MATHEMATICS. (2.) 1. If a straight line be divided into two equal parts, and also into two unequal parts, the rectangle contained by the unequal parts together with the square on the line between the points of section is equal to the square on half the line. State and explain Euclid's corollary to this proposition. 2. In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle, by twice the rectangle contained by the side upon which, when produced, the perpendicular falls, and the straight line intercepted without the triangle, between the perpendicular and the obtuse angle. Show that the sum of the squares on the two diagonals of a parallelogram is equal to the sum of the squares described on its four sides. 3. The angles in the same segment of a circle are equal to one another. Find the segment of a given circle so that the angle in the segment shall be always equal to two-thirds of a right angle. 4. If from any point without a circle two straight lines be drawn, one of which cuts the circle and the other touches it, the rectangle contained by the whole line which cuts the circle and the part of it without the circle shall be equal to the square on the line which touches it. Prove this only in the case in which the cutting line does not pass through the centre of the circle. 5. Describe a circle about a given obtuse-angled triangle. If the triangle be isosceles and the obtuse angle double one of the angles of an equilateral triangle, show that the radius of the circumscribing circle is equal to one of the equal sides of the given triangle. 6. Show how to describe an equilateral and equiangular hexagon about a given circle. 7. Equal triangles which have one angle of the one equal to one angle of the other, have the sides about the equal angles reciprocally proportional. If the sides of the triangle be expressed numerically, show that their areas will be to each other as the product of the sides in each triangle containing the equal angles. 8. What test does Euclid give to determine when the first of four Post 8vo, 90 pp., 2s. 6d. A SHORT SKETCH OF THE PENINSULAR WAR. Intended chiefly for the Use of Candidates for the BY WALTER W. NORTHCOTT, OF TRINITY COLLEGE, CAMBRIDGE. With Twenty-two Plans of Sieges and Battles. LONDON: EDWARD STANFORD, 55, CHARING CROSS, S.W. 1880. 8. What test does Euclid give to determine when the first of four magnitudes has to the second the same ratio which the third has to the fourth? Prove that in equal circles, angles whether at the centres or at the circumferences have the same ratio which the circumferences on which they stand have to one another. If the circles be unequal, how may the ratio of the angles be expressed? 9. Two circles touch externally in a point A, from A a common tangent is drawn which cuts, in the point (B), another line touching both circles; prove that AB is a mean proportional between the radii of the circles. 10. Is any unit of angular measurement referred to in the first book of Euclid? What units of angular measurements are commonly used in English treatises on Plane Trigonometry? Examine the relation of these units to each other. Find the circular measure of an angle of an equilateral and equiangular pentagon. 11. Define the tangent and cotangent of an angle. Express the values of cot 90°, tan 180°, tan 270°. Show that the tangent of any angle will have the same sign as its cotangent. 12. Prove by means of a geometrical construction— cos (AB) = cos A cos B+ sin A sin B, and explain the result if B = A. Find cos 15°. 13. Find tan 3A in terms of tan A, and from the formula obtained, determine the numerical value of tan A, if 3A = 90°. Also, if (a) and (b) are the sides of a triangle subtending the angles (A) and (B) respectively, prove either of the above expressions equal to a+b α 15. In any triangle prove (1) a sin B b sin A 0. = (2) a cos B+b cos A = c. Hence find cos A in terms of the sides a, b, c. 16. The side of an equilateral triangle is 20 feet, find the numerical value of the radius of the circle circumscribing the triangle. 17. Find the least angle of the triangle whose three sides are 200, 250, 300 feet respectively. 18. Two straight railroads are inclined at an angle of 20° 16'. At the same instant from their point of intersection an engine starts along each line; one travels at the rate of 20 miles an hour; at what rate must the other travel so that after 3 hours the engines shall be at a distance from each other of 30 miles? Show that the question admits of two solutions. SET II. MATHEMATICS. (1.) 1. State and prove the rule for pointing in the division of decimal fractions. Divide 4.8 by 12 and by 12. 2. Find the value of (x + y) √ x2 + y2 - (x − y) √ ‡ ( x2 — y2) when x = 4 and 3. Divide x3 - c, and (a + b + c)x2 + (ab + bc+ca)x — abc by x divide the quotient by x-b. To the product of what three factors is the first dividend equal? 4. Simplify (x − y)3 + (x + y)3 + 3 (x − y)2 (x + y) +3 (x + y)2 (x − y). 5. Show that if 28 = a+b+c (8 − a)2 + (8 − b)2 + ( s − c)2 + s2 = a2 + b2 + c2. 6. Find the greatest common measure of 7. State and prove the rule for the extraction of the square root, and find the square root of 9. A, B, and C travel from the same place at the rate of 4, 5, and 6 miles an hour respectively; and B starts two hours after A. How long after B must C start in order that they may overtake A at the same instant? |