Manipal University 2010 B.E Computer Science and Engineering University: ; : ; Title of the : Engineering Mathematics-I - exam paper
MANIPAL INSTITUTE OF TECHNOLOGY
MANIPAL UNIVERSITY, MANIPAL - 576 104
FIRST SEMESTER B.E DEGREE END SEMESTER EXAMINATION- NOVEMBER 2010
SUB: ENGG. MATHEMATICS I (MAT – 101)
(REVISED CREDIT SYSTEM)
Time : three Hrs. Max.Marks : 50
Summary: This is a regular term ques. paper of 2010 of the subject "Engineering Mathematics-I" which will help the students to expertise their knowledge and skills on this subject.
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MANIPAL INSTITUTE OF TECHNOLOGY MANIPAL UNIVERSITY, MANIPAL - 576 104
FIRST SEMESTER B.E DEGREE END SEMESTER EXAMINATION- NOVEMBER 2010
SUB: ENGG. MATHEMATICS I (MAT - 101) (REVISED CREDIT SYSTEM)
Time : 3 Hrs.
Max.Marks : 50
Note : a) Answer any FIVE full questions. b) All questions carry equal marks
1A. Find the nth derivatives of
3x2 - 5x -1 2x3 - 3x2 +1
(ii) xe2xsin22x
i)
0cota
A radius vector intersects the curve r =ae at consecutive points
P0, Pi.....,Pn If Pm and pn denotes the radii of curvature at Pm and Pn, then
1
1B.
m
f \ p
show that
is independent of m and n for all m n.
-log
m-n
V Pn J
Find the reflection of the point ( 1, 3, 4) through the plane 2x - y + z + 3 = 0.
(4 + 3+ 3)
1C.
2A.
2 2 x y
Find the evolute of + = 1.
a2 b2
2B. Evaluate :
2ar -1/
(i) Jx4 2ax-x2 dx
dx
(io J
- 2 . 2 o a +x
0
dn n If y =- x2 -1 , then prove that
dx"
(1 - x2)y2 - 2xy! + n(n+1)y = 0.
2C.
(3 + 4+ 3)
Find the angle between the curves r2 sin20 = 4, and r2 = 16sin29
Test the Nature of the following series
... 3 3.6 3.6.9
00 t + -
3A.
4 4.7 4.7.10
-i r +
-2 r +
y__3_ 23 2
3C. Trace the following curve with explanation
y (1 - x2)=x2
4A. State Cauchys mean value theorem and verify it for
f(x) = Vx and g(x)= \= in [a,b]
Vx
4B. Find the magnitude and equations of the line of shortest distance between the x-3 y-5 z-7 , x +1 y + 1 z + 1
Also find the points where it intersects the lines.
4C. Obtain the first three nonzero terms in the Maclaurins series expansion of
f(x) = ==- (3+4 + 3)
2 2 2
5A. If u = f(x +y +z ) where x = rcos0cos()), y = r cosGSim)), z = r SinG find
d\\ OU
and.
50 dty
5B. Evaluate the following limits
tanx V2
(i) It - (ii) It
x->a xX ax x
x
5C. A plane passes through a fixed point (a, b, c). Show that the locus of the foot of the perpendicular from the origin on to the plane is the sphere, x2+y2 +z2 - ax - by - cz = 0.
(3 + 4+ 3)
6A. Find the region of convergence of the following power series.
i 3 3.6 2 3.6.9 3
(l) 1 + -X +-X +-X +....
7 7.10 7.10.13
f-Y
v4 j
/4\3
v5y
.... 1 2 (ll) I x + 2 3
x2 +
x3+...
2 3
6B. Find the volume of the solid obtained by revolving the curve y (2a - x) = x about its asymptote.
6C. If the sides of a plane triangle ABC vary in such a way that its circum - radius remains a constant, then prove that
8a 5b 8c .
-+-+-= 0
cos A cosB cosC
(4 + 3+ 3)
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