Hemchandracharya North Gujarat University 2004 B.Sc Computer Science 102 : Advance Mathematics - Question Paper

part – II

three (a) (I) describe limit of a function. 2
(II) Evaluate lim 3x – 2x/ x. 2
x->0
(b) explain the continuity of a function 2

f (x)=|x|/x, x?0
=0 x=0
at point x=0.
(c) obtain dy / dx: 6
(1) y=X² sin x + e³x.
______
(2) y=log(x+vx²-a²
(3) x=at² and y=2at
(d) The total cost function is 2
C(x)=x³/3 – 10x² + 300x, where x is the output obtain the output at which the
marginal cost is minimum.
OR

3 (a) Evaluate: 4
(1) lim x (ex-1)/1-cos x
x->0
_________
(2) lim vx²+x+1-x.
x->8
(b) explain the continuity of a function 2
f (x)={3x-2, if x=0
{x+1 if x>0
at point x=0
(c) (I) y=(cos x)x then obtain dy / dx. 2

(II) If y log x=x-y then prove that 2
dy / dx= log x / (1+log x) ².
(III) If y = e³x + sin x-1 then obtain d²y / dx².
(d) obtain the maximum and minimum values of 2
y=x³/3 + x² -15x +2.

4 (a) Evaluate: 6
(I) ? x³ +5x² -3x +4/x dx
(II) ?2x + five / x² + 5x +3 dx
(III) ? x log x dx
(b) obtain the area bounded by curve y=x² - x + 3, x-axis and the lines 3
x=1, x=3.
(c) describe various formula. Determine the degree and the order of 3
The differential formula
(d²y / dx²) ³ + 5(dy / dx)^4 +3y+4=0
(d) Solve any one: 2
(I) (x-2)dy + y dx=0
(II) dy / dx +2y=e-x.

OR
4 (a) Evaluate: 6
p/2
(I) ?sin6 x cos5 x dx
0
(II) ?ex (sin x + cos x) dx
(III) ?x ex dx
(b) The marginal cost of production is obtained to be MC=2000-40x+3x² 3
where x is the number of units produced. The fixed cost of production
is rs.18000 obtain the cost function.
(c) describe the subsequent terms with suitable example: 3
(I) Homogeneous equations
(II) General solution of a differential formula
(III) Particular solution of a differential equations
(d) Solve any one: 2
(I) x²y dx – (x³ + y³) dy=0
(II) (ey + 1) cos x dx +ey sin x dy =0




five (a) describe the subsequent terms with suitable examples: 3
(I) Parametric formula of a lien.
(II) Slope of line
(III) Intercepts of line
(b) (I) Prove that the point (4,3) , (7,1) , (9,3) are the vertices of an 2
an isosceles triangle.
(II)If the area of the triangle with vertices (2,3) , (4,5) and (k,3) is 2
five units then obtain k.
(c) obtain the co-ordinate of point which divides the point A(8,9) and B(-7,4) 3
internally in the ratio 2:3 and externally in the ratio 4:3
(d) obtain the formula of a line passing through the point (3,4) and makes 4
an angle of 45º with line 3x+y+2=0

OR


five (a) find the formula to line of the form 3
x / a + y / b = 1, where ab ? 0
(b) (i) If (4,3) is centroid of the triangle whose vertices are 2
(3,5), (m, 2) and (-2,n) then obtain m and n.
(ii) obtain the co-ordinate of circum center of a triangle whose vertices 2
are (1,2), (3,4) and (2,1).
(c) A (0,0), B (4,2), C (3,-3) and D(k,-2) are provided points. obtain 3

<--> <--> <--> <-->
k if AB || CD and AB - CD .
(d) obtain the formula of line parallel to the line 3x+2y+1=0 and 4
passes through the point of inter part of lines x+y+1=0 and x-y-3=0.

Earning:   Approval pending.