Veer Narmad South Gujarat University 2011-1st Year B.Sc Mathematics SB-0616 - 1 ( New ) - Question Paper
SB-0616
First Year B.Sc./B.A. Examination
March/April - 2011 Mathematics : Paper - I
(Algebra, Trigonometry & Vector Calculas) (New Course)
[Total Marks : 105
Time : 3 Hours]
nLal PuiKhloft [cpiffl SMcft.
Fillup strictly the details of signs on your answer book.
Name of the Examination :
F. Y. B.Sc./B.A.
Name of the Subject:
Mathematics : Paper - 1 (New)
Student's Signature
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-Subject Code No.: |
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-Section No. (1,2,.....): Nil |
(3) WSll %|Rei *1$ U&Vtl d.
M *lMd
*1 (*i) HLHL WH C-lHL :
(*l) i HL5l H.HH C-IHL
(0 tan a i a dl UUVilHl pRdl <HHl.
(3) Log x -ft &Hd <HHl.
(y) Htalldl hRsUH *U[ctdl (3K1 d Hl ?
Oh) ddldl oiSldl Md WIH *UUl :
*lo
HIS p()
4
12
(0 A dl eil&felS -2, -2 *id 4 <&H dl A
(3) x = cosh2 0 + sinh2 0, y = 2 sin h0. cosh 0 Hlfl 0
(x) ifcld sin2x + /sinh2j = 2sin(x + /j).cos(x-/j) .
(h) a = (x + 3_y)z + (_y-3z)j + (x-2z)k lddlei 9 ?
(h) cos w 0 Hd sin0 cos 0 Hd sin 0'{I US.W.I&Hi fedl 6 (w n HdlU d.)
(h) ifcld 3Ri tan-1 (cosh0-sinh0) = sin-1 (sec/z 0). 6
0 + 2 sin0 -sin30 25
U) aifad & lim :q = _T7. ?
e->0 0 +tan 0-tan 20 14
H*l<U
(*l) cos a a dl Ufciq.eG.Hi. fafcWSl Hiqi. e
(wi a d.) (h) lOid s$l I e
(2 cosh 0 -1) (2 cosh 20 -1) (2 cosh 40 -1) (2 cosh 0 +1) = 2 cosh 80 + 1
U) % xr = cosyy2rj + i sm\y2rj, r = 1,2,3..... 6LH dl RilOid 6
Xj.X2.X3........Hdd G. = COS 71 .
(*l) Log(a + i$) -ft &Hd #fl (viia,PeR). e
eiQ
(<h) ,_k =>Hd SKUfaS ?H?LLHL Pwtafd $l. 6
r -1
3 + i
($) ilOid & I Tan
v 2
= -{(4 + l)7T + /log3j _
H*l<U
3 (?h) UHH <HHl 5Hd lOid 5Hd d U**ft Cos0 Hd sin0d 6
Uidlih [iHdl U %|M.
(h) % a + / (3 = sin-1 (w + iv) 6LH dl *llfad iii I s[n a ?Hd 6
cosh2(3 fter Ah =*hs x2 ~(i+u2 + v2)jx+u2 =o d.
(is) RilOid log sin (x + iy) = j/ log CSh 2y CS 2*
7 z 2
+/ tan-1 (cotx- tanh j).
X (*l) lOid S$L I && *\\m A d ?Hd-H P + i Q 6
q.4 %iWl %i$ih d, vii P =*Hd Q d.
(*0
U)
M
U)
H (h)
M
U)
1/ r _ 3/ 72 /2
x-2y = \ 3y + z = 0
H*l<U
d*U
d 6R-%flHK %lWl.
1 3 0 3 -2 -1 0 -1 1
|
*lfel A = |
|
Hia P() Hiql. |
% /I = [//] Wl aij=ixjgin dl %lWl A PHd *lfel d.
5Hd % atj = i2- j2 61H dl %lWl i A feifad d.
n- ?H5d *U[ct?Hl Ax X = d*U A2 X = B2
WR S<klH ? %lWl
(*l) % A Hd B aftl *lfell <&H dl %lWl i A~1B 6
=*Hd BA_1di SlWlfelS <*CLl *RHl <&H d.
(0 *UPld I ?HLHei HlcH &[Ul X d *i3ld $5d <?f
3HkH-\6H UAH UH d.
(3HHPL Md Hiql.
"flat ?HLHC-il HUfrSllH (H) [dd 6M :
x + j-3z + 2w = 0 2x - j + 2z - 3w = 0 3x - 2j + z - 4w = 0 4x + y -3z + w = 0
HIS M-ted H.HH'j. &Hld
2x + 3y + 2z = 3 3_y + z = 0 x - 2y = I
(*l) y.LHd
x + y + z = 6 x + 2y + 3z = 10 x + 2y + Xz = \i
-ft *U[cW X H &Hdl Hia (*l) 6kl d*ft ?
(0 =*Hd-H 6kl d ?
-6
7
-4
2
-4
3
(H) *lfel =-4
di eilSlfelS 6ft< mil ?Hd dHLdl *M 6
dldl eilSlfelS 6ft<d *i3ld eilSlfelS fl[Ul ($) (l) Kl fllHl-H tfel eil&felS after 6LH dl UlRl i 6
*lfel adjA SlWlfeli ftfl.'V d.
(0 lOid *llAeU HlcH \Hd fl3Ld $U %U HlcH 6 fl[Ull Hl d.
(*l) a ?Hd /, =H ?H[Ul C-t / dl faK-tdlM. fl[Ul MHL 6LH dl 6
f - da ax dt
v y
d2 a
t// /l,il ~r\,.:\ . *\ *\ .-.n -..r>. . >-.*\ 1 6?/
'V/TJ ?Hd d HVfl lOid /j,
(h) Mi ?H &d l(xtHld UH & <M ddL VIK fl[Ul
r = cosU)t i + sinU)t j (wi. CO 9.) fclfcll "b
(*l) [H|-LdL =H r d ci<H {9.
(\) a|c r
(h) % f = x3 + y3 +3xyz 6LH dl grad f -ft &Hd mil d*U grad / UfWMlk-t B Mil it dl $l.
H*l<U
(*l) fl[Ul IM| Mh /dl ( curl) ?HLl lOid
cwr/ |(|)v j = (grad <()) x v + <() curl v .
(H) % f = xy2l+2x2yj-3yz2k 6LH dl (l,-1, l) divf
curlf HL.
= a;
dtz
U) %. r = a cost i + asintj+ at tanak dl &Hd mil.
-* 9_> 3_>
dr d r d r dt dt2 dt3
Instructions : (1) As per the instructions No. 1 of the page No. 1
(2) Figures to the right indicate marks.
(3) Follow usual notations.
1 (a) Answer the following : 5
(i) State De' Moivre's theorem.
(ii) Write expansion of tana in terms of a.
(iii) Write the principal value of Logx.
(iv) State the condition that non-homogeneous system of equations has no solution.
(b) Answer the following with calculation. 10
2-14 6 -3 12
(i) If A =
then find (A).
(ii) If characteristic roots of a matrix A are -2, -2 and
4 then find characteristic equation of A.
(iii) Eliminate 0 from x = cosh20 + sinh20 and y = 2sinh0cosh0.
(iv) Prove that sin2x + zsinh2j = 2sin(x + zj).cos(x-zj).
(v) Wether vector a = (x + 3y)i + (y-3z)j+ (x-2z)k is sole noidal.
(a) Obtain the expansion of cos0 and sin0 in 6 terms of cos 0 and sin 0 (where neN)
(b) Prove that tan-1 (cosh 0 - sinh 0) = J/ sin-1 (sec/z0). 6
0 + 2 sin0-sin30 25
(c) Prove that lim a , . : 7T = -T7. 6 w e->0 0 +tan 0-tan 20 14
2 (a) (b)
(c)
3 (a) (b)
(c)
3 (a) (b)
(c)
4 (a) (b)
(c)
6
6
6
6
6
3 + i
-1
Prove that an
6
6
Prove that every square matrix A can be expressed uniquely as P + i Q where P and Q are hermitian matrices.
If A = ctjj J where ajj = > x./ then show that matrix A 6
2 2
is symmetric and if o,j =i - j then show that A is skew symmetric.
When two linear systems AxX = and A2 X = B2 of equations in -unknowns are called equivalent. Show that following system are equivalent.
Obtain the expansion of cos a in terms of a (where a is in radian).
Prove that
(2 cosh 0 -1) (2 cosh 20 -1) (2 cosh 40 -1) (2 cosh 0 +1) = 2 cosh 80 + 1
having roots sin2a and cosh2(3 is
. , , 1/ (cosh 2y- cos 2x)
Prove that, log sin(x +iy) = y0 log-
' z 2
+i tan-1 (cot x tanh y)
State and prove Euler's theoem and hence express express cos0 and sin0 in terms of exponential function.
If a + i (3 = sin-1 (u + iv) then prove that the equation
If xr = cosjrj + i sinjrj, r = l,2,3..... then prove that 6
Xj .X2 .X3........ up to infinity = cos n .
Find the value of Log(a + i$) , a, (3ei?.
eiQ
Separate 1-k |> into its real and imaginary parts.
2 a io x2 ~(\ + U2 +V2)x + U2 = 0
= -{(4 n +1)71 + /'log 3 j
OR
6
1/ r _ 3/ 72 /2
x 2y = 1 3y + z = 0
OR
2x + 3y + 2z = 3 3 y + z = 0 x - 2y = 1
and
4 (a)
(b)
(c)
5 (a)
(b)
(c)
5 (a) SB-0616]
6
by
6
6
6
(i)
(ii)
6
6
Solve the following homogeneous system of linear equations :
x + y 3z + 2w = 0 2x - y + 2z - 3w = 0 3x - 2y + z - 4w = 0 -4x + y 3z + w = 0
Justify Cayley - Hamilton theorem for a matrix
'1 3 O'
3 -2 -0 -1 1
For non-singular matrices A and B show that eigen values of A-1B and BA-1
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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applying elementary row-operations. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
x + y + z = 6 x + 2y + 3z = 10 x + 2y + Xz = |i
find the value of % and |ifor which the system has
(i) No solution
(ii) Unique solution and
(iii) Infinitely many solutions.
7 [Contd...
are same.
Prove that there exist unique eigen value corresponding to a given eigen vector X. OR
The system of linear equation is
6
and obtain eigen vector corresponding to the smallest eigen value.
(c) (i) If eigen value of a non-singular matrix A is X 6
then show that eigen value of matrix adjA is
(ii) Prove that there exist inifinite eigen vectors corresponding to a given eigen value.
6 (a) If a and are differentiable vectors of scalar 6
variable t then find and hence prove that
d2a dt1
da
dt
d/
ax-
= a:
'dt
(b) A particle is moving such that its position vector r = cos cor i + smwt j (where co is constant) show that
(i) Velocity of a particle v, is perpendicular to r.
(ii) a\oc r .
(c) If f = x1 + y3 +3xyz then find grad f and check wether 6 grad f is irrotational.
OR
(a) Define the 'curl' of a vector point function ~f and prove that
6
6
6
curl |())vj = (grad <])) x v + (j) curl v.
(b) If f = xy2 i+ 2x2y j-3yz2k then find divf and curlf at point (1, -1, 1).
6
(c) If r = a cost i + asintj + at tmak then find the value of 6
dr d r d r dt dt2 dt3
SB-0616] 8 [2500]
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Attachment: |
| Earning: Approval pending. |
