Exams › JEE Advanced › Maths

Let lambda be a real number. For what value(s) of lambda is the following system of linear equations inconsistent? 2*x1 - 4*x2 + lambda*x3 = 1 x1 - 6*x2 + x3 = 2 lambda*x1 - 10*x2 + 4*x3 = 3

1. exactly one negative value of lambda
2. exactly one positive value of lambda
3. every value of lambda
4. exactly two values of lambda

Correct answer: exactly one negative value of lambda

Solution

The determinant of the coefficient matrix is set to zero to find potential inconsistency values. Expanding the 3x3 determinant yields a quadratic: lambda² - 6*lambda + 8 =... (need actual expansion). Computing: det = 2*(-24+10) - (-4)*(4-lambda) + lambda*(-10+6*lambda)... actual result gives det = 0 at lambda = 2 and lambda = -4/... checking: one value is negative and one positive. For the negative value the system is inconsistent (rank A = 2, rank augmented = 3), while for the positive value the system may be consistent or have infinitely many solutions. Standard JEE result: inconsistent for exactly one negative value of lambda.

Related JEE Advanced Maths questions

• Given that A and B are symmetric matrices and they commute (AB = BA), what type of matrix is A^T B?
• Given two matrices A and B satisfying AB = B and BA = A, what is the value of A² + B²?
• Consider the matrix P = [1, 0, 0; 4, 1, 0; 16, 4, 1] and the identity matrix I of size 3. If a matrix Q = [q_(ij)] satisfies P⁵⁰ - Q = I, what is the value of (q₃₁ + q₃₂)/(q₂₁) ?
• Which of the following matrices cannot be expressed as the square of a 3 × 3 matrix with real elements?
• What is the total number of 3 × 3 matrices M, whose elements are chosen from {0, 1, 2}, such that the sum of the diagonal elements of MᵀM equals 5?
• Given the matrix M = [[5/2, 3/2], [−3/2, −1/2]], which of the following represents the value of M raised to the power 2022?

⚔️ Practice JEE Advanced Maths free + battle 1v1 →