If the energy, E = Gᵖhᵃcʳ, 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, q and r are, respectively

The correct option is -1/2, 1/2 and 5/2.

To find the values of $p$, $q$, and $r$, we can use the method of dimensional analysis.

First, let us write down the dimensional formulas for each of the physical quantities involved:

1. Energy ($E$):
$\left[E\right]=\left[{\mathrm{ML}}^2{T}^{-2}\right]$

2. Universal gravitational constant ($G$):
From Newton's law of gravitation, $F=\frac{G{m}_1{m}_2}{d^2}$, we have:
$\left[G\right]=\frac{\left[F\right]\left[L^2\right]}{\left[M^2\right]}=\frac{\left[{\mathrm{MLT}}^{-2}\right]\left[L^2\right]}{\left[M^2\right]}=\left[M^{-1}{L}^3{T}^{-2}\right]$

3. Planck's constant ($h$):
From Planck's relation, $E=h\nu$, where $\nu$ is frequency:
$\left[h\right]=\frac{\left[E\right]}{\left[\nu \right]}=\frac{\left[{\mathrm{ML}}^2{T}^{-2}\right]}{\left[T^{-1}\right]}=\left[{\mathrm{ML}}^2{T}^{-1}\right]$

4. Velocity of light ($c$):
$\left[c\right]=\left[{\mathrm{LT}}^{-1}\right]$

Given the relation:
$E=G^p{h}^q{c}^r$

Substituting the dimensions of each quantity into the equation:
$\left[{\mathrm{ML}}^2{T}^{-2}\right]={\left[M^{-1}{L}^3{T}^{-2}\right]}^p{\left[{\mathrm{ML}}^2{T}^{-1}\right]}^q{\left[{\mathrm{LT}}^{-1}\right]}^r$

Simplifying the right-hand side by combining the powers of $M$, $L$, and $T$:
$\left[{\mathrm{ML}}^2{T}^{-2}\right]=\left[M^{-p+q}{L}^{3p+2q+r}{T}^{-2p-q-r}\right]$

By comparing the exponents of $M$, $L$, and $T$ on both sides, we obtain a system of linear equations:

For $M$:
$-p+q=1$ --- (Equation 1)

For $L$:
$3p+2q+r=2$ --- (Equation 2)

For $T$:
$-2p-q-r=-2$
Which simplifies to:
$2p+q+r=2$ --- (Equation 3)

Now, let us solve these equations. Subtract Equation 3 from Equation 2:
$(3p+2q+r)-(2p+q+r)=2-2$
$p+q=0$
$q=-p$ --- (Equation 4)

Substitute Equation 4 into Equation 1:
$-p+(-p)=1$
$-2p=1$
$p=-\frac{1}{2}$

Using $q=-p$:
$q=\frac{1}{2}$

Now, substitute the values of $p$ and $q$ into Equation 3 to find $r$:
$2(-\frac{1}{2})+\frac{1}{2}+r=2$
$-1+\frac{1}{2}+r=2$
$-\frac{1}{2}+r=2$
$r=2+\frac{1}{2}=\frac{5}{2}$

Therefore, the values of $p$, $q$, and $r$ are, respectively:
-1/2, 1/2 and 5/2.