There is no need to use captcha code in SignUp page. But as far as security reason is concerned, you need to captcha code where you have any form. Because some one puts a robot for signup in your page at that you have to face a problem with your website.
Should I use CAPTCHA for registration?
Captcha is quite effective as an anti-spam tool. It's free, easy to install, and provides websites with an extra layer of security in 3 areas: Protecting website registration from receiving useless information and bot accounts.
Should you use CAPTCHA?
CAPTCHA helps protect you from spam and password decryption by asking you to complete a simple test that proves you are human and not a computer trying to break into a password protected account.
Is CAPTCHA still effective?
But while attack technology has evolved, CAPTCHAs have not kept up with the times. Even though they have been improved to be less invasive (and don't require identifying traffic lights or crosswalks all of the time), they still introduce friction and are easy to bypass.