UPSC NDA Math Model Paper 1

For the given set of equations:
4x+ky+z=0
kx+4y+z=0
2x+2y+z=0
It can be observed that x = y = z = 0 is one solution of the system.
In order to have infinitely many (including non-zero) solutions, D must be zero.
⇒D=|

4	k	2
k	4	2
1	1	1

|=0
⇒4(4−2)+k(2−k)+2(k−4)=0
⇒8+2k−k2+2k−8=0
⇒k2+4k=0
⇒k(k+4)=0
⇒k=0‌OR‌k+4=0
⇒k=0‌OR‌k=−4
Hence, there are 2 possible values of k but non-zero solution is one.