An inquest is a hearing held after a party defaults to determine the amount and type of ancillary relief that should be granted.
In a divorce context it means that when you filed for your divorce and served the other party, they had a deadline by which to respond. They failed to do so, and your attorney moved for a default judgment on your cause of action seeking a divorce. That motion was granted.
Now the court has to decide about the other things in the divorce: equitable division, child support and/or custody. That decision is made at the inquest, which can amount to a mini-trial on those specific issues, at which the other side can, but may not, appear and present evidence to oppose that which you'll be putting in.
In the context of a divorce proceeding, an inquest will take place in open court when the parties have already agreed to get divorced, if not also the financial, custody and other aspects related to the divorce. During the inquest, the parties are sworn and are questioned (usually by their respective attorneys, but sometimes by the judge) regarding the basis for the divorce (e.g., abandonment, adultery, cruel and inhuman treatment), their understanding of the terms of the divorce, and the voluntariness of their decision to proceed in this manner. An inquest can take the place of a written separation agreement or settlement stipulation.
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