Evaluate the following determinants:

Solution:

(i)

$\Delta=x(5 x+1)+7 x=5 x^{2}+x+7 x$

$=5 x^{2}+8 x$

(ii)

$\Delta=\cos ^{2} \theta-\left(-\sin ^{2} \theta\right)$

$=\cos ^{2} \theta+\sin ^{2} \theta=1$

(iii)

$\Delta=\cos 15^{\circ} \cos 75^{\circ}-\sin 15^{\circ} \sin 75^{\circ}$

$=\cos 15^{\circ} \cos 75^{\circ}-\sin \left(90^{\circ}-75^{\circ}\right) \sin \left(90^{\circ}-15^{\circ}\right)$         $\left[\because \sin \left(90^{\circ}-\theta\right)=\cos \theta\right]$

$=\cos 15^{\circ} \cos 75^{\circ}-\cos 75^{\circ} \cos 15^{\circ}$

(iv)

$\Delta=a^{2}-i^{2} b^{2}-\left(i^{2} d^{2}-c^{2}\right)$

$=a^{2}-i^{2} b^{2}-i^{2} d^{2}+c^{2}$

$=a^{2}+c^{2}-i^{2}\left(b^{2}+d^{2}\right)$          $\left[\because i^{2}=-1\right]$

$=a^{2}+c^{2}+b^{2}+d^{2}$

$=\cos 15^{\circ} \cos 75^{\circ}-\cos 15^{\circ} \cos 75^{\circ}=0$