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Engineering Mathematics -M-II,Satyabama university

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- 1. Consolidated.M2 1.Define Symmetric function of roots. A function of the roots of an equation, which remains unaltered when any two of the roots are interchanged is called Symmetric function of the roots. If a,b,c are the roots of an equation of degree 3, then we use the following notation. ∑a = a + b + c + d ∑ 1/a =1/a + 1/b + 1/c + 1/d 2.Define Descarte’s rule of signs. Descartes’s rule of signs, in algebra, rule for determining the maximum number of positive real number solutions (roots) of a polynomial equation in one variable based on the number of times that the signs of its real number coefficients change when the terms are arranged in the canonical order (from highest power to lowest power). 3.If one of the roots of the equation x3 -6x2 +13x-10=0 is 2+i, find the remaining roots.(2) 4.Show that the equation x5 -6x2 -4x+5=0 has atleast two imaginary roots.(2) 5.Show that x7 – 3x4 + 2x3 – 1 = 0 has atleat four imaginary roots. 6.Solve the equation x3-19x2 +114x-216=0. 7.Find the condition that the roots of the equation.(2) 8. X3 + px2 + qx + r = 0 may be in geometrical progression.(2) 9.Find the condition that the roots of the equation x3 +px2 +qx+r = 0 may be in Arithmetical progression.(2) 10.Find the condition that the cubic x3 – lx2 + mx – n = 0 should have its roots in semetric progression. 11.Solve 24x3 – 26x2 + 9x – 1 = 0 if the roots are in Harmonic progression 12.If α, β and γ are the roots of x3 – 3ax + b = 0, show that Σ (α-β) (α-γ) = 9a. 13.If α, β, γ are the roots of the equation 3x3 + 6x2 – 9x + 2 = 0 find the value of .∑β α 14.If α,β,γ are the roots of x3 – 2x2 + 5 = 0, find the value of . 111 γβα ++ 15..If α, β and γ are the roots of x3 – 3ax + b = 0, show that Σ (α-β) (α-γ) = 9a. 16.If α,β,γ are the roots of x3 – 14x + 8 = 0, find ∑α2 and ∑α3 .
- 2. 18.If α, β, γ are the roots of the equation 3x3 + 6x2 – 9x + 2 = 0 find the value of .∑β α 19.If α, β, γ are the roots of x3 – 14x +8 = 0, find Σα2 . 20.If α, β, γ are the roots of the equation x3 + px + q = 0 form the equation whose roots are (α+β) (γ+β), (β+γ)(α+γ), (γ+α)(α+β). 21.Diminish the roots of the equation x3 + x2 + x – 100 by 4. 22.Diminish by 2 the roots of the equation x4 +x3 -3x2 +2x-4=0 23.Diminish the roots of x4 – 5x3 + 7x2 – 4x + 5 = 0 by 2 and find the transformed equation 24.Diminish by 3 the roots of x4 + 3x3 – 2x2 – 4x – 3 = 0. 25,State the relation between the coefficients and roots of the equation 0 0 =∑ = R n R R xa (where aR real and x complex) 26.Show that x7 – 3x4 + 2x3 – 1 = 0 has atleat four imaginary roots. 27.Solve the equation x3 + 6x + 20 = 0, are root being 1 + 3i. 28. Solve: p2 – 9p + 18 = 0 .
- 3. 1. (a) Solve x4 + 4x3 – 5x2 – 8x + 6 = 0 given that the sum of two of its roots is zero. (b) By suitable transformation, reduce the equation x4 – 8x3 + 19x2 – 12x + 2 = 0 to one in which the term x3 is absent and hence solve the original equation. (or) 12. (a) Solve x4 + 2x3 – 21x2 – 22x + 40 = 0 whose roots are in Arithmetic progression. (b) Solve x6 + 3x5 – 3x4 + 3x2 – 3x – 1 = 0 (a) Solve 012222 2456 =−−−++ xxxxx . (8) (b) Find k, if the equation 01292 23 =++− kxxx has a double root. (4) (or) 12. (a) If α, β, γ are the roots of 0332 23 =+++ xxx , find ∑ + 2 2 )1(α α . (b) Diminish the roots of 012883 23 =+++ xxx by 4.
- 4. (a) Find the condition that the roots of the equation x3 + px2 + qx + r = 0 may be in A-P. (b)Solve 4x4 – 20x3 + 33x2 – 20x + 4 = 0. Find the equation whose root are the reciprocal of the roots of x4 – 7x3 + 8x2 + 9x – 6 = 0. (or) 12. Transform the equation 2x3 – 9x2 + 13x – 6 = 0 into one in which the second term is missing and hence solve the given equation. Transform the equation x3 – 6x2 + 5x + 8 = 0 into another in which the second terms in missing .Hence find the equation of its squared differences. (or) 12. If α,β,γ be the roots of x3 + px + q = 0, show that (a) α5 +β5 +γ5 = 5αβγ(βγ,+ γα + αβ) (b) 3 ∈α2 ∈α5 = ∈α3 ∈α4 . 11.Solve x5 – 5x4 + 9x3 – 9x2 + 5x – 1 = 0. (or) 12.Diminish by ‘3’ the roots of x4 + 3x3 – 2x2 – 4x – 3 = 0. 13.Solve x5 – 5x4 + 9x3 – 9x2 + 5x – 1 = 0.(or) 14.Diminish by ‘3’ the roots of x4 + 3x3 – 2x2 – 4x – 3 = 0. 2. 3. If α, β, γ are the roots of x3 – 6x2 + 11x – 6 = 0 form the equation whose roots are α + 1, β + 1, γ + 1.(or) 4. Solve the equation 6x5 + 11x4 – 33x3 – 33x2 + 11x + 6 = 0.
- 5. (a) Solve 6x3 – 11x2 + 6x – 1 = 0 given the roots are in harmonic progression. (b) Transform the equation x4 – 8x3 – x2 + 68x + 60 = 0 into one which does not contain the term x3 . Hence, solve it. (or) (a) If α, β and γ are the roots of x3 – 14x + 8 = 0, find the value of Σα2 and Σα3 . (b) Solve 8x5 – 22x4 – 55x3 + 55x2 + 22x – 8 = 0 5. Solve the equation x3 -4x2 -3x+18=0 given that two roots are equal (or) 6. Solve = 6x5 + 11x4 – 33x3 – 33x2 +11x + 6 = 0. 11. If α,β,γ are the roots of the equation x3 -6x2 +11x-6=0 form the equation whose roots are α+1,β+1,γ+1. Or 12. Solve the equation x3 -9x2 +26x-24 = 0 given that one root is twice the other 1.Find the equation whose root are the reciprocal of the roots of x4 – 7x3 + 8x2 + 9x – 6 = 0. (or) 2. Transform the equation 2x3 – 9x2 + 13x – 6 = 0 into one in which the second term is missing and hence solve the given equation. (a) Find the equation whose roots exceeds by 2 the roots of the equation 4 3 2 4 32 83 76 21 0x x x x+ + + + = and hence solve the equation. (b) If , ,α β γ are the roots of equation 3 3 2 0x x+ + = form an equation whose roots are 2 2 2 ( ) ,( ) ,( )α β β γ γ α− − − . (or) 12. (a) Solve the equation 6 5 4 2 6 25 31 31 25 6 0x x x x x− + − + − = .
- 6. (b) Solve by Cardonn’s method 3 2 27 54 198 73 0x x x+ + − = . 11. (a) Prove that the radius of curvature at any point of the cycloid X = a (θ + sin θ), y = a(1 - cosθ) is 4 a cos θ/2. (b) Find the envelope of the family of lines = x/a + y/b = 1 where a2 + b2 = c2 and c is a constant. 11. (a) Solve x4 + 4x3 – 5x2 – 8x + 6 = 0 given that the sum of two of its roots is zero. (b) By suitable transformation, reduce the equation x4 – 8x3 + 19x2 – 12x + 2 = 0 to one in which the term x3 is absent and hence solve the original equation (a) Find the equation of the sphere that passes through the two points (0, 3, 0) (-2, -1, -4) and cuts orthogonally the two spheres x2 + y2 + z2 + x – 3z – 2 = 0, 2(x2 + y2 + z2 ) + x + 3y + 4 = 0. (b) Show that the shortest distance between the lines x + a = 2y= –12z and x = y + 2a = 6z – 6a is 2a. (a) Solve 012222 2456 =−−−++ xxxxx . (8) (b) Find k, if the equation 01292 23 =++− kxxx has a double root. (4) Find the equation of the circle of curvature of the curve . 4 , 4 =+ aa atayx (or) 12. Examine f(x, y) = x3 + 3xy2 – 15x2 – 15y2 + 72x for extreme values 11. (a) Find the condition that the roots of the equation x3 + px2 + qx + r = 0 may be in A-P. (b)Solve 4x4 – 20x3 + 33x2 – 20x + 4 = 0.
- 7. Find the equation whose root are the reciprocal of the roots of x4 – 7x3 + 8x2 + 9x – 6 = 0. (or) 12. Transform the equation 2x3 – 9x2 + 13x – 6 = 0 into one in which the second term is missing and hence solve the given equation. 13. Find the equation of circle of curvature of the parabola y2 = 12x at the point (3, 6). Transform the equation x3 – 6x2 + 5x + 8 = 0 into another in which the second terms in missing .Hence find the equation of its squared differences. (or) 13. Prove that the radius of curvature at any point of the asteroid 3 2 3 2 3 2 α=+ yx in three times the length of the perpendicular from the origin to the tangent at that point. Solve x5 – 5x4 + 9x3 – 9x2 + 5x – 1 = 0. (or) 12. Diminish by ‘3’ the roots of x4 + 3x3 – 2x2 – 4x – 3 = 0. 13. Find the centre and circle of curvature of the curve . 4 , 4 =+ aa atayx What is the radius of curvature at (3,4) on x2 +y2 = 25? 1. Find the radius of Curvature for the Curve xy = c2 at the point (c,c). 2. Find the radius of curvature of the circle of x2 + y2 = a2 . 2. Find the perpendicular distance of P(1, 2, 3) form the line 2 7 2 7 3 6 − − = − = − zyx 3. Find the curvature of the circle x2 + y2 = 49.
- 8. 4. Find the radius of curvature of the curve y = ex at x = 0. 5. Find the radius of curvature y2 = 2x(3 – x2 ) at the points where the tangents are parallel to x axis. 3. Prove that the points A (3,2,4), B (4,5,2), C(5,8,0) are collinear. 6.Prove that the planes 5x – 3y + 4z = 1, 8y + 3y + 5z = 4 and 18x – 3y + 13z = 6 contain a common line 7.Find the radius of curvature at (1, 0) on x = et Cost, y = et Sin t Define radius of curvature in Cartesion coordinates and its polar coordinates. 8.What is the radius of curvature at (3,4) on x2 +y2 = 25? 1. What is the radius of curvature at (3,4) on x2 +y2 = 25? 2. Find the radius of curvature of the curve xy = c2 at (c, c).

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