Find the sum of the infinite series 1 - 3 +225 21 2 3:2 4: --- by matching with a basic Macaurin series. e1/2 1/2 In(1 + 7/2) In(1-/2)
-2+31 (a) Express in the form a + ib 3-L41 (b) Calculate: (21)4
3. Consider 2 00 0 0 3 12 A=1-4 3 3-2 -2 21 0 You are given that the characteristic polynomial of A is XA (z) = (z 2). Find the Jordan form J of A and find a matrix P such that P-1AP J. (You do not need to find P-1.) (You may use an online RREF calculator, but remember you only have an ordinary calculator in the exams.)
21. The sequence a, -6,9. - 30. ., can also be written as 2 4, 6-3 e, -6.3.1 α. - 2-3" d, - 2.3.1 4 24. Evaluate cach of the following. Place any non-integer answer in simplest rational form. (a) Σ4 (6) Σ (2) +1) (4) Σ2 «) Σ-1)- 25. Which of the following represents the sum 2+5+10+ ... +82 +101? 10 (1) Σ(4j-3) (3) ΣΤ' +1) 103 (2) ΣΟ-2) (4) Σ(4 +1) 0
3. (2 points) The tableau r421 21 02 5 3 2 10 1 6 4 2 1 0 0 0 represents a solution to the linear programming problem Minimize z 41 22 + r3, subject to the constraints 31 +2a2+r3 6, that satisfies the optimality criterion but is infeasible. Use the dual simplex method to restore feasiblity and hence find an optimal solution.
1 -3 1 4 - 21 2. Let A= 1 1 3 06 0 1 -1 5-5 2 Do the columns of A span Rº? What does this mean for the general matrix equation Ax = b where be R*? 5 3. (a) Determine if A = 7 homogeneous equation. (8pts) -3 -4 5 2 1 has at least one free variable by solving the 0
Question 2 Find the general solution to 21 2 -2 X1 X2 1 4 32 Enter Answer D Question 3 Find the solution to the initial value problem X1 1 -2 21 2] [3 (2:0) = [] X1(0) 22(0) X2 3 6 X2 Enter Answer Question 2 Find the general solution to 21 2 -2 X1 X2 1 4 32 Enter Answer D Question 3 Find the solution to the initial value problem X1 1 -2 21 2] [3 (2:0)...
SORU 25 Find the intersection. x = -2 – 21, y = -5+21,2= -3+21; 1x - 3y - 7z=7 ОА 49 11 28 11 6 11 OB. (-4,-3,1) Ос. 82 11 - 60 11 9 11 OD (-4,-3,-1) ОЕ. (0, -7,-5)
13. + . Solve: 2. Suppose: 21 = 2+1, 2=3–21, and 23=- (a) 321 +421 (b) +-3 +421-8
E(s) = 5. 1. E (21) = 2. E (4r) = 3. B (6-) - 4. E (r) = 5. E(2r + 1) = 6. E (-+ 1) = 7.E (7 - r) =