Determinants Te...

The vectorial angle of a point $$P$$ on the line joining the points $$(r_{1}, \theta _{1})$$ and $$(r_{2},\theta _{2})$$ is $$\dfrac{\theta_{1} +\theta _{2}}{2}$$ then the length of radius vector of $$P$$ is

$$\mathrm{D}\mathrm{e}\mathrm{t} \left\{\begin{array}{lll}
1^{2} & 2^{2} & 3^{2}\\
2^{2} & 3^{2} & 4^{2}\\
3^{2} & 4^{2} & 5^{2}
\end{array}\right\}=\ldots$$.

$$\det \left[\begin{array}{lll}
18 & 40 & 89\\
40 & 89 & 198\\
89 & 198 & 440
\end{array}\right]=$$ 

If the entries in a $$3\times 3$$ determinant are either $$0$$ or 1, then the greatest value of their determinats is:

The value of $$\left|\begin{array}{ll}
2+i & 2-i\\
1+i & 1-i
\end{array}\right|$$ is:

$$\left|\begin{array}{lll}
1 & \omega & \omega^{2}\\
\omega & \omega^{2} & 1\\
\omega^{2} & 1 & \omega
\end{array}\right|=\ldots$$....(where $$\omega$$ is the cube root of unity)

If$$ A=\left\{\begin{array}{ll}
1 & 4\\
2 & 8
\end{array}\right\}$$ and $$B=\left\{\begin{array}{ll}
x & y\\
y & x
\end{array}\right\}$$ then the cofactor of $$\mathrm{a}_{21}$$ in $$\mathrm{A}\mathrm{B}$$ is:

lf $$A=\left\{\begin{array}{lll}
a & c & b\\
b & a & c\\
c & b & a
\end{array}\right\}$$ then the cofactor of $$a_{32}$$ in $$\mathrm{A}+\mathrm{A}^{\mathrm{T}}$$  is

If $$A=\left[\begin{array}{lll}
1^{2} & 2^{2} & 3^{2}\\
2^{2} & 3^{2} & 4^{2}\\
3^{2} & 4^{2} & 5^{2}
\end{array}\right]$$, then the minor of $$\mathrm{a}_{22}$$ is

$$\left|\begin{array}{lll}
\mathrm{a}+\mathrm{b} & \mathrm{a} & \mathrm{b}\\
\mathrm{a} & \mathrm{a}+\mathrm{c} & \mathrm{c}\\
\mathrm{b} & \mathrm{c} & \mathrm{b}+\mathrm{c}
\end{array}\right|=$$