How to clear an entrance exam for a PhD in Maths? How to clear an entrance exam for a PhD in Maths?
In a 5-6 months time there is a PhD exam coming up for me.It consists of questions from Real Analysis,Abstract Algebra,Linear Algebra,Functional Analysis,Measure Theory,Probability and Statistics,Topology.
Though I am quite familiar with these topics,I am finding it difficult to understand how should one prepare for the exam.
There are large amount of problems available  on these topics.How is it possible to solve so many of them?
Also the problem is though there is a particular book I am studying from but there are still enormous amount of books that I have failed to solve them.
What should I do ?It is impossible to solve many books on a single topic.
I know there are many on this site who have given exams for PhD.Please suggest some way to remember the enormous amount of information.
How should I prepare for the exam?
Is this question suitable for this site?If not please inform me without down voting.Thank you.
 A: There are no guaranteed methods, but here are some strategies that might be helpful. Everyone is different, so some suggestions might not be right for you.
Others who answer the question may have opinions that are better targeted
to your situation than mine. So no list of suggestions, including this one,
is likely to be entirely useful.
If questions from past exams are available, study them carefully. But
you are unlikely to see exactly these same questions recycled on your
exam. Consider the type of each question: Why was it put on the exam before?
What topics, methods, and approaches was it designed to test? 
If feasible, work in a group to answer as many recent past questions as
possible. Then get together to discuss the issues and explain them to each other.
Even the top students benefit from the thought processes of explanation. But
be sure these are serious work sessions, not primarily social events.
A good rule is that everyone in the group must participate actively, no auditors.
As @rschwieb suggests, talk to your adviser and other faculty. They may be
aware of strengths and weaknesses in your background that you have not
realized. 
Former exams are not always a good guide to future ones. Be especially
alert to changes in the direction of the emphasis within your department.
For example, about ten years ago one of the top ten departments in statistics
made a noticeable shift from emphasis on abstract probability to applied statistics, and
the PhD exams reflected that shift. If you believe new faculty members may
be joining the group who prepare the questions, be aware of their fields of
interest.
Read each question carefully (twice) and try to answer exactly what was asked. 
Sometimes there are choices which questions to answer, but
if you must answer a question to which you don't know the answer, take a little
time to show your knowledge of relevant topics (after you have attended to
the ones you can answer precisely). 
Get a couple of full nights of sleep just before you take the exam. Stay away from social media and TV news for a week before the exam. On the exam day, avoid
stimulants that make you feel more 'jittery' and less insightful.
