Download paper 1 case 1

EMAEM4A
Duration: 90 minutes
Cass 1
Question paper
05 October 2021
1.1 Determine the absolute error deviation and percentage error in the calculation of 𝑅 where
𝑇
3
𝑅 = ln (𝑇 −𝑇
) with measured values 𝑇1 = 50𝑐𝑚, 𝑇2 = 40𝑐𝑚, 𝑇3 = 30𝑐𝑚, and allowable
1
2
error deviations from the measured values being 2%, 3% and 1% respectively.
(6)
1.2. The data in the table illustrates the upward velocity of a rocket. Construct a finite divided
difference scheme and approximate the acceleration of the rocket when 𝑡 = 16 seconds.
𝑡 (𝑠)
𝑣(𝑡) 𝑚/𝑠
0
0
10
227,04
15
362,78
20
517,35
(7)
2. Use the finite difference equation to determine the solution at interior mesh points of
the differential equation 𝑦 ″ + 5𝑦 ′ + 4𝑦 = 0 , 𝑦(0) = 0 , 𝑦(1) = −1
, 𝑛=4
(10)
3. Solve the following differential equations
2
2
3.1. 2𝑥(𝑦𝑒 𝑥 − 1)𝑑𝑥 + 𝑒 𝑥 𝑑𝑦 = 0
𝑑𝑥
(5)
𝑥
3.2. 2 𝑑𝑦 − 𝑦 + 𝑥 3 𝑐𝑜𝑠𝑦 = 0 , 𝑦(1) = 1
(7)
4. A tank has pure water flowing into it at the rate of 10 𝑙𝑖𝑡𝑟𝑒𝑠/𝑚𝑖𝑛. The contents of the
tank are thoroughly mixed, and the contents flow out at 10 𝑙𝑖𝑡𝑟𝑒𝑠/𝑚𝑖𝑛. Salt is added to
the tank at the rate of 0,1 𝑘𝑔/𝑚𝑖𝑛. Initially, the tank contains 10 𝑘𝑔 of salt in 100 𝑙𝑖𝑡𝑟𝑒𝑠 of
water. Let the net rate of change of salt be
𝑑𝑆
𝑑𝑡
, formulate a differential equation describing
the process and determine what percentage of salt is left after 30 𝑚𝑖𝑛𝑢𝑡𝑒𝑠.
(9)
5. A 10𝑘𝑔 mass is attached to a spring hanging from a ceiling. This causes the spring to stretch
2𝑚 on coming to rest at equilibrium. At time 𝑡 = 0 an external force 𝑓(𝑡) = 20𝑐𝑜𝑠4𝑡 is
applied to the system. Let the damping constant 𝛽 = 3, formulate a differential
equation describing the system and use the method of undetermined coefficients to
determine the steady state solution of the system and sketch one cycle of the sine wave
form of the steady state.
(14)
𝑃𝑎𝑝𝑒𝑟 𝑡𝑜𝑡𝑎𝑙 = 58