If the energy, `E = G^p h^q c^r,` where G is the universal gravitational constant, h is the Planck's constant and c is the velocity of light, then the values of p are q and r are, respectively

Options

A. `-1//2,1//2 and 5//2`
B. `1//2,-1//2 and -5//2`
C. `-1//2, 1//2 and 3//2`
D. `1//2, -1//2 and -3//2`

Explanation

To solve the problem where energy \( E \) is expressed in terms of the universal gravitational constant \( G \), Planck's constant \( h \), and the speed of light \( c \) as \( E = G^p h^q c^r \), we need to find the values of \( p \), \( q \), and \( r \) using dimensional analysis.

### Step-by-Step Solution:

1. **Identify the Dimensions of Each Quantity**:
- The dimension of energy \( E \) is given by:
\[ [E] = [M][L^2][T^{-2}] \]
- The dimensions of the constants are:
- Gravitational constant \( G \):
\[ [G] = [M^{-1}][L^3][T^{-2}] \]
- Planck's constant \( h \):
\[ [h] = [M][L^2][T^{-1}] \]
- Speed of light \( c \):
\[ [c] = [L][T^{-1}] \]

2. **Express the Dimensions of \( E \) in Terms of \( G \), \( h \), and \( c \)**:
- Substitute the dimensions into the equation:
\[ [E] = [G^p][h^q][c^r] \]
- This expands to:
\[ [E] = [M^{-p}][L^{3p}][T^{-2p}][M^q][L^{2q}][T^{-q}][L^r][T^{-r}] \]

3. **Combine the Dimensions**:
- Combine the dimensions:
\[ [E] = [M^{-p + q}][L^{3p + 2q + r}][T^{-2p - q - r}] \]

4. **Set Up the Equations for Homogeneity**:
- Equate the dimensions from both sides:
- For mass \( M \):
\[ -p + q = 1 \quad \text{(Equation 1)} \]
- For length \( L \):
\[ 3p + 2q + r = 2 \quad \text{(Equation 2)} \]
- For time \( T \):
\[ -2p - q - r = -2 \quad \text{(Equation 3)} \]

5. **Solve the System of Equations**:
- From Equation 1:
\[ q = p + 1 \]
- Substitute \( q \) into Equation 2:
\[ 3p + 2(p + 1) + r = 2 \implies 3p + 2p + 2 + r = 2 \implies 5p + r = 0 \quad \text{(Equation 4)} \]
- Substitute \( q \) into Equation 3:
\[ -2p - (p + 1) - r = -2 \implies -3p - 1 - r = -2 \implies -3p - r = -1 \quad \text{(Equation 5)} \]

6. **Solve for \( p \) and \( r \)**:
- From Equation 4:
\[ r = -5p \]
- Substitute \( r \) into Equation 5:
\[ -3p + 5p = -1 \implies 2p = -1 \implies p = -\frac{1}{2} \]
- Now, substitute \( p \) back to find \( q \) and \( r \):
\[ q = p + 1 = -\frac{1}{2} + 1 = \frac{1}{2} \]
\[ r = -5p = -5\left(-\frac{1}{2}\right) = \frac{5}{2} \]

7. **Final Values**:
- The values are:
\[ p = -\frac{1}{2}, \quad q = \frac{1}{2}, \quad r = \frac{5}{2} \]

### Conclusion:
The values of \( p \), \( q \), and \( r \) are \( -\frac{1}{2} \), \( \frac{1}{2} \), and \( \frac{5}{2} \) respectively.