2004 Hemchandracharya North Gujarat University M Tech Automobile 102 : Advance Mathematics Question paper
DC-2502
First Year M.Sc. (CA & IT) Examination
March / April – 2004
Advance Mathematics: Paper – 102
Time: 3 hours] [TotalMarks:100
SECTION-1
1 (a) Define the following terms with suitable examples: 3
(1) Union of two sets.
(2) Symmetric difference of two sets.
(3) Cartesian product of two sets.
(b) Express the following sets in a Venn diagram. 4
P1=Ac n Bc n C, P2=AnBcnCc, P3=Ac n B n Cc, P4=Ac n Bc n Cc.
(c) If f, g:, IR->IR defined as f (x)=3-v2.x and g(x)=2x+v3 then find: 4
(1) gof
(2) f¯¹.
(d) Find the break-even point to the given data: 3
Cost function C=1200+120x number article to be produced. Selling
price rs.200 per unit.
OR
1 (a) State and prove the De Morgan's law for two sets A and B. 4
(b) Let P={x?IN | 3 Q={x?IN | x is even and x<15}
R=={x?N | x+4 <15}, Universal set U=IN
find the following sets.
(1) P×(Q?R)
(2) (P?Q) ×R.
(C) Define the following terms with suitable example: 4
(1) Function
(2) Invertible function
(3) Even function
(4) Reverse function
(D) Let f: IR-.IR define as f (x)=|x-2|. 3
Explain whether f¯¹ exists or not?
2 (a) Saturation properties of a determinant. 3
(b) Show that r a a²
r b b² = r(a-b)(b-c)(c-a)
r c c²
0 A 3
A= 1 A -3
-1 A 1 then show that ?³=A hence find A¯¹
(d) Solve the system of linear equations using matrix method. 4
2x+y+3z=6
4x –2y+5z=7
3x+y-2z=2
OR
2 (a) show that 1 1 1 3
X Y Z =(x-y)(y-z)(z-x).
X² Y² Z²
(b) Solve by Crammer's rule: 15/x + 6/y =3; 16/x + 5/y =3. 3
(c) Define the following terms with suitable example: 4
(1) Matrix of order m×n.
(2) Non-singular matrix.
(3) Skew-symmetric matrix.
(4) Null matrix.
(d) If 0 1 2 1 2 3
A= 3 4 2 and B= 2 0 1 then find AB and BA(if possible). 2
1 2 0
(e) If a b
A= c d then find AT and A¯¹. 2
SECTION – II
3 (a) (I) Define limit of a function. 2
(II) Evaluate lim 3x – 2x/ x. 2
x->0
(b) Discuss the continuity of a function 2
f (x)=|x|/x, x?0
=0 x=0
at point x=0.
(c) Find 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 find 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) Discuss 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 find 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 find d²y / dx².
(d) Find 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 + 5 / x² + 5x +3 dx
(III) ? x log x dx
(b) Find the area bounded by curve y=x² - x + 3, x-axis and the lines 3
x=1, x=3.
(c) Define different equation. Determine the degree and the order of 3
The differential equation
(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 found to be MC=2000-40x+3x² 3
where x is the number of units produced. The fixed cost of production
is rs.18000 find the cost function.
(c) Define the following terms with suitable example: 3
(I) Homogeneous equations
(II) General solution of a differential equation
(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
5 (a) Define the following terms with suitable examples: 3
(I) Parametric equation 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
5 units then find k.
(c) Find 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) Find the equation of a line passing through the point (3,4) and makes 4
an angle of 45º with line 3x+y+2=0
OR
5 (a) obtain the equation 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 find m and n.
(ii) Find 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 given points. Find 3
<--> <--> <--> <-->
k if AB || CD and AB - CD .
(d) Find the equation of line parallel to the line 3x+2y+1=0 and 4
passes through the point of inter section of lines x+y+1=0 and x-y-3=0.
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