North Maharashtra University 2008 B.Sc Mathematics S.YBSc MTH-211 (Calculus of Several Variables) - exam paper

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NORTH MAHARASHTRA UNIVERSITY, JALGAON

(New Syllabus w.e.f. June 2008)
Class- S. Y.B.Sc.
Subject : Mathematics
Paper MTH-211
(Calculus of Several Variables)

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I) Objective ques. ( two marks each)
1) If u =f(x,y) x= F (t) , y = ?( t )
Then state the formula for
dt
du
5
2) If w =f(u,v), u = F( x, y ) , v = ?(x, y)
Then state the formula for
x
w
?
?
.
3) If w =f(u,v) , u = F( x, y ) , v = ?( x, y )
Then state the formula for
y
w
?
?
4) describe homogenous function of x and y with degree n.
5) State Euler’s theorem for the homogenous function f(x,y) with degree n.
6) If u = ??
?
??
?
??
?
??
- ?
x
G one xn f y and G1( u ) ? 0
Then what is the value of x
y
y u
x
u
?
?
+
?
?
7)If u = f(x,y) is a homogenous function of degree n then what is the value of
xx xy yy x2u + 2xyu + y 2u
8) Using Mean value theorem complete the equality
f(a+h,b+k) =- - - - - - - - + h x f (a+?h, b+?k) + kfy (a+?h, b+?k)
9) If f(x,y) = a x2 + 2hxy + by2 then obtain the degree of the homogeneous function using definition.
10) If f(x,y) = x4 + y4 then obtain the value of
y
y f
x
f
x
?
?
+
?
?
11) If f(x,y) =( )2
1
x2 + y two then obtain the value of x
y
y f
x
f
?
?
+
?
?
12)If f(x,y) = x2 + y2 then obtain the value of
xx xy yy x2 f + 2xyf + y two f
13) If z = x2 + y2 , x = t two + 2y = 2t
find
dt
dz
14) If z =sin (x+y), x = t two +1,y = t 2
find
dt
dz
15) If z = u2 + v2 , u = x+y , v = x-y
Find
x
z
?
?
16)If z = u2 + v2 , u = x+y , v = x-y
Find
y
z
?
?
II ) Multiple option ques. ( one marks each)
select the accurate choice from the provided 4 choices.
1) If u = ?
??
?
? ??
?
-
- +
x y
tan x y
3 3
1 .x ? y
Then x y xu + yu = -------
a) cos2u b)sinu
c)sin2u d)None of these.
6
2) If f(x,y) = ( )2
1
2 2
-
x + y then x y xf + yf is
a) 0 b)-f(x,y)
c)f(x,y) d) None of these.
3) If z = x+y, x = 2t, y = 3t.
then
dt
dz is
a) 0 b) 3
c)1 d)5
4)If z=xy ?
??
?
? ??
?
y
f x then
x
y
y z
x
z
?
?
+
?
? = --------
a) z b) 0
c)
z
1 d)2z.
II ) Theory and Examples ( three – marks each)
1) If u =f(x,y) is a differentiable function of x and y ,
x= F( t ) ,y = ?( t ) are differentiable functions of t then prove that
dt
du
y
u
dt
du
x
u
dt
du
?
?
+
?
?
=

Earning:   Approval pending.