It is given that $P = \frac{V^{2}}{R}$ so P is inversely proportional to R. But it is also given that $P = I^{2} R$ and here P is directly proportional to R. Isn't this a contradiction?

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Solution

This is not a contradiction at all. In fact, it's just a matter of playing with variables.
$P = \frac{V^{2}}{R} B u t, V = I R Which means I is itself proportional to V S o, P = \frac{(I R)^{2}}{R} \Rightarrow P = I^{2} R$
So both the statement are in-line with each other.

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Q. The electric power of a device can be calculated using P=I²R or P=V²/R .
The first expression indicates that the resistance is directly proportional.where as in the second one the R is inversely proportional. How can the seemingly different dependence between P and R in these expressions be explained .

V=IR, from this equation it's clear that V is directly proportional to I and R..
I=V/R, from this equation it's clear that I is directly proportional to V and inversely proportional to R.
But when V is directly proportional to R and I is directly proportional to V then why is I not directly proportional to R?

Q. From Bernoulli's principle it was found that as area decreases velocity increases and pressure at that point decreases. But we also know that pressure is inversely proportional to area[P=F/A]. So it is found that pressure is proportional to the area[Bernoulli], but also pressure is inversely proportional to the area[F/A]. how is that possible?

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It is given that P=V2R so P is inversely proportional to R. But it is also given that P=I2R and here P is directly proportional to R. Isn't this a contradiction?

This is not a contradiction at all. In fact, it's just a matter of playing with variables. P=V2RBut,V=IRWhich means I is itself proportional to VSo,P=(IR)2R⇒P=I2R So both the statement are in-line with each other.

It is given that $P = \frac{V^{2}}{R}$ so P is inversely proportional to R. But it is also given that $P = I^{2} R$ and here P is directly proportional to R. Isn't this a contradiction?

This is not a contradiction at all. In fact, it's just a matter of playing with variables.
$P = \frac{V^{2}}{R} B u t, V = I R Which means I is itself proportional to V S o, P = \frac{(I R)^{2}}{R} \Rightarrow P = I^{2} R$
So both the statement are in-line with each other.