The number of distinct integer solutions \(( x, y )\) of the equation \( |x + y| + |x - y| = 2 \), is 

For some constant real numbers \( p, k \) and \( a \), consider the following system of linear equations in \( x \) and \( y \): \[ px - 4y = 2 \] \[ 3x + ky = a \]

A necessary condition for the system to have no solution for \( (x, y) \), is

If \( (a + b\sqrt{3})^2 = 52 + 30\sqrt{3} \), where \( a \) and \( b \) are natural numbers, then \( a + b \) equals

A circular plot of land is divided into two regions by a chord of length \( 10\sqrt{3} \) meters such that the chord subtends an angle of \( 120^\circ \) at the center. Then, the area, in square meters, of the smaller region is

In a group of 250 students, the percentage of girls was at least 44% and at most 60%. The rest of the students were boys. Each student opted for either swimming or running or both. If 50% of the boys and 80% of the girls opted for swimming while 70% of the boys and 60% of the girls opted for running, then the minimum and maximum possible number of students who opted for both swimming and running, are

If \( 10^{68} \) is divided by \( 13 \), the remainder is

The midpoints of sides \( AB \), \( BC \), and \( AC \) in \( \triangle ABC \) are \( M \), \( N \), and \( P \), respectively. The medians drawn from \( A \), \( B \), and \( C \) intersect the line segments \( MP \), \( MN \) and \( NP \) at \( X \), \( Y \), and \( Z \), respectively. If the area of \( \triangle ABC \) is \( 1440 \) sq cm, then the area, in sq cm, of \( \triangle XYZ \) is 

After two successive increments, Gopal's salary became \( 187.5\% \) of his initial salary. If the percentage of salary increase in the second increment was twice of that in the first increment, then the percentage of salary increase in the first increment was

Sam can complete a job in \( 20 \) days when working alone. Mohit is twice as fast as Sam and thrice as fast as Ayna in the same job. They undertake a job with an arrangement where Sam and Mohit work together on the first day, Sam and Ayna on the second day, Mohit and Ayna on the third day, and this three-day pattern is repeated till the work gets completed. Then, the fraction of total work done by Sam is

A certain amount of water was poured into a \( 300 \) litre container and the remaining portion of the container was filled with milk. Then an amount of this solution was taken out from the container which was twice the volume of water that was earlier poured into it, and water was poured to refill the container again. If the resulting solution contains \( 72\% \) milk, then the amount of water, in litres, that was initially poured into the container was