Hemchandracharya North Gujarat University 102 : Advance Mathematics model question papers



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Posted Date: 29 Dec 2008      Posted By:: Darshak     Member Level: Silver  Points: 5 (₹ 1)

2006 Hemchandracharya North Gujarat University M Tech Automobile 102 : Advance Mathematics Question paper



Course: M Tech Automobile   University/board: Hemchandracharya North Gujarat University





AA – 3402 Seat No.______
First Year M. Sc. (CA & IT) Examination
March / April – 2006
102 : Advanced Mathematics

Time : 3 Hours ] [ Total Marks :70


SECTION-1
1 (a) Do as directed: 6

(i) Define: subset
(ii) If A={1,2,3},b={2,3,4,5} and U={1,2,3,4,5,6,7,8,9,10}
(AUB)?=________.
(iii) Define: Determinant of order 2.
(iv) If f(x)=x2 + 2x then f(2) - f(0) =_________.
(v) Define: Limit of a function.
(vi) Define: Revenue and cost function.


(b) Do as directed: 6

(i) lim { 2 + 2x } =_________.
x->0 3x +4

(ii) lim ah -1 =___________ .
h

(iii) d (ex - xe ) =__________.
dx

(iv) d 1 =__________.
dx v x

(v) d (u · v) =_________.
dx

-1
(vi) a b =_________.
c d


2 (a) If A,B and three sets then prove that AU( BnC)=(AUB) n (AUC) 4

(b) Answer the following : 4

(i) If Rf ={3,8,13,18} of f(x)= 5x-2, find its Df .

(ii) If f(x) = 1 , x ? Z – {-1,0,1} , prove that f(x+1) – f(x-1)= 2
x 1- x2

(c) Answer the following : 4

(i) Prove that 1 x yz
1 y zx = (x-y) (y-z) (z-x)
1 z xy


0 4 3
(ii) If A = 1 -3 -3 then prove that A2 = I .
-1 4 4


OR

2 (a) If A={x | x ? N, x3-2 = 25} , B={y y ? N, 1 < y <5} , 4
C={z z ? N, z4 = 81} then verify the An(BnC) U (AnC) .

(b) Answer the following : 4

(i) If f(x)= x(x+1) (2x+1) then prove that f(x)-f(x-1)=6x2 .

(ii) find adjoint of the metrix , A= 2 -5 .
7 9

(c) Answer the following : 4

(i) Find lim x -2 + 3x -3
x->8 5x -2 + 7x -4

(ii) Find lim v x2 + 5x – x .
x->8

3 (a) If y = log ( 3x+4 ) e 2x+3 find dy . 4
dx

(b) Find dy
dx

(i) xy = yx

(ii) y = xex

1 -1 1 1 then prove that 4
(c) If A= 2 -1 , B= 4 -1

(A + B)2 = A2 + B2 .


OR

3 (a) Find lim ( x+2 )3/2 – ( a+2 )3/2 4
x->1 x – a

(b) Find the following : 4

(i) lim x5/2 – 1
x->1 x3/2 – 1


(ii) lim 5x2 + 17x + 14
x->-2 9x2 + 5x – 26


(c) Define Null metrix , Diagonal metrix , Symmetric metrix ,
Identity metrix .






SECTION – II

4 (a) Evaluate : 6

(i) ? 3x2 + 5x2 – 4x +7 dx
x
(ii) ? xex dx

(iii) ? (2x +3)(x2 + 3x +5 ) dx


(b) Define the following terms with suitable example : 3

(i) Homogeneous equation
(ii) General solution of a differential equation
(iii) Particular solution of a differential equation

(c) Solve any one : 3

(i) (x+2) dy + y.dx = 0
(ii) (x2+y2) dy = xy



OR



4 (a) Evaluate : 6

(i) ? x – 1 3 dx
x

(ii) ? ex (tan x + sec2 x) dx
(iii) ? log x dx

(b) Define differential equation. Determine the degree and the 3
Order of the differential equation.
__________
v d2y +3y +1 =2 dy
dx2 dx
(c) Solve any one : 3

(i) dy + y =ex
dx

(ii ) xy2 dy =(x3 + y3) dx



5 (a) Derive the equation of line y= mx + c where m is the slope 3
and c Is the intercept on y – axis.

(b) Answer the following :

(i) If three points (2,3) , (k,1) and (3,4) are collinear. Find K . 2
(ii) Find the co-ordinates of the circum center of a triangle
whose vertices are (1,2) , (3,4) and (2,1).


(c) Find the equation of linew passing through the point (3,4) and 3
Parallel and perpendicular to the line 4x - 3y + 2 = 0.

OR

5 (a) Derive the equation of line of the form x + y = 1 where ab?0. 3
a b

(b) Answer the following :

(i) If (3,5) centroid of the triangle whose vertices (4, -1),(k ,2) 2
and (0, m) then find k and m.
(ii) If (4 ,k) divides the line segment joining A (2,3) and B(5,-1) 3
Find the ratio of divison from A and value of k.
(c) Find the equation of line passing through the point (3,2) and 4
Making an angle of 45° with the line 3x + y – 5 = 0.




6 (a) Define with illustrations : 6

(i) Parallel graph
(ii) Even vertex
(iii) Walk.

(b) Answer the following:

(i) Draw the graph G(V,E) for which V={a,b,c} and 2
E= {( a,b ),( a,c ), ( a,a ),( b,c )} .
(ii ) Determine the degree of each vertax and all simple paths 2
from A to D for the given graph G.


A B




C D

Graph : G

OR

6 (a) Define with illustrations : 6

(i) Simple graph
(ii) Tree
(iii) Loop

(b) Determine the following for the given graph G . 4

(i) The set V(G) of the vertices of G
(ii) The set E(G) of the edges of G
(iii) The degree of each from vertex
(iv)All simple paths from A to E.








A

B E



C D

Graph : G


=================








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