If the function $f: R \rightarrow R$ is defined by $f(x)=|x|(x-\sin x)$, then which of the following statements is TRUE?
- $f$ is one-one, but NOT onto
- $f$ is onto, but NOT one-one
- $f$ is BOTH one-one and onto
- $f$ is NEITHER one-one NOR onto
The Correct Option is C
Solution and Explanation
The Correct Option is (C): $f$ is BOTH one-one and onto
Top Questions on cartesian products of sets
- Let $S =\{1,2,3,5,7,10,11\}$ The number of non-empty subsets of $S$ that have the sum of all elements a multiple of 3 , is _____
- JEE Main - 2023
- Mathematics
- cartesian products of sets
- If the equations \( x = t^2 + t + 1, \quad y = t^2 - t + 1 \) represent a curve \( C \) with parameter \( t \), then the Cartesian equation of \( C \) is:
- AP EAPCET - 2023
- Mathematics
- cartesian products of sets
- The Cartesian form of the curve given by \( x = \frac{a}{2}\left(t + \frac{1}{t}\right) \), \( y = \frac{a}{2}\left(t - \frac{1}{t}\right) \), where \( t \) is a parameter, is:
- AP EAPCET - 2023
- Mathematics
- cartesian products of sets
Let \(S ={ (\begin{matrix} -1 & 0 \\ a & b \end{matrix}), a,b, ∈(1,2,3,.....100)}\) and
let \(T_n = {A ∈ S : A^{n(n + 1)} = I}. \)
Then the number of elements in \(\bigcap_{n=1}^{100}\) \(T_n \) is- JEE Main - 2022
- Mathematics
- cartesian products of sets
- If \( A = \{1, 2, 5, 6\} \) and \( B = \{1, 2, 3\} \), then \( A \times B \cap B \times A \) is equal to:
- COMEDK UGET - 2021
- Mathematics
- cartesian products of sets
Questions Asked in JEE Advanced exam
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.
- JEE Advanced - 2025
- binomial expansion formula
- The total number of real solutions of the equation $$ \theta = \tan^{-1}(2 \tan \theta) - \frac{1}{2} \sin^{-1} \left( \frac{6 \tan \theta}{9 + \tan^2 \theta} \right) $$ is
(Here, the inverse trigonometric functions $ \sin^{-1} x $ and $ \tan^{-1} x $ assume values in $[-\frac{\pi}{2}, \frac{\pi}{2}]$ and $(-\frac{\pi}{2}, \frac{\pi}{2})$, respectively.)- JEE Advanced - 2025
- Inverse Trigonometric Functions
Let $ \mathbb{R} $ denote the set of all real numbers. Then the area of the region $$ \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x > 0, y > \frac{1}{x},\ 5x - 4y - 1 > 0,\ 4x + 4y - 17 < 0 \right\} $$ is
- JEE Advanced - 2025
- Coordinate Geometry
The center of a disk of radius $ r $ and mass $ m $ is attached to a spring of spring constant $ k $, inside a ring of radius $ R>r $ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following Hooke’s law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as $ T = \frac{2\pi}{\omega} $. The correct expression for $ \omega $ is ( $ g $ is the acceleration due to gravity):
- JEE Advanced - 2025
- Waves and Oscillations
- Consider the vectors $$ \vec{x} = \hat{i} + 2\hat{j} + 3\hat{k},\quad \vec{y} = 2\hat{i} + 3\hat{j} + \hat{k},\quad \vec{z} = 3\hat{i} + \hat{j} + 2\hat{k}. $$ For two distinct positive real numbers $ \alpha $ and $ \beta $, define $$ \vec{X} = \alpha \vec{x} + \beta \vec{y} - \vec{z},\quad \vec{Y} = \alpha \vec{y} + \beta \vec{z} - \vec{x},\quad \vec{Z} = \alpha \vec{z} + \beta \vec{x} - \vec{y}. $$ If the vectors $ \vec{X}, \vec{Y}, \vec{Z} $ lie in a plane, then the value of $ \alpha + \beta - 3 $ is ________.
Concepts Used:
Cartesian Products of Sets
Cartesian products of sets here are explained with the help of an example. Consider A and B to be the 2 sets such that A is a set of 3 colors of tables and B is a set of 3 colors of chairs objects, i.e.,
A = {red, blue, purple}
B = {brown, green, yellow},
Now let us find the number of pairs of colored objects that we can make from a set of tables and chairs in various combinations. They can be grouped as given below:
(red, brown), (red, green), (red, yellow), (blue, brown), (blue, green), (blue, yellow), (purple, brown), (purple, green), (purple, yellow)
There are 9 such pairs in the Cartesian product since 3 elements are there in each of the defined sets A and B. The above-ordered pairs shows the definition for the Cartesian product of sets given. This product is resembled by “A × B”.