(1) Test for consistency and if possible, solve the following systems of equations by rank method.
(i) x − y + 2z = 2, 2x + y + 4z = 7, 4x − y + z = 4 Solution
(ii) 3x + y + z = 2, x − 3y + 2z =1, 7x − y + 4z = 5 Solution
(iii) 2x + 2y + z = 5, x − y + z =1, 3x + y + 2z = 4 Solution
(iv) 2x − y + z = 2, 6x − 3y + 3z = 6, 4x − 2y + 2z = 4
(2) Find the value of k for which the equations kx − 2y + z =1, x − 2ky + z = −2, x − 2y + kz =1 have
(i) no solution (ii) unique solution (iii) infinitely many solution Solution
(3) Investigate the values of λ and m the system of linear equations 2x + 3y + 5z = 9 , 7x + 3y − 5z = 8, 2x + 3y + λz = μ , have
(i) no solution (ii) a unique solution (iii) an infinite number of solutions Solution
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