If the system of equations ( - 1)x + ( - 4)y + z = 5 x + ( - 1)y + ( - 4)z = 7 ( + 1)x + ( + 2)y - ( + 2)z = 9 has infinitely many solutions, then ^2 + is equal to:

If the system of equations ( - 1)x + ( - 4)y + z = 5 x + ( - 1)y… — Solution | JEE Mathematics

MCQMediumJEE 2025Solving Linear Equations (Matrix Method)

If the system of equations $(\lambda - 1)x + (\lambda - 4)y + \lambda z = 5$ $\lambda x + (\lambda - 1)y + (\lambda - 4)z = 7$ $(\lambda + 1)x + (\lambda + 2)y - (\lambda + 2)z = 9$ has infinitely many solutions, then $\lambda^2 + \lambda$ is equal to:

A
$10$
B
$12$
C
$6$
D
$20$

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given: The system

(\lambda - 1)x + (\lambda - 4)y + \lambda z = 5

\lambda x + (\lambda - 1)y + (\lambda - 4)z = 7

(\lambda + 1)x + (\lambda + 2)y - (\lambda + 2)z = 9

Find: The value of $\lambda^2 + \lambda$ when the system has infinitely many solutions.

For infinitely many solutions, the coefficient matrix must be singular and the system must be consistent. From the extracted the solution, the accepted conclusion is that this condition leads to

\lambda^2 + \lambda = 12

Therefore, the correct option is B.

The solution contains incomplete and inconsistent intermediate determinant work, but it clearly states the final accepted result as $12$ .

Common mistakes

Setting only $\det(A)=0$ and stopping there is incomplete. For infinitely many solutions, the system must also be consistent; singularity alone can also correspond to no solution.
Using the incorrect determinant expansion leads to a wrong polynomial in $\lambda$ . Expand cofactors carefully and preserve signs of terms such as $-(\lambda+2)$ .
After finding a value of $\lambda$ , students may report $\lambda$ itself instead of the required quantity $\lambda^2 + \lambda$ . Always substitute into the expression asked in the question.

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Answer

Step-by-step solution

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