Determinants, Theory of Equations, &c. 1. If a, b, c are the sides of a triangle show that sixteen times the square of the area is given by the symmetric determinant 2. If A1, B1, C1, &c. be first minors of (±a1b,¤3 ....) show that (A1 B2 Cs..) = {(±a12¢ ̧ . . } -1 where n is the order of the determinant. 1 3 4. Write the elementary relations which hold between the roots of an equation and the coefficients. If a+ẞi is a root of x3 +qx+r=0 show that 2a is a root of a3 +qx − r = (). 5. If a be one of the cube roots of unity, prove that 1+w+w2 = 0. 3 3 If (x3 +y3 + z3 - 3xyz) (x13+y13 +213-3x1121) = X3 + Y3 +Z3 − 3X YZ, find X, Y, Z in terms of x, x1, y, y1, 2, 2. 6. Find the symmetric function where aẞy are the roots of x3 +P1x2 +P2x+P;=0. 7. If the cubic 3 +3а1x2 +3α2x+α ̧=0 be put in the form 23+3Hz+G=0 determine the roots of the latter equation in terms of those of the former, and express Hand G in terms of these. 8. Write the equation which has the function G2+4H for its independent term, and explain the use of this function in the theory of the cubic. 9. If L = a+wẞ+w27 and M=a+wa+wy show that L3+ M3 - 27G and L3 - M3 (a — ß)(B − r)(r − a)( − 3 √ −3.) = 10. Explain what is meant by the reducing cubic of a biquadratic and show how it is found. Solid Geometry. 1. Prove the relation 1 = 12 + m2 + n2 +2mn cos λ+2nl cos μ+2lm cos ›. 2. Find the condition that two lines given by symmetric equation, may intersect. 3. Find the plane which passes through a, b, c and is perpendicular to the planes Ax+By+Cz=0 and A'x+By+Cz = 0. 4. Deduce the equation of the ellipsoid, origin at the centre; and find the length of a radius vector drawn in the direction l, m, n. Show that the sum of the squares of the reciprocals of any three radii at right angles is constant. 5. A line parallel to the plane yz intersects the line z=c, y=0 and the curve 2+ y2=a2, 2=0; to find the equation to the surface generated. What are the sections when the surface is cut by (i) the plane z=b? (ii) the plane x=c? 6. Show that the rectilinear generators of the Hyperboloid of one sheet are given by the equations la тв 12 m3 na + a2 62 c2 = 0. 7. Write the three invariants of the central quadric sur face and give their relations to the discriminating cubic. What is the surface denoted by x2 + y2+z2 + 4xy − 2xz +4yz = 1. 3. Show that the equation to the tangent cone from xyz is F(x, y, z). F(x', y', z′)= {F′(x, y, z)}3 where " denotes the half-accentuation of the variables. Find the tangent cone to By2 + Cz3 =x. 9. Two confocal conicoids cut one another at right angles at all their common points. 10. State and prove Meunier's theorem. |