CUET Mathematics Algebra > Relations & Functions PYQs

The relation R on the set of real numbers defined by $R = \{(a, b): a \leq b^2\}$ is

The function $f: [-1, 1] \rightarrow R$ is given by $f(x) = \frac{x}{x + 2}$

The range of function $f(x) = 4x^2 + 12x + 7, x \in \mathbb{R}$ is

For the relation $R = \{(a, b): a \leq b\}$ in $\mathbb{R}$, which of the following is correct?

The domain of $y = \cos^{-1}(x^2 - 4)$ is

Relation R on the set $A = \{1, 2, ..., 15\}$ defined as $R = \{(x, y): y - 4x = 0\}$ is

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = \frac{x}{x^2+1}$ is (where $\mathbb{R}$ is a set of real number)

The relation R on the set of real numbers defined by $R = \{(a, b): a \leq b^2\}$ is (A) Reflexive (B) Not symmetric (C) Neither reflexive nor transitive (D) Transitive Choose the correct answer from the options given below:

The function $f: \mathbb{R} \rightarrow \mathbb{R}, f(x) = |x|$ ($\mathbb{R}$ is the set of real numbers) is

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = 100x + 1$, where $\mathbb{R}$ is a set of real numbers, then

The relation R in the set $\{1, 2, 3\}$ given by $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (1, 3), (2, 3)\}$ is:

Assume that R is a relation on the set Z of integers and it is given by $(x, y) \in R \Leftrightarrow |x - y| \leq 1$. Then, R is

The function $f: [-1, 1] \rightarrow R$ (set of real numbers) given by $f(x) = \frac{x}{x+3}$ is

The function $f: \mathbb{R} \rightarrow [-1, 1]$ defined by $f(x) = \cos x$ is:

The domain of the function $y = \sin^{-1}(x-1) + \cos^{-1}\sqrt{x-1}$ is:

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x) = [x]$, where [x] denotes the greatest integer less than or equal to x. Then which of the following statements are correct? (A) f is one-one but not onto (B) f is not onto (C) f is not one-one (D) f is one-one and onto Choose the correct answer from the options given below:

Let A = {1, 2, 3}. The number of equivalence relations containing (1, 3) is

The value of $\left|\begin{array}{cc}\log_5 10 & 2 \\[4pt] 2 & \log_{10} 5\end{array}\right|$ is

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = \frac{x}{x^2+1}$ then

A relation R on the set $A = \{1, 2, 3, \ldots, 13, 14\}$ defined as $R = \{(x,y): 3x - y = 0\}$ is

The graph given below represents which of the following function? <img src="https://balti.afterboards.in/B69GKLFdaZzveBZ" width="300px"/>

A and B are two sets such that $n(A) = 5$ and $n(B) = 7$. The number of one-one functions from A to B is

The number of equivalence relation on the set $\{1, 2, 3\}$ containing $(1, 2)$ and $(2, 1)$ is

A relation $f: N \rightarrow N$ be defined by $f(x) = x^2$, $x \in N$ (Set of Natural numbers). Then $f(x)$ is

Let $R = \{(L_1, L_2): L_1 \perp L_2\ $ where $L_1, L_2 \in L$ (set of straight line in a plane)}, then

If R be a relation on the set of integers Z, given by R = {(a, b) : (a - b) is a multiple of 3}, then R is:

Let N be set of natural numbers, then the function $f: \mathbb{N} \rightarrow \mathbb{N}$, defined by $f(x) = \begin{cases} \frac{n+1}{2} & \text{, if } n \text{ is odd} \\ \frac{n}{2} & \text{, if } n \text{ is even} \end{cases}$, is

A function $f: \mathbb{R} \rightarrow \{x \in \mathbb{R}: -1 < x < 1\}$ is defined as $f(x) = \frac{x}{1+|x|}$, then $f$ is:

The relation R in $\mathbb{R}$ (set of real numbers) is defined by $R = \{(a,b): a \leq b^3\}$, then R is

Match List-I with List-II Where ℝ is set of real numbers | List-I | List-II | |---|---| | (A) f: ℝ → ℝ s.t f(x) = x⁴ is | (I) one-one, Into | | (B) f: ℝ → [0, ∞) s.t f(x) = x⁴ is | (II) many-one, into | | (C) f: [0, ∞) → ℝ s.t f(x) = x⁴ is | (III) one-one, onto | | (D) f: [0, ∞) → [0, ∞) s.t f(x) = x⁴ is | (IV) many-one, onto | Choose the correct answer from the options given below:

If R and S are two equivalence relations on a set A, then

Which of the following are correct? (A) If a ≡ b(mod n), then -a ≡ -b (mod n) (B) If a + b = c, then a(mod n) + b(mod n) ≡ (a + b + c)(mod n). (C) If a ≡ b (mod n), then ka ≡ kb(mod n), ∀ k ∈ I. (D) If a ≡ b (mod n), then (a + k) ≡ (b + k)(mod n), ∀ k ∈ I. Choose the correct answer from the options given below:

The domain of the function $\cos^{-1}(2x - 3)$ is

If $A = \{1, 2, 3, 4, ..., n\}$ and $B = \{x, y\}$, then the number of surjections from A to B is

A relation R in the set A = {1, 2,3, 4} is given by R = {(1,1), (2,2), (1,2), (2,3), (3,4), (4,4), (1,3), (2,4), (1,4)} is

Which of the following functions from $\mathbb{Z}$ to $\mathbb{Z}$ is a bijective function? (where $\mathbb{Z}$ is set of integers)

Let A= {1, 2, 3}, then the possible equivalence relations on A are: (A) {(1, 1), (2, 2) , (3, 3)} (B) {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} (C) {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3)} (D) {(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)} Choose the correct answer from the options given below:

Let a relation R = {(a, b) : a is a factor of b, a, b $\in$ N}. Then, R is ______.

The function $f: [0, \infty) \rightarrow \mathbb{R}$ defined by, $f(x) = 2x^2 + 3$, is

Let f: $\mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x) = 10x$. Then (Where $\mathbb{R}$ is the set of real numbers)

Let $A = \{1, 2, 3\}$. Then, the number of relations containing $(1, 2)$ and $(1, 3)$ which are reflexive and symmetric but not transitive, is

Match List-I with List-II Let $f: A \rightarrow B$ be a function given by $f(x) = x^2$ | List-I | List-II | |---|---| | **Domain and Co-domain** | **Kind** | | (A) $A = \mathbb{R}$ and $B = \mathbb{R}$ | (I) $f$ is both one-one and onto | | (B) $A = \mathbb{R}$ and $B = [0, \infty]$ | (II) $f$ is one-one but not onto | | (C) $A = B = [0, \infty]$ | (III) $f$ is not one-one but onto | | (D) $A = [0, \infty]$ and $B = \mathbb{R}$ | (IV) $f$ is neither one-one nor onto | Choose the correct answer from the options given below:

Let L be the set of all lines in a plane and R be the relation on set L defined by $R = \{(L_1, L_2): L_1 \perp L_2\}$ Then R is (A) an equivalence Relation (B) a symmetric Relation (C) not a transitive Relation (D) a reflexive Relation Choose the correct answer from the options given below:

Let $f: \mathbb{R}$ -> $\mathbb{R}$ be a function defined as $f(x) = x^4$. Which one of the following is true?

Let $A = \{a, b, c\}$. Then number of relations containing $(a,b)$ and $(b, c)$ which are reflexive and transitive but not symmetric is

If the function $f: \mathbb{N} \rightarrow \mathbb{N}$ is defined as $f(n)=\left\{\begin{array}{ll}n-1, & \text { if } n \text { is even } \\ n+1, & \text { if } n \text { is odd }\end{array}\right.$, then (A) f is injective (B) f is into (C) f is surjective (D) f is invertible

Let R be the relation over the set A of all straight lines in a plane such that $l_{1} \mathrm{R} l_{2} \Leftrightarrow l_{1}$ is parallel to $l_{2}$. Then R is :

The number of all onto functions from the set {1, 2, .......n} to itself is

The relation $R = \{(a, b) : a \leq b^2\}$ on the set of real numbers is:

Let $f(x) = x^3$ be a function with domain {0, 1, 2, 3} then domain of $f^{-1}$ is :

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | $f(x) = \frac{1}{x}, f : \mathbf{R} - \{0\} \to \mathbf{R} - \{0\}$ | (I) | neither injective nor surjective | | (B) | $f(x) = x^2, f : \mathbf{N} \to \mathbf{N}$ | (II) | surjective but not injective | | (C) | $f(x) = x^2, f : \mathbf{R} \to \mathbf{R}$ | (III) | injective but not surjective | | (D) | $f : \{1, 2, 3\} \to \{1, 2\}$ defined as $f : \{(1, 1), (2, 2), (3, 1)\}$ | (IV) | injective and surjective | Choose the **correct** answer from the options given below :

Let R be a relation on the set of natural numbers N defined by nRm if n divides m. Then R is : (A) Reflexive Relation (B) Symmetric Relation (C) Transitive Relation (D) Identity Relation Choose the **correct** answer from the options given below :

If $f(x) = \sqrt{x}$, $g(x) = 2x - 3$, then domain of $fog(x)$ is :

Relation R on Real Numbers is defined as $R = \{(a, b) : a \leq b\}$. The relation is :

Which of the following graphs represent a function ?

If a set P contains 5 elements and the set Q contains 8 elements, then the number of one-one functions from A to B is :

Given relation $R = \{(x, y) : y = x + 5, x < 4, x, y \in N\}$. Where N is a set of natural numbers then :

Let $f : R \to R$ defined by $f(x) = 2x^3 - 7$ for $x \in R$. Then : (A) $f$ is one-one function (B) $f$ is many to one function (C) $f$ is bijective function (D) $f$ is into function Choose the correct answer from the options given below :

The inverse of the function $f : R \to R$ given by $f(x) = 2x + 7$ is :

The modulus function $f : R \to R$, given by $f(x) = |x|$, is :

Let L be the set of all lines in a plane and R be the relation in L defined as $R = \{(l_1, l_2) : l_1 \text{ is perpendicular to } l_2, \text{ where } l_1, l_2 \in L\}$. Choose the correct answer :

The Relation $R = \{(x, y) : x \leq y^2\}$ defined on the set $\mathbf{R}$ of Real numbers is : (A) reflexive but not symmetric (B) neither reflexive nor symmetric (C) neither reflexive nor transitive (D) reflexive but not transitive (E) not reflexive but symmetric Choose the correct answer from the options given below :

Let $f : [2, \infty) \to \mathbf{R}$ be a function defined by $f(x) = x^2 - 4x + 5$. The range of $f$ is :

Which of the following relations on the set $A = \{1, 2, 3\}$ are equivalence ? (A) $R = \{(1,1), (2,2), (1,2), (2,1)\}$ (B) $R = \{(1,1), (2,2), (3,3)\}$ (C) $R = \{(1,1), (1,2), (2,1)\}$ (D) $R = \{(1,1), (2,2), (3,3), (1,2), (2,1), (2,3), (3,2), (1,3), (3,1)\}$ (E) $R = \{(1,1), (2,2), (3,3), (1,2)\}$ Choose the correct answer from the options given below :

Which of the following figures shows a bijective function from set $X_1$ to set $X_2$ : <img src="https://balti.afterboards.in/P356b70GmYqx15Z" width="500px"/> Choose the correct answer from the options given below :

If a relation R is defined on the set $X = \{1, 2, 3, 4\}$ as $R = \{(1,1), (2,2), (3,4), (4,3)\}$, then R is

Match List I with List II | List I | List II | |---|---| | A. Range of $\lvert x\rvert$ | I. $(-5, \infty)$ | | B. Range of $9x^2 + 6x - 5$ for all $x \geq 0$ | II. $[0, \infty)$ | | C. Domain of $\dfrac{1}{\sqrt{x+5}}$ | III. $\{(1,1), (2,2), (3,3)\}$ | | D. Smallest equivalence relation on Set $\{1,2,3\}$ | IV. $[-5, \infty)$ | Choose the correct answer from the options given below:

A function f: R $\to$ R is given by $f(x) = x^3 + 3$. If $f(x) = -24$, then the value of x is:

If R is a relation on $A = \{a, b, c\}$ such that $R = \{(a,a), (b,b)\}$, which element/elements should be included to make R an equivalence relation. A. (c, c) B. (c, c), (a, c), (c, a) C. (a, b), (b, c), (a, c) D. (b, c), (c, c), (c, a), (b, a) Choose the correct answer from the options given below:

A. A relation $R$ on a set $A$ is called an equivalence relation, if it is reflexive, symmetric and transitive. B. The function $f : R \to R$ defined by $f(x) = e^x$ is not one-one. C. The one-one function is also known as injective function. D. The onto function is also known as subjective function. E. A function $f : X \to Y$ is said to be many-one, if two or more than two elements in set $X$ have the different image in set $Y$. Choose the correct answer from the option given below:

If $R$ is a relation on $Z$ (set of all integers) defined by $xRy$, iff $|x - y| \leq 1$, then (a) $R$ is reflexive (b) $R$ is symmetric (c) $R$ is transitive (d) $R$ is not symmetric (e) $R$ is not transitive Choose the most appropriate answer from the options given below

<img src="https://balti.afterboards.in/1VZUCz5GIcv95ws" width="400px"/>Which of the following is true on the basis of above diagram?

If $f:\mathbb{R} \to [-5, \infty)$ is defined as $f(x) = x^2 - 5$, then the function f is A. one-one B. many-one C. onto D. into Which of the above statements are true? Choose the correct answer from the options given below:

A relation R = {(1,1),(2,2),(3,3),(1,2),(2,1),(1,3),(2,3)} on A = {1,2,3} will be an equivalence relation, if we delete: Choose the correct answer from the options given below:

Which of the following statements are true? (A) $f:\mathbb{R} \to \mathbb{R}$ given by $f(x) = 3x$ is one-one onto. (B) $f:\mathbb{R} \to \mathbb{R}$ given by $f(x) = x^4$ is one-one and onto. (C) $f:\mathbb{Z} \to \mathbb{Z}$ given by $f(x) = x^2$ is neither one-one nor onto. (D) $f:\mathbb{R} \to \mathbb{R}$ given by $f(x) = |x|$ is neither one-one nor onto. (Where $\mathbb{R}$ is the set of all real numbers and $\mathbb{Z}$ is the set of all integers) Choose the correct answer from the options given below:

Let $R : \mathbb{R} \to \mathbb{R}$, where $\mathbb{R}$ is the set of real numbers and R be a relation. An element $(x, y) \in R$ if $x + y - \sqrt{2}$ belongs to the set of irrational numbers. Then the relation R is :

If $n(A) = 3$, $n(B) = 2$, then number of all possible surjective function from set A to set B are :

Let A = {1,2,3}. Consider the relation R = {(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)}. Then R is

A manufacturer can sell $x$ items at a price of Rs $3x+5$ each. The cost price of $x$ items is Rs $x^2 + 5x$. If x is the number of items she should sell to get no profit and no loss, then:

If $f: R \to R$ is defined by $f(x) = \sin x + x$, then $f(f(x))$ is:

The range of the function $f(x) = \frac{1}{3 - \sin 4x}$ is: