(a)
$\forall x \left ( W\vee A\left ( x \right ) \right )\Leftrightarrow W\vee \forall x A\left ( x \right )$
$\Rightarrow$W=true $\Rightarrow \forall x \left ( True \right )\Leftrightarrow \forall x \left ( True \right )\Rightarrow Make Sense$
$\Rightarrow$W=false $\Rightarrow$$\forall x A\left ( x \right ) \Leftrightarrow \forall x A\left ( x \right )$(Hence a is valid)
(As the truth value totally dependent on $A\left ( x \right )$)
(b)
$\exists x \left ( A\left ( x \right )\rightarrow W \right )\Leftrightarrow \forall x A\left ( x \right )\rightarrow W$
if W= true$\Rightarrow \exists x\left ( true \right )\Leftrightarrow \forall x\left ( true \right )\Rightarrow$ make sense as it is saying there exists some element 'x' which is true if and only if there are all x which is 'true'
(as conclusion is true then whole statement is true)
if W=false $\Rightarrow$$\exists x{ \left( A\left ( x \right ) \right )}'\Leftrightarrow \forall x A{\left ( x \right )}'\Rightarrow$It is trivially valid as it says There exists some x which is not A(x) if and only if there are all x which is not A(x)
(As the truth value totally dependent on ${A\left ( x \right )}'$)
c)
$\forall x \left ( A\left ( x \right )\rightarrow W \right )\Leftrightarrow \forall x A\left ( x \right )\rightarrow W$
if W=true ,same as case b)
if W=false $\forall x {A\left ( x \right )}'\Leftrightarrow \forall x {A\left ( x \right )}'\Rightarrow TRUE$
d)$\forall x \left ( A\left ( x \right )\rightarrow W \right )\Leftrightarrow \exists x A\left ( x \right )\rightarrow W$
if W=True,same as case b) and c)
if w=False$\Rightarrow \forall x{ \left ( A\left ( x \right ) \right )}'\Leftrightarrow \exists x { \left ( A\left ( x \right ) \right )}'$
this does not makes sense as it says that for all x which is not A(x) if and only there exists some number which is not A(x).
Hence D) is answer.
