The value of the driving frequency at which the voltage across the capacitor becomes maximum in a series RLC ac fed circuit

As per the book voltage across the capacitor is maximum during resonance that is p=1/√(LC)

is an incorrect statement, rather when that condition is satisfied you have current resonance and the current through and the voltage across the resistor is a maximum.
You are asked about charge resonance when the charge stored on the capacitor and the voltage across the capacitor is a maximum.
You have found that it occurs at a different frequency.
This is a related link.

"As per the book voltage across the capacitor is maximum during resonance that is p=1/√(LC)"

EDIT: I looked closer after seeing R.W. Bird's answer. Updated below accordingly. You can learn something every day if you pay attention. ;--)

That is approximately correct statement for AC excitation when R is relatively small, but, apparently the OP equation is correct (and the exact solution).

Case 1: I ran a frequency scan for a series RLC circuit with these parameters: R = 1Ω, L = 26.5mH, and C = 10μF. A plot of the capacitor voltage amplitude vs. frequency is shown below. For a series circuit the resonance is a series resonance, |XC| = |XL|, so the current will be maximum at the resonant frequency.

$$ f = frac{1}{2pisqrt{LC}} = frac{1}{2pisqrt{(26.5E-3)(10E-6)}}= 309Hz$$ $$ f = frac{sqrt{frac{1}{LC}-frac{R^2}{2L^2}}}{2pi}= 309Hz$$

Case 2: I change R from 1Ω to 30Ω and re-ran. This time the peak voltage across the capacitor occured at 297Hz as per the OP's formula.

$$ f = frac{1}{2pisqrt{LC}} = frac{1}{2pisqrt{(26.5E-3)(10E-6)}}= 309Hz$$ $$ f = frac{sqrt{frac{1}{LC}-frac{R^2}{2L^2}}}{2pi}= 297Hz$$

Below is the same plot but i've added the voltage across the resistor too (red trace). Interestingly, the peak voltage across the R occurs at 309Hz - which jives with the link that Farcher gives in his answer.

I solved this problem many years ago and scanned the solution into my computer (messy derivative). My result agrees with yours. I also have a spreadsheet which solves RLC circuits (with numbers). I used “solver” to maximize the voltage on the capacitor, and found that the predicted angular frequency matched that suggested by your formula for several different sets of circuit parameters.