If $\theta \in \left(0,\frac{\pi }{2}\right)$ and $p=4{\mathrm{s}\mathrm{e}\mathrm{c}}^2\theta +9\mathrm{c}\mathrm{o}\mathrm{s}e{c}^2\theta$ , then the system of equations $3x-y+4z=3,x+2y-3z=-2,6x+5y+pz=-3$ have

An ordered pair $\left(\alpha , \beta \right)$ for which the system of linear equations
$\left(1+\alpha \right)x+\beta y+z=2$
$\alpha x+\left(1+\beta \right)y+z=3$
$\alpha x+\beta y+2z=2$

has a unique solution, is :

Which of the following statement is correct regarding homogeneous system?

Solve the following non-homogeneous equations by using the determinant method.

$2x+2y+z=5$

$x-y+z=1$

$3x+y+2z=4$

If the system of linear equations

$x+ky+3z=0$
$3x+ky-2z=0$
$2x+4y-3z=0$

has a non-zero solution $\left(x, y, z\right),$ then $\frac{xz}{y^2}$ is equal to:

To find the number of coins, in each category, write the suitable system of equations for the given situation:

"A bag contains $3$ types of coins namely $₹1, ₹2 \text{and} ₹5$. There are $30$ coins amounting to $₹100$ in total"

Let $\lambda$ be a real number for which the system of linear equations

$x+y+z=6,$

$4x+\lambda y-\lambda z=\lambda -2$ and

$3x+2y-4z=-5$

has infinitely many solutions. Then $\lambda$ is a root of the quadratic equation:

Discuss the solutions of the following system of equations for all values of $\lambda$.

$x+y+z=2,2x+y-2z=2,\lambda x+y+4z=2$.

If $a, b, c$  are non - zero real numbers and if the system of equations

$\left(a-1\right)x=y+z$

$\left(b-1\right)y=x+z$

$\left(c-1\right)z=x+y$

has a non - trivial solution, then $ab+bc+ca$  equals :