8/7/2019 MTE-3 - Dec 2009

1/10

MTE-3 I

BACH ELOR OF SCIENCE (B.Sc.)Term -End Examination

Decem ber, 200900MATHEMATICS

M TE-3 : M ATHEMATICAL M ETHODSTime : 2 hoursMaximum Marks : 50Note : Question no. 7 is compulsory. Do any four questions

from questions No. 1 to 6. Use of calculator is not allow ed.

.(a) If the equations x2 + ax + b = 0 and3x 2 + bx + a =0 have a common root, thenprove that a + b +1= O.For the equation ax 2 + 2hxy + by 2 = 1, verifythat dx_idx d y

Calculate mean, median, mode and standarddeviation for the da ta given below :

x: 1 2 3 4 5 6 7f(x ) : 8 2 8 30 2 6 1 4 52 2

2

5

MTE-31P.T.O.

8/7/2019 MTE-3 - Dec 2009

2/10

2 .(a) If the sum of three numbers in G.P. is 384and their product is 1 7 28 , find the numbers.Verify Euler's theorem for the function3xy ; (x + y)A die is thrown twice and the sum of3numbers appearing is observed to be 7.What is the conditional probability that thenum ber 2 h as appeared at least once ?

3.(a) Find the distance between the point3

(-1 , 5, 1 0) and the point of inter sectionx-2 _y+ 1 z-2of the line3412 with the

plane x y + z = 5 .In eight throw s of a die, 5 or 6 is considered3a success. Find the mean and standarddeviation of num ber of successes.

1 4Evaluatei oxi. 7 xd01-x 2(d) Find the middle term in the binomial2

( 2 23 )1 0expansion of x --32 x4 .(a) Solvethedifferentialequation6

( x +y + 1 ) dy =1dx

M T E - 32

8/7/2019 MTE-3 - Dec 2009

3/10

(b) In a certain experim ent to com pare two ty pe4of animal foods A and B, the followingresults of increase in w eights were ob servedin anim als :

Animal No. : 1 2 3 4 5 6 7 8Increasein weightin gms.

Food A 4 9 53 51 52 4 7 50 52 53Food B 52 55 52 53 50 54 54 53

For both ty pe of foods, sam e set of anim alsw ere tes ted. Can w e conclude a t 5% levelof significance that there is no significantdifference in both ty pe of foods A and B .Y ou m ay like to use the follow ing values :( t14,0 .05 1.76,t7,0.05 = 1 .90 , t 1 5 , 0 . 0 5 = 1.75).

5 .( a ) The d if ferenc e of tw o num bers is 1 00 . The3square of larger one exceeds 5 times thesquare of the smaller one by an amountw hich is maximum . Find the numbers .

(b) C onsider a po pulation of 6 units w ith values71 , 2 , 3 , 4 , 5 and 6 . Wri te dow n a ll poss iblesamples of size 2 from this population.Obtain the sampling distribution of thesample m ean. Com pute m ean and varianceof the sampling distribution obtained.

00C

M T E - 33P.T.O.

8/7/2019 MTE-3 - Dec 2009

4/10

6 .(a) Find the angle between vectors u and v ,2Where a=2 i j+4k and v=3i+j+k.

(b) The weekly wages of 2000 workmen are3normally distributed with mean wage ofRs. 70 and wage standard deviation ofRs. 5. Estimate the number of w orkerswhose w eekly wages are :

B etween Rs. 7 0 and Rs. 7 1More than Rs. 72

Y ou m ay l ike to use the fol low ing values4:1) (0.4) = 0.6554, 43. (0 .2 ) = 0 .57 93 ,4 (0 ) = 0 .5 , ( 1 ) (0.6) = 0.7257 where

Z 2( 1 ) ( 2 )- f 1.7r e 2 dz00

(c) The following data relate to marketing5expenditure X (in thousand rupees) and theCorresponding sales Y (in ten thousandrupees) :

X 1 0 1 2 1 5 2 0 1 4Y 1 4 1 5 2 0 2 5 1 0

Find the coef ficient of correlation betwee nthe marketing expenditure and thecorresponding sales.

MTE - 34

8/7/2019 MTE-3 - Dec 2009

5/10

7 .State whether the following statements are trueor false. Give reasons for your answer :2 x 5 = 1 0

The rule given below is a function,f : {2 , 3} --> {4 , 7) : f(2)= 4, f(2) = 7, f(3) = 7.For two m utually exclusive events A and B ,P(A U B) = P(A) + P(B) P (A ( 1 B ) .P robability of co m m itting a type II - error iscalled the level of significance.Equation of the normal to the curvey = 4 x x 2 at (1 , 3) is 2x y +1 = O.

(e) For two non-negative integers rand ns.t rnn Cr_n+lcr`-

-o 0 -

M T E - 35P.T.O.

8/7/2019 MTE-3 - Dec 2009

6/10

aqui t-imenTwit )f l41 . 1 trftaTrfir, 2009

4 1 u i c i: 41ruinlq fatirce

qgq:2./ u231fETWffq 37W : 50*F: 30-1qtsell 7 31V ITIct/WeitW 9d i/16#W 7Tso/--7-f--4q/ cil, c ,?ejTOT civ , ?3 . 7 5 7 7 -t/1 .(a)zi .f1 :1-4 1T-71 .x2 + ax + b = 0 33173x 2 + bx + a =0kiief 4-0#1Zfta+b+1=0.chtul ax 2 +2hxy+by2 = 1 ,.4c-t11 4cf2w r - - -A 7 -rw crYxd, cc 14 -=i .

c ie g o 31-t-*--m-1111 r, -rrrfizw, grict)5Hittalcii 4 tc hftr-A-R :x: 1 2 3 4 5 6 7

f(x ) : 8 28 30 2 6 1 4 5 2 2

MTE-37P.T.O.

8/7/2019 MTE-3 - Dec 2009

7/10

2 .(a) ljul'ff T Oft -4d1 ^i tl ( s .e.offw-bo38 t4ttu l - v - i- K 4 1 7 2 8 , t 1( - 1 ( sq l7 TU 4 1 7 R I41 01 xy / (x +3*i077473

,fic ti t 3zull 3743t-itra q 344 1 + 1 n -hrt 7 4 1 2 1 1t I titkll 2

c hHc i - 0 4- ) a 1 t 3 1 1 4 1 1 - 5 1 - 1- f f 4 t T(=Kilt?3 .(a) Wick." ' x y + z = 5 47tgix-2 _ y + 1z-2341 2

f(1 , 5, 10) t18 alt t"*-4 ITT 5 Zir 6 *ti ttnicti IT F T3A R V t It-P-hcic-1144tseifail f 14Taiglic hroNcti gild-w1 - 1 7 7 11dxr 1-1IchlIv 223N W

(d)A 7 2tcrq 37rrTT Iv:ttrq2MTE-38

8/7/2019 MTE-3 - Dec 2009

8/10

c l y4 .(a)7414.1.17 (x+y+ 1) dx 1 -WI 60647I

(b )UchRtk-lkja4T-6t41311TBW(-44mi 4 1"f* cfi Tr zfi I TRIAEl#491 PirtgcltfftuT P4M R - 1 F :.: 1 2 3 4 5 i 6 7 83-1MT A 49 53 51 52 47 1 50 52 53co. i .{ -113 T ro TZ B 52 55 52 53 50 54 54 53ci 4 7. Ti

triafft, -qvTT- 1 1 - 5 rw r T3Tru4A T t i/IIIw fi 5 % 1 4 1 247T - ?1 - 7 TI T4-1 1fivE F E fT T 7:1*t3 T M T A * BItwrffr r q - ft I3 1 -itT -1 4 - 1( V I 1 - 1 1 - 1 .-*T 5i4)+1'*-77:1*--a( t 1 4 , 0 . 0 51.76, t7,0.05=1.90, t15,0.05= 1.75).5.(a) t t i 4 . . sql43 - 1 -a7 100 * 174T9 14 '$-1 ti t If3'44 -0-ttrn 74 31fl-rT t-zff 3 3-ifqwdr-f t Iq" ) .- i i 4 c ? 4 1- * 1 f 7 7 I(b) 11T9 1, 2, 3, 4, 5 *6 -dM1 .06T*-1 -PTI-11gfa7I74 W I F E3 T F E T 11 2 *ZipTfilfq.gradFriiWq I TffdPi TutziTITff*-1 r 7 7 I T I M U T C - 1 1 4 4-1 eie.1 WI -I:WI 35 1 t i t u 1MTE-39P.T.O.

8/7/2019 MTE-3 - Dec 2009

9/10

6 .( a)Tik7 uvifT gild,m 62tt=2ij+4k 3v=3i+j+k I(b)2 0 0 0 c hi ..)4fdff3t ITq." 4 .I I T Taf 1 : 1-40. 70 T. atl-{ TF4 0 4 - 1 1 - 1 4 )Fc 1 74 ( 1 l 5 T . t IT1704tse4kitih o b 1 :17tql

70 T. 37 1 T72 T. -A 3 - 1 f W . f3 - 1 7 4dI .1 17 4 ) 1 1 - TT:4 (0.4) = 0.6554, 14:,(0.2) = 0.5793,

4 (0) = 0.5, (1) (0.6) = 0.7257 where

Z1Z 2(I)( z )= f n 57Tre2 dz_00

(c)P - 1 4 - 1 1 C 1r(5cf afr*1 -fd 1 4 1 7 1 . 9 .- 1 -4 X ( s 3-iit A') at5T iT rff "fff Y ( q (-1 1R T. 4) 74 4 41 -trut :

X 1 0 1 2 15 2 0 1 4Y 1 4 1 5 2 0 25 1 0

fq-q-71 Fluatc ik . 1 1 .4 % . 1 ..7 1 1 .W f i c - rMTE-310

8/7/2019 MTE-3 - Dec 2009

10/10

7 .- q d r 4 7z n . 3-71-mI altrk d-cR 2x5=10chivir,T RP if : (2, 31> (4,: f (2) = 4 , f (2) = 7 , f(3) = 7i c t . )410* 14 t k - L R3 3 177iT3 A 3Th B i FriP ( A U B ) = P ( A ) + P ( B ) P ( A ( 1 B ) .mcnR IIrd mta311r1ZH?t m .7rT(1 , 3 ) 1 T act) y = 4 x x 2 t 3 - T fird'qcivoi2xy+1=0 t

(e).Att;IT Ic r TW rath nrnnCrnCr-1=n+1Crt

-00

MTE-31 1