 Hi and welcome to the session. I am Deepika here as this is a question using the property of determinants and without expanding prove that One bca into b plus c one C a b into C plus a one a b into C into a plus b is Equal to zero. Let's start the solution solution Let Delta is equal to our given determinant that is one one one bcca a b a into b plus c b into C plus a C into a plus b is equal to We have to prove that this Delta is equal to zero Again, we can rewrite our Delta as Delta is equal to one one one bc C a a b now C column 3 we can write as a b plus a C bc plus b a and C a plus bc By applying 3 goes to C 3 plus C 2 we get Delta is equal to one one one bc C a a b now C 3 is C 3 plus C 2 that is equal to a b plus bc plus C a a b plus bc Plus C a and this is again a b plus bc plus C a or Delta can be written as Delta is equal to a b Plus bc plus C a one one one bc C a a b one one one By using the property Each element of a row or column of a determinant is multiplied by a constant k Then its value gets multiplied by k So using this property we get Delta is equal to a b plus bc plus C a into One one one bc c a a b and one one one again we have Delta is equal to a b plus bc plus C a into zero Because in this determinants Two columns are identical and we have a property if any two rows or columns of a determinant are Identical that is all corresponding elements are same then value of the determinant is zero Therefore Delta is equal to a b plus bc plus C a into zero so we have Delta is equal to zero hence proved Because in the given question we have to prove that our given determinant is equal to zero Given determinants are Delta so Delta is equal to zero hence we have proved I hope the question is clear to you. Bye and have a good day