Question

Let function f be defined as

$f(x) = [\sqrt{n^2 + 4}] - [n+ \sqrt{x}]; n \in N$,

where [x] = max{a ∈ Z: a ≤ x}. Now, consider the following sets

A = {x ∈ Z : f(x) ∈ R},

Based on above information, answer the following questions

Number of elements in set A can be

• A. 1
• B. 2
• C. more than 2
• D. less than 2

Answer 1

Answer: (A)

1

Answer 2

For $n \ge 2$, $f(x) = n - [n+\sqrt{x}]$. Set $A = \{x \in Z : f(x) \ge 0\}$. $f(x) \ge 0 \implies [n+\sqrt{x}] \le n \implies n+\sqrt{x} < n+1 \implies \sqrt{x} < 1$. Since $x \ge 0$, $0 \le x < 1$. Integers in this range are $x=0$. So $A=\{0\}$, $|A|=1$. Thus, 1 is a possible number of elements in A.