The formula for freezing point depression #ΔT_text(f)# is

#color(blue)(bar(ul(|color(white)(a/a)ΔT_text(f) = iK_text(f)bcolor(white)(a/a)|)))" "#

where

#i color(white)(ll)=# the van't Hoff #i# factor

#K_text(f) =# the molal freezing point depression constant of the solvent

#b color(white)(ll)=# the molality of the solute

We can rearrange the above formula to get

#b = (ΔT_text(f))/(iK_text(f))#

In this problem,

#ΔT_text(f) = "1.17 °C"#

#i = 1# because β-carotene is a nonelectrolyte

#K_text(f) = "37.7 °C·kg·mol"^"-1"#

∴ #b = (1.17 color(red)(cancel(color(black)("°C"))))/(1 × 37.7 color(red)(cancel(color(black)("°C")))·"kg·mol"^"-1") = "0.031 03 mol·kg"^"-1" = "0.031 03 mol solute"/"1 kg solvent"#

Our solution contains

#"0.0250 g solute"/"0.00150 kg solvent" = "16.67 g solute"/"1 kg solvent"#

∴ #"0.031 03 mol" = "16.67 g"#

and

#"Molar mass" = "16.67 g"/"0.031 03 mol" = "537 g/mol"#