## Question

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

### Solution

0

For infinitely many solutions, Δ = 0, and Δ_{1} = 0, Δ_{2} = 0

Δ = 0 ⇒ (*k* + 1) (*k* + 3) – 8*k* = 0

*k*^{2} – 4*k* + 3 = 0 ∴ *k* = 3,1

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

Δ_{2} = 0 ⇒ *k* = 1

Hence *k* – 1 is only value for which each Δ, Δ_{1}, Δ_{2} is zero.

#### SIMILAR QUESTIONS

The system of equations

* ax* + *ay* – *z* = 0

* bx* – *y* + *bz* = 0

– *x* + *cy* + *cz* = 0

(where *a*, *b*, *c* ≠ – 1 )has a non – trivial solution, then value of

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

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.