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The system of equations x+y+z=5, x+2y+lamdaz=mu, x+2y+3z=9 has (i) unique solution of lamda!=3 (ii) infinitely many solutions if lamda=3, mu=9 (iii) no solution if lamda=3, mu!=9 From the above statements, number of correct statements is |
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Answer» 1 `x+y+z=5` `x+2y+lamdaz=mu,x+2y+3z=9` is EQUIVALENT of `AX=B`, where `A=[(1,1,1),(1,2,lamda),(1,2,3)],X=[(x),(y),(z)],B=[(5),(mu),(9)]` Augmented matrix `[A|B]=[(1,1,1),(1,2,lamda),(1,2,3)][(5),(mu),(9)]` `=[(1,1,1),(0,1,lamda-1),(0,1,2)][(5),(mu-5),(4)]` `{:((R_(2)toR_(2)-R_(1)),(R_(3)toR_(3)-R_(1))):}` `=[(1,1,1),(0,1,lamda-1),(0,0,3-lamda)][(5),(mu-5),(9-mu)]` `:.` Rank A=` rank `[A|B]` if lamda!=3`, `:.` The system has UNIQUE solution if `lamda!=3` `:.` (i) is correct if `lamda=3,mu=9,` rand `A=` rank `[A|B]=2` `lt` number of unknowns. `:.` The sytem has infinitely many solutions if `lamda=3,mu=9` If `lamda=3, mu!=9` rank `A=2`, rank `[A|B]=3` `:.` The system has no solution. |
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