Which of the following are correct in respect of the system of equations `x+y+z=8` `x-y+2z=6` and `3x-y+5z=k` 1)They have no solution if k=15 2)Theyhave infinitely many solutions if k=20 3)They have a unique solution if k=25 Select the correct answer using the code given below a) 1 and 2 only b) 2 and 3 only c)1 and 3 only d)1,2 and 3

Options

A. 1 and 2 only
B. 2 and 3 only
C. 1 and 3 only
D. 1, 2 and 3

Answer

A

Explanation

Here `[(1,1,1),(1,-1,2),(3,-1,5)][(x),(y),(z)]=[(8),(6),(k)]orAX=B.`
`|A|=0" "(because" the system does not have a unique solution")`
Now, `(AdjA)=[(-3,-6,3),(1,2,-1),(2,4,-2)]and B=[(8),(6),(k)]`
For `k=15,(AdjA)B=[(-24-36+15),(8+12-15),(16+24-30)]=[(-45),(5),(10)]ne0`
`(because" system is inconsistent i.e., it has no solution")`
For `k=20,(AdjA)B=[(-24,-36+60),(8+12-20),(16+24-40)]=[(0),(0),(0)]=0`
`(because " system has infinitely many solutions")`