If Planck’s const A. Mass of electron (me)

B. Universal gravitational constant (G)

C. Charge of electron (e)

D. Mass of proton (mp )

According to plank, the energy of photon is given by the expression,

Where,

f= frequency of light

h= Planck’s constant

Dimension of E=

Dimension of f=

So, dimension of h,

As we can see in the question, we have to relate M, L and T in terms of h and c as fundamental quantities.

There is no charge or ampere dimension so option (c) is eliminated.

We have to check for option a, b and d.

Dimensions of m_e, m_p and G are , and respectively.

For m_e and m_p,

For mass, let

m

By substituting the dimension for each quantity

Now using principal of homogeneity, which states that the dimensions on both sides of a dimensional equation is same, we get

=

Equating like terms and solving above we get,

a+b=1, 2b+c=0, -b-c=0

a=1, b=0, c=0

Substituting the values in equation,

m

m=m_e

Similarly for length,

l

=

Solving we get,

and for time,

This holds true for both m_e and m_p.

And similarly for G,

Also we know,

Dimension of G=

Let,

m ...........(1)

By substituting the dimension for each quantity

Now using principal of homogeneity, which states that the dimensions on both sides of a dimensional equation is same, we get

=

Equating like terms and solving above we get,

a-c=1, 2a+b+3c=0, -a-b-2c=0

a=1/2, b=1/2, c=-1/2

Substituting the values in equation,

m

Similarly for length and time,

and