Hemchandracharya North Gujarat University 2002 B.Sc Computer Science 102 : Advance Mathematics - Question Paper

(b) Show that (-1, 3), (4,- 7),(14,-2) are the vertices of an right 3
angled triangle.
(c) obtain the area of the triangle whose vertices are (2, 4), (6, 3) and 2
(-1, -2).
(d) obtain the formula of the straight-line through the points (3, -1) 4
making an angle 450 with the line 6x +5y-1=0.

OR

3 (a) prove that A(3, -2), B(1, 2), C(-7, -2) and D(-5, -6) are the vertices of rectangle.
(b) obtain the formula of the straight line passing through the point (-3, 1) and
(1) Perpendicular to the line 5x-2y+7=0
(2) Parallel to the line 2x+3y+4=0.
(c) obtain the point of intersection of the subsequent lines :
2x+3y-5=0, 7x+11y-13=0.
(d) If the sides of a triangle on the line 3x+y+4=0, 3x+4y-15=0 and
24x-7y-3=0. then, prove that the triangle is isosceles.
4 (a) obtain the derivatives with respect to x of the subsequent functions : 6
(1) f(x)= xn + an + xn/an + an/xn + xa + ax , where a>0 is constant.
(2) g(x)=x x + e x .
(b) Solve : 6
(1) dy/dx + y/x = x5.y4
(2) (x+y+5)dx+(x-y2+2)dy=0.

OR
4 (a) obtain dy/dx : 4
___
(1) y= v1-x3
(2) y=x2. logx+4x.sinx+2.
(b) If x3 +y =xy+ex then evaluate d2y/dx . 2
(c) solve : 6
(1) dy/dx+2y=x3
(2) (x2 +y2 )dy/dx=xy.
five (a) Evaluate : 6
(1) ? x3 –x+1/x dx

_____
(2) ? dx/v2x2 -3
(3) ? dx/x-x3

(b) let the cost functionof a firm be provided by the formula 3
C=300x-10x2 +1/3x3, where C stands for cost and x for output.
Calculate :
(1) Output at which marginal cost is minimum
(2) Output at which avg. cost is minimum
(c) Find the area enclosed by the curve y=3x-x2 ,X-axis and the lines x=0 and
x=3. 2

OR
5 (a) Evaluate :
(1) ? x2.log x dx (2) ? x dx/(x2 +1) (x2 +2)
(3) ? (x3 ex +x2 ) dx
(b) obtain the maximum and minimum values of the function f(x) = x3 - 6x+9x+6.
(c) obtain the quantity of the solid generated when the region bounded by y = x2 ,
y=4x-x2 is rotated about X-axis.
6 (a) discuss the terms : (with illustration) 3
(1) Row matrix
(2) Non-singular matrix and
(3) symmetric matrix

-2 3 -1
(b) If A= 5 4 -1 then express it as a sum of the symmetric and a
1 -3 2
skew symmetric matrix.
1 1 1
(c) Prove that x y z =(x-y) (y-z) (z-x).
x2 y2 z2


1 1 3
(d) obtain the adj A, if A = 1 3 -3
-2 -4 -4

OR
7 (a) Write the properties of determinants.
0 1 2 1 -2
(b) If A = 1 2 3 and B = -1 0 then obtain (if possible)
2 4 3 2 -1

the product AB and BA.
0 4 3
(c) If A = 1 -3 -3 then prove that A2 = I.
-1 4 4
(d) Solve the system of linear equations (by using matrix):
x+2y+3z=6
2x+4y+z=7
3x+2y+9z=14



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Earning:   Approval pending.