7. Give a short discussion, with illustrations, on the solution of the homogeneous equation and its connection with the linear equation with constant coefficients. 8. (a) Obtain a test for determining whether a given linear equation is an exact differential equation. (b) Find values of m that will make r" an integrating factor for 9. (a) Show that if u=c, and v=c, are solutions of the simultaneous set dx/P=dy/Q=dz/R, then u=c1, v=c2, and (u, v)=0 are solutions of дх де (1-2) +(2-x) +(x-y) =0. dy FINAL HONOUR MATHEMATICS. TRIGONOMETRY II. Seven questions to be attempted. 1. Expand cos nx/cos" r in ascending powers of tan r. Show that tan2 7/16+tan2 3π/16 +tan2 5/16+tan2 7π/16=28. 2. If x is real, and n tends to ∞, show that (i) cos" r/n tends to 1, (ii) "C, cos"-r x/n. sin' x'n tends to "/r. 3. If e, sin r, and cos r are defined as the sums of powerseries, show that (ii) sin(x+y)=sin x cos y+cos r sin y, (iii) sin is periodic. 4. Define the logarithm of a complex number, and write it in the form A+iB. Write the principal logarithms of i, and sinh (a+iẞ) in this form. 5. Prove that tan1r=x/1-x3/3+. Calculate to four decimal places. 6. Prove that sin yr sin 2y+. ❘r being<1. r sin y/(1-2x cos y+x2), and sin r+1/3 sin 3x+1/5 sin 5x+.../4. For what values of r are these results valid? 7. Prove that x/(e-1)+x/2 can be expanded in even powers of r; and find the first four coefficients. Expand cot by means of the coefficients in the above expansion. 8. Show how to express sin x as an infinite product of linear factors. Deduce a series for 1. FINAL HONOUR MATHEMATICS Quaternions. 1. (a) Find a necessary and sufficient condition that three vectors should (i) be complanar, (ii) be coinitial and terminate collinearly. (b) By vector methods prove that the mid points of the diagonals of the complete quadrilateral are collinear. 2. Obtain the product aß in the standard form Ta TB(-cos 0+ sin 0), by (i) expressing a and ẞ in terms of the unit vectors i, j, k, (ii) any other method. 3. Prove the following formulae for quaternions: (i) Sqq'=Sq'q, (ii) Sqq'q"=Sq′q"q, (iii) qKq=(Tq)2, (iv) Kqq'=Kq'. Kq, (v) T(qq')=Tq.Tq', 4. (a) Prove Vaẞy Vyẞa. (b) If a, ẞ, y are the vectors along the sides of a triangle, then Vaẞy is a vector along the tangent to the circumcircle at the vertex (y, a). 5. Find the inverse, with respect to the circle p2=-a2, of (i) the point y, (ii) the line p=ẞ+ta, (iii) the circle. p2-2Spy=t2. 6. (a) Find the equation of the ellipse in the form Spop-1, and show that the equation remains the same in form for the hyperbola. (b) State and prove the distributive and commutative properties of pp. 7. Find the equation of the normal to the parabola in the form o=(x+2at)i+y(1—t)j. Thence find the vector from the origin to the centre of curvature. |