BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination

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1 No. of Printed Pages : 12 MTE-03 BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination 1,4* :ert June, 2015 ELECTIVE COURSE : MATHEMATICS MTE-03 : MATHEMATICAL METHODS Time : 2 hours Maximum Marks : 50 (Weightage 70%) Note : Question no. 7 is compulsory. Attempt any four questions from questions no. 1 to 6. Use of calculators is not allowed. 1. (a) The product of first three terms of a G.P. is If 6 is added to its second term and 7 added to its third term, the terms then form an A.P. Find the G.P. 4 (b) Discuss the continuity of the function at x = 2. 2 x ; x<2 f(x) = 2+x ; x> 2 MTE-03 1 P.T.O.

2 (c) Number of goals scored by a team in the matches are given below : No. of goals No. of matches Find the mean, median, mode and standard deviation of the given data (a) Find a vector of magnitude 9 which is perpendicular to both the vectors 4i j + 3k and 2i + j 2k. 3 (b) Evaluate : x3 dx. x8 (c) Among 64 offsprings of a certain cross between guinea pigs, 34 were red, 10 were black and rest were white. According to genetic model these numbers should be in the ratio of 9 : 3 : 4. Test whether the data is consistent with the model at 5% level of significance , 2 = 5.99, x.05, = 7.815, 4.05, 1= 3.841] 4 MTE-03 2

3 3. (a) Consider the following data : X Y I Find the line of regression of Y on X for the data. (b) Trace the following curve by stating the properties used : x=y (a) Find, if z = ln(x2 + y2), where x = dt y = et. (b) Find the equation of a plane through the point (1, 4, 2) and parallel to the plane 2x + y 3z = 7. MTE-03 3 P.T.O.

4 (c) If X is a Poisson variate such that 6 P(X = 3) + P(X = 1) = 240 P(X = 5), find the mean and the variance of the distribution (a) Solve the following differential equation : 4 x in x dy + y = 2 in x. dx (b) For the given p.d.f. f(x) = k(x 1); 1 x 3 find the value of k, mean and c.d.f. of the random variable having above p.d.f. (c) In how many ways can a committee of 5 persons be formed out of 6 men and 4 women so that the committee should have at least two women? 3 6. (a) X is a normal variate with mean 1 and standard deviation 0-5. Obtain P(X > 0), P( I X 1 I > 0-6) and P( 1-8 < X < 5 Given that P(Z < 2) = , P(Z < 1.2) = , P(Z < 3) 1. MTE-03

5 (b) A study was conducted to test the effects of growth hormone on the rate of growth of 10 children. Growth rates were measured before and after the subjects were given growth hormones three times a week for a year. Based on the data given below, check whether or not the sample shows a significant difference in the growth rate at 5% level of significance : Subject Measured before treatment Measured after treatment [t11, 0.05 = 2.201, t9, 0,05 = 2-262, t9, 0.1 = , t 10, 0.05 = 2.228] MTE-03 5 P.T.O.

6 7. State whether the following statements are true or false by giving reasons in support of your answer : 5x2=10 (i) If A and B are such that P( A ) P(A nt)= 0-5. two independent events = 0.7, P(B) = 5 then 7 (ii) If f : R > R : f(x) = x2 and g : R > R : g(x) = x2 + 1 then h : R > R : h(x) = (x + 1)2 + 1 is a composite function off and g. (iii) The function y = x6 has its maximum at x = 0. (iv) For a normal distribution mean, median, mode and variance are all equal. (v) Two regression lines for bivariate (X, Y) intersect at the point (X, Y). MTE-03 6

7 71:1."6.t.-03 tai loch 311Tiir cnitistoi "ffftd T T4, 2015 Wow Icti 71.t.t.-03 : 41 : nfarff I flq HHet : 2 4u2 arre ---dxt31w : 50 (25e1 W' 70%) q")e : 7 3 f 7 " I WV :1 A7-e W7F I 3RT er-47 / 4e-g,c,?eti YT ch4 3757:10 1. () Tew -PR Alq Ira Sub,t)ci 1000 t I zfr 4, col 6 * ciita tfq -4 7 \1 Tr4 kviicik 441 q-r4 74uT ker wa WAR 4 (13) x = 2 W{ theq f(x) 2-x ; x<2 2+x ; x2 fflcicel 11-4RI 2 MTE-03 7 P.T.O.

8 (IT) -cr- Tt-r -9-rqr,:t> TR 4101 Alio) Q. kitseo cil Al..i.k.s.e-o TR afrw ftirar, 1Trrurwr, 1 ccb afik imich fa4eq Ta. W-47 I 2. () trftillut 9 *T TAT 4oi f X i - j + 3k afit - 2i + j - 2k * t I 3 (U) 1 0 5x 3 lb.- x8 dx T itc.-toch-r 11-A-R 3 (Tr) -F4Tr 9hm A Amer 64 Tiedzil M, 10 chid) atik ff I a-tt0171* 1-40 (Hisci) * attr argtra. 9: 3: 4 >1klTīg7 I tr77 5% Trri-th dr F'( TR alitt r c4 *tip c1 t I 4 [4.05, 2 = 5.99' X0.05,3 = 7.815, X0.05,1= 3.841] MTE-03 8

9 3. () alkt #rf* : X Y ati-*-41 rr1 x Y ITITEW11 T W *11-4R I d atituir ttr-a-r arsliirot srasffi TIM 1ft cialw : 6 x = y 3 dz 4. (q) dt Tff R 7:fft z = ln(x2 + y2), A e x e-t alt{ y = et. 3 (11) (1, 4, - 2) cbt arr 3 (-moo - 2x + y - 3z = 7 tv-ii-cit ki&fichtui -Jrff =tf* 3 MTE-03 9 P.T.O.

10 zrr x taa a-cit P(X = 3) + P(X = 1) = 240 P(X = 5), 4-e4 wriar 3 SIWul Tff WA7 I 5. () fll-nchtul : 4 x In x dy +y=2 In x dx NO eichor 4zq Th--eq f(x) = k(x 1); 1 x 5. 3 TKT TlitiTERIff t, k I:FR, TrIt21. afik 4-dR 4bc.1-1 (c.d.f.) Ti c (TT) atik 4 Tiroiall 14 5 ftzil 1141 ITRAf IT t 17-ER4 -k Trr -ra 1Trgfffq 31-4FT? 6. () X, IfTur 1 AK 1-111Ich idqaq 0.5 aicii rcrilk t I P(X > 0), P( I X 1 I > 0.6) 3.11K P( 1.8 < X < 2.0) V. I 5 c411 TrziT P(Z < 2) = , P(Z < 1.2) = , P(Z < 3) 1. MTE-03.10

11 *'r.11:g 7 IR 6+.1 NiTrql TrftaTur 1Z-R aiverr Trin qei 1:Rrr tq 11: 1/45 tr-0 af1t*a-rq.341 -NwrEr wr lug roeil Trzrr I 41.d r afṯ413 Tt tr, R sgriu 3f-d-qi 5% Trr2*dr ;Err fawru Try& ai-dt atrat : 5 341:IR * TO TIN 34111k TR [t11, 0.05 = 2.201, t9, 0.05 = 2.262, t9, 01 = , t 10, 0.05 = MTE P.T.O.

12 7. 3-7(4 3tIk tfat chltul eici 1:11--1R1RSId TIFT t 1T ar{:p1 5x2=10 (i) 7TR A AK B P(A)=0.7,P(13). 75,P(A11E)=0.5t eir f : R > R : f(x) = x2 at( g : R R : g(x) = x2 + 1 h : R > R : h(x) = (x + 1)2 + 1, f g th-07 t I (iii) 'TM y = x6 W aftwdl:f x = 0 1:Ft in I (iv) W e4 4-d-R R1L TavEr, -grrr t7rwr, 5mtui (v).kk (X, Y)* fz1 Trrrrwrur i-urq q-s (R, 7f) cmcil t MTE ,500

ELECTIVE COURSE : MATHEMATICS MTE-08 : DIFFERENTIAL EQUATIONS

No. of Printed Pages : 8 MTE-08 BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination December, 2014 ELECTIVE COURSE : MATHEMATICS MTE-08 : DIFFERENTIAL EQUATIONS Time : 2 hours Maximum Marks : 50 (Weightage