List of top Mathematics Questions on Relations and Functions

The number of relations defined on the set $ \{a, b, c, d\} $ that are both reflexive and symmetric is equal to:

If the domain of the function \[ f(x)=\frac{1}{\ln(10-x)}+\sin^{-1}\!\left(\frac{x+2}{2x+3}\right) \] is \[ (-\infty,-a]\ \cup\ (-1,b)\ \cup\ (b,c), \] then find the value of $ (b+c+3a) $.

A relation $R$ on set $A=\{1,2,3\}$ defined as $R=\{(1,2),(2,1),(2,2)\}$ is

Consider two sets \[ A = \{ x \in \mathbb{Z} : |(|x-3|-3)| \le 1 \} \] and \[ B = \left\{ x \in \mathbb{R} - \{1,2\} : \frac{(x-2)(x-4)}{x-1}\,\log_e(|x-2|) = 0 \right\}. \] Then the number of onto functions $ f : A \to B $ is equal to

Let $ A = \{0,1,2,\ldots,9\} $. Let $ R $ be a relation on $ A $ defined by $(x,y) \in R$ if and only if $ |x - y| $ is a multiple of $3$. Given below are two statements:
Statement I: $ n(R) = 36 $.
Statement II: $ R $ is an equivalence relation.
In the light of the above statements, choose the correct answer from the options given below.

Show that the relation $ R $ in the set of natural numbers $ N \times N $ defined by $ (a, b)\, R\, (c, d) $ if $ a + d = b + c $ is an equivalence relation.

Let $ A = \{1,2,3,\ldots,9\} $ and $ R \subset A \times A $ be defined by \[ (x,y) \in R \iff |x-y| \text{ is a multiple of } 3. \] Consider the following statements:
S$_1$: Number of elements in $ R $ is 36.
S$_2$: $ R $ is an equivalence relation.
Which of the following is correct?

Let the sets \[ A = \{x : |x - 3| - 3 \le 1,\; x \in \mathbb{Z}\} \] \[ B = \left\{ x : x \in \mathbb{R},\; x \ne 1,2,\; \frac{(x-2)(x-4)}{(x-1)} \log_e |x-2| = 0 \right\} \] Then the number of onto functions from $ A $ to $ B $ is:

Let $ M = \{1,2,3,\ldots,16\} $ and $ R $ be a relation on $ M $ defined by $ xRy $ if and only if $ 4y = 5x - 3 $. Then, the number of ordered pairs required to be added to $ R $ to make it symmetric is

Let $ R $ be a relation defined by the teacher to plan the seating arrangement of students in pairs, where $ R = \{(x, y) : x, y \text{ are Roll Numbers of students such that } y = 3x \} $. List the elements of $ R $. Is the relation $ R $ reflexive, symmetric, and transitive? Justify your answer.

Assertion (A): Let $f(x) = e^x$ and $g(x) = \log x$. Then $(f + g)(x) = e^x + \log x$ where the domain of $(f + g)$ is $\mathbb{R}$.
Reason (R): $\text{Dom}(f + g) = \text{Dom}(f) \cap \text{Dom}(g)$.

Edge of a variable cube increases at the rate of 5 cm/s. The rate at which the surface area of the cube increases when the edge is 2 cm long is :

If $f(x) = 2x^8$, then the correct statement is :

If $ f(x) = -2x^8 $, then the correct statement is :

If $ R $ be a relation defined as $ a \, R \, b $ iff $ |a - b|>0 $, $ a, b \in \mathbb{R} $, then $ R $ is :

Let $ R $ be a relation defined by $ R = \{(x, y) : x, y \text{ are Roll Numbers of students such that } y = x^3 \} $. List the elements of $ R $. Is $ R $ a function? Justify your answer.

A carpenter needs to make a wooden cuboidal box, closed from all sides, which has a square base and fixed volume. Since he is short of the paint required to paint the box on completion, he wants the surface area to be minimum.
On the basis of the above information, answer the following questions :
Find a relation between $ x $ and $ y $ such that the surface area $ S $ is minimum.

Let $ R $ be a relation on the set of real numbers $ \mathbb{R} $ defined as \[ R = \{(x, y) : x - y + \sqrt{3} \text{ is an irrational number}, x, y \in \mathbb{R} \}. \] Verify $ R $ for reflexivity, symmetry, and transitivity.

A school is organizing a debate competition with participants as speakers and judges. $ S = \{S_1, S_2, S_3, S_4\} $ where $ S = \{S_1, S_2, S_3, S_4\} $ represents the set of speakers. The judges are represented by the set: $ J = \{J_1, J_2, J_3\} $ where $ J = \{J_1, J_2, J_3\} $ represents the set of judges. Each speaker can be assigned only one judge. Let $ R $ be a relation from set $ S $ to $ J $ defined as: $ R = \{(x, y) : \text{speaker } x \text{ is judged by judge } y, x \in S, y \in J\} $.

Let $ f : \mathbb{R} \to \mathbb{R} $ be defined as $ f(x) = 10 - x^2 $, then:

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

According to recent research, air turbulence has increased in various regions around the world due to climate change. Turbulence makes flights bumpy and often delays the flights. Assume that an airplane observes severe turbulence, moderate turbulence or light turbulence with equal probabilities. Further, the chance of an airplane reaching late to the destination are $55\%$, $37\%$ and $17\%$ due to severe, moderate and light turbulence respectively.
On the basis of the above information, answer the following questions:
(i) Find the probability that an airplane reached its destination
(ii)If the airplane reached its destination late, find the probability that it was due to moderate turbulence.

The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding vehicle from a distance of 300 m and even function in the dark. A camera is installed on a pole at the height of 5 m. It detects a car travelling away from the pole at the speed of 20 m/s. At any point, $x$ m away from the base of the pole, the angle of elevation of the speed camera from the car C is $\theta$.
On the basis of the above information, answer the following questions:
(i)Express $\theta$ in terms of the height of the camera installed on the pole and x.
(ii) Find $\frac{d\theta}{dx}$.
(iii) (a) Find the rate of change of angle of elevation with respect to time at an instant when the car is 50 m away from the pole.
(iii) (b) If the rate of change of angle of elevation with respect to time of another car at a distance of 50 m from the base of the pole is $\frac{3}{101} \, \text{rad/s}$, then find the speed of the car.

The function $ f : \mathbb{R} \to \mathbb{R} $ defined by $ f(x) = e^x + e^{-x} $ is:

Let $f : \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = \frac{e^{|x|} - e^{-x}}{e^x + e^{-x}}$, then: