The correct answers are:
Cortés:
-Spread of smallpox from soldiers to the Aztec
In the year 1520, smallpox appeared among the Aztec people for the first time due to the contact with the Europeans. This infection caused a demographic collapse in the native people of the Americas.
-Capture of the territory around Veracruz.
In 1519, Hernán Cortés and his soldiers went on an expedition on the continental part of the Americas. They entered through the South of what today is Mexico and founded a settlement under the name of Villa Rica de la Vera Cruz. After this event, he started the conquest of the Aztec Empire.
Pizarro:
-Execution of the last Inca emperor, Atahualpa.
Atahualpa was the last Inca emperor. He was facing a civil war against his brother Huascar when in 1532 Francisco Pizarro entered in the Andean territory. Pizarro captured Atahualpa and killed him.