UPSC 2013 Maths Optional Paper 2 Q7c — 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

The velocity of a train which starts from rest is given in the following table. The time is in minutes and velocity is in km/hour.

$t$	2	4	6	8	10	12	14	16	18	20
$v$	16	28.8	40	46.4	51.2	32.0	17.6	8	3.2	0

Estimate approximately the total distance run in 30 minutes by using composite Simpson’s $\tfrac{1}{3}$ rule.

Technique

Composite Simpson’s 1/3 rule on 10 equal intervals; unit conversion for mixed-time-unit integrand.

Solution

Step 1 — Augment the table with $v(0)$

The train “starts from rest” ⇒ $v(0)=0$ . Augmented table (11 points, 10 intervals of $h=2$ min):

$t$ (min)	0	2	4	6	8	10	12	14	16	18	20
$v$ (km/h)	0	16	28.8	40	46.4	51.2	32.0	17.6	8	3.2	0

Note that $v(20)=0$ : the train returns to rest at $t=20$ minutes. From $t=20$ to $t=30$ , the train remains stationary, so the additional 10 minutes contribute zero distance. Total distance in 30 minutes = distance in 20 minutes.

Step 2 — Apply composite Simpson’s 1/3 rule on $[0,20]$

10 intervals of width $h=2$ (even count — Simpson’s 1/3 applies):

\int_{0}^{20}v\,dt\approx\frac{h}{3}\bigl[f_0+f_{10}+4\sum_{i\text{ odd}}f_i+2\sum_{i\text{ even (interior)}}f_i\bigr].

Indices, role	Values
Endpoints $f_0,f_{10}$	$0+0=0$
Odd indices $f_1,f_3,f_5,f_7,f_9$ (×4)	$16+40+51.2+17.6+3.2=128.0$
Even interior $f_2,f_4,f_6,f_8$ (×2)	$28.8+46.4+32.0+8=115.2$

Weighted sum: $0+4(128.0)+2(115.2)+0=512.0+230.4=742.4$ .

\int_{0}^{20}v\,dt\approx\frac{2}{3}(742.4)=494.93\;\text{(km/h)}\cdot\text{min}.

Step 3 — Convert units

Since $v$ is in km/h and $t$ in minutes, the integral has units (km/h)·min = (km)·(min/h). Convert: 1 h = 60 min, so divide by 60:

\text{Distance}=\frac{494.93}{60}\approx 8.249\text{ km}.

Step 4 — Total over 30 minutes

The train is at rest from $t=20$ to $t=30$ , contributing 0 km. So total distance in 30 minutes = 8.249 km.

Answer

\boxed{\;\text{Total distance}\approx 8.25\text{ km (in 30 minutes; train at rest after }t=20\text{).}\;}

UPSC 2013 Maths Optional Paper 2 Q7c — Step-by-Step Solution

Question

Technique

Solution

Step 1 — Augment the table with v(0)v(0)v(0)

Step 2 — Apply composite Simpson’s 1/3 rule on [0,20][0,20][0,20]

Step 3 — Convert units

Step 4 — Total over 30 minutes

Answer

This solution is part of the Maths Coverage Map — 13 years, mapped. Get the take-away PDF free.

Step 1 — Augment the table with $v(0)$

Step 2 — Apply composite Simpson’s 1/3 rule on $[0,20]$