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**Hint:**First find the distance Robert travelled to reach point A. After that find the time taken by him to reach point A at the speed of 10 kmph. Then, subtract 1 hour from that time to reach the destination at 1 P.M. Now, calculate the speed of the cycle by dividing the total distance to travel by the time calculated to reach at 1 P.M.

**Complete step by step answer:**

Given: - Robert is travelling on his cycle and has calculated to reach point A at 2 P.M. if he travels at 10 kmph, he will reach there at 12 noon if he travels at 15 kmph.

Let the distance travelled by Robert be $x$ km.

Time taken at the speed of 10 kmph,

$ \Rightarrow \dfrac{x}{{10}}$

Time taken at the speed of 15 kmph,

$ \Rightarrow \dfrac{x}{{15}}$

Since the time taken by Robert on travelling at the speed of 15kmph is 2 hours less than the time taken on travelling at the speed of 10kmph. So,

$ \Rightarrow \dfrac{x}{{10}} - \dfrac{x}{{15}} = 2$

Take L.C.M. on the left side,

$ \Rightarrow \dfrac{{3x - 2x}}{{30}} = 2$

Subtract the values in the numerator,

$ \Rightarrow \dfrac{x}{{30}} = 2$

Multiply both sides by 30,

\[ \Rightarrow x = 2 \times 30\]

Multiply the values on the right side,

$ \Rightarrow x = 60$ kmph

Now, the time taken by Robert to travel at 10 kmph,

$ \Rightarrow \dfrac{{60}}{{10}} = 6$ hr

To reach point A at 1 P.M., subtract 1 hour from the time taken at the speed of 10 kmph,

$ \Rightarrow 6 - 1 = 5$ hr

Now, calculate the speed to reach point A in 5 hours,

$ \Rightarrow \dfrac{{60}}{5} = 12$ kmph

Thus, the speed must be 12 kmph to reach point A at 1 P.M.

**Hence, option (C) is the correct answer.**

**Note:**The formula for speed distance time is mathematically given as:

Speed = Distance/Time

where, x = Speed,

d = Distance travelled,

t= time taken.

Distance travelled formula is $d=xt$.

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