Euclid. 1 If a side of a triangle be produced the exterior angle is greater than either of the interior and opposite angles. 2. Define a square. Describe a square on a given straight line. What is Euclid's corollary to this proposition ? 3. If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the two parts, together with twice the rectangle contained by the parts. 4. What is the angle in a segment of a circle ? The angles in the same segment of a circle are equal to one another. VOLUNTARY PORTION. 1. Give Euclid's construction for describing an isosceles triangle having each of the angles at the base double of the angle at the vertex. Show that the base of the triangle thus described is equal to the side of a regular decagon inscribed in the circle whose centre is the vertex of the isosceles triangle and radius one of its equal sides. 2. If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or those produced, proportionally. Show that the straight lines which join the points of bisection of the adjacent sides of any trapezium form a parallelogram, and compare the area of this parallelogram with the area of the trapezium. 3. Equiangular parallelograms are to each other in the ratio compounded of the ratio of their sides. When a parallelogram is represented arithmetically by the product of its base and altitude, how is the unit of area related to the unit of length? Show, by the application of algebra, that equiangular parallelograms are to each other as the product of their conterminous sides. 4. Divide 03 - px? + qx — r by x - a, and express the whole remainder from which (w) has disappeared after the division. Find (a) in terins of (p) and (q), so that x?-pr+g may be divisible by (x – a) without any remainder. Solve the following equations ac. 2? + bcx=adx + abcd. 1 1 1 + y a-y=xy? +16 5. In the expansion of (1+x)", prove that the second term =nx, whether n be positive, negative, whole, or fractional. Assuming the law of expansion, write down the (r+1) term of (1+x)", and find (r) when the (r+ 1 )th coefficient is equal to the (r+3)", and (n) is a positive integer and even. 6. Show how to transform the fraction into a continued fraction, and hence obtain a law for determining a series of fractions converging to 7 7. Investigate the formula sin (A+Bi)=sin A cos B+cos A sin B and deduce from it the corresponding expression for cos A-B. cos A= Prove 1 (1) A 1+tan A . tan 2 and adapt this to radius (r). A B С (2) cos A +cos B+cos C=4 sin sin sin +1 2 2 if A+B+C=180°. 8. When two sides and an included angle of a triangle are given, show how to solve the triangle. Example C=112° 14' AC=356.1 BC=435.2 9. Find the area of a regular polygon inscribed in a given circle, and hence obtain an expression for the area of the circle. The diameters of two concentric circles being 12 and 8 feet, find the area of the ring contained between their circumferences. 10. To find the height of a mountain above a horizontal plane, a base line of 2764 feet was measured on the plane, and at each extremity of this base the angles formed by the other extremity and the top of the mountain were found to be 59° 22' and 108° 35' respectively, also at the extremity from which the greater angle was observed, the elevation of the mountain was 10° 13'; find the height of the mountain. 11. Investigate the condition of equilibrium, when two forces acting at any angles on the arms of a bent lever are in equilibrium. The lengths of the arms of a lever at right angles to each other, are as 3 : 2; find the ratio of weights suspended at the extremities of the arms when the longer arm is inclined to the horizon at an angle of 30°. 12. If a body descend from rest down any smooth curve in a vertical plane, show that the velocity at any point is the same as if the body had fallen freely down the same vertical height. A body descends down the quadrant of a circle, the radius of which is 64.4 feet; find the velocity at the lowest point. MINERALOGY AND GEOLOGY. WARINGTON SMYTH, Esq., M.A., F.G.S. Maximum Number of Marks, 600. 1. What is meant in speaking of mineral bodies by the terms crystalline and amorphous ? Give examples of each. 2. Among the substances commonly called porphyry, marble, prussiate of potash, alabaster, which of them are properly termed minerals, and why are the others not included under the same term? 3. Which of the component parts of gunpowder belong to the mineral kingdom, and whence are they principally obtained ? 4. State the principles on which the specific gravity of bodies is determined, and the most convenient methods for its investigation. 5. How is the comparative hardness of solid minerals ascertained ? 6. Describe the minerals employed for the production of the metal lead. 7. How does the phenomenon of cleavage in rock masses differ from that which is known by the same term in certain minerals ? 8. What is meant by the terms dip and strike of stratified rocks? 9. Enumerate in true order of succession the various members of the carboniferous system as developed in the British Islands. 10. What are the proofs of the vegetable origin of coal ? 11. Mention the leading fossils which are characteristic of the chalk formation. 12. Describe the nature of faults, and their influence on springs of water. |