Which of the following properties show a similar trend on moving from Li to Cs within the group?
a.Ionic mobility in aqueous solution
b.Reactivity towards water
c.Solubility of bromide salt
d.Thermal stability of carbonate salt

Hint: The metals for which the trends are asked are in the first group of periodic tables called as alkali metals and in general, all the elements follow a trend in periodic table i.e. moving from top to bottom and from left to right. So, the properties and their trend are required to answer the question.

Complete step by step answer:
First property is ionic mobility, which basically means movement of ions in aqueous solution. Now if we have\[L{i^ + },{\text{ }}N{a^ + },{\text{ }}{K^ + },{\text{ }}R{b^ + },{\text{ }}C{s^ + }\] and they are in aqueous solution (water). So here \[L{i^ + }\]is smallest in size over here and when it is in aqueous solution, lot of H2O molecule will surround \[L{i^ + }\] and therefore \[L{i^ + }\]will be bound by water molecule and could not move.
Similarly, the concept applies and among this \[C{s^ + }\] will be one which will be surrounded by lesser \[{H_2}O\] molecules hence it will move faster and will have higher mobility.
So Ionic mobility in aqueous solution increases from Li to Cs
Option b: - Group 1 elements reacts with water to form metal hydroxide and Hydrogen
Example: - \[2Na\left( s \right){\text{ }} + {\text{ }}2{H_2}O{\text{ }} \to {\text{ }}2NaOH{\text{ }} + {\text{ }}{H_2}\left( g \right)\]
So, this reactivity towards water increases from Li to Cs because as we move down, size increases and outer electrons become farther from the nucleus and therefore outer electrons can easily be lost and therefore this ease makes it react faster.
So, the reactivity towards water increases from \[Li{\text{ }}to{\text{ }}Cs\]
Option c: - Generally solubility depends on lattice energy (inversely proportional) and hydration energy (directly proportional). Since hydration energy decreases from Li to Cs because size increases so water molecules will be less bound to it so hydration energy will decrease.
This in turn will decrease solubility of salt (bromide salt) from\[Li{\text{ }}to{\text{ }}Cs\].
Option d: - Carbonates are \[C{O_3}^{2\_}\]and group 1 elements combine with carbonates and form stable salt. Carbonate stability increases down the group, because down the group size increases so charge density will be lesser and greater charge density means greater polarization which further requires more heat to break carbonates.
So, polarization decreases down the group and more heat will be required to break carbonate salt.
Hence stability of carbonates also increases down from \[Li{\text{ }}to{\text{ }}Cs\]

So correct options are a, b, d, among these which increases as we move from Li to Cs

Note:
So these were the few properties and their trend as we move from top to bottom and there are other properties also like size, ionization, polarization, hydration energy, lattice energy.