2002 Hemchandracharya North Gujarat University M Tech Automobile 102 : Advance Mathematics Question paper
First Year m. sc. (CA & IT) Examination March/April – 2002 Advanced Mathematics : Paper  102 Time : 3 Hours] [Total Marks : 55 Instruction : (1) All question are compulsory. (2) figures to the right index the marks of the corresponding question.
1 (a) state and prove demorgan rules. 3 (b) if a={1,2,3,4,5} and b={4,5,6,7} then obtain n(A),n(B),n(AnB) and n(AnB) 3 (c)Evaluate lim v3  x 1 . 2 x?2 ¯¯ ¯¯¯¯¯¯¯¯¯¯ x2 (d) Examine of continuity of the function 2 { x ; if x !=0 { x at x=0 f(x) 0; if x=0;
(e) Solve the following system of linear Equations by elimination method of variables: 2 3x – 4y = 7 2x + 5y =10
OR 1 (a) Define (with ilusration) the following terms : (1) Cartesian product of two sets (2) Power set. (b) (By Venn diagram) prove that : A(B U C)=(A  B)n(A – C) (c)Evaluate lim tanxsinx x?0 ¯¯¯¯¯¯¯¯¯¯¯¯ x3 (d) Let a function F be defined as {x; if 0=x f(x)= {0; if x=1/2 {x1; if ½
(e) Solve the following system of linear equations by elimination method of variables : x+yz=2,2x+z=3,xy+4z=1
2 (a) The salary of an employee in In1995 was Rs.8000. In 1997 it will be 3 10500.Express salary S, as a linear function of time t and estimate his salary in 2002. (b)Determine whether the function f(x)=x3 is even or odd. Also 3 determine whether it is increasing or decreasing. (c) Trace the curve (Cubical parabola) : y=x3. 4 (d) Draw the graph of the linear inequalities : 2 2x + y = 4, x = 0, y = 0 OR 2 (a) A function f is defined as : 3 (b) For the first year the fixed cost for setting up new electronic pocket 3 calculators company is Rs.3,00,000. The variable cost for producing a calculator is Rs.70.The company expects the revenue from the sales of the calculators to be Rs.270 per calculator. Then, (1) construct the revenue function and the cost function (2) Fine the break – even point. (C) Find the point of inflexion of the curve : 4 y = 3x4 – 4x3 + 1. (d) Draw the graph of the linear inequalities : 2 x + y = 0, x = 0, y = 1. 3 (a) In what ratio does the Xaxis divide the segment joining 3 A(1,2) and B(4,5) from A and at which point ? (b) Show that (1, 3), (4, 7),(14,2) are the vertices of an right 3 angled triangle. (c) find the area of the triangle whose vertices are (2, 4), (6, 3) and 2 (1, 2). (d) find the equation of the straightline through the points (3, 1) 4 making an angle 450 with the line 6x +5y1=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) find the equation of the straight line passing through the point (3, 1) and (1) Perpendicular to the line 5x2y+7=0 (2) Parallel to the line 2x+3y+4=0. (c) Find the point of intersection of the following lines : 2x+3y5=0, 7x+11y13=0. (d) If the sides of a triangle on the line 3x+y+4=0, 3x+4y15=0 and 24x7y3=0. then, prove that the triangle is isosceles. 4 (a) Find the derivatives with respect to x of the following 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+(xy2+2)dy=0. OR 4 (a) Find dy/dx : 4 ___ (1) y= v1x3 (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. 5 (a) Evaluate : 6 (1) ? x3 –x+1/x dx _____ (2) ? dx/v2x2 3 (3) ? dx/xx3 (b) let the cost functionof a firm be given by the equation 3 C=300x10x2 +1/3x3, where C stands for cost and x for output. Calculate : (1) Output at which marginal cost is minimum (2) Output at which average cost is minimum (c) Find the area enclosed by the curve y=3xx2 ,Xaxis 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) Find the maximum and minimum values of the function f(x) = x3  6x+9x+6. (c) Find the volume of the solid generated when the region bounded by y = x2 , y=4xx2 is rotated about Xaxis. 6 (a) Explain the terms : (with illustration) 3 (1) Row matrix (2) Nonsingular 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 =(xy) (yz) (zx). x2 y2 z2
1 1 3 (d) Find 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 find (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
_____________________________________________
Return to question paper search
