I have met such sentence in my tutorial text:
"Today my brother has announced that he is going to enter the university next year. I wondered if he had thought it over properly. He would be able to pass all the exams providing he studies at full speed, wouldn't he?"
I wonder if we can use "would" with present tense in the conditional clause. I feel that it may have something with subjunctive uncertainty, but I'm not sure. All grammar rules I've found prescribe to use "will" instead of "would" in this example, as it is 1st conditional. Can you tell me, please, if such usage is ok and how can it be explained grammatically?
Another example I've met on this forum while trying to browse the answer to my question (It was a part of the answer to a related question, but it didn't clarify the usage rules):
"If you are a mathematician, and understand the conjugal relationship between real and imaginary numbers, you would see that the subjunctive is the imaginary conjugate of the real world.
Best Answer
Preamble
The passage you cite has various verb form inconsistencies and the ungrammatical use of 'the' in the phase 'enter the university next year'. You state that the passage comes from a tutorial text. Perhaps you could give a little more information about both the passage and the tutorial text, together with the task that is based on them.
Analysis
That aside, the main issue here is how to construct conditional sentences. This is a tricky issue for non-native English learners, and many pedagogic grammars try to simplify it by reducing the possibilities to three - which they call the First, Second and Third Conditionals. Certainly, these are three common patterns, but I would be very wary of any grammar rules 'prescribing' the use of the present tense in the if-clause (protasis) when using 'will + infinitive' in the main clause (apodosis) - or vice versa.
As Michael Lewis points out in the section on Conditional Sentences in The English Verb: An Exploration of Structure and Meaning (p148):
Applying this explanation to your example sentence, the speaker is probably using 'would' (He would be able to pass all the exams) rather than 'will' (He will be able to pass all the exams) to express doubt that 'he' has thought it over and is going to pass the exams.
As to the second clause, 'studied' is the more likely form in this context, and would conform to the common Conditional 2 pattern. But perhaps the speaker is expressing the simple fact that exam success is assured if studying is done at full speed.
Examples of 'mixed conditionals'
Mixing the so-called first and second conditionals in this way is not uncommon. Below are several authentic examples from Google. That said, if you are studying for English exams, you are probably better advised to stick to the 'prescribed' patterns.