6198599946

r-£

f id ^rU C/\f, r₇.c\

i S

A ^ I * L

(3 liours)

N.B. (1) Qucstion no. 1 is Coinpulsory

(2) Solvc any thrcc from the rcmoining. i)(a) If 5 sinh x - cosh x = 5 fmd tanhx .

Moi- ■    <5 o i

^l'l» f)jf (W

Q. P. Codę: 27175

Total Marks: 80

(b) If u n e' * " * ' prove that

d'u

dxCydz

•- $xyzu .

(c)    If u =--—, v ^ tan ' x + tan ' y find ^    .

\-xy    0(x,y)

(d)    By Maclaurins scrics cxpand log (l 4 <r^ł) in powcrs of x uplo .

! sfŹ    Ol

I

I

X

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(3)

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^    1 ¹    /•    r*

(c) Show that the niątrix a — Iw 2 V2 o | is unitary andhence find A ^:

2 '    I

L ^{0    0 2}j

(f) Find the «' derivative of y

^or

(x+ 2)(2.v + 3)

(a)    Solve x^f - i t i and find the cominued produet o1‘the rools.

(b)    Find the nonsińgular matrices P and Q such that PAQ is in norma! form also

(4)

(4)

(6)

(6)

fmd the rank of A

T2 1	4 1
X»	^2! to i i
1 7 4 i
t' ^0	^6 j

(c) State and provo Euler*s theorcm for homogencous functions on tlirce variab!cs.

a)    Invcstigatc for what values of A and u the eouations

x + y + z = 6,.v + 2y i 3a - i o, x + 2y i Az - // havc i) no Solutions, ii) a uniąiic solution.    iii) infinite number of Solutions.

b)    Find the stationar>' valucsof / (.v, y) = x² + xy¹ + 2\x-l2x* -2y *

(8)

(6)

(6)

(6)

:) If sin [O + i$) - cos a + i sin u Prove that cos* 0 s i r.^? a sinh'tf

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