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UPSC 2018 Maths Optional Paper 2 Q5d — Step-by-Step Solution

10 marks · Section B

Simpson's 1/3 and 3/8 rules · Numerical Analysis · asked 5× in 13 yrs · Read the full method →

Question

Starting from rest in the beginning, the speed (in Km/h) of a train at different times (in minutes) is given by the table below. Using Simpson’s $\tfrac13$rd rule, find the approximate distance travelled (in Km) in 20 minutes from the beginning.

Technique

Prepend the rest point $v(0)=0$ to get 10 even sub-intervals; Simpson’s $\tfrac13$ rule with step converted to hours ($h=\tfrac{1}{30}$ h).

Solution

Setup. Distance $=\displaystyle\int_0^{20\text{ min}} v\,dt$. The train starts from rest, so at $t=0$ the speed is $0$. Adding this base point gives 11 ordinates at $t=0,2,4,\dots,20$ min, i.e. 10 sub-intervals (even — Simpson’s $\tfrac13$ applies directly).

Unit care. Speed is in km/h but time in minutes. Convert the step to hours:

Then the integral $\int v\,dt$ comes out in km directly.

The 11 ordinates ($y_0,\dots,y_{10}$):

Step 1 — Simpson’s 1/3 rule

Step 2 — Group the ordinates

End ordinates:

Odd-indexed (coefficient 4):

Even-indexed interior (coefficient 2):

Bracket sum:

Step 3 — Multiply by $h/3$

Answer