Find the number of non-negative integer values of k for which the quadratic equation x² − 5x + k = 0 has only integer roots.(CAT 2025  Slot 1)

For what range of values of c is the minimum value of f(x) = x² − 4cx + 8c strictly greater than the maximum value of g(x) = −x² + 3cx − 2c?(CAT 2025 Slot 1)

If m and n are integers such that (m + 2n)(2m + n) = 27, then the maximum possible value of 2m − 3n is:(CAT 2025 Slot 2)

If a, b, c, and d are integers such that their sum is 46, then the minimum possible value of (a−b)² + (a−c)² + (a−d)² is:(CAT 2025 Slot 2)                                                                                                                                                                

Suppose a, b, c are three distinct natural numbers such that 3ac = 8(a + b). The smallest possible value of 3a + 2b + c is:(CAT 2025 Slot 2)                                                                                                                                                

Let p, q, and r be three natural numbers such that their sum is 900, and r is a perfect square whose value lies between 150 and 500. If p is not less than 0.3q and not more than 0.7q, find the sum of the maximum and minimum possible values of p.(CAT 2025 Slot 3)                                                                                                   

If (a + b√n) is the positive square root of (29 − 12√5), where a and b are integers and n is a natural number, the maximum possible value of (a + b + n) is:(CAT 2024 Slot 1)                                                                                         

If x is a positive real number such that 4log₁₀x + 4log₁₀₀x + 8log₁₀₀₀x = 13, then the greatest integer not exceeding x is:(CAT 2024 Slot 1)                                                                                                                                                                

If m and n are natural numbers such that n > 1, and mⁿ = 2²⁵ × 3⁴⁰, then m − n equals:(CAT 2024 Slot 2)

If (a + b√3)² = 52 + 30√3, where a and b are natural numbers, then a + b equals: (CAT 2024 Slot 3)