Both cases refer to the rule that articles work as the direct complement so when referring to the noun without mention it directly, the article used in the noun remains in the phrase as a substitute. And, the article have to be in coherent with the gender and number of the noun.
First answer: Yo la miré.
Step-by-step explanation: The bolded phrase uses the article “la” to refer to “la cadena”. In this sense, when one omits the use of the noun (“cadena”, in this case) the same unchanged article can be used to refer to a noun that has been mentioned before. In this case, as "cadena" (chain) is a singular femenine noun in Spanish, then the articule also needs to be singular and femenine, to be coherent and to be used correctly as a substitute, so the second option using the article "la" is the correct option, because "la" is also singular and femenine.
The articles in the other options are for singular masculine: "lo", plural masculine: "los", and plural femenine: "las", respectively. None applies to the singular femenine noun that "cadena" requires.
Second answer: Jaime los buscó.
Step-by-step explanation: The bolded phrase uses the article "los" to refer to "los chicos" and is coherent in gender and number to the noun "chicos", for "los chicos" (the boys) is plural masculine noun, so the coherent article needs to be also plural and masculine, as the one indicated in the third option: "los".
The first option is also a masculine article, but singular: "lo"; the second option is femenine and singular: "la"; and the fourth option is plural and femenine. But "los chicos" is not singular nor femenine, and the article needs to be both to be coherent to the noun, and to be correct.