MATHEMATICS. (2.) 1. In every triangle, the square on the side subtending either 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 acute angle and the perpendicular let fall upon it from the opposite angle. If the side BC of a triangle ABC is bisected in D, prove that AB2 + AC2 = 2AD2 + 2BĎ2. 2. The angles in the same segment of a circle are equal to one another. Prove that of all triangles which have a given base and vertical angle, the greatest is that in which the two sides are equal. 3. If two straight lines cut one another within a circle, the rectangle contained by the segments of one of them is equal to the rectangle contained by the segments of the other. Divide a given straight line so that the rectangle contained by its segments shall be equal to a given square, not greater than the square on half the line. 4. Inscribe a circle in a given triangle. If in any triangle the centres of the inscribed and circumscribed circles coincide, the triangle is equilateral. 5. In equal circles, angles, whether at the centres or the circumferences, have the same ratio which the circumferences on which they stand have to one another. When are two magnitudes said to be of the same kind? When are two magnitudes of the same kind said to be incommensurable? Does Euclid's definition of the sameness of ratios apply to incommensurable magnitudes? 6. Define compound ratio; and prove that equiangular parallelograms have to one another the ratio which is compounded of the ratios of their sides. Show that any two parallelograms have to one another the ratio which is compounded of the ratios of their bases and altitudes. 7. Each of three circles cuts the other two; prove that the three common chords, or these chords produced, pass through one and the same point. Given a circle, and two points in its plane; describe a circle which shall pass through the two given points, and meet the given circle in two points diametrically opposite to one another. 8. Show that there are in general two different angles between 0° and 360° which have a given sine; or a given cosine; but that there is only one angle between 0° and 360°, which has both a given sine and a given cosine. Given the tangent of an angle, express in terms of it the other trigonometrical ratios of the angle. and employ (2) to obtain the values of cos 18° and sin 18°. = tan A; (AB) (2.) sin (BC) + sin (C – A) + sin (A (B - C) sin † (C – A) sin † (A – B) = 0; and solve the equation sin (A + 30°) + cos (A + 30°) = sin (A — 30°) 11. Prove that in any triangle— 12. Given the three sides of a triangle; find the logarithmic sines of its half-angles. The sides of a triangle are 525 feet, 650 feet, and 777 feet respectively. Determine its three angles. 13. A, B, C, are three points in a straight line on a level piece of ground. A vertical pole is erected at C; the angle of elevation of its top, as observed from A. is 5° 30'; and, as observed from B, is 10° 45'; the distance from A to B being 100 yards. Find the distance BC, and the height of the pole. 14. Prove that the area of a circle is equal to the product of its semicircumference by its radius. Three circles, each of radius 1 foot, touch one another externally. Find the area of the curvilinear figure included between them. SET IV. MATHEMATICS. (1.) 1. Prove that a+b” is divisible by a+b without remainder, when n is an uneven number. Resolve into their simple factors the expressions 2. State and prove the rule for finding the highest common divisor of two algebraical quantities. Find the highest common divisor of 9x3 +53x2 9x and x3+10x2+19x 30. 18, 4(1-x)2 8(1 4. Prove that if the sum of two square numbers be multiplied by the sum of two other square numbers, the product is the sum of two square numbers. 6. Establish the relations between the coefficients and the roots of a quadratic equation. = 0 Under what circumstances are the roots x2+2px+q both real, and both positive? If a and ẞ are the two roots of this equation, express a2 - aßß2 and a3 + ß3 in terms of p and q. 8. Prove that if N is a whole number, which is divisible by any prime number other than 2 and 5, the decimal fraction 9. A rectangular court is 20 yards longer than it is broad, and its area is 4524 square yards. Find its length and breadth. 10. The weights of two spheres, which are solid and made of the same material, are 512 lbs. and 729 lbs. respectively. If the radius of the first sphere is 16 inches, what will it cost to gild the surface of the second sphere at a penny and three 22 11. The perimeters of a circle, a square, and an equilateral triangle are each of them one foot. Find the area of each of these figures to the nearest hundredth of a square inch. 12. The sides of a triangular field are respectively 10 chains, 8 chains, and 12 chains. The chain being 22 yards, find the acreage of the field, and the perpendicular distance of its longest side from the opposite corner. 13. A cubical cistern contains, when completely full, 2000 cubic feet of water. Find the length of one of its sides to the nearest tenth of an inch. 14. Define the logarithm of a given number to a given base. The logarithm of 2 to base 10 is 30103; find the logarithms of 2 15 of ⚫002, of 5 and of 6250. 15. Prove that the logarithm of a product of two factors is the sum of the logarithms of the two factors. Solve the equation 2* = 5. 16. The mantissas of the logarithms of 79531 and 79532 are respectively 9005364 and 9005419; find the logarithm of 795.314, and find the number of which the logarithm is 2.900539. MATHEMATICS. (2.) 1. The sum of the squares of two right lines is equivalent to twice the squares of half their sum and of half their difference. 2. AB is a given straight line. On one side of it a point P moves so that the angle APB is constant, and on the other so that the sum of the angles PAB, PBA are together equal to the former angle. Find the locus of P. 3. Draw a straight line from a given point without the circumference, which shall touch a given circle. Describe a circle passing through two given points, and touching a given straight line. 4. Upon a given straight line describe a segment of a circle, which shall contain an angle equal to a given rectilineal angle. 5. Describe an isosceles triangle having each of the angles of base double of third angle, and hence Divide a right angle into five equal parts. How may an isosceles triangle be described upon a given base, having each angle at the base one-third of the angle at the vertex? 6. Inscribe an equilateral and equiangular hexagon in a given circle. Compare the magnitudes of the angles of an equilateral triangle, pentagon, and hexagon. 7. Give Euclid's definition of ratio. When is the first of four magnitudes said to have the same ratio to the second which the third has to the fourth? 8. The sides about the equal angles of equiangular triangles are proportional. If a straight line touch a circle, and from the point of contact two chords be drawn, and if from the extremity of one of them a straight line be drawn parallel to the tangent meeting the other chord (produced, if necessary), then will the two chords, and the segment intercepted between the parallels, be proportional. 9. Trace the changes of sign of sin 0, as increases through four right angles. 10. Find the values of sin 30° and sin 18°, and prove that a value of 0, which satisfies the equation 2 tan = cos lies between 18° and 30°. |