If $0<A<\frac{\pi }{2}$  and $\sin A+\cos A+\tan A+\cot A+\sec A+\cos ecA=7$ ,$\sin A\&\cos A$  are the roots of the equation $4x^2-3x+\alpha =0$   then the value of $\left|\frac{25\alpha }{4}\right|$

$4x^2-3x+\alpha =0$ $\Rightarrow$ $\sin A+\cos A=\frac{3}{4}$ $\Rightarrow$ $\sin A.\cos A=\frac{\alpha }{4}$

$\Rightarrow$ $\left(\sin A+\cos A\right)+\left(\tan A+\cot A\right)+\left(\sec A+\cos ecA\right)=7$

$\Rightarrow$ $\left(\sin A+\cos A\right)+\left(\frac{{\sin }^{2}A+{\cos }^{2}A}{\sin A\cos A}\right)+\frac{\sin A+\cos A}{\sin A\cos A}=7$

$\Rightarrow$ $\frac{3}{4}+\frac{4}{\alpha }+\frac{3}{\alpha }=7$   $\Rightarrow$ $\frac{7}{\alpha }=\frac{25}{4}$    $\Rightarrow$ $\frac{25\alpha }{4}=7$