It is given that the distance between Delhi and Agra is $233\;\text{km}.$
Let the speed of $Q$ be $x\;\text{km/hr}.$ Then the speed of $P$ will be $(x+10)\;\text{km/hr}.$
We know that, $\text{Speed (S)} = \dfrac{\text{Distance (D)}}{\text{Time (T)}}$
In $1$ hour, distance travelled by $P= (x+10) \times 1 = (x+10)\;\text{km}$
Now the remaining distance between $P \& Q= 233 – (x+10)\;\text{km}$
Both trains are going in the opposite direction and so the relative speed will be their sum of speeds.
$i.e., x+x+10 = (2x+10)\;\text{km/hr}$
The two cars crossed each other $75$ minutes after car $Q$ started.
So, $\dfrac{233-(x+10)}{2x+10} = \dfrac{75}{60}$
$\implies \dfrac{233-x-10}{2x+10} = \dfrac{5}{4}$
$\implies 892– 4x = 10x + 50$
$\implies 14x = 842$
$\implies x = 60.142\;\text{km/hr}$
Now the distance travelled by $Q=60.142 \times \dfrac{75}{60} = 60.142 \times \dfrac{5}{4} = 75.177 \approx 75.2\;\text{km}.$
$\therefore$ Car $Q$ had travelled $75.2\;\text{km}$, when the cars crossed each other.
So, the correct answer is $(B).$