Roll No. ...........................
1~~~ ~\~'\ . -
T~al No. of Questions: 09]
[Total No. of Pages: 02
,
Paper ID [AM tOt] (Plense fill this I'nper ID in O:\IH. Sheet)
B.Tech. (Sem. ENGINEERING
- 1st/2nd)
MATHEMATICS
- I (Al\l - 101) ..
Time: 03 Hours
Maximum Marks: 60
Instruction to Candidates: 1)
.
Section - A is Compulsory.
2)
Attemptany Five questionsfrom Section - B & C.
3)
Select atleast Two questions from Section - B & C. Section - A
QI)
(10 x 2
= 20)
a) Find the radius of curvature of the curve x2 + y2 = a2 at (x, y). b) Find mean square value ofj{x) = sin x in the interval (0, 1). . c) Find df/dt at t = 0, wher.ej{x, y) = xcos y + el:sin y, x = P + 1,Y = t3 + t. .
.
d) Find the app.roximatevalue of(4.05)li2 (7.97)1/3,using derivatives. e) Write the expansion of the Taylor's seriesj{xo + h, Yo+ k) up to second order.
.
f) Write the equation of Ellipsoid and draw a rough sketch of it. g) Write two applications of double and triple integral each. I
h) State the integral test of convergence of infinite series. i)
How the cqnvergence of alternating series is checked.
j)
Separate into real and imaginary parts exp (5+ i;)
Section - B (Marks: 8 Each) -
1
Q2) Sketchthe graph of the curve y =x + -.x
R-21/2058j
RT.O. ~
-
.
Q3) Find centre of gravity of a lamina in the shape of a quadrant of the curve .... 2~
2/
X /3
Y
( a ) + ( b)
/3
~
.
'
= 1, the density being P = kxy, where k is a constant.
Q4) "Ifj{x,y) is a homogeneous function of degree n inx andy and has continuous first and second order partial derivatives, then show that 2a2f x -+2xy2 .
Q5)
a2f
aa x~y
ax
2a2f +y -=n(n-.Df 2 '
.
ay
Using Lagrange's method find the minimum value of x2 + y2 + Z2subject to the condition xyz = a3.
Section
-C (Marks: 8 Each)
Q6) Find the equation of the right circular cylinder'having for its base the circle x2 + y + Z2 = 9, x - y + Z = 3. I
Q 7) Express
integral
,
f0
xlll
(1- xp
r dx in terms of Beta function and hence evaluate the .
f0 ~% (1 - ~
dx.
)>i
Q8) Find the radius of convergence and circle of sonvergence of the power 2 n
.
.
1 ~ ( n. ) Z . senes £.J .' . (2n)!
.
.
Q9) Find the sum of.the trigonometric series ?
sina+ xsin (a + /3)+
~~
sin (a+ 2/3) +:..00 .
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