The number of integral value(s) of k such that the system of equations kx-2y-z=x, ky-z=z+3x and 2x+kz=2y-z has non-trivial solution, is/are

Question

The number of integral value(s) of k such that the system of equations kx-2y-z=x, ky-z=z+3x and 2x+kz=2y-z has non-trivial solution, is/are (A) 0 (B) 1 (C) 2 (D) 3

Tardigrade Admin · Accepted Answer

Rearranging all the equations, we get, (k - 1)x-2y-z=0 3x-ky+2z=0 2x-2y+(k + 1)z=0 For a non-trivial solution, | (k - 1) -2 -1 3 -k 2 2 -2 k+1 |=0⇒ (k - 1)[- k2 - k + 4]+2[3 k - 1]-1[- 6 + 2 k]=0 ⇒ -k3-k2+4k+k2+k-4+6k-2+6-2k=0 ⇒ -k3+9k=0⇒ k=0,-3,3

Q. The number of integral value(s) of $k$ such that the system of equations $k x - 2 y - z = x, k y - z = z + 3 x$ and $2 x + k z = 2 y - z$ has non-trivial solution, is/are

$0$

$1$

$2$

$3$

Q. The number of integral value(s) of k such that the system of equations kx−2y−z=x,ky−z=z+3x and 2x+kz=2y−z has non-trivial solution, is/are

0

1

2

3

Rearranging all the equations, we get, (k−1)x−2y−z=0 3x−ky+2z=0 2x−2y+(k+1)z=0 For a non-trivial solution, ∣∣​(k−1)32​−2−k−2​−12k+1​∣∣​=0⇒(k−1)[−k2−k+4]+2[3k−1]−1[−6+2k]=0 ⇒−k3−k2+4k+k2+k−4+6k−2+6−2k=0 ⇒−k3+9k=0⇒k=0,−3,3

Q. The number of integral value(s) of $k$ such that the system of equations $k x - 2 y - z = x, k y - z = z + 3 x$ and $2 x + k z = 2 y - z$ has non-trivial solution, is/are

$0$

$1$

$2$

$3$