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.

The values of λ for which the system of equations

(λ + 5)x + (λ – 4)y + z = 0

(λ – 2)x + (λ + 3)y + z = 0

                 λx + λy + z = 0

has a non – trivial solution is (are)

Number of real values of λ for which the system of equations

(λ + 3)x + (λ + 2)y + z = 0

        3x + (λ + 3)y + z = 0

                 2x + 3y + z = 0

has a non – trivial solution is

The values of λ for which the system of equations

                2x + y + 2z = 2,

                  x – 2y + z = – 4

                  x + y + λz = 4

has no solution is

Evaluate the determinant without expansion as for as possible.

Then f (100) is equal to

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

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