Illa quidem dum te fugeret per flumina præceps What would be the difference in sense between dum te fugit and dum te fugeret ? 10. Give the derivations of: Isis—Danube-Pelasgi Alms-Poltroon-Tribulation-pantaloon-vixen sergeant-page. 11. Give the meaning of: HS–S.V.B.E.-Heres ex dodrante. For the other Subjects, Candidates for Honours take the papers for Students not Cadidates for Honours. For Honours in Mathematical and Physical Science. I. PURE MATHEMATICS. (I.) 1. Three points D, E, F are taken on the sides BC, CĂ, AB of the triangle ABC, and circles are described about the triangles AEF, BFD, CDE; if a, b, c be the centres of these circles, then the triangles abc, ABC are equiangular. 2. Prove geometrically that any plane section of a cone is a conic section, and that a parallel section through the vertex wiil give lines parallel to the asymptotes of this section. Having given a point within a cone shew how to draw the two sections which have that point as focus. 3. If A + B.C + Cam + Deco + .... + K2" be an algebraic quantity that vanishes for more than n values of x, then all the coefficients are identically zero. Hence, or otherwise, prove that 2.3+3. 8+4. 15+ ...... Em (7–1) (n 1). n. (n + 1). (n + 2). e 4. If n be a prime number and N be prime to n, then will N-L 1 be divisible by n. Shew that 23– 1 is divisible by 168. 5. Find the general term and sum to n terms the recurring series 1 + 4 + 14 + 46 + 146 + 6. Assuming the truth of Demoivre's Theorem, prove that cose + i sin =eli, where i =v1. . Hence shew a? + qi + arqi = 2 ap. cos (q log, a). 7. Sum the series cos 0 + c. cos 2 0 ca cos 30 + + .... ad inf. 2 3 8. If a rational integral equation f (x) = 0, have equal roots, prove that these roots are given by equating to zero the H. C. D. of f (x) and its first derived function. Solve the equation x* + 2x + 8x + 5 = 0. 9. Find the equation of the normal to a parabola in terms of its inclination to the axis. Shew that in general through a given point within a parabola there can be drawn Three normals to the curve ; also that if these normals meet the parabola in P, Q, R, the centre of gravity of P Q R lies on the axis. 10. Determine the nature and position of the curves 12xy + 8y+ 4ax + 12ay — 72a* = 0, (2) 422 12xy + 9y + 2ax + 10ay + 3a = 0; in each case find the latus rectum. 11. Find the polar equation of the tangent to a conic section, the focus being the pole. Tangents are drawn at the extremity of a chord which subtends a constant angle at the focus; prove that they intersect on another (1) 4X2 conic having the same focus and directrix, and that the tangent to this second conic intersects the corresponding chord on the directrix. 12. If a, b, y.. be the trilinear co-ordinates of a point, then will I a + m B + ny represents a straight line. If in a triangle a point o be taken, and A0, BO, CO intersect the opposite sides in D, E, F; and if also EF, BC intersect in G; FD, CA in H; DE, AB in K, prove that GHK is a straight line. Also prove that if the point o be always situated on a fixed straight line through A, then will GHK pass through a fixed point G. II. PURE MATHEMATICS. (II.) 1. Define differential coefficient, and from your definition deduce the value of dy, when y = a*. 3 = . dx ♡ 2ax a Differentiate (logx)? x, (sin x) 2. Enunciate and prove Taylor's Theorem for the expansion of f(a+ x) in ascending powers of x. Hence expand sin x in powers of x. 3. If u be a function of y, z; and y, z are functions du du of x. then will du dy dz + dy' da dz dut Xy ca 4. Shew how to determine th asymptotes of the curve f (x, y) = 0. Trace the curve yê (a + x) = 4x3. 5. Investigate the locus of the ultimate intersec tions of the series of curves f (x, y, a) = 0, ). a x2 a2 + a being the variable parameter; show that the curve so obtained touches each curve of the series. A straight line is drawn through a fixed point meeting the axes in H,K; HP, KP are drawn parallel to the axes. Shew that the envelope of the polar of P with respect to the ellipse y? = 1 is a parabola. 72 da 6. Integrate tan x. dx, (a + bx2) dx, e* tex' dx 23 -1 7. Find an expression for the area of a plane curve in Cartesian co-ordinates. Determine by the aid of the eccentric angle the area of an ellipse. If a circle concentric with the ellipse, and radius vab, be described cutting the ellipse; the area of that portion of each curve that lies without the other is the same. 8. Shew that the equation Ax + By + C + D=0, represents a plane. Interpret the equation distance of this plane from the origin is p. 9. Find the equation of the tangent plane at any y za a 73 = 1. 10. Define rectilinear generators ; shew that on the hyperboloid of one sheet there are two systems of such lines ; and that every line of one system intersects every line of the other system, but that no two lines of the same system intersect one another. Shew that all points at which the two gene rators are at right angles lie on a sphere. 11. Find the locus of the middle points of parallel chords of a conicoid. If (x,y,z,), (x2Y222), (x34323) be the extremities of three conjugate diameters in an ellipsoid ; shew that *;? + 2,? + xz? = a?, y? +42? + 3? = 62,2,2 + ,? + 2,2 = ca. 12 +2 = If P, Q, R be the three extremities, O the pole of the plane PQR, C the centre of the ellipsoid, s the middle point QR; then will OP be parallel to CS. 12. Trace the surfaces : 2y? - - 5x2 + 222 + 4y2 + 4y +162 + 18 = 0; x2 - y2 + z2 4yx + 6x2 2yz = f. . 2 3 2 = 2 2 3 2 2 +23 III. PURE MATHEMATICS. (III.) 1. Define reciprocal polar ; find that of a circle with regard to another circle. Points D, E, F are taken on the sides BC, CA, AB of a triangle; shew that three parabolas can be drawn to touch the sides of the triangles AEF, BFD, CDE, having a common focus ; also that the three directrices form a triangle whose sides subtend at the focus s angles equal to those subtended by the sides of the original triangle. |