Set X contains \( 3n \) elements and set Y contains \( 2n \) elements, and they have \( n \) elements in common. How many elements does \( (X - Y) \times (Y - X) \) have?
Let \( A = \{-3, -2, -1, 0, 1, 2, 3\} \) and \( B = \{0, 1, 4, 9\} \). How many elements does the subset of \( A \times B \) corresponding to the relation \( R = \{(x, y) : |x| < y\} \) have, where \( x \in A \) and \( y \in B \)?
Consider the following statements: Statement-I: If \( X \) is an \( n \times n \) matrix, then \( \det(mX) = m^{n} \det(X) \), where \( m \) is a scalar. Statement-II: If \( Y \) is a matrix obtained from \( X \) by multiplying any row or column by a scalar \( m \), then \( \det(Y) = m \det(X) \).
Which one of the following is correct in respect of the above statements?
A.
Both Statement-I and Statement-II are correct and Statement-II explains Statement-I
B.
Both Statement-I and Statement-II are correct but Statement-II does not explain Statement-I
C.
Statement-I is correct but Statement-II is not correct
D.
Statement-I is not correct but Statement-II is correct
A. Both Statement-I and Statement-II are correct and Statement-II explains Statement-I
45.
Consider the following statements about the matrix \( M = \begin{bmatrix} 71 & 23 & 48 \\ 57 & 28 & 29 \\ 65 & 17 & 48 \end{bmatrix} \): Statement-I: The inverse of \( M \) does not exist. Statement-II: \( M \) is non-singular.
Which one of the following is correct in respect of the above statements?
A.
Both Statement-I and Statement-II are correct and Statement-II explains Statement-I
B.
Both Statement-I and Statement-II are correct but Statement-II does not explain Statement-I
C.
Statement-I is correct but Statement-II is not correct
D.
Statement-I is not correct but Statement-II is correct
How many values of \( \theta \), where \( -\pi < \theta < \pi \), satisfy both the equations \( \cot \theta = -\sqrt{3} \) and \( \mathrm{cosec} \theta = -2 \) simultaneously?
