At what time between 5.30 and 6 will the hands of a clock be at right angles ?

At 5 O'clock, the hands are25 min. spaces apart. 

To be at right angles and that too between 5.30 and 6, 

The minute hand has to gain$(25 + 15)=40$ min. spaces 

55 min. spaces are gained in 60 min. 40 min. 

spaces are gained in

$\left(\frac{60}{55}\times 40\right)\min =43\frac{7}{11}\min .$

$\therefore$Required time $43\frac{7}{11}\min$. past 5