I don't know otherwise, but that seems unlikely to me, so I'm interested if you have any evidence that poor kids pass these exams at the same rate.
Some will do, for sure. However, when tests are challenging enough, teachers and talent become dominating factors. Good teachers make sure that students work hard and have access to illuminating exercises to realize their full potential, and students with insufficient talent will not excel in those tests no matter how much tutoring they have. After all, tutoring is not education. Of course, I may have been biased per my own observation: the distribution of student performance in my schools hardly changed due to tutoring because my teachers simply gave challenging enough homework and exams, which did a good enough job to remove the advantages of tutoring, if there's any. Using an extreme example: yes you can find the best coach for you to prepare Math Olympiad, but you'd be able to cross a threshold first before you can even understand what the coach talks about. Similarly on the level of JEE or national entrance exams, I may spend 10 times as much time memorizing some patterns while my classmates whoop my ass by simply attending classes -- this is also why I think the progressives are doing a deservice to poor kids or minority groups. They could've raise standards and build better public schools to help the talented stand out, yet they choose to lower standards and really damage the perception on the otherwise truly talented group.
On the negative side, we also have reservations depending on your caste, so whether or not you get admission into a prestigious institute depends on your family history.
With a harder test like the JEE, we can separate these two outcomes