9. The first two terms of an arithmetical series are 11 and 23, find the 10th term. How many terms must be taken that the sum may be 171 ? Prove that 3 - 1) (x - 1) 3 (2 — 1). + 1 1 2 1 + (x − 1)2 = x2 + 1 10. Find the number of words which can be formed out of 8 letters taken all together, each word being such that 3 giveu letters are never separated. 11. What is meant by one quantity varying (1) directly, (2) inversely as another? The value of diamonds varies as the square and of rubies as the cube of their weight, a diamond and a ruby weighing two carats each are of equal value, a diamond and a ruby each weighing three carats are together worth Rs. 450, find the value of each. 12. Expand (1 — x)" by the Binomial Theorem, and write down the (x + 1)th term. 1 1.3 Prove that ✓=1+ 2+ 1.3.5 + &c. 2.24 13.28 GEOMETRY. Examiner-MR. W. GRIFFITHS, M. A. 1. In any triangle A B C, if E F be the points where the per. pendiculars from the opposite angles meet the sides AC, AB; prove that BC2 = AB, BF + AC, CE. 2. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles. Two circles intersect in A, B ; PAP', QBQ' are drawn equally inclined to AB to meet the circles in P, P, Q, Q,: prove that PP' is equal to Q Q'. 3. Inscribe a circle in a given triangle. If lines be drawn from the angles of a triangle ABO to the centre of the inscribed circle, cutting the circumference in P, Q, R; prove that the angles at P, Q, R of the triangle formed by joining these T + A + B + + O points are respectively equal to 4 4 4 4. Similar polygons are to one another in the duplicate ratio of their homologous sides. Two similar polygons whose areas are A and B are inscribed and circumscribed about the same circle. Prove that B - A = area of a similar polygon inscribed in a circle whose diameter is a side of B. 5. If two planes cut one another their common section is a straight line. A, B, C are the three points on a horizontal plane at which the tops of three trees appear to be two and two in the same direction. Show that A, B, C must be in the same straight line. 6. If from a point 0, a pair of tangents 0 Q and O Q' be drawn to a parabola, the triangles OS Q, OS Q' will be similar, and O S will be mean proportional bet ween Q S and Q-S. If a parabola be described touching the sides of a triangle MNO, and if the focus S be joined to any angular point O by a straight line cutting the opposite side in D; prove that the triangles MDO, ND S are similar. 7. If the points Q, Q? in question 6 be joined, the area of the figure bounded by Q Q' and the curve is two-thirds of the triangle QOQ? The latus-rectum of a parabola is 41; show that if a focal chord of length f cut off an area A then 9 A2 =1f3. 8. If from the foci S sé of an ellipse SY and S'Y' are drawn at right angles to the tangent at P, then Y and Y are on the circumference of the auxiliary circle, and SY. S'Y? = BC%. If the normal at P meets CY in R, prove that RY SP. 9. If QV be any ordinate to the diameter PCP of an ellipse, and CD be conjugate to CP; then QV9: PV.P?V:: CD2: CP2. 10. If the tangent at any point P of a hyperbola meet the asymptotes in L and l; then the area of the triangle LCI is equal to the rectangle contained by AC and BC. TRIGONOMETRY AND MENSURATION. Examiner-MR. W. GRIFFITIS, M. A. 1. Define the sine and cosine of an angle. Show that cos (180° — A) = cos A, sin (90° + A) = cos A. Determine the values of sin 675°, sin 815° cos 1050°, cos 1230°. 2. Prove the formula :Cos (A + B) = cos A, cos B sin A, sin B. tan A + tan B Tan (A + B)=ī tan A, tan B Cos 2 A, cos 3 A cos A + sin 3 A, sin 2 A=0. And find sin 143° 7' 30" having given sin 286° 15' = - .96. 3. In any triangle when A, B, C are the angles, and a,b,c the sides respectively opposite to them, show that (i.) ao = 12 + ca — 2 b c cos A. A B C (ii.) Sin A + sin B + sin C = 4 cos 2 A + B A + B If (a? + 6%) cot = (a? — 12) cot shew that the tri. 2 angle is isosceles. 4. The sides of a triangle are 1717, 1919, 2424; find the angle opposite to 1919 Givenlog 3030 =3.4814426; log 1313=3.11826 17; L tan 25°33'=9.6865768. log 11l1=3.0157141; log 606 =2.782426; Diff. for 1' = 3213. COS 2 5. In the triangle ABC, a =345, b=234, and A=123.0 Find B Given log 234 = 2.3692159; L sin 34°40' 9.7549604. - 2.5378191; L sin 34°41' 9.7551431, 6. At noon a column in the S. E. cast upon the ground a shadow, the extremity of which was in the N. E. ; the angle of elevation of the column being 30°, and the distance of the extremity of the shadow from the column 180 feet, find the height of the column. 7 A railway which for some distance has been laid in a straight line at a certain point takes a circular bend for 484 yards, and then proceeds again in a straight line, which deviates from the former by an angle of 16° 30°; find the radius of the curve. Explain how the radius of the curve can be determined without observing any angles. (The value of in this and the next 3 ques22 tions is to be taken as 7 8. A piece of paper 4620 yards long is rolled into a solid cylinder; find the diameter of the cylinder supposing 360 leaves of such paper to have a thickness of one inch. 9. A cylindrical boiler, 16 feet long, and 5 feet in diameter, with hemispherical ends, in addition to the above length, has to be covered with felt; what will it cost at 2 as. per square foot ? 10. A tub in the form of a frustrum of a cone is made of wood one inch thick every where. The internal diameters at the top and bottom are 4 and 2 feet respectively, and the slant height on the inside is 14 feet. Find (1) the number of cubic feet of water it contains ; (2) the area of the inner curved surface ; (3) the area of the outer curved surface. a STATICS AND DYNAMICS. Examiner-MR. W. GRIFFITHS, M.A. 1. Prove the triangle of forces and its converse. Forces act on a triangular lamina ABC along and proportional to AC, CA, BA. Find the direction and magnitude of their resultant. 2. When three forces act on a body, state the conditions of equili. brium. A uniform heavy bar of length I rests suspended by two strings of Jengths a and b fastened to the ends of the bar and to two fixed points in the same horizontal line at a distance d apart. If the directions of the strings being produced meet at right angles, prove that the ratio of their tensions is al + bd : 61 + ad. 3. When a body is placed on a horizontal plane, show that it will stand or fall according as the vertical line drawn through its centre of gravity falls within or without the base. A cone surmounted by a globe is placed on a perfectly rough inclined plane ; if the axis of the cone = diameter of the base = diameter a 2 of the globe, and if the cone is on the point of toppling over, show that the inclination of the plane is tan-1 4. Find the ratio of the weight to the power in the system of pullies in which a separate string goes round each pully and is attached to the weight, and all the strings are parallel. ABCD is a rectangle; E and F are the middle points of A B and CD; and AB: B C:: 3:2. If A B be a punkah pole which is sus pended by a rope having one end fastened at E and the other at F, and passing through smooth hooks at F, B, C, E, D, A respectively; prove that the tension of the rope is w, where w is the weight of the punkah, the weight of the rope being neglected. 5. A uniform beam AB lies horizontally upon two others at points A and C; if AB 2a, AC b, and w = weight of the beam, find the pressures at A and C. And prove that the least horizontal force applied at B in a direction perpendicular to A B which is able to move the beam is the u W (b − a) W less of the two forces and where u is the co-effici. 2 a b ent of friction. 6. Define velocity and acceleration, and explain how they are measured. If f be the measure of an acceleration when a foot and a second are the units of length and time, and so the measure of the same acceleration when n feet and m seconds are the units, prove that ma f f. If the acceleration of a heavy body falling freely be the unit of acceleration, and the velocity acquired in 8 seconds be the unit of velocity, find the unit of length. 7. Prove that the velocity, acquired by a heavy particle sliding freely down a smooth inclined plane of a given height, is indepen. dent of the inclination of the plane to the horizon. The base AB of a vertical triangle ABC is horizontal: if t, ť be the times which a particle takes to slide down CA, CB respectively, show that t:ť:: sin B : sin A. 8. Find the time of flight, and the range of a projectile on a plane inclined at a given angle to the horizon. Two heavy particles are projected from a point with equal velo. cities, their directions of projection being in the same vertical plane; t, t' are the times taken by the particles to reach the other point where their paths intersect, and tu ta' are the times taken to 2u2 reach the highest points. Shew that t. ty + t'i ta' ga where u is the velocity of projection, and g the acceleration of gravity. 9. Two weights are connected by a string which passes over a fixed smooth pulley : find the tension of the string, and the acceleration of motion, If the string can only bear a strain of one-fourth the sum of the n : g weights shew that the least acceleration possible is 10. Two imperfectly elastic balls, whose centres are moving in tho same straight line, impinge on one another; determine their velocities after impact. a where P P' HYDROSTATICS AND OPTICS. Examiner-MR. W. GRIFFITIS, M.A. 1. Shew that the pressure of a liquid at rest is the same at all points of the same horizontal plane, and prove that if two liquids that do not mix together meet in a bent tube, the heights of their upper surfaces above their common surface will be inversely proportional to their densities. 2. Find the resultant vertical pressure of a liqnid on any surface. A double funnel formed by joining two equal hollow cones at their vertices stands upon a horizontal plane with the conmon axis vertical, and liquid is poured in until its surface is one-third up the axis of the upper cone. If the liquid be now on the point of escaping between the lower cone and the plane, prove that the weight of either cone is to that of the fluid it can hold as 40: 27. 3. Find the conditions of equilibrium of a floating body. Two equal uniform rods CA, CB are joined together by a hinge at C; the ends A, B are connected by a string. If the rods float with C downwards in a liquid aud AB horizontal and out of the liquid ; prove that the tension of the string = WP-p? tan 2p 2' are the densities of the liquid and rods respectively, W=weight of either rod, and a=the angle between the rods where the string is stretched. 4. Explain the method of finding the difference of altitude of two stations above the earth's surface by means of the barometer. Assuming that a change from 30 inches to 27 inches in the barometer corresponds to an altitude of 2,700 ft., adopt the formula for numerical application; and determine the altitude corresponding to 21.87 inches of the barometer. 5. Explain the construction and action of the condenser. If the greatest and least volumes between the valves be V, V and if the valves be open or closed according as the difference of pressure on the two sides is greater or less than p, show that the V limiting pressure in the receiver will be (a — P) · P where V is the pressure of the atmosphere. 6. State the laws of refraction. A stick is immersed in a liquid; the stick and its image are in. clined to the horizon at angles P and Qrespectively ; prove that tan o u tan P', where u is the index of refraction from air into the Trace the pencil of light by which an eye sees a point of the stick not vertically below it. 7. Find the position and magnitude of the image, formed by a concave mirror, of a small object at a distance from the mirror liquid. |