If \(m\) is a positive integer then the values of \(k\) for which \(6m + k\) cannot be a perfect square are

Let \(S = \{1, 2, \ldots, 180\}\). Define \(A\) as the set of all multiples of 4 in \(S\), \(B\) as the set of all multiples of 6 in \(S\), and \(C\) as the set of all multiples of 9 in \(S\). The number of elements in \(S\) that belong to exactly one of \(A, B, C\) is

A fair die is rolled repeatedly. The probability that the cumulative sum is at least 17 in the third trial is

If \(\log_{18} 24 = p\), then \(\log_{96} 108\) equals

From the passage: A nutritionist is designing a daily diet plan of a person using five f...

When one serving of each of the five items is included in the diet plan, the nutrient with the least percentage of MDR satisfied is

The number of integer solutions \((x, y)\) of the inequality \(x^2 + y^2 \leq 10\) is

If \(a\), \(b\), and \(c\) are three prime numbers such that \(abc = 23(a+b+c)\), then the maximum possible value of \(a+b-c\) is

The number of ways in which coins of denominations of rupees 2, 5 and 10 can be combined to make a value of rupees 50 is

Positive reals \(x, y\) satisfy \(x \neq y\) and \(\frac{x^2 + y^2}{xy} = k\). If replacing \(x\) by \(x+y\) and \(y\) by \(|x-y|\) leaves the value of \(k\) unchanged, then \(k\) equals