 Hello and welcome to the session. I am here to help you with the following problem. Using properties of determinants prove that determinant 1, 1 plus p, 1 plus p plus q, 2, 3 plus 2p, 4 plus 3p plus 2q, 3, 6 plus 3p, 10 plus 6p plus 3q is equal to 1. Now here is why the solution. Consider LHS which is equal to the determinant 1, 1 plus p, 1 plus p plus q, 2, 3 plus 2p, 4 plus 3p plus 2q, 3, 6 plus 3p, 10 plus 6p plus 3q. Now applying r2 tends to r2 minus twice of r1 and r3 tends to r3 minus twice of r1. Applying row operation on these two rows we get which is equal to 1, 1 plus p, 1 plus p plus q, 0, 1, 2 plus p, 0, 3, 7 plus 3p. Now expanding it we get eliminating this row and this column we get which is equal to 7 plus 3p minus 3 multiplied by 2 plus p which is equal to 1. As we can see this can be written as 7 plus 3p minus 6 minus 3p. Now here this and this get cancelled and 7 minus 6 gives us 1 which is equal to rHS. Therefore LHS is equal to rHS hence proved. I hope you understood the problem. Bye and have a nice day.