W AutoSave Off ひ MAT 291 (FINAL EXAM, NOV 2010) - Compatibility Mode Saved . Search File Home Insert Draw Design Layout References Mailings Review View Help QUESTION 3 Page 3 of 4 (a) Use partial differentiation to determine expressions for in the implicit equation sin²x-5 sin x cos y + tan y = 0 [5] (b) Use Maclaurin's series to obtain the expression of sin X and hence determine sin x-x lim (c) For the series 5- 5 5 5 248 (-1)0-15 - +...+ 2- find an expression for S the sum of the first n terms. Also if it converges, find the sum to infinity. (d) Test whether each of the following series is convergent or divergent. [5] [5] (i) 2 2 2x3 3x4 4x5 5x6 2 + 2 + (11) 1+3 2+n² [10] QUESTION 4: 2xy (a) If V tan prove that av av +y = 0 [10] (b) If I = (1) J-1- esinx.cosxcx ecos" xax where n≥2, prove that (ii) (n+1)=1+(n−1)I.-2 263-144e-2 Hence show that I [15] 629 332 words English (United States) Text Predictions: On Accessibility: Unavailable CK Comments Editing Share Focus + 80% W AutoSave Off ひ MAT 291 (FINAL EXAM, NOV 2010) - Compatibility Mode Saved Search File Home Insert Draw Design Layout References Mailings Review View Help QUESTION 1 Page 2 of 4 (a) Using Cramer's rule, find the solution for x, only in the linear system x+2x+3x-5=0 3x,-x+2x-8=0 4x-6x-4x+2=0 x+2 7 5 (b) Solve the equation: 5 x+1 1 =0 -3 -4 x-2 (c) Solve the set of equations below by Gaussian Elimination method. 2x+4y-5x=-7 x-3y+2=10 3x+5y+3z=2 QUESTION 2 (a) Compute all eigen values and the eigen vector corresponding to the smallest eigen value for the matrix (1-4-2) A=0 3 1 1 2 4 (b) Find the radius of curvature and centre of curvature for the curve y=2 tan x 3 seco at the point = 45° (c) The formula a² is used to calculate z for observed values of x² + y² −a² [7] [8] [10] x and y. If x and y have the same percentage error p, show that the corresponding percentage error in z is approximately -2p(1+2). 332 words English (United States) Text Predictions: On Accessibility: Unavailable [6] བ [12] CK Comments Editing Share [7] Focus + 80%