List of top Mathematics Questions on Linear Systems and Determinants

Consider the system of linear equations in x, y, z:
x+2y+tz = 0,
6x + y + 5t z = 0,
3x + y + f(t) z = 0,
where f: R$\rightarrow$ R is a differentiable function. If this system has infinitely many solutions for all t $\in$ R, then f

The sum of all possible values of \( \theta \in [0,2\pi] \), for which the system of equations : \[ x\cos3\theta - 8y - 12z = 0 \] \[ x\cos2\theta + 3y + 3z = 0 \] \[ x + y + 3z = 0 \] has a non-trivial solution, is equal to :

If the system of linear equations \(x + y + z = 6\), \(x + 2y + 5z = 10\), \(2x + 3y + \lambda z = \mu\) has infinitely many solutions, then the value of \(\lambda + \mu\) equals:

Let the system of equations $ x + 5y - z = 1 $ $ 4x + 3y - 3z = 7 $ $ 24x + y + \lambda z = \mu $ where $ \lambda, \mu \in \mathbb{R} $, have infinitely many solutions. Then the number of the solutions of this system, if $x, y, z$ are integers and satisfy $7 \leq x + y + z \leq 77$, is:

If the system of linear equations: \[ x + y + 2z = 6, \] \[ 2x + 3y + az = a + 1, \] \[ -x - 3y + bz = 2b, \] where \( a, b \in \mathbb{R} \), has infinitely many solutions, then \( 7a + 3b \) is equal to: