Hemchandracharya North Gujarat University 2001 B.Sc Computer Science 102 : Advance Mathematics - Question Paper

F (x)=e1/x-1/ e1/x+1 for x?0 and f (0)=0
(b) Find:
(I) lim e4x-2e2x+2x
x->0
(II) lim (n+4/n+1)n+3.
n->8
© Evalute dy/dx if,
___ __
(i) y=3v3x2-1/4v4x3
_ _
(ii) y=vx-1/vx+1
_____
(iii) y=log(x+vx2-a2).
OR

2. (a) (i) Show that lim f(x)=f(a) where f(x)=a0+a1x+a2x2+…..+anxn. 3
x->a
(ii) Show that lim |x-a|/x-a does not exists.
x->a
(b) obtain :
(i) lim 1/x 1-x
x->1
(ii) lim x27+1/x17+1.
x->-1
(c) Diffrentiate the followings: 6
(i) ex+_ey=ex+y
(ii) xvx
(iii)x2/3+y2/3=d2/3.
3. (a)Consider the fnction f(x)=x3/3-5x2/2+6x+12 for x?5 with f(5)=0. 3
(i) sketch the graph of the function.
(ii) obtain the local maxima or minima ,if any.justify your ans.
(b) obtain
(i)? x3+5x2-4/x2 dx.
____
(ii)? x2/4vx3+2 dx.
(iii) ? dx/x(1+x3)2
OR
3.(a) obtain all local and global maxima and minima for the subsequent function. 3
(i) f(x)=3x+5 for 0=x=7.
(ii) f(x)=x3/3-2x2-6x+5 for -10=x=0.
(b) Find, _ _
(i) ?(vx+1/vx)2dx
____
(ii) ?3xv1-2x2 dx.
(iii) ? dx/ex+1.
4. (a) (i) Derive the formula to obtain the area of a triangle ABC whose vertices are a(x1,y1) B(x2,y2) and C(x3,y3). 3
(ii) obtain the co-ordinates of the circum center of a triangle whose vertices are
(-2,3),(2,1) and (1,2).

(b) (i) A(00) B(4,2) C(3,-3) and D(K,-2) are provided points.Find K if
<--> <--> <--> <-->
AB||CD and AB|CD. 2
(ii) if P(4,5) divide the line segment joining A(1,2) and B in the ratio 3:2 from
A Fin d the co-ordinate of B.
(c) obtain the formula of a line passing through (2,5) and makes an angle of 45 with the line
x- 3y+6=0 3
OR
4.(a) (i) find the formula of the line of the form x/a+y/b=1 where ab?0. 2
(ii) obtain the formula of line passing through the poits A(3,2) and B(5,2).Also obtain its slope and intercepts. 3
(b) (i)if the area of the triangle with vertices (k,3),(-1,4) and (-5,-2) is 11 units.find k. 2
(ii) if(-3,4) is the centroid of a triangle whose vertices are (6,2),(a,3) and (0,b).find a and b. 2
(c) obtain the formula of the line passes through the point of intersection of lines 2x+y-6=0 and x-y-3=0 and making equal intercepts on the axes. 3
5(a) describe differential formula.Determine the degree and the order of the differential formula. 2
__________
3v d2y/dx2 +3y=dy/dx.
(b) Solve: 4
(i) (2x+7)dy+y dx=0.
(ii) dy/dx+2y=e-x.
(c) (i) obtain the quantity of sphere of radius r using the definite integral. 3
(ii) Draw the graph of the linear equalities. 3
y=0;
x=0;
x=0;
x+y= 1;
x+2y=10.
OR
5. (a) discuss the method of the solving a linear formula dy/dx+Py=Q. 2
where P and Q are function of x only.
(b) Solve: 4
(i) x(dy/dx)=y(logy-logx+1).
(ii) dy/dx+xy=xy3.
© (i) Determine the cst of producing 200 pens if the marginal cost 3
(in rupees per unit) is
MC=x/200+3.50.
(ii) Draw the graph of the linear equalities.x=0,y=0,x+3y=9;x+y=2;x=5. 3
6 (a) (i) Solve the formula by cramer’s rule. 3
X+6y=2xy.
3x+2y=2xy.
(ii) Prove that: 2
1 a bc
1 b ca =(a-b)(b-c)(c-a)
1 c ab
(b)Define every with example. 2
(i) Row matrix.
(ii)Skew symmetric matrix.

(c) if one two one 3
A= 0 one -1
three -1 one then prove that A3-3A2-A+9I=0.
(d) obtain the maximum and minimum value of f(x)=x+4/x. 3
OR
6. (a) (i) Solve the formula by cramer’s rule. 3
2/x+7/y=8
3/x-1/2y=1
(ii) Prove that: 2
a+x a-x a-x
a-x a+x a-x = 4x2(3a-x).
a-x a-x a+x _
(b) if A=[1 two three ] and four then obtain AB and BBT. 2
b= five
six

© Solve using matrix method. 3
X+y+z=3
2x-y-z=3
x-y+z=9
(d) The total cost of producing x items in a factory is provided by C(x)=x3/10-3x2+50x at which out-put x the avg. cost is minimum? At this level show that the marginal cost is equal to avg. cost. 3

Earning:   Approval pending.