A few questions to stretch candidates for university entrance exams

Here is a taster of some questions you may like to think about if you are aiming to stretch yourself with one or more of the various university entrance exams. Enjoy!

(If you’re not aiming for university entrance exams, don’t worry, my GCSE, A level and IB tuition aren’t as tricky as these puzzles!)

Solve the following equations. log of a cubed times b squared equals 6 log of a squared times b cubed equals minus 1. Logs are to base 2.
A complicated expression involving logs to be simplified.
Which of the following numbers are square numbers: 1, 11, 111, 1111, 11111, ...
What is the smallest number with nine factors? Which number from 1 to 100 has the most factors?
The square matrix M has elements 1, -1, 1, 1. What is M raised to the power 2021?

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