Download AKTU B-Tech 1st Sem 2018-2019 Mathematics 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) 2018-2019 Mathematics Question Paper

Printed Pa cs: 03 1" Sub Code: KA8103
Paperldzl 19910. _] RollNo.I LI I I I I I I I I
B.Tech.
(SEM?I) THEORY EXAMINATION 2018?19
MATHEMATICS-I
T ime: 3 Hours Toral Marks: 100
Note: Attempt all Sections. If require any missing data; then choose suitably.
SECTIONA
1. Attempt all questions.
Qno. Question Marks (?0
a. _ . 2 2 2 2 l
Fmd the rank of the mamx 2 q 7
3 2 2
b. .. ,, . , . _ 3 ,3 a 2 3
Find the stdt10nanpmnt 0f f(x, y) ?x +y +3axv,a>0
C- If 21210039, y: rsinB, z: Zthen ?nd 5L? 6. LEAR? 2 3
6(x y?\
d. De?ne delV operator and gradient (2? 2 5 (?15?
9? If?=3x y- 1-32 ,?ndgr ?pointtl o. -2) 2 ?52!?
f.
, \3 ? 4
1 x 3. .
Evaluate J- E'; ex dxd?>5E~ ?(?63
g. If the eigen Values Oofig?f? A are 1 1 1. then ?nd the eigen ?vKIhe of 2 1
A + 2A + 31.
h. De?ne Rolle s Theo m 2 2
1' Ifu=x'_\ sin (y/\) then?ndx-?-+'1? . .V: 2 3
ax 8? ax
m?j
j. In RI? ? E and possible error in P. and I are 20%??d 10? /0 respectively, 2 3
then ?nd the error in R.
k. State the Tavlor 3 Theorem for two variql?s;\ 2 3
(.2
$10}; B
2. Attempt any rhree of the qu?h?iing:
Qno. \ Question ' Marks (70
a. Using Cayley- Hamilton theorem ?nd the inverse of the matrix A: 10 l
1 2 3?]
2 4 5?
3 5 6J
Also express the polynomial B: A8-11A7-4A"+A5+A4-11A5-3A3+2A+I
as a quadratic polynomial in A and hence ?nd B.
f?a??????v?r?; Ki??y?m?i JHA i ??vQ?C-Bfi??i? 128157140 ? 117.55.2=?~12,?13?1

"X
b. If y = Sin(m sin"x), prove that : (1 - x2) ymz ? (2n + l)x yn+| ? (n2 - IO
m2)yn = 0 and ?nd yn at x = 0.
C. If u. v, w are the roots of the equationtx ?a)3 +(x ? b)? +(x ? c)3 = 0, '0?
then ?nd M
6(a,b.c)
d? 0000 2 2 IO
Evaluate I je?(x +y )dxdy by changmg to polar coordinates.
O 0
(D
2
Hence show that Ie_x dx = "?15-.
0 _
eh Verify the divergence theorem for 10
1:" :(x3 ?y:)-i +(y3 -zx)j+(z3 -xy)?, taken over the cube bounded by
planesx=0,y=0,z=0,x= l,y= 1,2: 1?
SECTION C
3. Attempt any one part of the following:
Q no. Question Marks
:1. 3 ?3 4 10
Find inverse employing elementary transformation A = 2 _3 4
' 0 ?1 1
b. 1 2 -I 4 to
Reduce the matrix A to its normal form when A = 2 4 3 4
I 2 3 4
V -l -2 6 -7
Hence ?nd the rank ofAA ,
4. Attempt any one part of the following:
Q nu. Question Marks
a. Ifsin" y = Zlogtx + n show that 10
(x + Uly?+2 +(2n + ?(x + Hy"+l +(n2 +4)yn = 0
b. Verify Lagrangc?s Mean vaIUe Theorem for the function f(x)=x? in '0
[? 2.2]
5. Attempt any one part of the following:
Q no Question Marks
at Find the maximum or minimum distance of the point (1. 2. -I) from the 10
sphere x 1 +y 2 +2 1 =24.
b' Ifu=cos"( x+y )thenshowthat x?ai+y?u?+lcotu=0 10
x 4h]; ax 6y 2

Qno.
Attempt any one part of the following:
Change the order of integration and then evaluate: I Ix y dydx,
Calculate the volume of the solid bounded by the surface x=0, F0?
x+y+z=l & z=0?
Attempt any one part of the following:
Prove {had}2 - :2 + 3y: ? 2x)? + (3x1 + 2xy)] + (3w - 2x2 + 22M} is both
Solenoidal and lrrotational.
Find the directional derivative of d) = 5ny ? Syzz + ?zzx at the point
D
910.
Question
Question
P(l, l, l) in the direction ofthe line
2 3?1:
0 5i
4
Marks
[0
Marks
CO
CO