Question

Sum of all integral values of $a\in [1,500]$ for which the equation $[x{]}^3+x-a=0$ has a solution ($[.]$ denotes the greatest integer function) is

• A. $1342$
• B. $512$
• C. $784$
• D. $812$

Answer 1

The correct option is A $1342$

$\begin{array}{c}{\left[x\right]}^3+x-a=0\\ x^3+x=a\\ x\left(x^2+1\right)=a\\ Given,\\ a\in \left[1:1500\right]\\ Then2\le x\le 11\end{array}$

If $x=2$ then $a=10$

If $x=3$ then $a=3.10=30$

If $x=11$ $a=11.({11}^2+1)=1342$

Hence,

Option $A$ is correct answer.