Gujarat University 2007 B.C.A Computer Application F-04 il- Advanced Mathematics (Old ) - Question Paper

FBCA-04
April-2007
Advanced Mathematics
(Old Course)

Seat No. :




Time : three Hours] [Max. Marks : 50

Instructions : (1) Figures to the right indicate full marks. (2) Scientific Calculator is not allowed.




1. (a) If A, B and C be any 3 sets, then prove that A – (B n C) = (A – B) ? (A – C).


(4)

(b) Attempt any 2 parts : (6)

(1) (i) If A = {1, 2, 3, 4} and B = {4, 5,}, obtain A?B and A ? B.
(ii) If A = {a, b, c}, B = {b, d}, C = {b, c}, then verify that A ? (B?C) =
(A ? B) ? (A ? C).

(2) If the daily cost of production for x units of a manufactured product is provided
by c(x) = 15x + 15,000. ans the subsequent :

(i) If every unit is sold for Rs. 25, determine the minimum number of units that should be produced and sold to ensure no loss.

(ii) If the selling price is reduced by Rs. 5, per unit what would be the break–even point ?

(3) If f (x) = x5 – 2x + one , prove that f(x) + f (– x) = 0.





2.
(a) Find maximum and minimum value of the function f (x) = 2x3 + 9x2 – 60x + 25.
(4)
(b) Attempt any 2 parts : (6)


(1) Evaluate :


(i) lim
x? 3


(ii) lim
n ? 8



x3 – 27
x – 3

n2 + 2n – one (n + 1) (2n + 1)




(2) Show that the function

?? two




, x < 3

f(x) = ?
??

5 – x is
5 – x , x = 3


(i) discontinuous from the left at x = 3.

(ii) Continuous from the right at x = 3. (3) Differentiate the subsequent w.r.t. x.
e2x

(i) y =

x2 + 2x + 1


(ii) y = ex [(4x –1)2]





?/2
3. (a) Write the reduction formula of (
)
0



Sinn

?/2
x dx. Hence evaluate (
)
0



sin8



x dx. (4)


(b) Evaluate the subsequent integrals (any three) : (6)




(i)

( x7/2 + x8 + 1
? dx

) x5/2



(ii) (
)

x · log x dx


1
? dx

(iii) )

(x + 1) (x – 2)





(iv)

–1
( ? 1
? ?x2 –
–2


one ?
x3? dx



4. (a) Find the formula of a straight line which makes intercepts of a and b on x–axis
and y–axis respectively. (4)
(b) Attempt any 2 parts : (6)

(i) Find the formula of lines passing through the intersection of 4x – 3y – one = 0
and 2x – 5y + three = 0 and perpendicular to 5x + 4y = 6.



(ii) In what ratio is the line joining the points A(4, 4) and B(7, 7) divided by
P(–1, –1) ?

(iii) Show that the points (4, –5), (8, 1), (14, – 3) and (10, –9) are the vertices of a square.



5. (a) Obtain the order and degree of the subsequent differential equations (any two) : (4)


d3y dy

2
?dy?

(i) (2x + 3)

+ = ? ?





(ii)




d2y dx2

dx2 dx


dy
= five dx

?dx?



?d4y?5

?d2y?3

(iii) ?

? + ?

? = 3y

?dx3?

?dx2?



(b) Solve the subsequent differential equations (any two) : (6)

dy

(i)

dx + 5y = e–x


(ii) (2x + 3y + 5) dx + (3x + 5y + 6) dy = 0


(iii) (x2 + y2) dy


= xy

Earning:   Approval pending.