Uttar Pradesh Technical University (UPTU) 2011-1st Sem B.Tech Computer Science and Engineering theory -12 Mathematics-I - Question Paper
beneath attachement is the theory paper of mathematics-I semester one paper
EAS103
Printed Pages4
((Following Paper ID and Roll No. to be filled in your Answer Book) Roll No.
PAPER ID : 9601
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B. Tech.
(SEM. I) THEORY EXAMINATION 2011-12 MATHEMATICSI
Time : 3 Hours Total Marks : 100
SECTIONA
1. All parts of this question are compulsory: (2x10=20)
(a) Find the n,h derivative of x"'1 log x. _ \
a <
(bkrdtheTaylor's series expansion of: V f(xy) = x3 + xy2 about point (2, 1). u = ex sin y and v = ex cos y, evaluate :
<3(u,v)
3(x,y)
,A$) Find the minimum value of x2 + y2 + 6x + 12 = 0.
1 1 1 1 1 1 1 1 1
Find the eigen values of the matrix
Calculate the inverse of the matrix :
A
1 2
5 7
EAS103/KIH-26107 1 [Turn Over
1 23
Evaluate the area enclosed between the parabola y = x2 9 and the straight line y = x.
vv Find the magnitude of the gradient of the function f = xy z3 * at (1,0,2).
(j) Write the statement of divergence theorem for a given vector field F.
SECTIONB
2. / " three parts of the following : (10x3=30)
C e eigen values and eigen vectors of the following
2 -1 1 -1 2 -1 1-2 2
(b) If y = a cos (log x) + b sin (log x). Find (yn)0.
(c) The angles of a triangle are calculated from the sides
a, b, c. If small changes 8a, 8b and 8c are made in the sides, find 8A, 8B and 8C where A is the area of the triangle and A, B, C are angles opposite to sides, a, b, c respectively. Also show that 8A + 8B + 8C = 0.
(d) Find the volume bounded by the elliptic paraboloids z = x2 + 9y2 and z = 18 - x2 - 9y2.
(e) If A = (x - y)i + (x + y)j, evaluate <j*A dr around the
c
curve C consisting of y = x2 and y2 = x.
EAS103/KIH-26107 2
SECTIONC
Attempt any two parts from each question. All questions are compulsory. (5x2x5=50)
. A a + x
3. y = tan 1
, prove that:
va-x,
(a2 + x2) yn+2 + 2(n + l)x yn+, + n(n + 1) yn = 0. u(x, y, z) = log (tan x + tan y + tan z), prove that
_ <3u . _ du . 3u
sin 2x + sin 2y + sin 2z = 2 . dx dy dz
(c) Trace the curve :
r2 = a2 cos 20.
4. (a)[f yj = XzX3 , y2 = - and y3 = - 1 2 , find the value xi x2 x3
nf S(yuy2,y-i) d (xj, x2, x3)
Find the extreme values of: f(x, y) = x3 + y3 - 3 axy
(c) If the base radius and height of a cone are measured as 4 cm and 8 cm. with a possible error of 0.04 and
0.08 inches respectively, calculate the percentage (%) error in calculating volume of the cone.
5. Define curl of a vector. Prove the following vector
identity:
Div(u xv) = Curlu v - Curlv-u .
If r = (x2 + y2 + z2)1/2, evaluate V2 (log r).
(c) Find the surface area of the plane x + 2y + 2z = 12 cut off by x = 0, y = 0 and x2+ y2 = 16.
EAS103/KIH-26107 3 {Turn Over
6. (a) Express the Hermitian Matrix:
1 -i 1+i
i 0 2-3i 1i 2+3i 2
as P + iQ where P is a real symmetric and Q is a real skew symmetric matrix.
A =
(b)/tjsing elementary row transformations, find the inverse of the following matrix:
2 3 4
A =
4 3 1
1 2 4
State and verify Cavley-Hamilton theorem for the following
matrix
2 -1 1 -1 2 -1
A =
1 -1 2
7. (a) Find the mass of a plate which is formed by the co-ordinate
x y z
x y z
planes and the plane + + = 1, the density is given
+ + a b c
dx
4 .
1 + x
by p = k xyz.
jOsing Beta and Gamma functions, evaluate
Evaluate the integral (x2 + y2)dxdy by
changing into polar coordinates.
EAS103/KIH-26107 4 113925
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