INTERMEDIATE HONOUR MATHEMATICS. Determinants and Theory of Equations. 1. Define a determinant; and prove from the definition that a1+ lb1+mc1, b1, c1 C3 a2 + lb 2+mc2, b2, C2 sin B sin Y in its simplest form. = a1 1 a1 b1 c1 3 a 3 2. Use determinants to find the eliminant of s, representing a+b+c". 3. Prove the rule for expressing the product of two third order determinants as a third order determinant. Show that $4 $3 $2 S3 $2 $1 is a perfect square, S2 $1 So 4. Prove that a skew symmetric determinant of even order is a perfect square; and that its reciprocal is also skew symmetric. 5. Prove Descartes' rule of signs. What information does it give as to the equations x"=1 and x" +ax"−3 + bx2-4 + . . . . =o? 6. Show how to find the repeated roots of multiplicity in an algebraic equation. Find the values of k for which x3 +3x2+6x+k=0 has a repeated root. 7. Find intervals in which real roots of 3x3 -3x+1=0 lie separately; and find a real root to three decimal places. 8. Show that the roots of x1+6Hx2+4Gx+K=0 are given by the formula u+v+w, where u2, v2, w2 are the roots of a certain cubic. Discuss the ambiguous signs. Solve the equation x4 - 20x2-8/30x-24=0. INTERMEDIATE HONOUR MATHEMATICS SPHERICAL TRIGONOMETRY AND ASTRONOMY. (Any seven questions). 1. (a) Show how to find the remaining parts in the following triangles, given (1) A, b, c; (2) A, B, c; (3) A, B, C. 2. (a) Deduce Napier's Analogies from Delambre's formulae. (b) Given A=44° 13′, B=57° 27', c=24° 18', find a and b, using Napier's Analogies and logarithms through out. 3. Given a 13h 45m, 8=32° 24', lat. 44° 13' N., sidereal time 15h 22m, find the azimuth and altitude of the star. 4. Show how to survey a parallel of latitude. Find the deflection at every 10 miles in surveying the 45th parallel. 5. Explain: aberration, parallax of a star, horizontal parallax of the moon, geocentric latitude, tropical year, loxodrome, libration of moon. 6. (a) How is standard time determined? Explain fully. (b) Explain how the equation of time depends upon the eccentricity of the earth's orbit and also upon the obliquity of the ecliptic. 7. B Cassiopeiae, 8=58° 40', is observed to cross the outside and middle wires at the times 0 3m 51.78 and 0h 4m 30.28 respectively. If ŋ Ceti, d=—10° 38′ crosses the same outside wire at 1h 4m 12.6a, find the equivalent time of crossing the middle wire. η 8. (a) Explain the phases of (1) the moon, (2) Venus, (3) Mars. (b) What is meant by the synodic and sidereal periods of a planet? Show how the two are connected for (1) an inferior planet, (2) a superior planet. 9. (a) Show how to find the distance of (1) an inferior planet, (2) a superior planet in terms of the distance of the earth from the sun. (b) Find the sidereal time at Kingston 5 5m 55 W., on April 14th at 8 p.m. (standard time), given G=R.A. of mean sun at Greenwich mean noon, 1h 28m 12.48, μ=gain of sidereal time on mean time in one mean hour, 98.8565. INTERMEDIATE HONOUR MATHEMATICS. -Modern Synthetic Geometry. 1. O is the incentre, R the circumradius, r the inradius of the triangle. Prove (a) aAO2+bBO2 +cCO2=abc, and (b) AO.BO.CO=4Rr2. 2. For any three circles, prove (a) the six centres of similitude are the vertices of a complete quadrangle, (b) the three circles of similitude are coaxal, (c) the three circles of antisimilitude are coaxal. 3. Prove that the ratio of the rectangle on the diameters of two circles to the square on a common tangent is unchanged by inversion. Indicate how Ptolemy's theorem on the concyclic quadrangle may be deduced from this. 4. Prove Pascal's theorem for a concyclic hexagram; and deduce Brianchon's theorem by polar reciprocation. 5. An isosceles triangle revolves about a complanar line through its vertex. Find the volume of the figure generated; and deduce an expression for the volume of a sector of a sphere. 6. Use Pappus' theorems to find the following for a thin semi-circular plate: (a) its centre of gravity, (b) the centre of figure of its perimeter, (c) the volume and surface area of the solid generated when it revolves about a tangent which makes 60° with the diameter. |