Evaluate The Determinant Without Expansion As Far As Possible.                  

The number of values of k for which the system of equations

                (k + 1) x + 8y = 4k

                kx + (k + 3) y = 3k – 1

has infinitely many solution is

For what real values of k, the system of equations x + 2y + z = 1; x + 3y + 4z = k; x + 5y + 10z = k² has solution? Find the solution of each case.

Let aij denoted the element of the ith row and jth column in 3 × 3 determinant (1 ≤ i ≤ 3, 1 ≤ j ≤ 3) and let aij – aij for every I and j. then the determinant has all the principle diagonal elements as

If each element of a determinant of third order with values A is multiplied by 3, then the value of newly formed determinant is

then x is equal to

Find the value of the determinant

For a fixed positive integer n, let D =