List of top Mathematics Questions on Linear Equations asked in IIT JAM MA

Let \(a,b\in\mathbb{R}\). If the system of linear equations

Consider the following system of linear equations: 
\[ \begin{cases} x + y + 5z = 3, \\ x + 2y + mz = 5, \\ x + 2y + 4z = k. \end{cases} \] 

The system is consistent if 
 

Suppose \( \alpha, \beta, \gamma \in \mathbb{R} \). Consider the following system of linear equations.
\[ x + y + z = \alpha, x + \beta y + z = \gamma, x + y + \alpha z = \beta. \] If this system has at least one solution, then which of the following statements is (are) TRUE?
Let \( v_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \) and \( v_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \). Let \(M\) be the matrix whose columns are \(v_1,\; v_2,\; 2v_1 - v_2,\; v_1 + 2v_2\) in that order. Then the number of linearly independent solutions of the homogeneous system \(Mx = 0\) is ...........
Let \( P_3 \) denote the real vector space of all polynomials with real coefficients of degree at most 3. Consider the map \[ T: P_3 \to P_3 \quad \text{given by} \quad T(p(x)) = p'(x) + p(x). \] Then