If Y, K and \eta are the values of Young's modulus, bulk modulus and modulus of rigidity of any material respectively. Choose the correct relation for these parameters.

K = Y\eta \over 9\eta - 3YN/m^2

Y = 9K\eta \over 3K - \eta N/m^2

Y = 9K\eta \over 2\eta + 3KN/m^2

Properties of Matter PYQ — JEE Main 2021 · 24 Feb · Shift 1 · Q56 with Solution

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Correct answer: D

For an isotropic material, the standard relation between Young's modulus $Y$ , bulk modulus $K$ , and modulus of rigidity $\eta$ is

Y = \frac{9K\eta}{3K+\eta}

Option A:
$Y = \frac{9K\eta}{3K-\eta}$
This is incorrect because the denominator should be $3K+\eta$ , not $3K-\eta$ .
Option B:
$Y = \frac{9K\eta}{2\eta+3K}$
This is also incorrect because the correct denominator is $3K+\eta$ , not $3K+2\eta$ .

Starting with

Y = \frac{9K\eta}{3K+\eta}

Multiply both sides:

Y(3K+\eta)=9K\eta

3YK+Y\eta=9K\eta

Rearranging for $K$ :

3YK=9K\eta-Y\eta

3YK=\eta(9K-Y)

Bring $K$ terms together:

K(3Y-9\eta)=-Y\eta

K=\frac{-Y\eta}{3Y-9\eta}=\frac{Y\eta}{9\eta-3Y}

This matches Option D.

Y(3K+\eta)=9K\eta

3YK=\eta(9K-Y)

\eta=\frac{3YK}{9K-Y}

But Option C gives

\eta=\frac{3YK}{9K+Y}

which is incorrect.

\boxed{D}

Hence, the derived answer agrees with the stored answer.