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

Updated On: Nov 24, 2025

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If \( a, b, c \) are all positive integers, with \( 4a>b \), then which of the following conditions is BOTH NECESSARY AND SUFFICIENT for the expression \[ \sqrt{(3)^a (21)^{3a-b} (49)^{2b+c}} \] to be a positive integer?
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The market value of beams, made of a rare metal, has a unique property: the market value of any such beam is proportional to the square of its length. Due to an accident, one such beam got broken into two pieces having lengths in the ratio 4:9. Considering each broken piece as a separate beam, how much gain or loss, with respect to the market value of the original beam before the accident, is incurred?
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In an 8-week course, the teacher conducts a test every week, and the scores are in the range of 1–4. There are only two students enrolled in the course, \(R\) and \(S\). The following conditions are given:
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The total of the first three test scores of \(R\) equals the total of the first two test scores of \(S\).
From the fifth test onwards, \(S\) scored the same as \(R\).
The scores of \(S\) from the first two tests and all the remaining tests follow a geometric progression.
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