2001 Hemchandracharya North Gujarat University M Tech Automobile 102 : Advance Mathematics Question paper
B-12702 Seat No.---------
First Year Master of Science Examination
May/June –2001
Advance Mathematics: Paper-102
(Computer Application & informationTechnology)
Time: 3 Hours] [Total Marks: 70
Instructions: (I) All questions are compulsory
(II) Figures to the right indicates marks of the corresponding questions
1 (a) Let S=Set of all people, 3
B=Set of all IT men,
C= Set of all people who smoke cigarettes,
D= Set of all pipe smokers,
E= Set of all wine drinkers.
Express the following statements in the set notations.
(I) Some IT men pipe smokers are wine drinkers.
(ii)An IT man is neither a wine drinker nor a pipe smoker.
(b) State and prove the distributive laws. 3
(C) (1) Find the largest interval or IR on which the formula f (x)=x2 defines a one-one
Function.
(2) If f (x)=log a x and f (x)=ax then show that F (f (x)) = f (F(x)) for all suitable x.
(3) Explain the following terms with suitable illustrations (A) convex function (B)
Concave function.
OR
1 (a) Examine whether the following sets are finite or infinite: 2
(I) The set of points on the circumference of a circle of radius 2 c.m
(II) The set of all lines parallel to a given line.
(III) The set of all real numbers between zero and one.
(IV) The set of all citizens of India.
(b) (I) which of these sets are equal: 2
{n, g, u}, {n, g, n,u},{n,g,u,g},{n,u,g,u}
justify your answer.
(II) Let M={ ?, { ? }, { ?, { ? }}} Then , explain with justification, which of the statements are correct or incorrect.
(A) {?} ?M (B) {{ ? }} ? M
(C) { ?,{ ? }} ? M
(c) (I) Determine whether f (x)=2x3,0=x (II) patan Soap company is interested in determining the break-even production to their new janta toilet soap. It would cost rs.2, 00,000 for advertising, promotion and other connection with the introducing of the new product. The variable cost rs.6.50 and selling price is rs.10.00. Determine the break-even production.
_
2 (a) (I) Show that 3v2 is an irrational number. 3
(II) Show that the function defined as follows is discontinuous at x=0.
F (x)=e1/x-1/ e1/x+1 for x?0 and f (0)=0
(b) Find:
(I) lim e4x-2e2x+2x
x->0
(II) lim (n+4/n+1)n+3.
n->8
© Evalute dy/dx if,
___ __
(i) y=3v3x2-1/4v4x3
_ _
(ii) y=vx-1/vx+1
_____
(iii) y=log(x+vx2-a2).
OR
2. (a) (i) Show that lim f(x)=f(a) where f(x)=a0+a1x+a2x2+…..+anxn. 3
x->a
(ii) Show that lim |x-a|/x-a does not exists.
x->a
(b) Find :
(i) lim 1/x 1-x
x->1
(ii) lim x27+1/x17+1.
x->-1
(c) Diffrentiate the followings: 6
(i) ex+_ey=ex+y
(ii) xvx
(iii)x2/3+y2/3=d2/3.
3. (a)Consider the fnction f(x)=x3/3-5x2/2+6x+12 for x?5 with f(5)=0. 3
(i) sketch the graph of the function.
(ii) find the local maxima or minima ,if any.justify your answer.
(b) find
(i)? x3+5x2-4/x2 dx.
____
(ii)? x2/4vx3+2 dx.
(iii) ? dx/x(1+x3)2
OR
3.(a) Find all local and global maxima and minima for the following function. 3
(i) f(x)=3x+5 for 0=x=7.
(ii) f(x)=x3/3-2x2-6x+5 for -10=x=0.
(b) Find, _ _
(i) ?(vx+1/vx)2dx
____
(ii) ?3xv1-2x2 dx.
(iii) ? dx/ex+1.
4. (a) (i) Derive the formula to find the area of a triangle ABC whose vertices are a(x1,y1) B(x2,y2) and C(x3,y3). 3
(ii) Find the co-ordinates of the circum center of a triangle whose vertices are
(-2,3),(2,1) and (1,2).
(b) (i) A(00) B(4,2) C(3,-3) and D(K,-2) are given points.Find K if
<--> <--> <--> <-->
AB||CD and AB|CD. 2
(ii) if P(4,5) divide the line segment joining A(1,2) and B in the ratio 3:2 from
A Fin d the co-ordinate of B.
(c) Find the equation of a line passing through (2,5) and makes an angle of 45 with the line
x- 3y+6=0 3
OR
4.(a) (i) Obtain the equation of the line of the form x/a+y/b=1 where ab?0. 2
(ii) Find the equation of line passing through the poits A(3,2) and B(5,2).Also find its slope and intercepts. 3
(b) (i)if the area of the triangle with vertices (k,3),(-1,4) and (-5,-2) is 11 units.find k. 2
(ii) if(-3,4) is the centroid of a triangle whose vertices are (6,2),(a,3) and (0,b).find a and b. 2
(c) find the equation of the line passes through the point of intersection of lines 2x+y-6=0 and x-y-3=0 and making equal intercepts on the axes. 3
5(a) Define differential equation.Determine the degree and the order of the differential equation. 2
__________
3v d2y/dx2 +3y=dy/dx.
(b) Solve: 4
(i) (2x+7)dy+y dx=0.
(ii) dy/dx+2y=e-x.
(c) (i) Find the volume of sphere of radius r using the definite integral. 3
(ii) Draw the graph of the linear equalities. 3
y=0;
x=0;
x=0;
x+y= 1;
x+2y=10.
OR
5. (a) Explain the method of the solving a linear equation dy/dx+Py=Q. 2
where P and Q are function of x only.
(b) Solve: 4
(i) x(dy/dx)=y(logy-logx+1).
(ii) dy/dx+xy=xy3.
© (i) Determine the cst of producing 200 pens if the marginal cost 3
(in rupees per unit) is
MC=x/200+3.50.
(ii) Draw the graph of the linear equalities.x=0,y=0,x+3y=9;x+y=2;x=5. 3
6 (a) (i) Solve the equation by cramer's rule. 3
X+6y=2xy.
3x+2y=2xy.
(ii) Prove that: 2
1 a bc
1 b ca =(a-b)(b-c)(c-a)
1 c ab
(b)Define each with example. 2
(i) Row matrix.
(ii)Skew symmetric matrix.
(c) if 1 2 1 3
A= 0 1 -1
3 -1 1 then prove that A3-3A2-A+9I=0.
(d) Find the maximum and minimum value of f(x)=x+4/x. 3
OR
6. (a) (i) Solve the equation by cramer's rule. 3
2/x+7/y=8
3/x-1/2y=1
(ii) Prove that: 2
a+x a-x a-x
a-x a+x a-x = 4x2(3a-x).
a-x a-x a+x _
(b) if A=[1 2 3 ] and 4 then find AB and BBT. 2
b= 5
6
© Solve using matrix method. 3
X+y+z=3
2x-y-z=3
x-y+z=9
(d) The total cost of producing x items in a factory is given by C(x)=x3/10-3x2+50x at which out-put x the average cost is minimum? At this level show that the marginal cost is equal to average cost. 3
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