2004 Hemchandracharya North Gujarat University M Tech Automobile 102 : Advance Mathematics Question paper
DC2502 First Year M.Sc. (CA & IT) Examination March / April – 2004 Advance Mathematics: Paper – 102 Time: 3 hours] [TotalMarks:100
SECTION1 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)=3v2.x and g(x)=2x+v3 then find: 4 (1) gof (2) f¯¹. (d) Find the breakeven 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)=x2. 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(ab)(bc)(ca) 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+y2z=2
OR 2 (a) show that 1 1 1 3 X Y Z =(xy)(yz)(zx). 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) Nonsingular matrix. (3) Skewsymmetric 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+vx²a² (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 (ex1)/1cos x x>0 _________ (2) lim vx²+x+1x. x>8 (b) Discuss the continuity of a function 2 f (x)={3x2, 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=xy then prove that 2 dy / dx= log x / (1+log x) ². (III) If y = e³x + sin x1 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, xaxis 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) (x2)dy + y dx=0 (II) dy / dx +2y=ex.
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=200040x+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 coordinate 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 coordinate 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 xy3=0.
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