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15MATll
USN
m
First Semester B.E. Degree Examination, June/July 2017 Engineering Mathematics - I Time: 3 hrs.
Max. Marks: 80 questions, choosing onefull questionfrom each module.
co
Note: Answer FIVEfull
Module-l
x (x-VCx+2) b. Find the angle of intersection of the curves r = a(1 +sin 8) and r = a(l-sin 8). c.
Obtain the nth derivative of
(32a, 32a)on the curve x
Find the radius of curvature at the point
OR J
a.
3
+
i
= 3axy.
J
ik
b. Obtain the pedal equation of the curve r" = an cos n8. c. Find the derivative of arc length ofx = a (cost + log tan (.!.)) and y = a sin t.
b. Ifz
a.
=
X2+
l
w w
w
£ o Z E
t!o
0..
.§
5
+ Z2 ,v
=
Ifu =
= a2_.
&2
xy + yz + zx, w
(06 Marks) (05 Marks) (05 Marks)
2
=
02Z
(05 Marks)
()y2
x + Y+ z, then find oCu, v, w)
oCx,y,z) OR
Ifu(x+ y) ~ x' +.;, then prove that (:
b. Evaluate Lt ( x-+o
c.
02Z
sin (ax + y)+ cos (ax-y), prove that _
.p
4
If u
=
ed
C.
(05 Marks)
Module-2 Expand Iog,x in powers of (x - I) and hence evaluate loge (1.1) , correct to four decimal places. (06 Marks)
w
a.
ia
3
(05 Marks)
If y~ + y-~ = 2x, then prove that (x2 - 1) Yn+2+ (2n + 1) XYn+1+ (n2 - m2)Yn= O.
ib
2
(06 Marks)
g.
a.
lo
1
J~
{I-
- :
~ - :).
(05 Marks)
(06 Marks)
aX + bX + CX + dX),v. (05 Marks)
4
tf'ly~,l.,~) ,then prove z x
that x u, + y uy + z u, =
o.
(05 Marks)
Module-3 a. A particle moves on the curve x = y= 4t , z = 3t - 5, where t is the time. Find the ~ ~ ~ components of velocity and acceleration at time t = 1 in the direction i - 3j + 2k. (06 Marks)
2e,
e-
b. If f= (x + y + az) i + (bx + 2y - z) J + (x + cy + 2z) k , find a, b, c such that f is irrotational. (05 Marks) c. Find the angle between the surfaces X2 + y2 + Z2 = 9 and z = X2 + l - 3 at the point P(2, -1, 2).
(05 Marks)
lof2
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15MATll
a. b.
OR Find the directional derivative of xy' + yz3 at (2, -1, 1) in the direction ofthe vector ~ ~ ~ i +2j +2k . (06 Marks) 2 If U = x2i + y2J + z k and v = yzi + zxJ + xyk , show that ii x v is a solenoidal vector.
m
6
c.
ik ib lo g. co
(05 Marks)
For any scalar field ~ and any vector field
f,
prove that curl
(~f) = ~ curl f
+ (grad ~)
x
f.
(05 Marks)
Module-4
7
a.
Obtain the reduction formula for
r
coso xdx .
o
b. c.
f coso x dx, where n is a positive
integer, hence evaluate
(06 Marks)
y
Solve: (x2 + + x) dx + xydy = O. Find the orthogonal trajectories of the family of circles r = 2 a cos parameter.
(05 Marks)
e,
where 'a' is a (05 Marks)
OR
Evaluate
b.
Solve xy ( 1 + x
a.
y)
io'c
Module-5
Solve the following system of equations by Gauss Elimination Method. (06 Marks) X+ 2y + z = 3 , 2x + 3y + 2z = 5 , 3x - 5y + 5z = 2. Find the dominant eigen value and the corresponding eigen vector by power method 6 A = - 2 [
2
-2 21 3
-1,
-1
3
perform 5 iterations, taking initial eigen vector as [I 1 1]I. (05 Marks)
Show that the transformation YI = 2x+ Y+ z , Y2= X+ Y+ 2z down the inverse transformation.
w
w
w
c.
(06 Marks)
dy = 1. (05 Marks) dx Water at temperature takes 5 minutes to warm upto 20De in a room temperature 40De. Find the temperature after 20 minutes. (05 Marks)
.p e
b.
% dx .
(1+X2) 2
aw
c.
9
fo
a.
di
8
x6
""
10
, Y3= X- 2z is regular. Write
a.
OR Solve the following system of equations by Gauss - Seidel method. 10x+2y+z=9 , x+ 10y-z= -22 , -2x+ 3y+ IOz=22.
b.
Reduce the matrix
c.
Reduce 8x2 +
A = [-19 7] -42 16
7y + 3z2 -
to the diagonal form.
12xy + 4xz - 8yz into canonical form.
***** 20f2
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(05 Marks)
(06 Marks)
(05 Marks) (05 Marks)
