## Question

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

### Solution

∴ Solution is not unique

The system will have infinite solution if

Δ_{1} = 0, Δ_{2} = 0, Δ_{3} = 0

Δ_{1} = 0 ⇒ *k*^{2} – 3*k* + 2 = 0 ∴ *k* = 1, 2

For these values Δ_{2} and Δ_{3} are also zero.

Now take

#### SIMILAR QUESTIONS

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

λ*x* + λ*y* + *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

3*x* + (λ + 3)*y* + *z* = 0

2*x* + 3*y* + *z* = 0

has a non – trivial solution is

The values of λ for which the system of equations

2*x* + *y* + 2*z* = 2,

* x* – 2*y* + *z* = – 4

* x* + *y* + λ*z* = 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* + 8*y* = 4*k*

*kx* + (*k* + 3) *y* = 3*k* – 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