 Hi, and welcome to the session, I am Deepika here. Let's discuss the question by using properties of determinants, show the determinant 1, 1, 1, a, b, c, a, q, b, q, c, q is equal to a minus b into b minus c into c minus a into a plus b plus c. So let's start the solution. We have on the left hand side a, q, 1, b, 1, c, c, q. Interchanging goes with column. The left hand side is equal to 1, 1, a, c, q. Minus r1, 3 goes to r3 minus r1. We get side is equal to a, q. Now r2 goes to r2 minus r1. So 1 minus 1, 0. This is b minus a, and this is b, q, minus a, q. Again, r3 goes to r3 minus r1. So 1 minus 1 is 0. This is c minus a, and this is c, q, minus a, q. q minus y, q is equal to x minus y into x square plus xy plus y square. So we will apply the formula here in b, q, minus a, q, and c, q, minus a, q. So our left hand side will become, so this is 1, 0, 0, a, b minus a, c minus a, a, q. Now this is b minus a into b square plus b, a, plus a square. And this is c minus a, c square plus c, a, plus a square. b minus a is the common factor in r2, and c minus a is the common factor in r3. So by taking out b minus side is equal to b minus a into c minus a. Now rho1 is as it is. This is 1, a, a, q. And b minus a is common from r2. So we get 0, 1, b square plus b, a, plus a square. And here c minus a is common from r3. So we get 0, 1, c square plus c, a, plus a square. Side is equal to into c minus a, rho1 is as it is. This is 1, a, a, q. So this is 0, b square, plus a, b, plus 3 minus r2. So this is 0, 0. Now c square plus c, a, plus a square, minus b square, minus b, a, minus a square. So we will get c square minus b square, minus b, a. So we will get this is equal to into c minus a, 0, 1, b square plus a, b, plus a square. Now this is 0, 0. Now c square minus b square is c minus b into c plus b, minus b, a. So this is plus a into c minus b. So by taking this is equal to minus a into c minus a into c minus b. So this is equal to 1. So this is 0, 1, b square plus a, b, plus a square. This is 0, 0. Now c minus b is common here. So we are left with c plus b plus a. That is a plus b plus c. We can take a plus b plus c common factor from r3. So by taking out common from, we get our left hand side is equal to, this is equal to b minus a. Now b minus a and c minus b can be written as a minus b, b minus c into c minus a, into a plus b plus c. So this is r1 is as it is 1, a, aq, 0, 1, b square plus a, b plus a square. And this is 0, 0, 1, equal to a minus b into b minus c into c minus a into a plus b plus c into, so this is equal to minus b, b minus c, c minus a into a plus b plus c. So which is our right hand side. Hence we have proved that our given determinant is equal to a minus b into b minus c into c minus a into a plus b plus c. Hence left hand side is equal to right hand side. I hope the question is clear to you. Bye and have a nice day.