If $a=1+2+4+\cdots$ up to $n$ terms,$b=1+3+9+\cdots$ up to $n$ terms and $c=1+5+25+\cdots$ up to $n$ terms,then $\Delta=\left|\begin{array}{ccc}a & 2b & 4c \\ 2 & 2 & 2 \\ 2^n & 3^n & 5^n\end{array}\right|=$

The product of a matrix and its transpose is an identity matrix. The value of the determinant of this matrix is:

Using properties of determinants,prove that:
$\left|\begin{array}{ccc}1 & 1+p & 1+p+q \\ 2 & 3+2 p & 4+3 p+2 q \\ 3 & 6+3 p & 10+6 p+3 q\end{array}\right|=1$

If ${I_1} = \int\limits_1^{\sin \theta } {\frac{x}{{1 + x^2}}} \,dx$ and ${I_2} = \int\limits_1^{\csc \theta } {\frac{{dx}}{{x\left( {{x^2} + 1} \right)}}}$; then the value of $\left| {\begin{array}{*{20}{c}} {{I_1}}&{I_1^2}&{{I_2}} \\ {{e^{{I_1} + {I_2}}}}&{I_2^2}&{ - 1} \\ 1&{I_1^2 + I_2^2}&{ - 1} \end{array}} \right|$ is

Using the property of determinants and without expanding,prove that:
$\left|\begin{array}{lll}1 & bc & a(b+c) \\ 1 & ca & b(c+a) \\ 1 & ab & c(a+b)\end{array}\right|=0$

If $\Delta = \begin{vmatrix} x & y & z \\ p & q & r \\ a & b & c \end{vmatrix}$,then $\begin{vmatrix} x & 2y & z \\ 2p & 4q & 2r \\ a & 2b & c \end{vmatrix}$ equals