No values of \(\alpha\) can be found for which A² = B is true

We have \(y={ \left\{ x+\sqrt { { x }^{ 2 }+{ a }^{ 2 } } \right\} }^{ n }\) ....(1)
\(\frac { dy }{ dx } =n{ \left\{ x+\sqrt { { x }^{ 2 }+{ a }^{ 2 } } \right\} }^{ n-1 }\frac { d }{ dx } \left\{ x+\sqrt { { x }^{ 2 }+{ a }^{ 2 } } \right\} \)
\(=n{ \left\{ x+\sqrt { { x }^{ 2 }+{ a }^{ 2 } } \right\} }^{ n-1 } \)
\(\left[ 1+\frac { 1 }{ 2\sqrt { { x }^{ 2 }+{ a }^{ 2 } } } (2x+0) \right] \)
\(=n{ \left\{ x+\sqrt { { x }^{ 2 }+{ a }^{ 2 } } \right\} }^{ n-1 }\left\{ \frac { \sqrt { { x }^{ 2 }+{ a }^{ 2 } } +x }{ \sqrt { { x }^{ 2 }+{ a }^{ 2 } } } \right\} \)
\(=\frac { n{ \left\{ x+\sqrt { { x }^{ 2 }+{ a }^{ 2 } } \right\} }^{ n } }{ \sqrt { { x }^{ 2 }+{ a }^{ 2 } } } =\frac { ny }{ \sqrt { { x }^{ 2 }+{ a }^{ 2 } } } \)
which is true.

Here \(I_{ N }\) is onto. [Given]
But \(I_{ N }+I_{ N }\) is not onto 
[∵ We can find 3 in the co−domain N, where 3 is not a multiple of 2]

\(A={1\over2}\begin{vmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1 \end{vmatrix}\)
But in equilateral triangle of each side equal toa, area = \(\sqrt{3}a^2\over4\)
= \({\sqrt{3}a^2\over4}={1\over2}\begin{vmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1 \end{vmatrix}\)
\(\Rightarrow\begin{vmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1 \end{vmatrix}={\sqrt{3}a^2\over2}\)
Hence,\(\left| \begin{matrix} x_1 &y_1 &1 \\x_2 &y_2 &1 \\x_3 &y_3 &1 \end{matrix} \right| ^2={3\over4}a^4\) is true.

Let 'M' be the mass at time t.
By the question,  \(\frac { dM }{ dt } =-k M\Rightarrow \frac { dM }{ M } -k\quad dt\)
|Variables Separable
Integrating, \(\int { \frac { dM }{ M } } =-k\int { 1.\quad dt+c } \)
\(\Rightarrow\)  \(log|M|=-kt+c\) ...(1)
When \(t=0, M={m}_{0}, \ \therefore log|{m}_{0}|=0+c\)
\(\Rightarrow c=log|{m}_{0}|.\)
Putting in  (1),\(log|M|=-kt+log|{ m }_{ 0 }|.\)
When  \(M=\frac { { m }_{ 0 } }{ 2 } ,log|\frac { { m }_{ 0 } }{ 2 } |=-kt+log|{ m }_{ 0 }|\)
\(\Rightarrow\)   \(log|\frac { 1 }{ 2 } |=-kt\Rightarrow log\quad 2=-kt\)
\(\Rightarrow\)  \(t=\frac { 1 }{ k } log\quad 2,\)