Is the niṣāb calculated based on the price of 21-karat gold, or 24-karat gold?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
Zakāh is obligatory on cash and on gold designated for saving or for trade, provided the total reaches the niṣāb (minimum threshold liable for zakah), and a full ḥawl (lunar year) has elapsed over it. The niṣāb is (85) grams of 24-karat gold; for 21-karat gold, the niṣāb equals (97) grams; and for 18-karat gold, it equals (113) grams.
The relied-upon position is to calculate the zakāh based on the buying price [i.e., the price the gold merchant would pay the owner to purchase the gold from him] on the day the zakāh obligation is due.
The one paying zakāh may assess/calculate the niṣāb using any of the aforementioned gold karats [i.e., he may choose whichever karat standard he wishes for the purpose of the calculation]. And Allah, the Exalted, knows best.
What is the ruling on X-ray imaging while fasting in Ramadan?
X-ray imaging itself does not break the fast. However, if a person ingests a contrast agent or medication to enhance the imaging process through an open passage to the body cavity (jauf)—such as the mouth or rectum—then their fast is invalidated.
Does burping (belching) break the fast?
Burping does not break the fast. However, if anything comes up from the stomach, the fasting person must spit it out and rinse their mouth to cleanse it.
If they swallow it intentionally while being able to expel it, or if they swallow their saliva before rinsing their mouth, their fast is invalidated.
Is it permissible to offer prayer at home, or should it be offered in the mosque?
All perfect praise be to Allah, The Lord of The Worlds. It isn`t preferable for the man to pray at home as praying in the mosque is twenty seven times more rewarding. Therefore, this should motivate him to offer prayers in the mosque. And Allah Knows Best.