In a class of 45 students, 34 like to play cricket and 26 like to play football. Further, each student likes to play at least one of the two games. How many students like to play exactly one game?
If
\( \left(\frac{1 - i}{1 + i}\right)^{m}\left(\frac{1 + i}{1 - i}\right)^{n} = 1 \)
where \( i = \sqrt{-1} \), then what is the smallest positive value of \( (m - n) \)?
In obtaining the solution of the system of equations \( x + y + z = 7 \), \( x + 2y + 3z = 16 \) and \( x + 3y + 4z = 22 \) by Cramer's rule, the value of \( y \) is obtained by dividing \( D_{2} \) by \( D \), where
\( D = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 4 \end{vmatrix} \)
What is the value of the determinant \( D_{2} \)?
Consider the following in respect of non-singular matrices \( A \) and \( B \):
I. \( (AB)^{-1} = A^{-1}B^{-1} \)
II. \( (BA)(AB)^{-1} = I \), where \( I \) is the identity matrix
III. \( (AB)^{T} = A^{T}B^{T} \)
How many of the above are correct?
C. \( \begin{vmatrix} p & q & r \\ a & b & c \\ l & m & n \end{vmatrix} \)
17.
Let \( 1, \omega, \omega^{2} \) be three cube roots of unity. If \( x = a + b \), \( y = a\omega + b\omega^{2} \), \( z = a\omega^{2} + b\omega \), then what is \( x^{2} + y^{2} + z^{2} \) equal to?
If \( a, b, c \) are the sides of a triangle ABC and \( p \) is the perimeter of the triangle, then what is
\( \begin{vmatrix} p+c & a & b \\ c & p+a & b \\ c & a & p+b \end{vmatrix} \)
equal to?
