If $u\sin t+v\cos t=5\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}u\cos t-v\sin t=7$, where $u_1,u_2$ denote the first and second derivatives of  $u$ with respect to $t$ and  $v_1$and  $v_2$ have similar meaning then the value of $|u_1{v}_2-u_2{v}_1|=$

$u\sin t+v\cos t=5$

$u.\cos t+u_1\sin t+v_1\cos t-v\sin t=0$

$u\cos t-v\sin t+u_1\sin t+v_1\cos t=0$

$7+u_1\sin t+v_1\cos t=0\cdots \left(1\right)$

$u\cos t-v\sin t=7$

$u(-\sin t)+\cos t.u_1-(v\cos t+\sin t.v_1)=0$

$-5+u_1\cos t-v_1\sin t=0\cdots \left(2\right)$

Solve $\left(1\right)\&\left(2\right)$

$u_1{v}_1 1$

$\begin{array}{cccc}\cos t& 7& \sin t& \cos t\\ -\sin t& -5& \cos t& -\sin t\end{array}$

$\frac{u_1}{-5\cos t+7\sin t}=\frac{v_1}{7\cos t+5\sin t}=\frac{1}{-1}$

$u_1=5\cos t-7\sin t$

$v_1=-[7\cos t+5\sin t]$

$\Rightarrow u_2=-5\sin t-7\cos t=-[7\cos t+5\sin t]$

$v_2=-[-7\sin t+5\cos t]=7\sin t-5\cos t$

$|u_1{v}_2-u_2{v}_1|=|-{(5\cos t-7\sin t)}^2-{(5\sin t+7\cos t)}^2|$

$=|-(25+49)|$

$=|-74|$

$=74$