Question

The set of all values of `lambda` for which the system of linear equations : `2x_1-2x_2+""x_3=lambdax_1` `2x_1-3x_2+""2x_3=lambdax_2` `-x_1+""2x_2=lambdax_3` has a non-trivial solution, (1) is an empty set (2) is a singleton (3) contains two elements (4) contains more than two elements
A. is an empty set
B. is a singleton set
C. contains two elements
D. containes more than two elements

Answer 1

Correct Answer - C
Given system of linear equations
`2x_(1)-2x_(2) + x_(3) = lambda x_(1) " "…(i)`
`rArr (2-lambda) x_(1)-2x_(2) + x_(3) = 0 " "…(ii)`
`2x_(1)- 3x_(2) + 2x_(3) = lambdax_(2)" "...(iii)`
`rArr 2x_(1) -(3 + lambda)x_(2) + 2x_(3) = 0`d
`-x_(1) + 2x_(2) = lambda x_(3)`
`rArr -x_(1) + 2x_(2) - lambdax_(3) = 0`
Since, the system has non-trivial solutions.
`therefore [{:(2-lambda, -2, 1), (2, -(3+lambda), 2), (-1, 2, lambda):}] = 0`
`rArr (2-lambda)(3lambda + lambda^(2)-4) + 2(-2lambda + 2) +1(4-3)-lambda) = 0`
`rArr (2-lambda)(lambda^(2) + 3lambda-4) + 4(1-lambda) + (1-lambda) = 0`
`rArr (2-lambda)(lambda + 4)(lambda-1) + 5(1-lambda) =0`
`rArr (lambda-1)[(2-lambda)(lambda+4)-5] =0`
`rArr (lambda-1)(lambda^(2) + 2lambda-3) =0`
`rArr (lambda-1)[(lambda-1)(lambda + 8)] =0`
`rArr (lambda -1)^(2)(lambda + 3) = 0`
`rArr lambda = 1, 1,-3`
Hence, `lambda` contains two elements.