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Let \( A = \{1,6,11,16,\ldots\} \) and \( B = \{9,16,23,30,\ldots\} \) be the sets consisting of the first \( 2025 \) terms of two arithmetic progressions. Then \( n(A \cup B) \) is:

Let \( R=\{(1,2),(2,3),(3,3)\} \) be a relation defined on the set \( \{1,2,3,4\} \). Then the minimum number of elements, needed to be added in \( R \) so that \( R \) becomes an equivalence relation, is:

Define a relation \( R \) on the interval \( \left[0, \frac{\pi}{2}\right) \) by \( x \,R\, y \) if and only if \( \sec^2 x - \tan^2 y = 1 \). Then \( R \) is:

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

The number of relations on the set \( A = \{1,2,3\} \), containing at most \( 6 \) elements including \( (1,2) \), which are reflexive and transitive but not symmetric, is ________________.

Let \( A = \{0, 1, 2, 3, 4, 5\} \). Let \( R \) be a relation on \( A \) defined by \( (x, y) \in R \) if and only if \( \max\{x, y\} \in \{3, 4\} \). Then among the statements:
\( (S_1): \) The number of elements in \( R \) is \( 18 \), and
\( (S_2): \) The relation \( R \) is symmetric but neither reflexive nor transitive

The number of non-empty equivalence relations on the set \( \{1, 2, 3\} \) is:

The relation \( R = \{(x, y) : x, y \in \mathbb{Z} \ \text{and} \ x + y \ \text{is even}\} \) is:

Let \( X = \mathbb{R} \times \mathbb{R} \). Define a relation \( R \) on \( X \) as: \( (a_1,b_1) \, R \, (a_2,b_2) \Leftrightarrow b_1 = b_2 \).
Statement I: \( R \) is an equivalence relation.
Statement II: For some \( (a,b) \in X \), the set \( S = \{(x,y) \in X : (x,y) R (a,b)\} \) represents a line parallel to \( y = x \).
In the light of the above statements, choose the correct answer from the options given below:

Let \( A = \{1, 2, 3, 4\} \) and \( B = \{1, 4, 9, 16\} \). Then the number of many-one functions \( f : A \to B \) such that \( 1 \in f(A) \) is equal to:

Let \( A = \{1, 2, 3, \ldots, 10\} \) and \( B = \left\{ \dfrac{m}{n} : m, n \in A,\, m < n \text{ and } \gcd(m,n) = 1 \right\}. \) Then \( n(B) \) is equal to:

Let \( A = \{-3,-2,-1,0,1,2,3\} \). Let \( R \) be a relation on \( A \) defined by \( xRy \) if and only if \( 0 \le x^{2} + 2y \le 4 \). Let \( l \) be the number of elements in \( R \) and \( m \) be the minimum number of elements required to be added in \( R \) to make it a reflexive relation. Then \( l + m \) is equal to:

Let \( A = \{(x,y) \in R \times R : |x + y| \ge 3\} \) and \( B = \{(x,y) \in R \times R : |x| + |y| \le 3\}. \)
If \( C = \{(x,y) \in A \cap B : x = 0 \text{ or } y = 0\} \) then \(\displaystyle \sum_{(x,y)\in C} |x + y| \) is:

Let \( A=\{-2,-1,0,1,2,3\} \). Let \( R \) be a relation on \( A \) defined by \( xRy \) if and only if \( y=\max\{x,1\} \). Let \( l \) be the number of elements in \( R \). Let \( m \) and \( n \) be the minimum number of elements required to be added in \( R \) to make it reflexive and symmetric relations, respectively. Then \( l+m+n \) is equal to:

Let \( A = \{-3,-2,-1,0,1,2,3\} \) and \( R \) be a relation on \( A \) defined by \( xRy \) if and only if \( 2x - y \in \{0,1\} \). Let \( l \) be the number of elements in \( R \). Let \( m \) and \( n \) be the minimum number of elements required to be added in \( R \) to make it reflexive and symmetric relations, respectively. Then \( l + m + n \) is equal to:

For \( n \ge 2 \), let \( S_{n} \) denote the set of all subsets of \( \{1,2,\ldots,n\} \) with no two consecutive numbers. For example \( \{1,3,5\} \in S_{6} \) but \( \{1,2,4\} \notin S_{6} \). Then \( n\!\left(S_{5}\right) \) is equal to ____________.

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JEE Main 2025 23rd Jan Evening Shift

Let \( A = \{(x,y) \in R \times R : |x + y| \ge 3\} \) and \( B = \{(x,y) \in R \times R : |x| + |y| \le 3\}. \)
If \( C = \{(x,y) \in A \cap B : x = 0 \text{ or } y = 0\} \) then \(\displaystyle \sum_{(x,y)\in C} |x + y| \) is: