UPSC NDA Math Model Paper 3

Given: The points (2, - 1, 2), (1, 2, - 3) and (3, k, 7) are collinear
As we know that, if the points (x1,y1,z1),(x2,y2,z2) and (x3,y3,z3) be collinear then |

x1	y1	z1
x2	y2	z2
x3	y3	z3

|=0
Here, x1=2,y1=−1,z1=2,x2=1,y2=2, z2=−3,x3=3,y3=k and z3=7
⇒|

2	−1	2
1	2	−3
3	k	7

|=0
⇒ 2 × (14 + 3k) + 1 × (7 + 9) + 2 × (k - 6) = 0
⇒ 28 + 6k + 16 + 2k - 12 = 0
⇒ 32 + 8k = 0
⇒k = - 4
Hence, option D is the correct answer.