If P²varies as R and Q²varies as R, (P ≠ Q), then which of the following are correct?

1. P²+ Q²varies as R.

2. PQ varies as R.

3. P²- Q² varies as R.

Select the correct answer using the code given below:

Correct option is D

Given:

P²varies as R and Q²varies as R, (P ≠ Q).

Concept:

If a variable X varies as Y, it implies X = kY for some constant k.

Solution:

Let P²= k₁R and Q²= k₂R, where k₁and k₂are constants.

1. To check if P²+ Q²varies as R:

 P²+ Q²= k₁R + k₂R

 P²+ Q²= (k₁+ k₂)R

Since k₁+ k₂is a constant, P²+ Q²varies as R.

2. To check if PQ varies as R:

 P = √(k₁R) and Q = √(k₂R)

 PQ = √(k₁R) × √(k₂R)

 PQ = √(k₁k₂)R

Since √(k₁k₂) is a constant, PQ varies as R.

3. To check if P²- Q²varies as R:

 P²- Q²= k₁R - k₂R

 P²- Q²= (k₁- k₂)R

Since k₁- k₂is a constant, P²- Q²varies as R.

∴ All three statements are correct.