 Hello and welcome to the session. I am Ashima and I am there to help you with the following problem. If A, B, C are in AP, then the determinant X plus 2, X plus 3, X plus 2A, X plus 3, X plus 4, X plus 2B, X plus 4, X plus 5, X plus 2C is A0, B1, CX, D2X. Now let us write the solution. Given to us the determinant X plus 2, X plus 3, X plus 2A, X plus 3, X plus 4, X plus 2B, X plus 4, X plus 5, X plus 2C. Now it will be equal to X plus 2, X plus 2 plus 1, X plus 2A because X plus 3 can be written as X plus 2 plus 1. Similarly, X plus 3, X plus 3 plus 1, X plus 2B, X plus 4, X plus 4 plus 1, X plus 2C. Now which can be written as X plus 2, X plus 2, X plus 2A, X plus 3, X plus 3, X plus 2B, X plus 4, X plus 4, X plus 2C, plus X plus 2, 1, X plus 2A, X plus 3, 1, X plus 2B, X plus 4, 1, X plus 2C. Now let this be 1 and this be 2. Now first consider first matrix which is X plus 2, X plus 2, X plus 2A, X plus 3, X plus 3, X plus 2B, X plus 4, X plus 4, X plus 2C. Now here we can see that column 1 and column 2 have identical terms so it is equal to 0 because C1 and C2 are identical. Now consider C2 that is the second matrix which is X plus 2, 1, X plus 2A, X plus 3, 1, X plus 2B, X plus 4, 1, X plus 2C. Now applying R2 tends to R2 minus R1 and R3 tends to R3 minus R1. So we get it is equal to first row as it is and in the second row 1, 0, 2B minus 2A, 20, 2C minus 2A. Now expanding it along this so eliminating this column and this row we are left with 2C minus 2A minus 2 multiplied by 2B minus 2A which is equal to 2C minus 2A minus 4B plus 4A which is equal to 2C minus 4B plus 2A which is equal to twice of A plus C minus 4B. Now it is given to us that A, B, C are in AP therefore it implies B is equal to A plus C by 2 which implies A plus C is equal to 2B. Now substituting this value of A plus C in this equation so this equation becomes twice of 2B minus 4B which is equal to 4B minus 4B which is equal to 0. Now we see that this matrix that is A becomes it is equal to matrix 1 plus determinant 2. Now we can see that determinant 1 is equal to 0 which we have calculated above. So it is equal to 0 and similarly determinant 2 which we have calculated here is equal to 0. So 0 plus 0 which is equal to 0 therefore value of determinant is equal to 0 hence the correct option is A. I hope you understood the problem. Bye and have a nice day.