G Tewani Solutions for Chapter: Determinants, Exercise 1: Exercises

Let $\left\{D_1, D_2, D_3,\dots ..D_n\right\}$ be the set of third-order determinants that can be made with the distinct non-zero real number $a_1, a_2,\dots ..a_9.$ Then

If $\left|\begin{array}{lll}x& 3& 6\\ 3& 6& x\\ 6& x& 3\end{array}\right|=\left|\begin{array}{lll}2& x& 7\\ x& 7& 2\\ 7& 2& x\end{array}\right|=\left|\begin{array}{lll}4& 5& x\\ 5& x& 4\\ x& 4& 5\end{array}\right|=0,$ then $x$ is equal to

If $\left|\begin{array}{ccc}{x}^n& x^{n+2}& x^{2n}\\ 1& x^n& a\\ x^{n+5}& x^{a+6}& x^{2n+5}\end{array}\right|=0 \forall x\in R,$ where $n\in N,$ then value of $\text{'}a\text{'}$ is

If $f\left(x\right)=\left|\begin{array}{lll}x& a& a\\ a& x& a\\ a& a& x\end{array}\right|=0,$ then

The number of positive integral solutions of the equation $\left|\begin{array}{ccc}{x}^3+1& x^{2}y& x^{2}z\\ x{y}^2& y^3+1& y^{2}z\\ x{z}^2& y{z}^2& z^3+1\end{array}\right|=11$ is

${\Delta }_1=\left|\begin{array}{ccc}{y}^5{z}^6\left(z^3-y^3\right)& x^4{z}^6\left(x^3-z^3\right)& x^4{y}^5\left(y^3-x^3\right)\\ y^2{z}^3\left(y^6-z^6\right)& x{z}^3\left(z^6-x^6\right)& x{y}^2\left(x^6-y^6\right)\\ y^2{z}^3\left(z^3-y^3\right)& x{z}^3\left(x^3-z^3\right)& x{y}^2\left(y^3-x^3\right)\end{array}\right|$ and ${\Delta }_2=\left|\begin{array}{ccc}x& y^2& z^3\\ x^4& y^5& z^6\\ x^7& y^8& z^9\end{array}\right|$, then ${\Delta }_1{\Delta }_2$ is equal to

The value of the determinant $\left|\begin{array}{llll}{\left(a_1-b_1\right)}^2& {\left(a_1-b_2\right)}^2& {\left(a_1-b_3\right)}^2& {\left(a_1-b_4\right)}^2\\ {\left(a_2-b_1\right)}^2& {\left(a_2-b_2\right)}^2& {\left(a_2-b_3\right)}^2& {\left(a_2-b_4\right)}^2\\ {\left(a_3-b_1\right)}^2& {\left(a_3-b_2\right)}^2& {\left(a_3-b_3\right)}^2& {\left(a_3-b_4\right)}^2\\ {\left(a_4-b_1\right)}^2& {\left(a_4-b_2\right)}^2& {\left(a_4-b_3\right)}^2& {\left(a_4-b_4\right)}^2\end{array}\right|$ is

If $c<1$ and the system of equations $x+y-1=0, 2x-y-c=0$ and $-bx+3by-c=0$ is consistent, then the possible real values of $b$ are