II. Translate the following passage into Hebrew : Stultissimus sum virorum, et sapientia hominum non est mecum. Quis ascendit in caelum atque descendit? quis continuit spiritum in manibus suis? quis colligavit aquas quasi in vestimento? quis suscitavit omnes terminos terrae? quod nomen est eius, et quod nomen filii eius si nosti? Omnis sermo Dei ignitus clypeus est sperantibus in se; Ne addas quidquam verbis illius et arguaris, inveniarisque mendax. Duo rogavi te, ne deneges mihi antequam moriar; Vanitatem et verba mendacia longe fac a me; Mendicitatem et divitias ne dederis mihi; tribue tantum victui meo necessaria, Ne forte satiatus illiciar ad negandum et dicam: Quis est Dominus ? aut egestate compulsus furer, et periurem nomen Domini mei. Nec accuses servum ad dominum suum, ne forte maledicat tibi, et corruas. Generatio, quae patri suo maledicit, et quae matri suae non benedicit; Generatio quae sibi munda videtur, et tamen non est lota a sordibus suis. Generatio cujus excelsi sunt oculi, et palpebrae eius in alta surrectae ; Generatio, quae pro dentibus gladios habet, et commandit molaribus suis, ut comedat inopes de terra, et pauperes ex hominibus. 1. Through a focus of an ellipse let there be drawn, intersecting at right angles, two chords of the circle described with the major axis as diameter; construct geometrically a pair of conjugate diameters which are respectively equal to these chords. 2. A normal to an ellipse at a point whose eccentric angle is meets the ellipse again at a point the eccentric angle of which is ; if two diameters make with the major axis angles respectively equal to and 4+4 2 , prove that these diameters will be conjugate. 3. A circle passing through the focus of a conic intersects it; show that the continued product of the distances from the focus to the points of intersection has a constant value if the diameter of the circle is given. 4. A normal at a point P of an ellipse whose centre is O is produced externally to Q, PQ being equal to the radius of curvature at P; express OQ in terms of the diameter conjugate to that passing through P, and hence show that the circle having PQ as diameter cuts the director circle orthogonally. 5. Let r be a semidiameter of an ellipse, and p the perpendicular let fall from the centre upon the tangent at the extremity of r; if p and r make with the major axis angles respectively equal to and, it is re4+4 quired to express tan in terms of the rectangle pr. 2 6. P is a point on an ellipse whose centre is O, and Q is the foot of the perpendicular let fall from O upon the tangent applied at P; show that PQ is a maximum when P is the point of intersection of the ellipse with the confocal hyperbola which passes through the intersection of the angents at the extremities of the axes; also find the value of PQ in this case. 7. Being given a system of confocal ellipses, find the equation of the locus of a point P on any one of them when the diameter conjugate of that passing through P is of given length. 8. Let a common tangent be drawn to two equilateral hyperbolas having the same centre O, and intersecting in a point Q; if P and R are the points of contact, find the relation between the lengths OP, OQ OR. 9. P, Q, R are three points on an equilateral hyperbola whose centre is 0, such that OQ bisects the angle POR; M is the middle point of the chord PQ, and N the middle point of the chord QR. It is required to OM. PQ ОР express the ratio in terms of the ratio ON. QR OR* 10. Let p, p' be the radii of the osculating circles at two points on a given ellipse; it is required to express in terms of the distance, d, between the centres of these circles, and of the perpendicular, p, dropped from the centre upon their radical axis. II. Two parabolas intersect, which are both touched by a given right line, and which have a given point for focus; find the locus of their intersection when the angle included between their axes is constant. 12. Two tangents to a given parabola make angles with the axis such that the product of the tangents of their halves is constant; prove that the locus of the intersection of the tangents is a confocal parabola. 9. Eliminate by differentiation the constants a, b, from the equation = ayb. 1. Find the equation whose roots are the squares of the differences of the roots of x3 + 8x2 + 16x + 16 = 0. 2. If a, B, y are the roots of 3 − px2 + qx − r = 0, find the relation between the coefficients if Σ a5 B = Σ at B2. x1 — y1 + z1 — u1 + 4y2xz + 4u2xz — 4x2uy — 4%3uy — 2x2x2 + 2u2y2. - 5. If the equation x2 + px2 + qx + r = o is put under the form x4 - 61x2-8√ 13 + m3 + n3 3 lmn . x 3 (12 — 4mn) = 0, find the quadratic on which the determination of m and n depends, and give the roots in terms of l, m, n. 6. Express the roots of the equation 43 – 3 9gʻu = = (27 rr' ±√/ (27 x2 + 4 23) (27 p22 + 4 9′3)) in terms of the roots of the equations and the reducing cubic in Euler's method of the biquadratic ax1 + 4 bx3 + 6 cx2 + 4 dx + e = 0; and express the roots of the above cubic in terms of the roots of the biquadratic. 8. If λ, μ, v, are the roots of the cubic in last question, express (c − aλ)2 (μ − v)2 + (c − aμ)2 (λ − v)2 + (c − av)2 (λ – μ)2 in terms of the roots of the biquadratic. 1. If a uniform beam, resting on a horizontal plane and against a vertical wall, be kept in equilibrium by a cord attached to both its extremities, and passing through a fixed ring placed above the beam; find the position of the ring so that the tension of the cord may be the least pos sible (a). If the wall and plane be smooth. (b). If the wall and plane be equally rough. (c). In the second case find the limiting value of the coefficient of friction which gives a determinate answer. (d). In these questions it is assumed that the vertical plane through the beam is perpendicular to the wall; show that when the planes are smooth this condition is necessary to equilibrium. 2. If a weight, attached by strings to two fixed points on a smooth horizontal table, hang over the edge of the table; show that the two parts into which each string is divided by the edge make equal angles with the edge. (a). Hence show that if the table were taken away, the fixed points being retained, the tension of the strings would not be changed. |