UPSC NDA Math Model Paper 8

If the system of equation a1x+b1y+c1=0,a2x+b2y+c2=0, a3x+b3y+c3=0 be consistent, then
|

a1	b1	c1
a2	b2	c2
a3	b3	c3

|=0
Given, the system of equation 2x + 3y + 5 = 0, x + ky + 5 = 0, kx - 12y - 14 = 0 be consistent.

|

2	3	5
1	k	5
k	−12	−14

|=0
⇒2(−14k+60)−3(−14−5k) +5(−12−k2)=0
⇒−28k+120−42+15k−60 −5k2=0
⇒−5k2−13k+18=0
⇒5k2+13k−18=0
⇒5k2−5k+18k−18=0
⇒5k(k−1)+18(k−1)=0
⇒(k−1)(5k+18)=0
⇒k=1,

−18

5

Hence, if the system of equation 2x + 3y + 5 = 0, x + ky + 5 = 0, kx - 12y - 14 = 0 be consistent, then the values of k are 1, −

18

5