 Hello friends, and how are you all doing today the question says using properties of determinants solve x now here We are given the determinant as In the first row we have a plus x a minus x a minus x in the second row We have a minus x a plus x a minus x and in the third row We have a minus x a minus x and then a plus x and the value of this determinant is equal to Zero it's given to us in the question. So let us proceed with our solution Let us rewrite the given determinant once again my hair on applying Let us add all the three rows In row one so it would be equal to row one plus row two plus row Three so we will have if we add these three rows. We'll have three a Minus x because one positive x will get cancelled out with the negative x and we are left with a negative x in the same way here We have three a minus x then three a minus x and then the second and third row will remain the same Now on taking out Three a minus x from the first row common. We are left with one a Minus x a minus x in the first column then one a Plus x a minus x and then again one a minus x a plus x Equal to Zero now if we subtract column three firstly by column one and then by column two that is a apply Column one column one minus column three and the column two Subtract column three from column two and on doing so we will have Three a minus x now on subtracting column three from column one. We will get Zero then again zero because a minus x minus a plus x will give us zero and here we will have Minus two x right then Here also on subtracting column three from column one. We are left with Zero two a x and here we have minus two x and we will have column three as it is on expanding Row one We will have minus x Now here if you observe these two terms are zero so anything getting multiplied by zero will give us zero So we are left with the third term. So here One zero in two minus two x will give us zero Minus minus will become Plus so we have plus four x square Equal to further we have three Minus x into Four x square equal to zero so by this we can write down that the value of x is equal to three or value of x is equal to Zero right so this is a required answer to the given question Hope you understood it well and have a nice day You