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

Consider the real vector space \( \mathbb{R}^3 \). Let \( T : \mathbb{R}^3 \to \mathbb{R} \) be a linear transformation such that \[ T(1, 1, 1) = 0, \quad T(1, -1, 1) = 0, \quad T(0, 0, 1) = 16. \] Then, the value of \( T \left( \frac{1}{2}, \frac{2}{3}, \frac{3}{4} \right) \) is equal to ............... (rounded off to two decimal places).

Let \( T, S : P_4(\mathbb{R}) \to P_4(\mathbb{R}) \) be the linear transformations defined by \[ T(p(x)) = xp'(x), \quad S(p(x)) = (x + 1)p'(x) \] for all \( p(x) \in P_4(\mathbb{R}) \). Then, the nullity of the composition \( S \circ T \) is ................

Let \( f_1, f_2, f_3 \) be nonzero linear transformations from \( \mathbb{R}^4 \) to \( \mathbb{R} \) and \[ \ker(f_1) \subset \ker(f_2) \cap \ker(f_3). \] Let \( T : \mathbb{R}^4 \to \mathbb{R}^3 \) be the linear transformation defined by \[ T(v) = (f_1(v), f_2(v), f_3(v)) \quad \text{for all } v \in \mathbb{R}^4. \] Then, the nullity of \( T \) is equal to:

Define \( T : \mathbb{R}^3 \to \mathbb{R}^3 \) by \[ T(x, y, z) = (x + z, 2x + 3y + 5z, 2y + 2z), \quad \text{for all } (x, y, z) \in \mathbb{R}^3 \] Then, which one of the following is TRUE?