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UPSC 2022 Maths Optional Paper 2 Q7b — Step-by-Step Solution

15 marks · Section B

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

Question

Velocity of a train (km/h) at times (min): table given. Estimate distance traveled in 20 min using Simpson’s 1/3 rule.

Technique

Simpson’s 1/3 rule needs even intervals; insert $v(0)=0$ at the start; standard $(h/3)(endpoints+4\cdot odd+2\cdot even)$.

Solution

Data:

Issue: Train starts from rest, so $v(0)=0$. The table starts at $t=2$. To apply Simpson’s 1/3 rule, we need data at $t=0$ as well (11 equally-spaced points for 10 intervals).

Augmented data (including $t=0,v=0$):

11 points → 10 intervals → applicable to Simpson’s 1/3 (needs even number of intervals).

Step 1 — Unit conversion

$v$ in km/h, $t$ in min. To get distance in km, convert $v$ to km/min: divide by 60. Or convert step to hours: $h=2$ min $=1/30$ hour.

Using $h=2$ min:

Then divide by 60 to convert.

Step 2 — Compute Simpson sum

Coefficients (Simpson 1/3): endpoints × 1, odd × 4, even × 2.

Odd indices (1,3,5,7,9): $v_1=16, v_3=40, v_5=51.2, v_7=17.6, v_9=3.2$. Sum × 4: $4(16+40+51.2+17.6+3.2)=4\cdot 128=512$.

Even indices (2,4,6,8): $v_2=28.8, v_4=46.4, v_6=32, v_8=8$. Sum × 2: $2(28.8+46.4+32+8)=2\cdot 115.2=230.4$.

Endpoints: $v_0+v_{10}=0+0=0$.

Simpson sum: $0+512+230.4=742.4$.

Simpson rule (with $h=2$ min): $\dfrac{2}{3}\cdot 742.4=494.93\overline{3}$ km·min/h.

Step 3 — Convert units

$494.93\overline{3}$ km·min/h ÷ 60 min/h = $8.249$ km.

Answer