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2. Prove (geometrically, if you can) that
cos + cos 20+ + cos no = cos
Evaluate the sum
sec sec 20+ sec 20 sec 30 +
in a single expression.
(n + 1) 0 . sin ‡n✪ . cosec 10.
a + b + c − 2b3c3 cos A - 2c3a3 cos B-2a3b3 cos C
= a2b2c2 (1 - 8 cos A cos B cos C).
4. ABCD is a quadrilateral whose sides AB, BC, CD, DA touch a circle at P, Q, R, S, respectively; prove that
AB. BC. sin2 (B) = CD. DA. sin2(D),
5. Find from first principles the differential coefficient of 2", where n is any rational number.
quoting, in full, any rules that you use.
6. A normal is drawn to the curve y = ax2 + bx3 at its point of intersection with the line = c, find where the normal cuts the axis of y.
Show that the function 2x/(p + qx2) has one maximum and one minimum provided that pq is positive; but that it has neither maximum nor minimum values if pq is negative. What will happen if pq = 0 ?
7. Construct a triangle of given species with one vertex at a given point and the other two vertices lying on two given straight lines. Under what circumstances would there be an infinite number of solutions?
8. A circle U and a point A are inverted from a point 0, the inverses being a circle V and a point B. If AP and BQ are the tangents from A and B to U and respectively, prove that AP : BQ :: OA: OT, where OT is the tangent from 0 to the circle V.
9. Show how to describe circles co-axal with two given circles which do not intersect.
Describe a circle belonging to a given co-axal system of this kind, and bisecting the circumference of a given circle. What happens when the centre of the given circle is on the radical axis?
10. The middle points of the sides BC, CA, AB, of a triangle are D, E, F, and the points of contact with the inscribed circle are L, M, N. If the straight lines EF and MN intersect in X, FD and NL in Y, and DE and LM in Z, prove that YZ passes through A, ZX through B, and XY through C.
11. Four parallel straight lines AP, BQ, CR, DS, are drawn from the vertices of a tetrahedron ABCD terminated by the opposite faces. Prove that, with the usual convention of sign,
12. Prove the formula for the area of the curved surface of a slice of a sphere of radius r intercepted by two parallel planes whose distance apart is d.
Show that the volume of the segment of a sphere cut off by a plane whose distance from the centre is x, is
[Full credit will be given for answering FOUR-FIFTHS of
(1−x + x2)" = (X- Y)+(X- Y) (Y-Z) + (Y-Z)2,
3. A sum of £400 is to be repaid in the form of an annuity lasting for 20 years, the first payment of which is to be made a year hence. The first ten payments are to be of equal amount, and are each to be twice as much as each of the remaining ten payments. Calculate the amounts of the payments to the nearest shilling, allowing compound interest at the rate of 24 per cent. per annum.
4. If a, ẞ, y are the roots of the cubic
5. Solve the equation
19x44x3+ 78x2 - 36x + 27 = 0,
having given that its roots are in harmonical progression.
6. If a1, a2, az a,, are the roots of the equation
Find, in terms of P1, P2 ・・・ Pn, the value of the ratio of the absolute term to the coefficient of " in the resulting equation.
ANALYTICAL GEOMETRY, &c.
7. Find the area of the triangle formed by the feet of perpendiculars from the point (h, k) to the sides of the triangle whose vertices are
8. Two fixed parallel lines equidistant from the centre of a given circle meet a variable tangent to the curve in G, H; the other tangents from G, H meet in P. Find the locus of P.
9. Prove that the equation y2 = ax + by + c represents a parabola, and find its focus and directrix.
Two parabolas whose axes are at right angles cut one another in four points. Prove that these four lie on one circle.
10. Chords of an ellipse in a fixed direction are divided in a constant ratio. Prove that the points of division lie upon another ellipse.
11. Prove that the diameter bisecting any chord of a conic passes through the intersection of the tangents at the ends of the chord.
Four tangents to an ellipse form a quadrilateral. Prove that the line joining the middle points of its diagonals passes through the centre.
12. Show that the focal distances of a point on a hyperbola are equally inclined to the tangent at the point.
Given the asymptotes and a tangent, construct the foci.
1. Explain how the connexion between two physical quantities can be shown by means of a curve. Take, as illustrations, the variation of the pressure of a mass of gas with (a) its volume, (b) its density.
2. What is the measure of the energy stored in a stretched elastic string? Two masses, connected by an elastic string, are moving in any manner on a smooth plane. Show that a shortening of the string during the motion must be accompanied by an increase in the speed of at least one of the masses.
3. Explain the origin of the backward force exerted on a vessel from which a jet of fluid is issuing, quoting illustrative experiments.
heavy again as air at the The ratio of its specific Under what condition
4. Carbonic acid gas is half as same temperature and pressure. heats is 1.3, the value for air 1.4. will sound travel at the same rate in air and carbonic acid