Mathematics. Each paper is designed to prepare for the particular examination named, but more especially in the parts of the subject peculiar to each ; therefore students are advised not to confine themselves to one paper, but to make use of the whole set. Junior. (a) 1. If two triangles have two sides of the one equal to two sides of the other, each to each, and have also the angles contained by those sides equal to one another, they shall also have their bases or third sides equal; and the two triangles shall be equal, and their other angles shall be equal, each to each, namely those to which the equal sides are opposite. Upon the sides A B, BC, and C D of a parallelogram ABCD, three equilateral triangles are described, that on BC towards the same parts as the parallelogram, and those on A B C D towards the opposite parts. Prove that the distances of the vertices of the triangles on A B, C D from that on BC, are respectively equal to the two diagonals of the parallelogram. 2. If a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior and opposite angle on the same side; and also the two interior angles on the same side together equal to two right-angles. Through two given points draw two lines, forming with a line given in position, an equilateral triangle. 3. The opposite sides and angles of a parallelogram are equal to one another, and the diagonal bisects the parallelogram, that is, divides it into two equal parts. If from the base to the sides of an isosceles triangle three lines be drawn, making equal angles with the base, namely, one from its extremity, the other two from any point in it, these two are together equal to the first. 4. If there be two straight lines, one of which is divided into any number of parts, the rectangle contained by the two straight lines is equal to the rectangles contained by the undivided line and the several parts of the divided line. Show how this proposition is a geometrical illustration of a Law governing the addition and subtraction of quantities. Зах 4 (6) 23 3 22 5. Multiply + 2 x + 1 by 16 DC + 1, and 2 4 203 За2 divide the product by + 1. a? a3 7. If one quantity measure two others, show that it will measure the difference of any multiples of those quantities. Find the H.C.D. of 4x4 - 4 xy + 3 xya - ays - 6yo and 2 ac* + 202 y + 4 x y: – 3 y*. 8. Find the two times between 6 and 7 o'clock when the hands of a watch are separated by 15 minutes. a (©) Forces are represented by the sides AB, AC of a triangle A B C. If their resultant passes through the centre of the circle described about the triangle ABC, prove that the triangle is either isosceles or right-angled. 10. Explain how the magnitude and direction of the resultant of any number of forces acting at a point may be found by resolving them along two lines at right angles to each other. ABCDEF is a regular hexagon, and forces are represented by A B, AC, AD, A E, AF: find their resultant. 11. In what time will a force which would support a 5 lb. weight move a mass of 10 lbs. weight through 50 feet along a smooth horizontal plane, and what will be the velocity acquired ? 12. Upon what experimental evidence do we base the assertion that the attraction of the earth upon any body is proportional to its mass, and independent of the nature of the material of which it is formed ? Supposing the attraction of gravitation at the equator to be 0.995 of its value in London, if a person sell goods in London by a spring balance accurately graduated at the equator, how much per cent, on the selling price does he gain in excess of his fair profit? D Mathematics. Each paper is designed to prepare for the particular examination named, but more especially in the parts of the subject peculiar to each ; therefore students are advised not to confine themselves to one paper, but to make use of the whole set. Senior. : (a) EUCLID; (b) ALGEBRA; (c) MECHANICS. (a) 1. If a straight line be divided into any two parts, the square on the whole line is equal to the squares on the two parts together with twice the rectangle contained by the parts. Divide a straight line into two parts so that the sum of their squares may be the least possible. 2. In every triangle, the square on the side subtending any of the acute angles is less than the squares on the sides containing that angle, by twice the rectangle contained by either of these sides and the straight line intercepted between the perpendicular, let fall upon it from the opposite angle, and the acute angle. In any quadrilateral the squares on the diagonals together equal twice the sum of the squares on the lines joining the middle points of opposite sides. 3. If two circles touch one another externally, the straight line which joins their centres shall pass through the point of contact. If two circles touch each other, and parallel diameters be drawn, then lines which join the extremities of these diameters, will pass through the point of contact. а 4. Show how to inscribe a circle in a given triangle. Describe a circle touching one side of a triangle and the produced parts of the other two. а 17 = C (6) 5. Solve the equations : 4 3 3 10: seca + xy = 10. 3. 6. Prove that the product of any four consecutive even integers increased by 16 is a perfect square. 7. Prove that: 3 (a + b + c)2 – (a + b)3 – (b + c)2 – (c + a)3 – ai – 83 - 03 = 6 (ab + bc+ca)(a + b + c). 8. Find the Least Common Multiple of: a (a – y), wo y (a + y)2, x y (x2 - y2), 203 – y + xy?, 22 y + x ya + y), and 26 – yo. (c) 9. What is meant by the moment of a force about a point ? Prove that the moments of two parallel forces about any point in the line of their resultant are equal in magnitude and opposite in direction. The resultant of two parallel and similar forces is 121 lbs., and acts at a point 2 feet distant from one force and 3 feet distant from the other. Find the forces. 10. What is meant by the efficiency of a machine ? Show that the efficiency of a lever when the weight acts at its middle point is the same as that of a lever whose fulcrum is between the power and the weight, and twice as far from the power as from the weight. 11. If the acceleration caused by gravity be the unit acceleration, and the velocity of a mile in 5 minutes the unit of velocity, find the unit of length. 12. Find the time occupied by a particle in sliding through 20 feet down a rough plane inclined 60° to the horizon, the coefficient of friction being one-half: and its velocity at the end of the time. a Each paper is designed to prepare for the particular examination named, but more especially in the parts of the subject peculiar to each ; therefore students are Women (Pass). (a) 1. The angle at the centre of a circle is double of the angle at the circumference, subtended by the same arc. If two straight lines A EB, CED in a circle intersect in E, the angles subtended by A C and B D at the centre are together double of the angle A E C. 2. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles made by this line with the line touching the circle must be equal to the angles, which are in the alternate segments of the circle. Through a given point without a circle draw a chord such that the difference of the angles in the two segments, into which it divides the circle, may be equal to a given angle. 3. Describe a circle about a given triangle. If two circles be described, one without, the other within, a right-angled triangle, the sum of their diameters is equal to the sum of the sides containing the right-angle. 4. Describe an isosceles triangle having each of the angles at the base double of the third angle. Divide a right-angle into five equal parts. a (6) 5. Solve: at y + ytact = 78 |