Solution Task 1: (a) Consider 12lim,1Rt tLteR Since1R, let’s consider a particular case 3 2R, then 12 3122 1 22 2 lim lim limlim 1limlim Rt t t t t tt t tt Lte te te et Simplify further, 12lim 1 0 Rt t Lte e Hence 12limRt tte (b) Consider the limit 23limRtt tLttee Since 0e So, 23 2 2 lim 0 0 Rtt t Rt Rt Lttee ttee tte Hence 23lim0Rtt tttee Solution Task 2: (a) Consider the definite integral 0 1 F aFxsxKeedx where ,sa and F are positive constants. Now, 0 0 0 0 1 1 1 F aFxsx F sxaFax F sxaFax F saxaFsx Keedx eeedx eeedx eeedx Simplify further, 00 00 1 111 111 FF saxaFsx aFFFsaxsx aFsaFsF sFaFsF Keedxedx eeesas eeesas eeesas Simplify further, sFseK aFsFsese 11 sF sFaF aesa ssa aese ssa
Hence 11sFaFaeseKssa (b) To show 11sFaFssaKaese Last result from part (a) 11 11 sFaF sFaF aeseKssa ssaKaese This completes the proof. Solution Task 3: Consider the system of equation, 1233 21 14 0112 11 0 x y za (a) The augmented matrix is, 1233 21 14 0112 11 0a Now let’s perform elementary row operations. Perform 2213312,RRRRRR 1233 05510 0112 0333a 2 25 RR 1233 0112 0112 0333a 332442,3RRRRRR 12 33 0 1 12 0 0 00 0 0 03a (a) Since system has four equations and three unknown that is system has either infinitely many solutions or no solution. Now, If 30a that is for 3a system has infinitely many solutions and if 3a0 that is if a3 then the system is inconsistent (b) If 3a then system has infinitely many solutions that is 233 2 xyz yz Suppose zt then 1,2 and xtytzt Where tR Solution Task 4: Given the matrices, 0 04341 3 0 0 and 121 0 123 tsAB ts (a)
End of preview
Want to access all the pages? Upload your documents or become a member.
Related Documents
Solution1: Given Let’s simplify.lg...
|4
|205
|26
Solved problems on calculus, mechanics, differential equations, linear algebra and matrixlg...
|6
|610
|369
Differentiation and Integration Examples with Solutionslg...
|6
|591
|378
Assignment Maths Problem and Solutionslg...
|8
|685
|19
Solution 1: To prove for an integer.lg...
|2
|274
|42
Deriving Posterior Distribution for Poisson Distribution with Gamma Priorlg...