Determinants

If thesystem of linear equations, $x+y+z=6$ $x+2y+3z=10$ $3x+2y+\lambda\,z=\mu\,$ has more than two solutions, then $\mu\,-\lambda\,^{2}$, is equal to.

Let for any three distinct consecutive terms $a,b,c$ of an A.P, the lines $ax+by+c=0$be concurrent at the point $P$ and $Q(\alpha\,,\beta\,)$ be a point such that the system of…

The system of linear equations$3x-2y-kz=10$ $2x-4y-2z=6$ $x+2y-z=5m$ is inconsistent if :

For the system of linear equations: $x-2y=1,x-y+kz=-2,ky+4z=6,k\in\,R$ Consider the following statements: (A) The system has unique solution if $k\neq\,2,k\neq\,-2.$ (B) The system has unique solution if $k=-2.$ (C) The…

Let the system of linear equations $x+y+kz=2$ $2x+3y-z=1$ $3x+4y+2z=k$ have infinitely many solutions. Then the system $\left(k+1\right)x+\left(2k-1\right)y=7$ $\left(2k+1\right)x+\left|k+5\right|y=10$has :

For the system of linear equations $2x-y+3z=5$ $3x+2y-z=7$ $4x+5y+\alpha\,z=\beta\,$, which of the following is NOT correct?

Consider the following system of equations: $x+2y-3z=a$ $2x+6y-11z=b$ $x-2y+7z=c$ where $a,b$ and $c$ are real constants. Then the system of equations :

Let$p$ and $p+2$be prime numbers and let $\Delta\,=\left|\begin{matrix} p! & \left(p+1\right)! & \left(p+2\right)! \\ \left(p+1\right)! & \left(p+2\right)! & \left(p+3\right)! \\ \left(p+2\right)! & \left(p+3\right)! &…

If $\Delta$ is the value of the determinant $\left|\begin{array}{lll}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right|$ then what is the value of the following determinant?…

If $\left[\begin{array}{ccc}x & -3 i & 1 \\ y & 1 & i \\ 0 & 2 i & -i\end{array}\right]=6+11 i$, then what are the values of $x$ and y respectively?

Paragraph: Question: What is $\lim _{x \rightarrow 0} \frac{f(x)}{x^2}$ equal to?

Under which of the following conditions does the determinant $\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|$ vanish? $a+b+c=0$ $a^3+b^3+c^3=3 a b c$ $a^2+b^2+c^2-a b-b c-c a=0$ Select the…