CAT 2023

Slot 1

QA

Question & Solution

Let us consider 3 cases:

1) x = 0, This is a solution, as both L.H.S and R.H.S will be equal (0) when x = 0. (1 solution)

2) x > 0

=> $2x\left(x^2+1\right)=5x^2$

=> $2\left(x^2+1\right)=5x$

=> $2x^2-5x+2=0$ => $2x^2-4x-x-2=0$

=> $2x\left(x-2\right)-1\left(x-2\right)=0$

=> $\left(x-2\right)\left(2x-1\right)=0$ => x = 2 or 1/2 => (1 integer solution)

3) x < 0

=> $-2x\left(x^2+1\right)=5x^2$

=> $2x^2+5x+2=0$

=> $2x^2+4x+x+2=0$

=> $2x\left(x+2\right)+1\left(x+2\right)=0$

=> $\left(x+2\right)\left(2x+1\right)=0$ => x = -2 or -1/2 => (1 integer solution)

So, the total number of integer solutions are 0, 2, -2 => 3.