Question
If $$\left[ x \right]$$ = the greatest integer less than or equal to $$x,$$ and $$(x)$$ = the least integer greater than or equal to $$x,$$ and $${\left[ x \right]^2} + {\left( x \right)^2} > 25$$ then $$x$$ belongs to
A.
$$\left[ {3,4} \right]$$
B.
$$\left( { - \infty , - 4} \right]$$
C.
$$\left[ {4, + \infty } \right)$$
D.
$$\left( { - \infty , - 4} \right] \cup \left[ {4, + \infty } \right)$$
Answer :
$$\left( { - \infty , - 4} \right] \cup \left[ {4, + \infty } \right)$$
Solution :
$$\eqalign{
& {\text{If }}x = n \in Z,{n^2} + {n^2} > 25.\,{\text{So, }}{n^2} > \frac{{25}}{2} \cr
& \therefore \,\,x = n = 4,5,6,.....\,\,{\text{or, }} - 4, - 5, - 6,..... \cr
& {\text{If }}x = n + k,n \in Z,0 < k < 1\,\,{\text{then }}{n^2} + {\left( {n + 1} \right)^2} > 25 \cr
& {\text{or, }}{n^2} + n - 12 > 0 \cr
& \therefore \,\,n < - 4\,\,{\text{or, }}n > 3 \cr
& \therefore \,\,x < - 4 + k\,\,{\text{or, }}x > 3 + k,\,{\text{where }}0 < k < 1 \cr
& \therefore \,\,x \leqslant - 4\,\,{\text{or, }}x \geqslant 4. \cr} $$