UPSC NDA Math Model Paper 7

kx+y+z=1,x+ky+z=k and x+y+kz=k2
⇒A=[

k	1	1
1	k	1
1	1	k

],B=[

x‌y

z

] and C=[

1

k

k2

]
⇒ For the given equations to have no solution, |A| = 0
⇒[

k	1	1
1	k	1
1	1	k

]=0
⇒k(k2−1)−1(k−1)+1(1−k)=0
⇒k3−k−k+1+1−k=0
⇒k3−3k+2=0
⇒(k−1)(k−1)(k+2)=0
⇒k=1,−2
If we put k = 1 in the above given equations, then all the equations will become the same.
Hence, the given equations have no solution if k = - 2.