IPM Indore 2019

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

The $$x^{2} + y^{2}- 6x - 10y + k = 0$$ 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

If a 3 X 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

Let the set = {2,3,4,..., 25}. For each k $$\epsilon$$ P, define Q(k)= {x ∈ P such that x > k and k divides x}. Then the number of elements in the set $$ P - U_{k = 2}^{25}$$ Q(K) 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 tiles and plate are of uniform thickness, and there is no loss of material in the melting and recasting process) is ...........

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

Assume that all positive integers are written down consecutively from left to right as in 1234567891011...... The $$6389^{th}$$ digit in this sequence is

The number of pairs of integers whose sums are equal to their products is

You have been asked to select a positive integer N which is less than 1000 , such that it is either a multiple of 4, or a multiple of 6, or an odd multiple of 9. The number of such numbers is