Determinants Te...

$$\begin{vmatrix}
1 &4 &20 \\
1 & -2& 5\\
1 &2x & 5x^{2}
\end{vmatrix}=0$$ find x

I. If A,B,C are angles of angle and $$\begin{vmatrix}
1 & 1 & 1\\
1+sinA& 1+sinB&1+sinC \\
sinA+sin^{2}A & sinB+sin^{2}B & sinC+sin^{2}C
\end{vmatrix}$$ =0  then triangle is isosceles
 II. lf $$a=1+2+4+---$$ upto $$\mathrm{n}$$ terms $$b=1+3+9+---$$ up to $$\mathrm{n}$$ terms $$c=1+5+25+----$$up to $$\mathrm{n}$$ terms  then $$\Delta \begin{vmatrix}
a &2b &4c \\
2& 2& 2\\
2^{n} & 3^{n} & 5^{n}
\end{vmatrix}$$ =0

if $$a\neq 6 , b, c $$ satisfy $$\begin{vmatrix} a & 2b & 2c\\  3& b & c\\  4& a & b \end{vmatrix} = 0$$, then $$abc=$$

$$\begin{vmatrix}
1 & cos\alpha & cos\beta \\
cos\alpha & 1 & cos\gamma \\
cos\beta &cos\gamma & 1
\end{vmatrix}$$ = $$\begin{vmatrix}
0 & cos\alpha & cos\beta \\
cos\alpha & 0 & cos\gamma \\
cos\beta &cos\gamma & 0
\end{vmatrix}$$ then 

If $${A}=\begin{bmatrix}
cos\theta & sin\theta\\
-sin\theta & cos\theta
\end{bmatrix}$$ then $$\displaystyle \lim_{n\rightarrow\infty}\frac{1}{n}|A^{n}|=$$

$$|f(x)|={\begin{bmatrix}{}
\mathrm{s}\mathrm{i}\mathrm{n}x & \mathrm{c}\mathrm{o}\mathrm{s}ecx & \mathrm{t}\mathrm{a}\mathrm{n}x\\
\mathrm{s}\mathrm{e}\mathrm{c}x & x\mathrm{s}\mathrm{i}\mathrm{n}x & x\mathrm{t}\mathrm{a}\mathrm{n}x\\
x^{2}-1 & \mathrm{c}\mathrm{o}\mathrm{s}x & x^{2}+1
\end{bmatrix}}$$ then . $$.\displaystyle \int_{-a}^{a}|f(x)|d$$ equals 

Adj $$\begin{bmatrix}
1 & 0 & 2\\
-1 & 1 & -2\\
0& 2 & 1
\end{bmatrix}$$= $$\begin{bmatrix}
5 & a & -2\\
1 & 1 & 0\\
-2& -2 & b
\end{bmatrix}$$ $$\Rightarrow [a\, \, \, \, \, b]=$$ 

The value of $$\begin{vmatrix}
-a^{2} &ab &ac \\
ab& -b^{2} &bc \\
ac & bc& -c^{2}
\end{vmatrix}$$ is 

If A =$$\begin{bmatrix}
0 &1 & 2\\
1& 2 & 3\\
3 & 1 & 1
\end{bmatrix}$$ then Adj (A) = 

$$1\mathrm{f}\mathrm{A}=\left[\begin{array}{lll}
1 & 5 & -6\\
-8 & 0 & 4\\
3 & -7 & 2
\end{array}\right]$$ then the cofactors of the elements $$3,-7,2$$ are p,q,r respectively their ascending order is