**(c) \(\frac{\pi}{2}\)**

Let PQ be the vertical pole of height h.

If the angle of elevation of the sun’s ray is α, then the length of the shadow QR = h cot α

In ΔQRP, for the first moment, we take α = α_{1}

Given, h cot α_{1} = h ⇒ cot α_{1} = 1 ⇒ tan α_{1} = 1

For the second moment, α = α_{2} (say)

Given, h cot α_{2} = 2h ⇒ tan α_{2} = \(\frac{1}{2}\)

Now, for the third moment, α = α_{3} (say)

h cot α_{3} = 3h ⇒ tan α_{3 }= \(\frac{1}{3}\)

Now, we need to find the value of α_{1} + α_{2} + α_{3}. tan (α_{1} + α_{2} + α_{3})

tan (α_{1} + α_{2} + α_{3}) = tan **\(\frac{\pi}{2}\)** ⇒ α_{1} + α_{2} + α_{3 }= **\(\frac{\pi}{2}\).**