7. A normal at point P on an ellipse meets the axes in and R. Show that PQ.PR=SP.PF, where S and F are foci. 8. The axes are at angle w, and PM and PN are perpendicular upon the axes. Find the locus of P when the area of the quadrilateral OMPN is constant. ALGEBRA. PRELIMINARY HONOURS. 1. Find a positive root of 3-3x2—x+2=0 to 4 decimal places; and state briefly how your method involves the remainder theorem. 1 2. Show that the expansion of (1+x) as x ap I x x as x ap proaches zero, is equal to that of (1+1) proaches infinity. From this expansion deduce the exponential series for aa. 3. Find a convenient series for calculating logarithms, and use it to find loge 2 to 4 places of decimals. Find the modulus of the system of logarithms to the base 8. 4. Find the fraction with two digits in numerator and denominator which most closely approximates to the value of √22—4. 5. Establish the principle of undetermined coefficients for functions of infinite dimensions. Use it to find the first 4 coefficients in the expansion of (1+x)" in terms of those of (1+x)". 6. Prove completely by induction that un-u2+"C111+.. for any difference series. 16 and 41, and one between 41 and 74, so as to leave the law of the series unchanged. (b) 1+3.3+7.32 +13.33 +21.34+... to n terms. 8. In the equations a1x+b1y+c1z=d1, a2x+b2v+c„z=d2, ax+by+c,z=d,, employ determinants to find 3. (a) the values of x, y, z. (b) the relation connecting the coefficients when d,=d, =d.=0, and x, y, z are not all zero. PRELIMINARY HONOURS. MODERN GEOMETRY. 1. A straight line APQ cuts a circle in A and P. Find the locus of Q when A is fixed, and AP.AQ constant. 2. (a) For a system of complanar points with multiples, Σ(a.AL)=Σ(a. AM)=0 for two lines L and M. Prove that for a concurrent line N, Σ(a. AN)=0. (b) is the inradius of the regular octagon ABCDEFGH. Prove AC2 + AD2 + AE2 + A F2 + A G2 = 2(6+v/2)r2. 3. (a) The angle of intersection of two curves is unchanged by inversion. (b) P and Q are the common points of one coaxal system of circles, and the limiting points of another. Invert the double system (i) with any point, (ii) with P, as centre of inversion. 4. (a) The locus of a point from which tangents to two given circles are in a constant ratio is a circle coaxal with the two. (b) Two circles subtend equal angles at any point on the circle of similitude. 5. The three circles of similitude of three given circles are coaxal, and also the three circles of antisimilitude. 6. A, B, C and D, E, F are two groups of three collinear points. Show that the intersections of AE and BD, AF and CD, BF and CE are collinear. Find the polar reciprocal of this theorem. 7. (a) Prove that the six lines determined by 4 given points cut any transversal in a six-point involution. What does the involution become when the transversal passes through one or more points of intersection of the six lines. (b) Given 5 collinear points, find a sixth point such that the six may be in involution. |