(x) = -x$

Answer: c) $f(x) = |x|$

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9. If A = {1, 2} and B = {3, 4}, what is A × B?

a) {(1, 3), (1, 4), (2, 3), (2, 4)}
b) {(1, 4), (1, 5), (2, 4), (2, 5)}
c) {(1, 1), (2, 2), (3, 3), (4, 4)}
d) {(1, 4), (1, 5), (2, 5), (2, 6)}

Answer: a) {(1, 3), (1, 4), (2, 3), (2, 4)}

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10. How many subsets does $A × B$ have if A = {1, 2} and B = {3, 4}?

a) 2
b) 8
c) 16
d) 4

Answer: c) 16

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11. The Cartesian product $R × R$ represents:

a) A single line
b) A three-dimensional plane
c) A two-dimensional plane
d) A one-dimensional line

Answer: c) A two-dimensional plane

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12. What is the total number of relations that can be defined from a set A to a set B if $n(A) = 3$ and $n(B) = 2$?

a) 6
b) 8
c) 64
d) 32

Answer: c) 64

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13. The set of all first elements in a relation R is called:

a) Codomain
b) Range
c) Domain
d) Function

Answer: c) Domain

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14. The domain of $f(x) = \frac{1}{x}$ is:

a) All real numbers except 0
b) All real numbers
c) Positive real numbers only
d) Non-negative real numbers

Answer: a) All real numbers except 0

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15. If $f(x) = x^2$, which of the following is a valid ordered pair?

a) (2, 2)
b) (3, 6)
c) (4, 16)
d) (5, 10)

Answer: c) (4, 16)

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16. The modulus function $f(x) = |x|$ is defined as:

a) $f(x) = x^2$
b) $f(x) = x$ for $x \geq 0$ and $-x$ for $x < 0$
c) $f(x) = x + 1$
d) $f(x) = 0$

Answer: b) $f(x) = x$ for $x \geq 0$ and $-x$ for $x < 0$

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17. Which of the following represents a linear function?

a) $f(x) = x + 3$
b) $f(x) = x^2$
c) $f(x) = 1/x$
d) $f(x) = |x|$

Answer: a) $f(x) = x + 3$

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18. For a function to be well-defined, each element in the domain must have:

a) At least one image
b) Exactly one image
c) More than one image
d) No image

Answer: b) Exactly one image

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19. If A = {1, 2, 3} and B = {4, 5}, what is $A × (B ∩ \{4\})$?

a) {(1, 5), (2, 5), (3, 5)}
b) {(1, 4), (2, 4), (3, 4)}
c) {(1, 3), (2, 3), (3, 3)}
d) {(4, 5), (5, 4)}

Answer: b) {(1, 4), (2, 4), (3, 4)}

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20. The graph of the identity function passes through:

a) (0, 0)
b) (1, 0)
c) (0, 1)
d) (-1, -1)

Answer: a) (0, 0)

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21. The range of $f(x) = x^3$ for $x \in \mathbb{R}$ is:

a) All real numbers
b) Non-negative real numbers
c) Positive real numbers
d) Integers

Answer: a) All real numbers

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22. In the relation $R = \{(x, y) : y = x + 2 \}$, what is the range if the domain is {1, 2, 3}?

a) {1, 2, 3}
b) {3, 4, 5}
c) {2, 3, 4}
d) {4, 5, 6}

Answer: b) {3, 4, 5}

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23. If $P = \{a, b\}$ and $Q = \{1, 2, 3\}$, what is the number of elements in $P × Q$?

a) 2
b) 3
c) 5
d) 6

Answer: d) 6

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24. The graph of $f(x) = x^2$ is called:

a) A straight line
b) A parabola
c) A circle
d) A hyperbola

Answer: b) A parabola

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25. If A = {a, b} and B = {1, 2, 3}, which of the following pairs belongs to $B × A$?

a) (1, a)
b) (b, 2)
c) (3, b)
d) (2, b)

Answer: a) (1, a)

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26. The domain of $f(x) = \sqrt{x}$ is:

a) $x \geq 0$
b) $x > 0$
c) All real numbers
d) All integers

Answer: a) $x \geq 0$

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27. The function $f(x) = [x]$, where [x] is the greatest integer less than or equal to x, is called:

a) Polynomial function
b) Rational function
c) Step function
d) Identity function

Answer: c) Step function

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28. If $A = {1, 2}$ and $B = {3, 4, 5}$, what is $(A × B) ∩ (B × A)$?

a) $\{(1, 3), (2, 4)\}$
b) Empty set
c) $\{(3, 1), (4, 2)\}$
d) $\{(1, 3), (3, 1)\}$

Answer: b) Empty set

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29. If $f(x) = 2x + 3$, find $f(0) + f(2)$.

a) 6
b) 9
c) 10
d) 12

Answer: c) 10

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30. Which of the following statements is true for the Cartesian product $A × B$?

a) $A × B = B × A$
b) $A × B = φ$ if $A = φ$
c) $A × B = {x: x ∈ A \text{ or } x ∈ B}$
d) $A × B$ is a union of sets A and B

Answer: b) $A × B = φ$ if $A = φ$

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31. A function $f: A \to B$ is not valid if:

a) All elements in A have a unique image in B
b) Two elements in A map to the same element in B
c) An element in A has no image in B
d) The domain of the function is A

Answer: c) An element in A has no image in B

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32. If $f(x) = \frac{1}{x - 1}$, which of the following values cannot be in the domain of f?

a) 0
b) 1
c) 2
d) -1

Answer: b) 1

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33. For a function $f: A → B$, the range is always:

a) A subset of A
b) A subset of B
c) Equal to the domain
d) Equal to the codomain

Answer: b) A subset of B

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34. The graph of $f(x) = x^3$ is symmetric about:

a) The x-axis
b) The y-axis
c) The origin
d) The line $y = x$

Answer: c) The origin

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35. What is the range of $f(x) = \frac{1}{x}$?

a) All real numbers except 0
b) All positive real numbers
c) All negative real numbers
d) All integers

Answer: a) All real numbers except 0

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36. If $A = {1, 2, 3}$ and $B = {4, 5}$, how many ordered pairs can be formed?

a) 4
b) 5
c) 6
d) 10

Answer: c) 6

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37. The range of $f(x) = x^2$ for $x \in [-3, 3]$ is:

a) [0, 9]
b) [-9, 9]
c) [0, 3]
d) [-3, 3]

Answer: a) [0, 9]

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38. Which of the following is a one-to-one function?

a) $f(x) = x^2$
b) $f(x) = x^3$
c) $f(x) = |x|$
d) $f(x) = x \cdot |x|$

Answer: b) $f(x) = x^3$

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39. The codomain of a function is:

a) The set of all possible outputs
b) The set of inputs
c) The set containing the range
d) The set of ordered pairs

Answer: c) The set containing the range

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40. If $f(x) = 3x + 1$ and $g(x) = 2x$, find $(f + g)(x)$.

a) $5x$
b) $5x + 1$
c) $6x + 1$
d) $5x - 1$

Answer: b) $5x + 1$

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41. The range of the signum function $f(x) = \text{sgn}(x)$ is:

a) $\{-1, 0, 1\}$
b) $\{0, 1\}$
c) $\{-1, 1\}$
d) All real numbers

Answer: a) $\{-1, 0, 1\}$

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42. If $f(x) = |x|$, then $f(-3) + f(2) =$:

a) -5
b) 5
c) 1
d) -1

Answer: b) 5

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43. If $A × B = φ$, then which of the following must be true?

a) $A = φ$ or $B = φ$
b) $A \neq φ$ and $B \neq φ$
c) $A = B$
d) $A ≠ φ$

Answer: a) $A = φ$ or $B = φ$

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44. Which of the following is not an example of a real-valued function?

a) $f(x) = x^2 + 1$
b) $f(x) = |x|$
c) $f(x) = \sqrt{-x}$
d) $f(x) = x^3 - 2x$

Answer: c) $f(x) = \sqrt{-x}$

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45. If $P = {a, b}$ and $Q = {1, 2}$, how many subsets does $P × Q$ have?

a) 4
b) 8
c) 16
d) 32

Answer: c) 16

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46. The function $f(x) = 0$ is called:

a) Constant function
b) Identity function
c) Polynomial function
d) Rational function

Answer: a) Constant function

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47. The equation $y = mx + c$ represents:

a) A quadratic function
b) A constant function
c) A linear function
d) A step function

Answer: c) A linear function

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48. For $f(x) = \frac{x^2}{x - 1}$, what is the domain?

a) All real numbers
b) All real numbers except 0
c) All real numbers except 1
d) Positive real numbers only

Answer: c) All real numbers except 1

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49. If $f(x) = x + 5$, then $f(2x) =$:

a) $2x + 5$
b) $x + 10$
c) $2x + 10$
d) $2x$

Answer: c) $2x + 10$

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50. The graph of the constant function $f(x) = c$ is a line:

a) Parallel to the y-axis
b) Parallel to the x-axis
c) Passing through the origin
d) Symmetric about the origin

Answer: b) Parallel to the x-axis

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51. The Cartesian product $R × R × R$ represents:

a) A one-dimensional space
b) A two-dimensional space
c) A three-dimensional space
d) Infinite-dimensional space

Answer: c) A three-dimensional space

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52. If $A = \{1, 2\}$ and $B = \{a, b\}$, which of the following pairs belong to $A × B$?

a) (b, 1)
b) (2, b)
c) (a, 2)
d) (3, a)

Answer: b) (2, b)

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53. Which of the following relations is NOT a function?

a) $\{(1, 2), (2, 3), (3, 4)\}$
b) $\{(1, 3), (1, 4), (2, 5)\}$
c) $\{(a, b), (c, d), (e, f)\}$
d) $\{(x, y), (y, z), (z, w)\}$

Answer: b) $\{(1, 3), (1, 4), (2, 5)\}$

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54. The function $f(x) = x^3 + 3x^2 + x + 1$ is an example of:

a) Linear function
b) Quadratic function
c) Polynomial function
d) Constant function

Answer: c) Polynomial function

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55. If $A = \{1, 2, 3\}$, then $A × A × A$ will have how many elements?

a) 6
b) 9
c) 27
d) 81

Answer: c) 27

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56. The total number of functions that can be defined from a set $A$ with 3 elements to a set $B$ with 2 elements is:

a) 2
b) 6
c) 8
d) 64

Answer: c) 8

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57. The set of all second elements in a relation is called the:

a) Codomain
b) Range
c) Domain
d) Cartesian product

Answer: b) Range

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58. The graph of $f(x) = |x|$ is:

a) A parabola
b) A straight line passing through the origin
c) V-shaped
d) A hyperbola

Answer: c) V-shaped

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59. The function $f(x) = 2x + 3$ is called a:

a) Rational function
b) Linear function
c) Constant function
d) Polynomial function

Answer: b) Linear function

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60. For the signum function $f(x) = \text{sgn}(x)$, what is $f(0)$?

a) -1
b) 0
c) 1
d) Undefined

Answer: b) 0

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61. Which of the following is the modulus of $-5$?

a) -5
b) 5
c) 0
d) Undefined

Answer: b) 5

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62. If $f(x) = x^2$ and $g(x) = 3x$, find $(f + g)(x)$:

a) $3x^2$
b) $x^2 + 3x$
c) $x^2 - 3x$
d) $3x - x^2$

Answer: b) $x^2 + 3x$

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63. A function that maps every element of the domain to the same value is called a:

a) Polynomial function
b) Constant function
c) Rational function
d) Step function

Answer: b) Constant function

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64. The Cartesian product $A × φ$ is always:

a) Empty
b) Equal to A
c) Non-empty
d) Equal to $φ$

Answer: a) Empty

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65. What is the range of $f(x) = x^2 + 2$ for $x \in \mathbb{R}$?

a) All real numbers
b) $[2, \infty)$
c) $(-\infty, 2]$
d) $(0, \infty)$

Answer: b) $[2, \infty)$

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66. If $f(x) = x^2$, what is the output when $x = -2$?

a) -4
b) 4
c) 0
d) -2

Answer: b) 4

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67. If $f(x) = \frac{1}{x}$, which of the following is NOT in the domain of $f$?

a) 1
b) 0
c) 2
d) -1

Answer: b) 0

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68. Which of the following functions is NOT one-to-one?

a) $f(x) = x + 1$
b) $f(x) = x^3$
c) $f(x) = x^2$
d) $f(x) = 2x$

Answer: c) $f(x) = x^2$

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69. If $A = {1, 2, 3}$ and $B = {a, b}$, what is the cardinality of $A × B$?

a) 3
b) 6
c) 9
d) 12

Answer: b) 6

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70. For $f(x) = x + 1$, what is $f^{-1}(x)$?

a) $x - 1$
b) $x + 1$
c) $-x + 1$
d) $-x - 1$

Answer: a) $x - 1$

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71. The graph of $y = x^3$ is symmetric about:

a) x-axis
b) y-axis
c) Origin
d) Line $y = x$

Answer: c) Origin

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72. If $f(x) = 2x^2 - 3x + 1$, what is $f(1)$?

a) 0
b) 1
c) 2
d) 3

Answer: c) 2

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73. What is the domain of $f(x) = \sqrt{x - 2}$?

a) $x \geq 2$
b) $x > 2$
c) $x \leq 2$
d) $x < 2$

Answer: a) $x \geq 2$

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74. The graph of $y = |x|$ passes through which of the following points?

a) (0, 0)
b) (-1, -1)
c) (1, -1)
d) (-2, 2)

Answer: a) (0, 0)

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75. Which of the following is the inverse of $f(x) = x^2$, where $x \geq 0$?

a) $f^{-1}(x) = x^2$
b) $f^{-1}(x) = \sqrt{x}$
c) $f^{-1}(x) = -\sqrt{x}$
d) $f^{-1}(x) = |x|$

Answer: b) $f^{-1}(x) = \sqrt{x}$

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