North Gujarat University question paper

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[Raman Vij]

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North Gujarat University B.Sc Computer Science Advance Mathematics question paper

1 (a) Write the De-Morgan’s lows and prove 3
any 1 of them .
(b) If A = {1,2,3}, B = {2,4} and C = {3,4} 4
then prove that
A × ( B ? C) = (A × B) ? (A × C).
(c) (i) If f : Z ->Z , x ? Z , f(x) = 4x + three 3
Is it one-one ? Is it onto? Is it even ?
(ii) If f : R -> R ,f(x) =5x + four and g : R->R, 4
g(x) = 4x + k and fog=gof then obtain k
and fog (-2).

OR
one (a) describe the subsequent terms : 3
(i) Intersection of 2 sets
(ii) Disjoint sets
(iii) Cartesian product of 2 sets

(b) If n(U)=100, n(A)=60, n(B)=50 and 4
n(A U B) = 90, then obtain n(A’ U B’),
n(A’ n B’) and n(A n B’).
(c) (1) If f : R -> R , x ? R, f(x) = 2x +3 then 3
prove that f has it’s inverse function and
find it.
(2) A company sells its product for Rs.5 4
per unit. Fixed cost for the company are
Rs.3500 and variable costs are estimates
to run 30% of the total revenue .
Determine :
(i) The total revenue functions
(ii) The total cost function
(iii) The break-even point.
two (a) (1) describe limit of a function . 2
_________
(2) Evaluate : lim v n2 + n + one – n 2
n->8

(b) explain the continuity of a function 2

F(x) = { 2/5-x : x < three , five – x ; x = three
at point x = three

(c) Find dy /dx 6
_______
(1) y = log(x + v x two - a two )

(2) y = (sin x)x

(2) y = e 3x – x sin x + x two /4
(d) obtain the maximum value of f(x) = x + 4/x. 2

OR

2 (a) Evaluate : 4

(1) lim x(ex - one )
x->0 -----------
1-cos x

(2) lim (1+ x)5 – one
x->0 --------------
x
(b) explain the continuity of a function : 2

F(x) = { |x| / x ; x ? 0 , one ; x = 0
at point x=0.

C(x) = x3 /3=3 x2-7z+16, where x is the
Output. obtain the output at which the total
Cost is minimum.

3 (a) (1) If every element of a row is multiplied 3
with K and added to the corresponding
element of a different row then prove that
the value of determinant is not changed.

(2)
one a a2
Prove that one b b2 =(a-b) (b-c) (c-a). 3
one c c2

one two two
(b) (1) If A = two one two then prove that
one two one

a2 -4A-5I=0

seven four
(2) If A = five three then prove that A + AT + A-1 3

(c) Solving usimg matrix method. 3
2x – 3y + 5= 0
3x +y -9 = 0

OR
3 (a) (1) Solving the formula by Cramer’s rule 3
x + 6y = 2xy
3x + 2y = 2xy

X p 3x – 5p 3
(2) Prove that y q 3y – 5q =0
Z r 3z – 5r

0 four three
(b) (1) IF A = one -1 -3 Prove that 2

-1 four four

A2 = I

(2) IF A = one two and B = four 0
three four two one

Prove that (AB)T = B2 AT
(c) describe every with example : 3
(1) Row matrix
(2) Null matrix
(3) Skew-symetrix metrix.

4 (a) obtain : 6
__ __
(1) ? (v x + 1/ v x )2
(2) x log x dx
two
(3) ? 2x + five / x2 +5x +3 dx
1

(b) Find tht area of region bounded by 3
Y = x2 and the line y=x +2
(c) describe differential formula . Determine the 3
degree and the order of the differential
___________
formula v d2 y/d x2 +2y=dy/dx
(d) Solve tany 1 : 2
(1) dy/dx = ex-y + x2 xey
(2) dy/dx +y = ex

OR
4 (a) obtain : 6
(1) ? x ex dx
(2) ? ex (sin x+ cos x) dx

two
(3) ? logx / x dx
one

(b) obtain the quantity of sphere of radius r 3
using the definite integral
(c) discuss the method of solving a linear three
formula dy/dx+Py = Q where P and Q are
the function of x only.

(d) Solve any 1 : 2
(1) (x+5) dy + y dx=0
(2) ( x2 + y2 ) dy/dx xy .

5 (a) (1) Derive the formula to obtain the area of 3
A triangle ?ABC whose vertices are
A(x1,y1), B(x2,y2) and C= (x3,y3)
(2) A( 0 , 0 ) ,B( four , two ), C( three , -3 ) and 3
D( k , -2 ) are provided points. obtain k if
<-----> <-----> <-----> <----->
AB || CD and AB | CD
(b) (1) Prove that points ( four , three ) ,( seven , one ) and two
( 9, three ) are the vartices of an isoseceles
triangle.
(2) If P( four , k ) divides the line segement 2
joining A( 2, 3) and B(5, -1). obtain the
ratio of division from A and value of k .

(c) obtain the formula of a line passing through 4
The point ( two , five ) and marking an angle of
45o with line x – 3y + two = 0

OR

5 (a) (1) find the formula of line of the 3
From x/a + y/b = one where ab ? 0
(2) If the area of the triangle with 3
Varitices (k ,3) ,(4,5) and (3,1) is
six unites ,find k.
(b) (1) Show that the 3 points 2
A(1,4), B(3,-2) and C(-3,16) are
Collinear.
(2) If (-3,4) is the centroid of the triangle 2
whose certices are (6,2),(x,3) and
(0,y) then obtain x,y.
(c) obtain the formula of line parallel to 4
3x + y + five = 0 and pass through the
point of intersection of lines
x + y – three =0 and x – y -5=0.