Manipal University 2007 A.M.I.E General Branch of Engineering Subjects FIRST SEMESTER B.E END SEMESTER - Question Paper
Engineering Mathematics
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MANIPAL INSTITUTE OF TECHNOLOGY
 RCS_files/52428-182240-Engineering Mathematics - I (MAT 101) RCS-1.png)
7TT
(A constituent college of Manipal University, Manipal -576
FIRST SEMESTER BE DEGREE END SEMESTER EXAMINATIONS - 2007
SUB: ENGG.MATHEMATICS - I (MAT - 101)
(REVISED CREDIT SYSTEM)
Max.Marks : 50
Time : 3 Hrs.
& Note : Answer any FIVE full questions.
x 1 y 2 z 3 Find the image of the line-= --=- in the plane 2x + y + z = 6.
1C.
2A.
1
(4 + 3 + 3)
If y = tan 1 x, show that
(1 + x )yn+2 + 2(n+1) xyn+1 + n (n +1) yn = 0
3 2
2B. Find the circle of curvature at the point (0, 1) on the curve y = x + 2x + x+ 1.
2C. Find the length of the parabola y = 4ax from the vertex to one extremity of the latus rectum
(3 + 3 + 4)
3A. Find the angle between two curves (in its simplest form) at the point of intersection given r2 = a2 cos20, r = a ( 1 + cos0).
3B. (i) State integral test
(ii) Test the nature of V, n xn
x > 0
V ' i Vn +1
Find the percentage error in r if 2% error is made in measurement of ri and r2
given - = i. (4 + 3 + 3)
2 2
4A. Sketch and find the area of the loop of the curve y (a + x) = x ( a - x)
4B. Find first three non zero terms in Maclurins series expansion of f(x) = tanx
4C. (i) State Cacuhys root test.
(ii) Test the convergence of
1 + i + aiaiil + a (a+'X;+2) +... (if a, b > o.)
b b (b +1) b (b + l)(b + 2)
(3 + 3 + 4)
5A. (j) Obtain the reduction formula for J cosn x dx hence evaluate J sinnx dx
x
o
1 x9
(ii) J /x 2 dx
o -\/1 - x
5B. (i) State Lagranges mean value theorem.
1
(ii) Verify Cauchys mean value theorem for f(x) = loge x, g(x) = in [ 1, e]
x
22
5C. Trace with explanation r = a cos20 and find its area.
(4 + 3 + 3)
6A. Find the equation of the right circular cone generated when the straight line 2y + 3z = 6, x = 0 revolves about z - axis.
6B. Evaluate :
lt
1
--cot x
v x j
(i)
lt
tanx
(ii) (sin x)
x 0V
6C. Find the points on the lines x - 6 = y - 7 = z - 4
3 = -1 = 1 x y + 9 z - 2
which are nearest to each other. Hence find the shortest distance between the lines. (3 + 4 + 3)
w
MANIPAL INSTITUTE OF TECHNOLOGY
 RCS_files/52428-182240-Engineering Mathematics - I (MAT 101) RCS-2.png)
(A constituent college of Manipal University, Manipal -576
FIRST SEMESTER BE DEGREE END SEMESTER EXAMINATIONS - 2007
SUB: ENGG.MATHEMATICS - I (MAT - 101)
(REVISED CREDIT SYSTEM)
& Note : Answer any FIVE full questions._
IA. Find the nth derivative of
4x
(i)-z--(ii) sinx. Sin2x.sin3x
(x -1)2 (x + 1)
2/ 2/ 2/
IB. Find the evolute of x3 + y3 = a'3, (a > 0).
IC. Find the image of the point (1, 2, 3) in the plane x + 2y + 3z = 21.
(4 + 3+ 3)
1/ -1/
2A. If y/m + y /m = 2x, prove that
(x2 - 1) yn+2 + (2n+1)xyn+1 + (n2 - m2)yn = 0
2B. If p be the radius of curvature at any point P on the parabola y = 4ax and S be
2 3
its focus, then show that p varies as (SP) .
23
2C. Find the volume of the solid formed by revolving the curve y (2a - x) = x about its asymptote.
(4 + 3 + 3)
3A. Find the angle of intersection of the curves r = a and r = b
1 + cos 9 1 - cos 9
(in its Simplest form).
3B. Define :
(i) Absolute convergence
(ii) Conditionally convergence Find the nature of the series
1 1.3 2 1.3.5 3
x + -x2 + -x3 + ...
2 2.4 2.4.6
3C. The pressure p and the volume v of a gas are connected by pv14 = K. Find the percentage increase in the pressure corresponding to a diminution of 1% in the
volume, if K is constant.
(4 + 3+ 3)
2 2 3
4A. Sketch and find the area enclosed by the curve a y = x (2a - x).
n
x--
V 4 y
4B. Expand tanx in powers of
up to three terms.
State DAlemberts ratio test.
Test the nature of the series
1 4 9 16
+-+-+-+.
4C.
4.7.10 7.10.13 10.13.16 13.16.19
(3 + 3+ 4)
5A. (i) Obtain the reduction formula for J sinn x dx
2a 3
x
(iii) Evaluate : J
dx
o J(ax - x2)
5B. (i) State Rolles theorem.
(ii) Verify Lagranges mean value theorem for f(x) = logx in [e, e ]
Trace and find the length of one arch of the cycloid x = a(0 - sin 0); y = a(1 - cos 0).
5C.
(4 + 3+ 3)
6A. The radius of a normal section of a right circular cylinder is 2 units; the axis lies
, , , x-1 y + 3 z-2 , . along the straight line ~ = _5, find its equation.
6B. Evaluate the following : lt
lt
x 0
1
x xtanx
1
tan2x
'(tanx)
(i)
(ii)
x + 4 _ y + 6 _ z -1
6C. Show that the lines
and
3 5 -2
3x - 2y + z + 5 = 0 = 2x + 3y + 4z - 4 are coplanar. Find their point of intersection and the plane in which they lie.
(4 + 3+ 3)
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