Would you mind if I use/used your mobile?
One of my friends said used is wrong here. But I think it's grammatical. Who is right?
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As @Kaiser Octavius says, "used" isn't wrong, as you can tell by replacing this verb with another in the following sentence: "Would you mind if I came to your party?" This implies that I'll come to your party, or use your mobile, at some point in the future.
"Use" is also fine. Applying the same switch above, you'd get: "Would you mind if I come?" This form implies more immediacy, as in you'll accompany me right away, or (using your original sentence), you'd use my mobile at once.
Both tenses of the verb are correct and either can be used. There's a subtle difference in meaning, though.
Would you mind if I use your mobile? -- May I use your mobile? (asking prior permission)
Would you mind if I used your mobile? -- What would you think if you find that I had used your mobile? (Using it on the presumption you would not mind.)
However, I cannot say it is grammar that brings about this distinction, rather, it's pragmatics.