Let \(\mathrm{R}\) be a relation in \(\mathrm{N}\) defined by \(\mathrm{R}=\left\\{\left(1+\mathrm{x}, 1+\mathrm{x}^{2}\right) / \mathrm{x} \leq 5, \mathrm{x} \in \mathrm{N}\right\\}\) which of the following is false? (a) \(\mathrm{R}=\\{(2,2),(3,5),(4,10),(5,17),(6,25)\\}\) (b) Domain of \(\mathrm{R}=\\{2,3,4,5,6\\}\) (c) Range of \(\mathrm{R}=\\{2,5,10,17,26\\}\) (d) (b) and (c) are true

Let's analyze the given relation: R = {(1+x, 1+x^2)/x ≤ 5, x ∈ N}. We are given that x is a natural number (x ∈ N). We want to find all pairs (1+x, 1+x^2) that satisfy the condition where x is a natural number, and the given expression is less than or equal to 5.

To find the elements of R, we will substitute x with natural numbers (starting from 1) and check if the condition (1+x, 1+x^2)/x ≤ 5 holds. Let's do that: x = 1: (1+1, 1+1^2)/1 = (2,2); This pair satisfies the condition as both 2 and 2 are less than or equal to 5. x = 2: (1+2, 1+2^2)/2 = (3,5); This pair also satisfies the condition as both 3 and 5 are less than or equal to 5. x = 3: (1+3, 1+3^2)/3 = (4,10); This pair does not satisfy the condition, as 10 is greater than 5. x = 4: (1+4, 1+4^2)/4 = (5,17); This pair does not satisfy the condition, as 17 is greater than 5. We can see that as we increase x, the second element of the pair is going to increase and not satisfy the condition. So, R = {(2,2), (3,5)}.

The Domain of R is the set of all the first elements in R: Domain (R) = {2,3}. The Range of R is the set of all the second elements in R: Range (R) = {2,5}.

Now, let's compare our results with the given options: (a) R = {(2,2),(3,5),(4,10),(5,17),(6,25)}; This is false, as we found R = {(2,2), (3,5)}. (b) Domain of R = {2,3,4,5,6}; This is false, as we found the Domain of R = {2,3}. (c) Range of R = {2,5,10,17,26}; This is false, as we found the Range of R = {2,5}. (d) (b) and (c) are true; This is false, as both (b) and (c) are false. Thus, all the given options are false.