Download AKTU B-Tech 1st Sem 2017-2018 RAS 103 Engineering Mathe Matics Question Paper

Download AKTU (Dr. A.P.J. Abdul Kalam Technical University (AKTU), formerly Uttar Pradesh Technical University (UPTU)) B-Tech 1st Semester (First Semester) 2017-2018 RAS 103 Engineering Mathe Matics Question Paper

Printed pages: 02 RollNo. l l l l l l l l l l l Sub CodezRAS-103
Paper Id: 9003
B. TECH
(SEM 1) THEORY EXAMINATION 2017-18
ENGINEERING MATHE MATICS
T ime: 3 Hours T otal Marks: 70
Note: 1. Attempt all Sections. If require any missing data; then choose suitably.
SECTION A
Attempt all questions in brief. 2 x 7 = 14
Find the nth derivative of X "?110g x.
2
Evaluate fol fox xey dxdy.
Ifx2 = au + bvy2 = au ? bv, then find (3?1;), (ax),
a
Evaluate the area enclosed between the parabola y = x2 and the straight line y = x.
What error in the logarithm of a number will be produced by an error of 1% in the number??
F79??
Find the value of m if F =mxi ? Syj + 22]? is a solenoidal vector.
WWFDP-O
Reduce the matrix l; i 1] in to normal form and ?nd its rank.
SECTION B
2. Attempt any threeparts 0f the following: 7 x 3 = 21
a) i) If u = sin nx + cosnx, then prove that ur ={nT1 + (?1)r sinan}1/2,where ur is the r?h differential
coefficient of u w. r. t. x.
3 3 3
ii) If u = sin?1 (?" +y +2
Bu Bu au
ax+by+cz) , prove that x ax +y ay +2 62 ? 2 tan u.
b) i) Using elementary transformations, find the rank of the following matrix:
2?13?1
= 1 2 ?3 ?1
A 1 0 1 1 '
0 1 1 ?1
0 1 2
ii) Compute the inverse of the matrix 1 2 3 by employing elementary row transformation.
3 1 1
c) i) If u1 = % 412% and u3= 9:962 find the value of W.
1 2 3 1. 2, 3
11)Ifu ?f (r,s,t),wherer ?y ,s ? Z,t x ,showthat x ax +y ay +2 62 0.
d) i) Change the order of integration in I = fol fxzz?x f (x, y)dy dx.
1"an
ii) Prove that: B (m,n ) 2 Km +71)!
m>0,n>0.
e) i) Determine the value of constants a, b, c if, ?= (x + 2y + a2)? + (bx ? 3y ? Z)i +
(4x + cy + 22)]? is irrotantional.
ii) If fl = (x ? 302 + (x + y)j , evaluate g?c fl . d?r, around the curve C consisting of y = x2 and
2
y = x.

(a)
(b)
(C)
(a)
(b)
(C)
(a)
(b)
(C)
(a)
(b)
(C)
(a)
(b)
(C)
SECTION C
Attempt any two parts of the following: 7 x 5 = 35
If y = emn?1 x, then prove that (1 + x2)y2+(2x ? 1)y1 = 0 and (1 + x2) yn+2+ [201 + 1)x ? 1] yn+1 +
7101 + 1)yn = 0.
azu _xZ?y2
6x631 x2+y2'
If u = x2 tom?1 (:1) ?y2 tom?1 (:7); xy qt 0 prove that
Trace the curvezy2(a + x) = x2(3a ? x).
Attempt any two parts of the following: 7 x 5 = 35
A balloon in the form of right circular of radius 1.5m and length 4m is surmounted by
hemispherical ends. If the radius is increased by 0.01m find the percentage change in the
volume of the balloon.
Using Lagrange?s method of Maxima and Minima, find the shortest distance from the
point (1,2, ?1) to sphere x2 + y2 + 22 = 24.
Express the function f (xy) = x2 + 33/2 ? 9x ? 9y + 26 as Taylor?s Series expansion
about the point (1, 2).
Attempt any two parts of the following: 7 x 5 = 35
Investigate for what values of 9? and ii, the system of equations x + y + z = 6, x + 2y + 32 = 10 and x +
2y + AZ = M, has:
(i) No solution
(ii) Unique solution
(iii) Infinite number of solutions.
2 ?1 1
Verify Clayey ? Hamilton theorem for the matrices A = ?1 2 ?1].
1 ?1 2
Sh that the matrix a + W ?3 + ?6] is nita ifaz + 32 + 2 + 62 ? 1
OW B + i6 a ? iy u ry y _ '
Attempt any two parts of the following: 7 x 5 = 35
F ind the mass of a plate which is formed by the co-ordinate planes and the plane g + %+ E = 1, the density
is given by p = k xyz.
Evaluate I = fol (?) ?d"
Evaluate fffR (x +31 + z) dxdydzwhere R: 05x ?1,1?y52,2?z?3.
Attempt any two parts of the following: 7 x 5 = 35
Verify Green?s theorem, evaluate IE (962 + xy) dx + (x2 + y2)dy where c is square formed by lines x =
i1, y = i 1
Verify Stoke?s theorem for I? = (x2 + y2)i ?2xyf taken round the rectangle bounded by the lines
x= :a,y=0,y=b.
If all second order derivatives of (1) and 17 are continuous, then show that
Curl (grad?) = 0 (ii) div (curl i7 ) = 0.