Determinants

Suppose $p, q, r \neq 0$ and system of equation $\begin{gathered} (p+a) x+b y+c z=0 \\ a x+(q+b) y+c z=0 \end{gathered}$ $a x+b y+(r+c) z=0$, has a non-trivial solution, then the value of…

The equations $x-y+2 z=4$ $3 x+y+4 z=6$ $x+y+z=1$ have

If $x$ is a complex root of the equation $\left|\begin{array}{ccc}1 & x & x \\ x & 1 & x \\ x & x & 1\end{array}\right|+\left|\begin{array}{ccc}1-x & 1 & 1 \\ 1 & 1-x & 1 \\ 1 & 1 & 1-x\end{array}\right|=0$, then…

If the system of linear equations $2 x+y-z=7$ $x-3 y+2 z=1 ; x+4 y+\delta z=k$ where $\delta, k \in R$ has infinitely many solutions, then $\delta+k$ is equal to:

If $p \neq a, q \neq b, r \neq c$ and the system of equations $\begin{aligned} & p x+a y+a z=0 \\ & b x+q y+b z=0 \\ & c x+c y+r z=0 \end{aligned}$ has a non-trivial solution, then the value of…

Given $2 x-y+2 z=2, x-2 y+z=-4$, $x+y+\lambda z=4$, then the value of $\lambda$ such that the given system of equation has no solution is

Suppose $\mathrm{p}, \mathrm{q}, \mathrm{r} \neq 0$ and system of equation $(\mathrm{p}+\mathrm{a}) \mathrm{x}+\mathrm{by}+\mathrm{cz}=0$ $a x+(q+b) y+c z=0$ $a x+b y+(r+c) z=0$ has a non-trivial solution, then value of…

Let $a, b, c∈{R}^{+}$ and the system of equations $(1-a)x+y+z=0, x+(1-b)y+z=0$ and $x+y+(1-c)z=0$ has infinitely many solutions, the minimum value of $abc$ is

If system of equation $ax+y+z=a, x+by+z=b$ and $x+y+cz=c$ is inconsistent, then which of the following is correct?

If ${x}^{2}=|\sinθ\cosθ0-\cosθ\sinθ1\sinθ\cosθ2|$ then the value of $4{x}^{2}+x\sin\frac{3π}{2}+5$ is

Let $\alpha_{1}, \alpha_{2}$ and $\beta_{1}, \beta_{2}$ be the roots of $a x^{2}+b x+c$ $=0$ and $\mathrm{px}^{2}+\mathrm{qx}+\mathrm{r}=0$ respectively. If the system of equations $\alpha_{1} \mathrm{y}+\alpha_{2}…

If the matrix $\left[\begin{array}{ccc}1 & 3 & \lambda+2 \\ 2 & 4 & 8 \\ 3 & 5 & 10\end{array}\right]$ is singular, then $\lambda=$