CO No. of Printed Pages : 8 MTE-0 BACHELOR'S DEGREE PROGRAMME (BDP) CO Term-End Examination CV CD December, 0 ELECTIVE COURSE : MATHEMATICS MTE-0 : NUMERICAL ANALYSIS Time : hours Maximum Marks : 50 (Weightage 70%) Note : Answer any five questions. All computations may be done upto decimal places. Use of calculator is not allowed.. (a) Obtain the 8th Taylor series expansion of the (c) function f (x) = sin x in [-77, 7/] about x0 = 0. Obtain a bound for the error R9 (x). (b) Find the approximate root of the equation x x += 0 upto three decimal places using Newton Raphson's method, starting with x0 = Y. If f (x) = e', show that On f(x), (eah -on ex.. (a) Find the inverse of the matrix _ - using Gauss Jordan method. MTE-0 P.T.O.
dy y x (b) Solve -- dx y+x / with y(0) = using Euler's method in the interval [0, 0.0] by dividing the interval into sub intervals. (c) Obtain the Newton's Backward difference polynomial for the following data : x 0. 0. 0. 0. 0.5 f(x).0.56.76.00.8 Also calculate the value of f (0.5).. (a) Determine the constants a, b, c and d such 6 that the interpolating polynomial y, = y(xo+sh) = ayo +byi +h (cyi; +di; ) becomes correct to the highest possible order. (b) Using fourth order classical Runge-kutta method solve y' = xy for x =., y () = and h = 0... (a) The iteration method [ 5N N xn+i = 5xn +, n = 0,,, 9 x X n n where N is a positive constant, converges to N IA. Find the rate of convergence of the method. (b) Perform four iteration of the power method 6 to find the largest Eigen value in magnitude for the matrix 0 0 0. Take the initial approximate Eigen vector as [,, ]T. MTE-0
5. (a) Solve the system of equations xi. x + x = x + x + x.--.x + 5x x = 7 by Gauss - elimination with partial pivoting. (b) The velocity of a vehicle beginning from rest is given in the following table for part of the first hour. Using Simpson's rule, find the 6 distance travelled by the vehicle in this hour. t = time in minutes V = velocity km/hr 0 80 0 60 0 70 0 75 50 70 60 80 6. (a) From the following table, find the number of students who obtained less than 55 marks, using interpolation. Marks No. of Students 0-0 0-50 50-60 60-70 0 70-80 MTE-0 P.T.O.
MTE-0 (b) Using Gerschgorin bounds, find the 5 estimates of the Eigen values of the matrix Draw a rough sketch of the bounds.. 7. (a) Find the interval of unit length which 6 contains the smallest positive root of the equation x x 0=0. Using the midpoint of this interval as initial approximation, perform two iterations of the Birge-Vieta method. (b) Determine a unique polynomial f (x) of degree 5 such that f(x0) =, f ' (x 0) =, f (x) =, f (x) =, where x xo = h.
7-0 tmi I ci et) ZEITFai ch I tich H :A- IT i st) 0 : + Laid 7.t.-0 : ti.tv-0-ich reik)c; u TP,P7 : Efor 7filWUPT iw : 50 (c 70%) TeiTe (95 a-dz. P.9,- 7-firq- (t7 V/Hcfq 70)(di*- TOT ar-jrnw. (a) - -dtm [-7, '/] f (x) = sin x T x = 0 TfSST Zrltfo'ft '7 -tfq-r d R9 (x) wfar I (b) xo= - -9.t wrk- 9-9. c ul x-x+=0 Wft da,.c I -;iche (c) f (x) = eax cktfitrf" An f(x) = (eah )n eax.. (a) +lid--, fqfq T& T - Via WW AR I MTE-0 5 P. T.O.
(b) -d-tt [0, 0.0] itzr d-t FaRT gitt dx y+x ' f-d7r y (0) = WE Srlqi Wir-- "R cfatd-tro -Nil-Cm. --r--a-7i (c) FrN t-iq*ck ATcrq qlc x 0. 0. 0. 0. 0.5 f(x).0.56.76.00.8 f (0.5) tf Tfri Trita. (a) --7 a, b, c d i TIT9-0 fql:r ict) 6 f-d--vri ys = y(xo+sh) ay0 +)Yi + (cio±dy; fqw9 wl-rz f-c"r "Tr77Z t I (b) h = 0. -*-7 qi 'WE Rmircirtdd -NDT grt yi = xy, y() = ti x =. t -5q "A"F- 7 I. (a) TiTTIth. -NftT : 5N N Xn+ = 9 -an + 5, 0,,, xn xn ziw\i kj,* 'ER -97 t, NIA - 7*u t I *ki -Nfq ct;) aifirt{t 70 MTE-0 6
(b) rcirtt iltuqrrigt t7t 9-6 0-0 0 _ 7-ffq qc TI 7ich dd TrtfiTW atr7rititvr [,,T. 5. (a) TMT ckcfrl Trfq TTTf rqc-. reirq t7t f- fifty -{TITTu-r fttzi T-t.f 6 xi x + x = x + x + x = x + 5x x =7. (b) fdl5t7t-t t-{ tt ATT t F- irciriad dirri) kir Trzfr #-:(zrff -wr f97- cit sft i grr ffzr Trt quo ----Ri t = frrit v =i fw.t/ea 0 80 0 60 0 70 0 75 50 70 60 80 MTE-0 7 P.T.O.
MTE-0 8 6. (a) -.--d-79. TfMt 55 aft c 5 -grqi -0T5t T f-.. -ur5 - -ibqi 0-0 0-50 50-60 60-70 0 70-80 (b) q I TTft -g-7 TT& t 5 7THITT9- TTTF--dttr--7 lq-w H clcq7-0-7 rk- Tft9 7. (a) (b) Teb ch -WM.c -i - 0u X x--0=0 7q chl taf-5z -W7 t I 7 at-d-{-r---grr f-s rnl -Trfq Irr9 W-f7-eT rolr c) t T-IlcirTicr car-7 I 5 T c atttizr f (x) oct 'Of-R 6 f7tia. f R f(x0 ) =, f '(x0) = f (xi) =, f ' (x) =, t zi$ x xo = h.,