If a \(3\times 3\) matrix is filled with +1's and -1's such that the sum of each row and column of the matrix is 1, then the absolute value of its determinant is...

The circle \(x^2+y^2-6x-10y+k=0\) does not touch or intersect the coordinate axes. If the point (1, 4) does not lie outside the circle, and the range of k is [a, b] then a + b is____

Let the set \(P = \{2, 3, 4, \ldots, 25\}\). For each \(k \in P\), define \(Q(k)=\{x\in P \text{ such that } x>k \text{ and } k \text{ divides } x\}\). Then the number of elements in the set \(P-\bigcup_{k=2}^{25}Q(k)\) is

The number of pairs (x, y) satisfying the equation \(\sin x+\sin y=\sin(x+y)\) and \(|x|+|y|=1\) is

The number of whole metallic tiles that can be produced by melting and recasting a circular metallic plate, if each of the tiles has a shape of a right-angled isosceles triangle and the circular plate has a radius equal in length to the longest side of the tile (Assume that the t…

If x, y, z are positive real numbers such that \(x^{12} = y^{16} = z^{24}\), and the three quantities \(3\log_y x\), \(4\log_z y\), \(n\log_x z\) are in arithmetic progression, then the value of n is

If \(|x| < 100\) and \(|y| < 100\), then the number of integer solutions of (x, y) satisfying the equation \(4x + 7y = 3\) is

The maximum distance between the point (-5, 0) and a point on the circle \(X^2 + Y^2 = 4\) is......

The average of five distinct integers is 110 and the smallest number among them is 100. The maximum possible value of the largest integer is