Plz fast exam tomorrow Q Without expanding prove that a + b x c + d x p + q x a x + b c x + d p x + q u v w = 1 - x 2 a c p b d q u v w

Dear student Consider, LHSa+bxc+dxp+qxax+bcx+dpx+quvwApply R1→R1-xR2=a+bx-xax+bc+dx-xcx+dp+qx-xpx+qax+bcx+dpx+quvw=a+bx-ax2-bxc+dx-cx2-dxp+qx-px2-qxax+bcx+dpx+quvw=a-ax2c-cx2p-px2ax+bcx+dpx+quvw=a1-x2c1-x2p1-x2ax+bcx+dpx+quvwTaking 1-x2 common from R1= 1-x2acpax+bcx+dpx+quvwApply R2→R2-xR1= 1-x2acpax+b-axcx+d-cxpx+q-pxuvw= 1-x2acpbdquvw=R H SHence Proved Regards

Plz fast exam tomorrow
Q. Without expanding prove that
$|\begin{matrix} a + b x & c + d x & p + q x \\ a x + b & c x + d & p x + q \\ u & v & w \end{matrix}| = (1 - x^{2}) |\begin{matrix} a & c & p \\ b & d & q \\ u & v & w \end{matrix}|$

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C o n s i d e r, L H S |\begin{matrix} a + b x & c + d x & p + q x \\ a x + b & c x + d & p x + q \\ u & v & w \end{matrix}| A p p l y R_{1} \to R_{1} - x R_{2} = |\begin{matrix} a + b x - x (a x + b) & c + d x - x (c x + d) & p + q x - x (p x + q) \\ a x + b & c x + d & p x + q \\ u & v & w \end{matrix}| = |\begin{matrix} a + b x - a x^{2} - b x & c + d x - c x^{2} - d x & p + q x - p x^{2} - q x \\ a x + b & c x + d & p x + q \\ u & v & w \end{matrix}| = |\begin{matrix} a - a x^{2} & c - c x^{2} & p - p x^{2} \\ a x + b & c x + d & p x + q \\ u & v & w \end{matrix}| = |\begin{matrix} a (1 - x^{2}) & c (1 - x^{2}) & p (1 - x^{2}) \\ a x + b & c x + d & p x + q \\ u & v & w \end{matrix}| T a k i n g (1 - x^{2}) c o m m o n f r o m R_{1} = (1 - x^{2}) |\begin{matrix} a & c & p \\ a x + b & c x + d & p x + q \\ u & v & w \end{matrix}| A p p l y R_{2} \to R_{2} - x R_{1} = (1 - x^{2}) |\begin{matrix} a & c & p \\ a x + b - a x & c x + d - c x & p x + q - p x \\ u & v & w \end{matrix}| = (1 - x^{2}) |\begin{matrix} a & c & p \\ b & d & q \\ u & v & w \end{matrix}| = R . H . S H e n c e P r o v e d .

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Plz fast exam tomorrow Q. Without expanding prove that a + b x c + d x p + q x a x + b c x + d p x + q u v w = 1 - x 2 a c p b d q u v w

Plz fast exam tomorrow
Q. Without expanding prove that
$|\begin{matrix} a + b x & c + d x & p + q x \\ a x + b & c x + d & p x + q \\ u & v & w \end{matrix}| = (1 - x^{2}) |\begin{matrix} a & c & p \\ b & d & q \\ u & v & w \end{matrix}|$