I've just now deleted many pages from my website (with 410 status) which were product pages and copied from respective companies.
Will now search engine rankings improve for my site?
I hope it should take 4-6 weeks for this.
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Search engines don't rank websites. They rank individual pages.
What made you decide to delete them? It's okay to have the same products as other websites. If you couldn't then only one website would be found when doing product searches and we know that isn't the case. And that is even with them all using the same product description. (It's very common to use the manufacturers product description and specs for a product's listing).
Assuming the pages were duplicate content, they either were already filtered out by Google or devalued as being duplicates so they weren't helping you to begin with (i.e. links from those pages didn't count, PageRank wasn't passed. etc). So removing them only makes Google's life easier but doesn't directly affect your other pages. The only benefit you may see is by not linking to those pages anymore you have fewer links to "bad" pages and thus have more links to "good" pages which makes those links a little bit stronger (pass more PageRank, stronger link value). But in all reality that gain will be barely noticeable if it is noticeable at all unless it was a very large number of pages.
Another thing to consider, if those pages weren't being penalized buy Google in anyway, this may actually hurt you. If those pages weren't filtered out, reduced in value, etc, then you had more pages that were linking to your important pages (i.e. home page, main product pages, etc) and now you have fewer which hurts those pages. It would be only a small hit but a hit nonetheless. Not to mention you have fewer pages in their index and thus have fewer opportunities to be found.