(Established
under section 3 of UGC Act, 1956)
Course
& Branch: B.E /B.Tech – Common to ALL Branches
(Except
to Bio Groups)
Title
of the paper: Engineering Mathematics -
II
Semester:
II Max. Marks: 80
Sub.Code:
6C0016 Time: 3 Hours
Date: 24-05-2008 Session: FN
PART – A
(10 x 2 = 20)
Answer All the Questions
1. Give the expansion of tanθ upto 5th
degree.
2. Separate the real and imaginary part of
cosh(x + iy).
3. Find the
equation to the plane through the point (1,2,3) and parallel to 3x + 4y + z + 5
= 0.
4. Find the
equation to the sphere with centre (1, 2, 1) and touching the plane z = 0.
5. Give two
integers such that their Gamma values are equal.
6. Write 
in terms of Beta
integral.
in terms of Beta
integral.
7. Find the
directional derivative of x2 + 2xy at (1, –1, 3) in the direction of
x axis.
8. 
is independent of the
path when?
is independent of the
path when?
9. Shade the
region of integration 
.
.
10. Evaluate 
.
.
PART – B (5
x 12 = 60)
Answer All the Questions
11. (a) Prove
that 
.
.
(b) If 
prove tha coshu = secθ.
prove tha coshu = secθ.
(or)
12. (a) If 
prove that θ is 1º 58' nearly.
prove that θ is 1º 58' nearly.
(b) If sin(θ+iф) =
cosα + i sinα prove that cos2θ = ± sinα.
13. (a) Find
the equation of one plane passing through the line of intersection of 2x + 3y – 4z = 8 and 4x – y + z = 7 and which is
perpendicular to the yx – plane.
(b) Show that the plane 2x – 2y + z = 9 touches the sphere touches the sphere x2 + y2 + z2
+ 2x + 2y – 7 = 0 and find the point of contact.
(or)
14. (a) Find
the shortest distance and its equation between the lines
.
.
(b) Find
the equation of the sphere that passes through the circle x2 + y2
+ z2 + x – 3y +2z = 1, 2x + 5y – z + 7 = 0 and cuts orthogonally the
sphere x2 + y2 + z2 – 3x + 5y – 7z – 6 = 0.
 |
15. (a) Prove
that β(m, n) =
(b)
Evaluate 
in terms of Gamma
function.
in terms of Gamma
function.
(or)
16. (a)
Evaluate 
(b)
Evaluate 
where A is the area
enclosed by x=0, y=0 and x + y = 1.
where A is the area
enclosed by x=0, y=0 and x + y = 1.
17. (a) Find the tangent plane to the surface xz2
+ x2y – z + 1 = 0 at (1, –3, 2).
(b) Find 
where 
where S is the surface
of the sphere having centre at (3, –1, 2) and radius = 3.
where where S is the surface
of the sphere having centre at (3, –1, 2) and radius = 3.
(or)
18. (a) Prove
that 
.
.
(b) Find 
, 
where S is the
upperhalf of the surface of the sphere 
, C is its boundary.
, where S is the
upperhalf of the surface of the sphere , C is its boundary.
19. (a)
Evaluate 
.
.
(b)
Change the order of integration and evaluate 
.
.
(or)
20. (a) If 
prove that 
.
prove that .
(b)
Evaluate 
.
.
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