,l§2.l§‘ .§“ !.£2 liniersiw QfsargodhaB.A/B.Sc 1” Annual Examination 2007

Math Genera.l.(I§Iew Course) Qaper- I;

Maximum Marks: 100 Time Allowed: 3 Hours

Note: Attempt any two questions from each section.

Section- I

1

Q-1- (21) If |g_| = |I3| = |g_ + _b_| , find the angle between the vectors g_ and 1_3 .

(b) Find the scalar function gp such that ’

Yrp = (yz - 2xyz3)§ + (3 + 2xy - x2z3)l + (623 - 3x2yz2)/_Q

Also show that Y >< §7_go = O `

.2. ' °°

Q (a) Prove that if a positive term series: Zan( Converges, then the series1

Z \/a,,a,,+, converges.

(b) 'Show that (-1 +i)""/3 = e" (Cosy + iSiny) where x = l/§ln2 - af- - Zmr

and y = %ln2 + -ii?-71' + 2\/§rm , where n is an integer.

Q~3- (3) If z, and zz are any complex numbers, then show that "Z, - lz2|| S |z, + 22|

(b) State Cauchy’s Root test for examining the convergence or divergence of an

infinite series. Use the test for convergence divergence of .

|. fl.

Q.4. (a) Find the sum of infinite series:

1+ 1-C0s29 - -1-Cos40 + C0s619~- ~ - ~ + - - -2 2.4 2.4.6

(b) If _ji(t)=acosl§+asinll+btl_g prove that _

I 2 I II 2

i. (z)` =a2+b2 ii. £(z)><L/i (1) =a2(a2+b2)'

Section- Il4

Q.5. (a) For a Skew symnietric matrix A, prove that (A")‘=(-1 )"A" for any positive

< integer n. Deduce that if n is even then A" is symmetric and if n is odd, then A"

is Skew symmetric.(b) a - b - c Za 2a

Show that 2b b - c -- cz 2b = (a + b J; c)3

2c 2c c - a - b

Q.6. (a) Reduce the following matrix into Echelon form and hence determine its rank.

1 2 2 3 4

2 l 3 2 1

0 2 1 1 3

4,/~L@' "9 3 1 3 4 2, gl r ¢ 2”

f-A /ilf~»7"` /1/ *,c

(3)

(9)

/

(9)

(3)

(9)

(3)

(8)

(9)

(3)

(8>

1

(3)

P.T.0

Q.7.

Q.8.

Q.9.

Q.l0.

Q.ll.

Q.l2.

(b) For what value of1 , the following system has Non-trivial solution.

ta)

(b)

(21)

(b)

(21)

(b)

(21)

(U)

(8)

(b)

(21)

(b)

(3-/l)x, -x2 +x3 =O

x,-(1-;t)x2+x,=o “x, -xz +(1--Z.)x, =0

The flow through a network is as shown in the diagrami 4

st, l Ka.

C

/ 8 X3

Solve the system. If x3 == 100, x, = 5O,x6 = 50 Find the flow. &

Let I/` be ,the set of all ordered pairs of real numbers. Check whether V is avector space or not over R (Set of real number) with respect to the indicatedoperations. (a,/J) + (c, d) = (a + c, b + d) and k(a,b) = (ka,0)Let T : R3 -> R4 be the linear Transformation' defined byT(x, ,x2,x3) = (xl + x3,x, + x2,x, ,x, -xl). Find the matrix of Tw.r.t. the baseB = {v,,v2,v3} for R3 and the base F = {w, ,w2,w3,w4}for R4 wherev, = (l,l,O), v; =(l,O,--1), v3==(0,l,O)w|=(1,-l,0,0), w2=(l ,0,1,0),w3=(O,1,0,0),w4==(0,0,l,1)

l a az a3 +bcd1 b 2 b” -dProve that b +C G =0l C Q2 C3 + délb

1 d d’ di’ +abcSection- III

Find the differential equation of which the function:x2 + yz + 2gx + Zfy + c = 0 is a solution ofit.

Solve: = Sin(x + y) -I5 Cos(x + y) `

x

Solve: + -J-L = -%,y(l) = 2dx 2x y

Solve: p3 - 4xyp + 8y2 = 0 where p = £-

3 2 2 ~-x dSolve: (D + D + 2D)y = x + e where D = -CTx

Solve by the Method of Undetermined Coefficients:2

“ ' Zix-gl-+3-3163+-yV== x2 +3Sinx

By the Method of Variation of parameters, find the solution of2

gg + y = Sec” x

2‘_

Solve: x2 gg? + Zxgxx -- 6y = l0x2 given that y(l) = 1, y'(l) = 6

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Mammy 0\‘Q University of SargodhaMan and maths

Mer mg B.A / B.Sc 1" Annual Examination 2008.

D

' o.

I

Math General (New Course) Pager: B

Maximum Marks: 100 Time Allowed: 3 Hours

Note: Attempt any two questions from each section.

Section- I

_ (a) lf Eiand I; are any vectors, thenshowthat |&7 I; + ll; |'-Zi isQi . _

nerpendicular to I5 | I; - ll; | Zi . »

(b) Find the condition for ahsolute convergence ,radius of convergence and interval of~ oo rl "

. 2 x - 3convergence of the series

n==l n

(2n+i!3"+1)t up

Q2 (a) es e convergence o e series E 4” +1T tth f th `

(b)Find the sum of the series cosz 9+ cosz 29 + cosz 30 +""" to n terms.

.3 . ` 1 ` 6?(a)l’rove that tan" (cos 19 _+ i sm H) == if + ilnifg-

4 4 1 - sin 6 ‘

2 A .-\ 2

(b) If 12 (t) is a unit vector , then show that ai + + = 1

(a) Evaluate' V x , where F = xi + yi + ZE and r = | F |

r(b) ‘lf Z; ,zz are complex numbers. , show that

|2»+22 |2+|_ 21-22 |2=2<l Z1 I2-rlzz IZ)Section II

1 ai af af1 coz C/14 az.

(a) Show that 3 4» = 125, where cu is a fifth root of unity (b) Finclga

1 a) ai 01

lcowca432basis and the dimension of R(T) and N('D where

T : R3 -> R4 is defined by'1l`(x|,x2,x3)=(2x1+x3 ’4X|+X2,X|+X] , x3 -¢4x2)

Where N(T) = {u e U : T(u) = 0} is the Null space andR(T) = {v e V : there exist ue U with T(u) = v } is the range of T

/lm

Q4 No Questions Mark.

Q.6 (a) For what value of 7L the following homogeneous equations (8)

Q.6 (1-7»)x1+ x2' + fx; =0° ' X| - XX; 4” X3 = 0 _

XV- X2 “l' (1-7l,)X3 _:O(have nontrivial solutions? Find these solutions.

x an cz "ra cz

a x a a a

(b) Show that a 61 x a a = 0 (8)

a a a x a

a a a a x

Q.7 (a) Set up a system of linear equations torepresent the network shown in the (8)

diagram and solve the system. RNA 'ua flaw if ,qgxgo

< ”TL , x

5 ’L\ \ 0 x

. 2 " 5

b)Let V be vector s ace of all functions defined on R to R . (8)( p 2 2

Check whether the vectors 2 , 4sin x , _cos x are linearly independenti in V

Q.8 (a) Find the inverse of the matrix (818

- 1 2 A3

2 l 0 over R

' J - _y 4 - 2 5

_(b) If 35,282 Q 44,759 2, 58,916 , 80,652 , and 92,469 are all multiples (8)

of 13 , show that 3 5 2 8 2

4 4 7 5 9

5 8 9 l 6 is also a multiple of 13 .

8 0 6 ’ 5 2

9 2 4 6 9 ‘

Section III(2.9 (a) Solve e " dx + (e " cot y + 2y cosec y dy = 0 (8)

(6) Solve. (134 + DZ) y = 3x2 +6 Sin X - 2cos X (9)

Q.10 dy 2y-x+5 (3)(a) Solvel E : ZX -y-4

2 (9)(b) Solve: ggi + y = sec3 x

Q.l1 (a)~Solve: y"-*8y'+15y=9xe2" , y(O)=5,y'(_0)=,l0 (8)

(b) Use the method of undetermined coefficients to solve the differential equation

y” -351 +2y=2x2 +2xé‘ (9)

Q-15-4 (a) Solve: p2 y + ( x - y) p -'x = O (3)dy

(b) Solve x(2+x)?+2(l+x)y=l+3x2 a y(_1) =]

x (9) _

Avai\a`6le at

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University of Sargodha _, l. l

B.A / B.s¢ 1" Annual Exam 2010 ll ' 3, \

General Math Paper: B Me'9'“9 wiv?

Time Allowed: 3 Hours Maximum Marks: 100

Note: Attempt any two questions from each section.Section- I

Q.1. a.(8)

|QlQ+IQ\l1 _

If g_ and Q are two non-zero vectors, show that the vector-T-F_-PDT IS equallyQ _

inclined with Q and Q.

b. State Cauchy’s integral test. Apply it to test the convergence or diverge of the p- (9)

. °° 1

series Z-n=l np

Q.2. a. Define alternating series and their absolute and conditional convergence. Determine

whether the series Z(-l)"" -Ei converges absolutely or conditionally or

n=l n(n + diverges.

b. - - "

Prove that -222-Jgi-9-qsii = cos n(£ - ac) + sin n(1r- - nc)

l+s1nx-icosx 2 2

Q-3- 3- If -tan(a + ip) = x+ iy, show that x2 + yz + 2xcot 20: =l and

xz +y2 -2ycoth2,B =-1b- If [_=acost§+asintl+attana_I$ thenshowthat

~ fir dz; d’£ 3 -- dr di 2

1. -x--7--7=a tana 11. -><-7=a secadt dt dt dt dt

QA' a' If 1" (1) = 4; and go) = Q, 15(0) = 41 show that the tip of io) traces a

parabola.b~ Find the sum ofthe infinite series: sina + csin(a + ,B) + cz sin(a + 2,B) + ...... ,\C| < l.

Section- II

Q.5. a. If A and B are symmetric matrices then prove that AB is symmetric if and only if A

and B commute.b- Let T = R 3 -> R2 be a linear transformation defined by

T (xl ,x2,x3) = (3x, + 2x2, x2 - 2x,) find the matrix of T with respect to the basis

{(l,0,l),(l,2,l),(O,l,2)} ofR3 and {(1,2),(1,1)} of R2.

Q.6. a. For what values of /l , the equations _

Ax, - x2 = 0

(fl - 3)x2 - 2x3 = 0 have the non trivial solution. Find the solution.

2x,+x2+(l+2)x,=0b. 1 x x2 xg

l 2 3

Evaluate: y y ya1 Z Z 2 Z

1 W W2 W3

Q.7. a. The traffic flow through a network is as shown in the diagram.

V- _

(8)

(9)

(9)

Q.8. a.

Q.9.

Q.1 0.

Q.ll.

Q.12.

zooA

X1 X4 X2

B xs is c

l00 X3 |90

Write the equations indicating the traffic flow in the diagram. Solve the system. If

x3 = 100, x5 = 50 , x6 = 50, find the flow.

Show that the set {(3, O, -3), (-1, 1, 2), (4, 2, -2), (2, 1, 2)} of R3 is linearly

dependent over R where r is the field ofreal numbers.Prove that `

12 22 32 4222 2 2 52

3 4 = 0 without solving.32 42 52 62

42 52 62 72

Find the multiplicative inverse of the matrix2 1 1

A= -1 1 -11 1 1

Section- III

Form the differential equation of which the function given below is a solution:

x2 +y2 +2gx+2fv+c=0 whereg,f, careconstants._

Solve: (DJ + l)y = 1+ e" + ez”

Solve: xsin(!-)dy = [y sin(Z-) - xi|dxx x

d 2y . .

Solve: -(ix-f + y = cos ec x by the method of variations of parameters.

Solve: (D’ - 3D2 -4- 4)y = O,y(0) = l,y'(0) = -8,y"(0) = -4Solve the differential equation by using the method of undetermined coefficients.

y"+y =12cos2 x2

Solve: x+l2iI-51+ x+l@)-+y=4cosln(x+1)2£1762 dx

3 2 2 2

Solve: 2x-jj?-%c-gi = - az LK-89-110 Available at

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M€llllE§Iy,0l`Q University_of SarggdhaMerging Man and maths ,_

B.A/B. Sc 15' Annual“Examination 2011

General Math Paper: B

Maximum Marks: ._A_ __ __‘__" _ Time Allowed: _3_ Hours

Note: Attempt any two questions from each section.

section# I

Q.1. af If M fv`§`<gQ +iy,)(x2'+`fy2) ..... ;§...(xn _1y_,)' (8)

prove that H tan” (fi) = tan" + tan" + ........ + tan" u x1~ x2 xn

b Deiine sequence, use the limit comparison test to determine whether series converges (8)»

_ ____w_n_4 _. ~_-_ordive es. ‘ __(_ _ __ __ _______fs_ __A_ __ _;n3+_n__3. __ _

Q¢2-* 3- "Separate into real and imaginaryparts' 7' »taii“`(x`+fy) (8)

b. An absolutely convergent series is convergent prove it. (8)

Q.3. a. Prove that the component ofa vector F parallel and perpendicular to E in plane of E ( )

and F are:

i. fic 11. -_--“>‘(”“)~_ .c.c_ c.c rrri s_»i -

_ fProve_that ` -f `tan”_. "=-75'+'i1n'--x+y __ ._ )

-~ ~' _ i- __ '“1;fiy. 4' 2 Xe-__y»

Q-4- 8- If Q(t) = tj, 91(0) = 4 and f_(0)_ = Q, Show that the tip of L(t) traces a parabola. (8)

b. Evaluate thesum-' ofthe infinite series; "

(8)

cosH~ lcos26+}-cos3¢9 -E-cos40 + .........2 3 4

‘_ -Section-Ili

Q~5- 21- Define linear -transformation. Let- T :Rs R2 be a linear transformation defined by (9)

T(xl,x2,x3)=(x1,-3x1+x5) then find matrix of T relative to the basis (l,0,3),

(l,2,5), (1,3,2) ofR3.b. ,Forwhat va1ue‘-of-lthepequations. :_ ‘_ _(8) __ i`-_ __t' '___ -g_f<' i__» ____ (5¥2._)xi¥|-4x13-2`x5=I0°

4x1 + (5 =- l)x2 + 2x3;= 0

'i .`2qc1+2x2+(2-2,)x3=0has non-trivial solution. Find the solution..

Q.6. a. If A, B and C matiices are conformable. for product then show that A(BC) = (AB)C (8)

b. Whatis determinant? Let A be a square matrix. Then A is non-singular if and only if (9)

PTTO

Q.7.

Q.s.

°Q°9f

Q.10.

Q.1_1.

Q.12.

._' 4. J !

detA ¢' 0 ifA' is non singular then <1¢f(A°‘) V; -L-det(A)

Provethatl+a 1 l _1'

1 1 b 1 1

»_ + _=abcdl+}-+l+l+iln _1 1+c _l_' ,Za be-q -d

1 1. _lid 1 -' f _~ 2 Show thatthe-system T

'2x, - x2 + 3x3 =a3x, + x2 - 5x3 =b

-Sx, -5x2 +2lx3 =cIs inconsistent if c af 2a - 3bShow thet I

25 S2 Y 2 1 1a a +2a 0+ :(a_1)6a 2a+1 0+2 1

1 3 3 1

For what value ofa and b, the system of equations‘2x+3y+5z=9 _. 7'x:ljV3y_-¢2'z¢=84

has: i. Unique solution ii. No solution iii. Infinite solution

Section- III

Solve _ (x-y)2 = az

2

Solve + 4y = sec 2xdx’

Solve and find the singular solution, if it exists. P2 + 2Px3 - 4x2 y = 0

Find the general solution (D2 + 4) y = 4sin2 x.

Solve -_ _ r(2x -If y ji-l)dx,+(4x_+_2yf1}dy = IfXisil funetion o-fx of aieonstant then pfove that ” 1'

I' ilX=e'"x _iXe°"“dx.D - m2

Solve <2x+1)2§x{-6(2x+1)-;fx¥+16y=s(2x+1)2

2 2

Find the solution of yi; + = Q dx_ dx' .idx-_»

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i i I BA/B. Se I"Annual Examination ZOIZ ,~< General Math Pager: B'_°“"*" .Man and maths

f.'I£lXiI1lUI\lAlI1|'k$Z l00 Mer "`9'!“1f""°"""'f' rs

,éotcz Attempt any two questions from each section.

.IP at If a, b are eonstainllvectors. if is a constant and F be a vector function ofseainr"varinble I, viven by (9)._ ._ _ ,,_ D

“ = cos wt a + sin wt b, then show that )7 X Q = wa X bV' Y _ _ dc _ _ rt

b 'Apply Cat|cl1y°s"i‘o`bl lest to determine whether the series converges or diverges XT (S)

Q‘2' 3 Test the series I`or convergt 'ICC or divergence EY - ii _

(9)

b If .ci"*‘/Z = (J: + iy)P+‘7 ,'a.> O prove that a = ip log,(xZ +5/2) - qt'nn"’ 5-logg (5)Q.3. Il Prove that >< (b X c)] X d] X e = (a.c)[e.l;d - e.db] - (a.b)[e.cd - e.dc] (9)

‘ob Find the square ol`all tifth roots of ' ' i , (8)

QA' U Prove that V(¢>(r)) =$F (9)b. Evaluate the sum ot` the series C0526 + 605226 + cos249 + -§,_eqsZn9 (8)

Section- II ' Q.5. a li` ii. it- are stibspaces ofa vector space v. then (11 + iv) is a subspace ol' v containing it and w. (8) \

Further ii + ti- is the smallest vector space containing both u and ua "* <\_.\__b Prove that the product ofmatrices (S) \\ ,

A = c0529 cost? sinl9]_ B = [ C0S2¢~ cost# 551109_

cosH sinH $17129 ' cosdl sincp sinzdvis the zero matrix when 6 and gb differ by an odd multiple of;-r. " _>\"¢\0

Q.6. a For what value oil, the equations (8) \(1"'/1)x1 ’|".x:"X3=O

_xl - /lxz - 2x3 = 0 have nontri\'ia|'solulions. Find these solutions. *Y

xl 'i' 212 "' /lx] = O __\rb 1 casa cos/I (S) \

Prove that casa 1 cos (cr + B) = 0,f _ cusp’ cos (af + B) 1 °

Q.7;.\ fa. The l`lo\v ol`lral`tic at Kal ma Chowk on Ferozepur road 30° ‘ (3)Lahore is shown below. ` "‘ ""'i. Solve the system ' too ;*`|UlI 5

ii. Find the traffic Flow when .vs = 300. _t-, 1,_ou

-ct tr - 1 cr + 1 (3)b_ For \\'l\at value ofa is the matrix A = 1 2 3 singn.lar_, `

Z ~ tr a + 3 a + 7Q.8. a. Find zi'basis and dimension ofthe subspace iv ofR‘ spanned by (l,4,-l, 3). (2_l,»3<_-I) (8)

and (O,2,l,-5)hf Find the solution ofsystem ol`linear equations (3)

2.`l`| "' .\': " X; = 3.\'| 'f' 5'i_\': -' = i I

3.\`| * lx; 'f' ~i.\'_\ = 1 i

_(,- Section- III

Q-9-D 3- Solve the dil`l`erential equation if = (S) ""b. Find the general solution (DJ -D: + D- lh== llsiu x (9)

Q.l0. a. Solve the diftbrentinl equation ' np* - 2_\p + #ix = O v (9) l

dl" Solve the dil`l`ei'ential equation dy + 5i53fctx = 0 i I; (3)

Q.ll. a. Solve the dillirrentidl equation by method ofU.C. y"' + y' = 2x2 + 4sin x (8)b. Find thc equation ot`orthogona~I trajectory of curve r = a(1 + sind) (9)

J IQ'l 2' 3° Solve the dil`t’erential equation x‘%-' + 2x33 - 9% + xy = 1 l97

b Solve sii1x%-cosx%+2ysinx=O (5) _

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Mntlielliaties ll./\/ll.Se. '|`ime Allowed: 3 Ilours(le||e|'a|- ll f ll-A/05 l\»'laximum l\1i\|`|(SZ ltltl ?

Attempt six questions in all, selecting two questions li'om~section- l, two qucstions=li'omsection- ll, one question from section- lll and one question li‘o|u section- lV.

(a)

(la)

(a)

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Define l’aral»ola, its locus clirectrix and latus rectum. Analyze and graph the

conic represented by the equation. _xl ~ 4.»qy + fly’ -t- 5\/§y »t- I = ll

Find the condition that the curves axz +/ryz ==I anal mx! + 11,3/2 =l

should intersect orthogonally.

Find the angle ol` intersection of the carcliotls 1'=u(l~i~cos0) and

I' = /i(| -~ cos (I)

Show that the pedal equation of the astroitl x =(lC(lS\10, y=usin’() is

rl =u2-3p2

lf f(I)is a vector l`unction of ‘1’ and H is a constant vector, differentiate

\v.r.t. ‘l’ the function tlelinecl by ' _/

Prove that if ¢(x,y.z) he a scalar point function and l7(x,,\.»,z)_ be a

vector point function, then ¢li\'(;/SF): gbdivlf + F`.grml¢ i

Iistahlish the identity: 26 = fx (Zi >< f) ~i~ >< (H >< _}) »i~ If >< (fi >< li)

lf 6 anal li are unit vectors and 1) is the angle hctween them,

show that Sin = 12|/i ~ 0|2 2

l

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Find the ratio in which the yz-plane divides the segment _joining the points

/\(-2, 4, 7)' and l!(3, -5, 8).

lfintl parametric equations of thc line containing the point (2, 4, -3) and

|>erpentlicular to the planei3x 4; 3y + 72 = 9

lixpress the given equation fx + _i-')Z - zz + 4 = 0 in cylindrical

and spherical eo-ordinales. i

lfintl thc equation ofthe sphere circumscrihing the tetrahethon whose laces

are .\' '= (lay = 0, z == () and lx +m_v + nz 't~ p = 0

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SC|)l\l`l\lC inlu rcnl aml llI\i\g,lll2\l`y parts lun '(x l~ iy)

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l',V£\lIl2\IC the mlmnlc serlcs I -9- -I ( n.s'2() --~--- (.n.s'¢l(] + -( ¢q.s'6H.........

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,_ /l+l? V, fl-h.Sm -I A~-. (I,os~-- - »~~

l’|'ovc that in il spl\cri<;al triangle ABC ~»-~ --%;»-~ == -»-~»-Z-C

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and a sulwspncc W ii S|)i`Illl\C(l by (I. 2, 3. 4), (-I, -l,I

lfiml ll\c ¢limc\\slnn nl` ll ll_|\(l IV.

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/Also wrllc an echclnn matrix row equivalent lu A.

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Zx, -xl 4- 3x3 ll

Shuw llml the syslcm: 3x, + X2 ~ Sx, = lr jf)

- Sx, ~ 5_\'z + Zlx, = c is in consistent il` c #2 Zu »-3/1

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Maximum Marks: 100 r

B.A/B.Sc 2 Annual Examination 2006

g Math General Pager- B

Time Allowed 3 Hours

Note: Attempt six questions in all, selecting two questions from section- I, two questions fromsection- Il, one question from section- III and one question from section- IV.

Q.l.

(b)

Q.2. (4)

(b)

Q.3. (H)

(b)

Q.4. (a)

(b)

ta)Q.5.

(b)

(H)

Section-_I

Analyze and graph the conic represented by the

4x2~4xy+7y2+l2x+6y~9=0ei

equation

If C P and CD are semi conjugate diameters of an ellipse with centre C

show that the eccentric angles of P and D differ by a right angle.

Find the pedal equation of r"' = a'"Cosm¢9.

Show that the normal at any point of the curve

x = aCosH + a6B`in6l

y'= aSin0 - at9Cos9

Is at a constant distance from the origin.

__ _ . - *I .If r = aSmaJt + bCoscut + £5-Smcurca

d Z _, 2prove that --gl + cuzr == -fi Cosrut,dt an

where 5,125 are constant vectors and ‘ an ’ being a constant scalar.

~ _ _ _ 1 _ _ .If V = a x r prove that a = §curlV' where cz is a constant vector.

Prove that for any three vectors fl',5 and 6 '

c7><(l;><E)+f;x(E><Z1`)+Ex(&><l;)=O

Prove that . i. |a >< EV + |zf.13|2 = |a|’ |13|2

ii. |a.z§|’ -|a><13|2 =a’b’C0¢~2@

Section- II

Find equations ofthe straight line passing through the point` P(2, 0,-2),and

perpendicular to each of the straight lines

3‘_;§=2i=E_“1 and 1:13:532 2 2 3 1 2

Find an equation of the plane that passes through the points (3, 2, -1) and (8)

(1,-3,4) and contains a line parallel to 2? - 4; + 3/E

P.T.O

610'/(1!JL|19

3

lQ.6.

Q.7.

Q.8.

Q.9.

Q.1o.

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Q-11

Q.12.

(H)

(b)

(8)

(b)

(21)

(b)

(8)

(b)

(H)

(b)

(H)

(b)

(H)

(b)

Find an equation of tl1e sphere passing through the points

(0,-2,-4); (2,-l,-l) and having its centre on the straight line

2x- 3y=0=5y+2z

Write the equation x2 - 9)/2 - 4z2 - 6x +18y + l6z + 20 = 0

referred to new set of parallel axes with the origin at (3, l, 2).

Prove that 64(C0s°9 + Sin°0)= Cos89 + 28Cos40 + 35

Separate (oz + i,H)""" into real and imaginary parts.

Evaluate the sum of Inlinite series.

CosH-}iCos2Q+}§Cos39-%Cos4H+ ......... _.

In any spherical triangle ABC, show that

2Cos Eg-I1 .Cos ig-I1 Tan-3 = SinbCosA + SinaC0sB

Section- III

Determine whether or not the set of vectors {(l,2,-1), ((),3,l), (I ,-5,3)} is

a basis for RJ.

a 1 1 l '

snowman I " I I =(a-l)°(a+3)l I a l

l l l a

1 2 1

Show that 3 _ l 2 R lg.

0 1 2

Find the _solution ofthe system of linear equations

2x, -x2 - x3 = 4

3x, + 4x2 - 2x3 = ll by Gauss-Jordan elimination Method.

3x,-2x1+4x3=llP section- IV

Solve the initial value _problem

(2xC0sy + Bxzi/)dx + (x’ -x2 sin y - y)dy = 0 y(o) = 2

Solve iv- = --3-dx ey - x

Solve and find tl1e singular solution (if any) 6 p2yz + 3px - y = 0

Find the General Solution of (Dzp - 2D - 3)y = 2e" - l0Sinx

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University of Sargodha Merging Man and maths B.A / B.Sc 2““ Annual Exam 2010

.Math General Paper: B

Time Allowed: 3 Hours Maximum Marks: 100

Note: Attempt any twoiquestions from each section.

Section- I

.1. __ 8

Q a If _q and Z3 are any two non-zero vectors, prove that the vector: fl-T+% rs ( ) _ _ __£l__equally inclined to g & »' it an -`

b. _ _ °° Lim _ (9)Prove that rf a series 2 a,, converges then n -> wan = 0. Gives an example ofn=l

a series which satisfies this condition but is not convergent.

Q.2. a. State the ratio' test _for an infinite series. Investigate the behavior of the series: (8)

, n!

b- Solve the equation: x6 +1 = \/§i (9)

Q.3. a. Separate into real and imaginary parts: tan`1(x + iy) _ (9)

b- If L(f ) is a vector frmction of t and Q is a constant vector, then differentiate (3)I ,_ _...ax p, I

the following w.r.t_ t + l'-ILI Q-1

Q.4. a. Find i(f) where f”(f) = cos wt§+ sin wtl subject to 5(0) = Q,£(0) = Q _ (8)

b~ _ _ . C’ _ c4 _ c” _ (9)Evaluate the sum of the miimte series gsm 20 - gsm 449 + -5 sm 66? - ._...

5 “Section-II, Q.5. a. If./1 is an m*m symmetric matrix and P is an m*n matrix, then prove that B = P'AP (8)

is a symmetric matrix.

b- Let T :R3 ->R’ bealinear transformation whose matrix is (8)

0 1 -1 ~O' »-1

~ f ""' 1 -1 »o_

Relative to the basis: {(1, 0, 0), (0,l,0), (0,0,1)}. Find the P matrix of T relative to

the basis: {(0, 1, 2), (l,l,l), (l,O,2)}.

Q.6. a. For what value -of /1 , the equations (8)

'_’b‘71Fx3 'E 9S' ~ _-l2x,+(2--/1)x2 »3x3 = 0

-xl +2x2 -(2+l)x3 =0

have nontrivial solution. Find the solution.

P.T.O

iQ.7.

Q.8.

Q.9.

Q.10.

Q.11.

Q.12.

b. Show that "

az (a +.1)2 (a+2)2 (a+3)2

if (b+1)2 (b+2)2 (b+3)2 = 0

6 ¢+n2 @+m2 @+n2

dz _(d+l)2 <dy+2>2 (d+3)»2 - M ,

a. The flow of at the Kalma Chowk on Feroie Pur Road, Lahore is shown

below: 200

X2 C X3

10 B D 100 __ ._ xr ,_ _A _M

200

i. Frame the system of equations and solve it.

ii. Find the traffic flow when x4 = 300.

b. Let u,`v and w be linearly independent vectors, prove that the vectors u_ + v - 3w,

u + 3v 4 w,-it -F are linearly independent. I

3- 2 3 4

Find the multiplicative inverse of A = 1 2 0

3 1 5

b' _bcd a u2_a3 414 413 a2 l” f ; < cdaf b` bi' Q'*"'1f“ b“ If If 1 ‘

Show that -~ =dab c cz ca cl c3 cz 1

abc d dz d3 d4 d3 dz l

Section- III

a. Form a differential equation of which the given function is a solution:

1 I _§~<x;y>»= f (?ff~f1y>+ g<x+ an 1 =

b- Solvef (D3i+D)y = 2x2 +3sinx

a' Solve; xQ+3y=x3y2 , y(l) =2dx

b. Solve by using the method of variation ofparameters:

" ~°`, _ ~?cg/`+_`y'=c0Sx

H- Solve: (D3 + 2D2 -D -2)y = 0, y(0) = y’(0), y"(0) = 2

b. Solve the differential equation by using the method of undetermined coefficients.

y"-3y'+2y=2x2+2xe"

a- soive; 5 x2y"'+2qcy' - 6y~;1Qx2 __ y(l)» = 1, 5/3(1) =--6b_ .- 2 3 __

Solve: xg- EX - iil}=0dx dx dx

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