Veer Narmad South Gujarat University 2010 B.Sc Nursing Mathematics : - 1 F.Y../B.A . - Question Paper
RR-0630
First Year B. Sc. / B. A. Examination March / April - 2010 Mathematics : Paper - I
(Algebra, Trigonometry & Vector Analysis) (Old Course)
Time : 3 Hours] :
[Total Marks
N Seat No.:
6silq<3i Pi*unkil SnwiA u* <KH=fl. 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 - 1 (Old)
-Section No. (1,2......): Nil
Student's Signature
-Subject Code No.:
(0 STHSfl. Hlgvll UMdl %|R d.
(3) IR&d
1 M HLHL WIH ?HLHl : H
(*l) IU 5HLHI.
(0 sin a j, fafcWSl <HHl. (wl a d)
(3) (Irrotational Vector) RUkI
M c-tHl.
(M) cos h 1 jc-ft Bnd wucfl.
(<h) -ft*WKl LSldl Md WIH =>HlUl : *lO
(*l) sin 90 dl fed*1 *iWl.
[cos 9 - i sin 9]5
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[cos 9 + i sin 9] 7 | ||||||||
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(3) % = |
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dl AB J>LlHl. | ||||||
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(y) Hdl<Cl *llAeU &[Uil HdtelH &[Uil d. | ||||||||
-> AAA
a = -2 i - 2j +4k
-> AAA
b=-2i+4j-2k
> A A A
c = 4 i-2j- 2k
(m) r =t2 i - tj + (2t + 1) dl
dt dt
{(cos 0 + cos (f>) + / (sin 0 + sin (|))j +
{(cos 0 + cos (sin 0 + sin (|))j =
2+1 cos cos n
(b) x = cos 0 + i sin 0 dl x8 &Hd *iWl. 6
x8
H*l<U
* W sin h "fl &Hd *iWl. e
(H) lOid & : 6
(<l) tan h (A + B) = tan (A + B)
(0 sin h (-0) = -sin h 0 U) cos 5 0 cos 0 -fl umeflHi fafcWSl 6
3 (?H) H.HH <HHl ?Hd lOid *l. 6
2 2 X V
(*0 -7t~+ . ,2 =1
cos /z u sin /z u
2 2
/ \ *_ , >* _1
(0 -2 2 sin w cos u
H*l<U
3 (?h) ifcld 3RI log e {l + cos 2 0 - i sin 2 0} = log e (2 cos 0) - i 0. (h) ifcld sin (a+n |3)-e'a sin n$ = e~m sin a.
(h) tan-1 (x + z j) ddl ?Hd &mHk*s HLlHL fa*ihfd
(*l) IMPHd d*U 6Kl*ld =*HLHl. dHdi LHhI <HHl.
(H) lOid *l M H51 *lfel A Hl
(*l) + le ?H 6KN.d d.
(0 ?H Jlft. d.
SlfelS A 6Kl*ld Ah dl 5Hd dtaf / = A.
(*l) *lfel A ?H Mi *lfel d d. =*HH 5H&
SklH *lWl :
2x + 3_y + 2z = 3 3_y + z = 0 x - 2y = 1
x = 3 x - 2y = 1 3_y + z = 0
dlL
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?H [d>U d. | ||||||||||||||||||||||||||||||||||||||||||
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(3HH>L Md 5>llHl. | ||||||||||||||||||||||||||||||||||||||||||
dl *lfel HRfoir
2 4 5 1 3 1 1 0 7
(3HH>L Md 5>llHl.
3 1 4
0 2 6
0 0 5
Hlai HlcH-SHd 0ld U) =*HH IlfeliWl eHlHl ?HLHl. lOid S$L I <H lifelildl 6
H Oh) lOid I SlfelMl ?HLrH-L Hlfctfai %lll Ah d. 6
HI *U<H-i6*iL *llHl. dH'tf A HIS
7 2-2 -6 -1 2 6 2-1
(H) *lfel -4 =
M-ted H.HH'j. &Hld iii.
6 Oh) % r = (l-cos f) i + (t - sin t) j
dt1
(it
b Ah dl Hdl<Cl i
|
% |
da * |
-> |
% |
db * | |
|
% |
-= r |
x a |
nd |
-= r | |
|
dt |
dt | ||||
|
d |
-> | ||||
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a x b \ |
= r |
X |
a |
x b \. | |
|
dt |
I J |
V |
J | ||
U) % ?Hd % &[Ul ad *WlHld Ah dl ?Hd dl <?f . d~a dt = 0
(h) %l f=xyi-2xzj + 2yzk dl Curl Curl f ?iWl.
f-
(H) lOid S$L : div
u x v = v, Cwr/ U-U. Curl V.
U) &[UlPi| PAHdl Vldl&SC-t (Solenoidal) A<U HLSdl
A A
%Rd WWl.d lOid iii I V = (x +y)i+ (y -z) j + (x-2 z) eidl&ei. (Solenoidal) &.
(1) As per the instruction no. 1 of page no. 1.
Instructions :
(2) Figures to the right indicate marks of the question.
(3) Follow usual notations.
(4) Use of "Scientific Non-programmable Calculator" is allowed.
Answer as directed :
(a)
(1) Give definition of Hermitian matrix.
(2) State the expansion of sin a; where a is in radian.
(3) Define Irrotational Vector.
(4) 'Divergance' Give definition.
(5) State value of cos h~l x.
Answer the following by computing :
(b)
10
(1) Find the last term in the expansion of sin 90.
[cos 9 - i sin 9]5
(2) Simplify :
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[cos 9 + i sin 9] i7 ' | |||||||||||||
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then find AB | ||||||||||||
matrix.
(4) Show that
A A
a = -2 i - 2j +4k
> AAA
b = -2 / + 4 j - 2k
> A A A
c = 4 i-2 j- 2k are coplannar vectors.
> A A A 2 ~J
(5) If r = t2 i - t j + (21 + l) k then find and r
dt
(a) State De'Moivre's theorem and prove it for positive and Negative integers.
{(cos 0 + cos <\>) + i (sin 0 + sin (|))j +
{(cos 0 + cos (sin 0 + sin (|))j =
2+1 coscos.fl
8 1
x8
OR
2 (a) Find the value of sin h~l x 6
(b) Prove that : 6
(1) tan h (A + B) = tan (A + B)
(2) sin h (-0) = -sin h 0
(c) Expand cos 5 0 interms of cos 0 . 6
3 (a) State and prove Euler's theorem. 6
(b) Separate cos (0 + i) into its real and imaginary parts. 6
(c) If x + i y = sin (u + i v) then prove that : 6
(1)
2 2 cos h u sin h u
= i.
(2)
sin2 u cos2 u
3 (a) Prove that: log e {l + cos 2 0-/ sin 2 0} = log e (2 cos 0)-/ 0. 6
(a) Define skew - symmetric and Hermitian matrices. 6 Write their properties.
(b) For any square matrix prove that : 6
(1) A + Ae is Hermitian
(2) A-Ae is Skew - Hermitian.
(c) Prove that : 6
Matrix A is Hermitian if and only if Ae = A.
OR
(a) When a matrix A is said to be 'row-equivalent' to 6
some matrix B ? Prove that the following systems of equations are equivalent
x = 3 x - 2y = 1 3y + z = 0
2x + 3y + 2z = 3 3y-z = 0 x-2y = I
and
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Echelon form. |
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elementary row operations. |
2 4 5 1 3 1 1 0 7
by applying 6
3 1 4 0 2 6 0 0 5
and 6
(a) Find the inverse of a matrix A = elementary row-operations.
(b) Find the eigen values of a matrix A =
obtain the eigen vectors of A corresponding to the Largest eigen-value among them.
(c) Define similar matrices. Prove that the eigen-values of two similar matrices are same.
(a) Prove that eigen-values of Hermitian matrix are real numbers.
6
(b) Find the eigen-values of a matrix A =
7 2-2 -6 -1 2 6 2-1
6
Also justify Cayley-Mamilton theorem for A.
(c) Prove that the eigen-values of a skew-Hermitian matrix are either zero or imaginary.
6
d2 r dt2
d r dt
6 (a) If r =(1-C0S t) / + (t - sin t) j then find
and
|
= r x | ||||||||||||||||||||||||||||||
ci a
b then show that
(c) Prove that a has constant magnitude iff a -
dt = 0
6 (a) If / = x2y i - 2x z j + 2 yz k then find Curl Curl f.
6
6
6
(b) Prove that : div
= v . Curl U-U. Curl V.
U X V V /
(c) State the condition for the vector to be Solenoidal.
Prove that : y = (x + y)i+ (y-z) j + (x-2 z) k is Solenoidal.
RR-0630] 8 [ 500 ]
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Attachment: |
| Earning: Approval pending. |
