If the velocity of light $$c$$, universal gravitational constant $$G$$ and planck's constant $$h$$ are chosen as fundamental quantities. The dimensions of mass in the new system is:

To find the dimensions of mass in terms of the fundamental quantities—velocity of light (c), universal gravitational constant (G), and Planck's constant (h)—we start by recalling their dimensional formulas.

The dimensional formula for velocity of light (c) is that of speed:

$$[c] = [L T^{-1}]$$

The dimensional formula for universal gravitational constant (G) is derived from Newton's law of gravitation, $$F = G \frac{m_1 m_2}{r^2}$$. Force F has dimensions $$[M L T^{-2}]$$, so:

$$[G] = \frac{[F] [r^2]}{[m_1] [m_2]} = \frac{[M L T^{-2}] [L^2]}{[M] [M]} = [M^{-1} L^3 T^{-2}]$$

The dimensional formula for Planck's constant (h) comes from the energy of a photon, $$E = h \nu$$, where $$\nu$$ is frequency. Energy E has dimensions $$[M L^2 T^{-2}]$$ and frequency $$\nu$$ has dimensions $$[T^{-1}]$$, so:

$$[h] = \frac{[E]}{[\nu]} = \frac{[M L^2 T^{-2}]}{[T^{-1}]} = [M L^2 T^{-1}]$$

We express mass (M) as a product of powers of h, c, and G:

$$[M] = [h]^a [c]^b [G]^c$$

Substituting the dimensional formulas:

$$[M] = [M L^2 T^{-1}]^a \times [L T^{-1}]^b \times [M^{-1} L^3 T^{-2}]^c$$

Expanding the right-hand side:

$$[M] = (M^a L^{2a} T^{-a}) \times (L^b T^{-b}) \times (M^{-c} L^{3c} T^{-2c})$$

Combining the exponents:

$$[M] = M^{a - c} L^{2a + b + 3c} T^{-a - b - 2c}$$

Since the left-hand side is $$[M] = M^1 L^0 T^0$$, we equate the exponents for M, L, and T:

For M: $$a - c = 1$$   ...(1)

For L: $$2a + b + 3c = 0$$   ...(2)

For T: $$-a - b - 2c = 0$$   ...(3)

We solve this system of equations:

From equation (1): $$a = c + 1$$

Substitute $$a = c + 1$$ into equation (2):

$$2(c + 1) + b + 3c = 0$$

$$2c + 2 + b + 3c = 0$$

$$b + 5c + 2 = 0$$

$$b = -5c - 2$$   ...(4)

Substitute $$a = c + 1$$ into equation (3):

$$-(c + 1) - b - 2c = 0$$

$$-c - 1 - b - 2c = 0$$

$$-b - 3c - 1 = 0$$

$$b + 3c = -1$$   ...(5)

Substitute $$b = -5c - 2$$ from equation (4) into equation (5):

$$(-5c - 2) + 3c = -1$$

$$-2c - 2 = -1$$

$$-2c = 1$$

$$c = -\frac{1}{2}$$

From equation (1): $$a = -\frac{1}{2} + 1 = \frac{1}{2}$$

From equation (4): $$b = -5(-\frac{1}{2}) - 2 = \frac{5}{2} - 2 = \frac{5}{2} - \frac{4}{2} = \frac{1}{2}$$

Thus, $$[M] = [h]^{1/2} [c]^{1/2} [G]^{-1/2}$$, or in dimensional notation:

$$[M] = [h^{1/2} c^{1/2} G^{-1/2}]$$

Comparing with the options:

A. $$[h^{1/2} c^{-1/2} G^{1}]$$ → Exponents do not match.

B. $$[h^{1} c^{1} G^{-1}]$$ → Exponents do not match.

C. $$[h^{-1/2} c^{1/2} G^{1/2}]$$ → Exponents do not match.

D. $$[h^{1/2} c^{1/2} G^{-1/2}]$$ → Exponents match: $$h^{1/2}, c^{1/2}, G^{-1/2}$$.

The correct answer is option D.