SET VI. EUCLID. 1. Construct a triangle of which the sides shall be equal to three given straight lines. What condition is necessary? Show how the construction would otherwise fail. Construct also a triangle, of which one angle is given, an adjacent side, and the difference of the other two sides. 2. If a side of any triangle be produced, prove that the exterior angle is equal to the two interior and opposite angles, and that the three interior angles of every triangle are equal to two right angles. State and prove also the corollaries of this proposition. 3. Describe a parallelogram equal to a given rectilineal figure, and having an angle equal to a given rectilineal angle. The squares of the diagonals of a parallelogram are together equal to the squares of the four sides. 4. In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square of the side subtending the obtuse angle is greater than the squares of 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. If one angle of a triangle be equal to two-thirds of two right angles, prove that the square of the side subtending it is equal to the sum of the squares of the sides containing it, together with the rectangle contained by those sides. 5. The opposite angles of a quadrilateral figure inscribed in a circle are together equal to two right angles. If two sides of this quadrilateral be produced to meet, the triangles so formed are equiangular. 6. 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 of the line which touches it. AB is a chord and AC a diameter of a circle. AC is produced to D and ED drawn perpendicular to it. If AB produced intersects ED in E, prove that the rectangle BA, AÊ is equal to the rectangle CA, AD. 7. Inscribe an equilateral and equiangular hexagon in a given circle. Compare the magnitudes of an equilateral triangle, a square, and hexagon in the same circle, whose radius is unity. 8. What are the loci of the following points? (1.) The vertices of equal triangles on the same base. (2.) The vertices of triangles on the same base with equal vertical angles. (3.) The vertices of isosceles triangles on the same base. When is a 9. When is the first of four magnitudes said to have a greater ratio to the second than the third has to the fourth? straight line said to be cut in extreme and mean ratio? Define alternando, componendo, and ex æquali. 10. If four straight lines be proportionals, the similar rectilineal figures similarly described upon them shall also be proportionals. 11. In equal circles angles, whether at the centres or circumferences, have the same ratio as the circumferences on which they stand. If the diameter of a semicircle be divided into any number of parts, and on them semicircles be described, their circumferences will together be equal to the circumference of the given semicircle. ་་་་ · рx + q 2 a = a 1 b 1 b x + = 4q. O has equal roots, p2 If the equation 2 + 2 (1 + k) x + k2 = 0 has equal roots, what must be the value of k? 7. If√ and √y are incommensurable quantities, and if Also find the square root of 16+6√7. 8. Show how to transform a number from the scale whose radix is (r) to that whose radix is (r2). Transform 25603 from the nonary to the septenary scale. 9. Define Geometrical and Harmonical Progression. If a b and c are in harmonical progression, show that The n α + 1th and n+1th terms of a geometrical series are the arithmetical and harmonical means between two quantities. Show that the nth term is the geometrical mean between the same two quantities. 10. Given the sum of n terms of an arithmetical progression, a the first term, and b the common difference; find the number of the terms. 1+2+3+4+. . . + n = [13 + 23 + 33+ · · · + n3]3. ... 11. Show that the number of combinations of n things taken r together is equal to that of n things taken nr together. If C, is the number of combinations of n things taken r together, show that n × n−1Cr_1 = r × „Cr. How many words of 4 consonants and 1 vowel can be made out of 12 consonants and 5 vowels? 12. If c, is the coefficient of x in the expansion of (1+x)", write down the values of c1, C2, C3, C4, and co, and show that co + C2+ C4 +...&c. = c12+cs + C5 +...&c. Find the middle term in the expansion of 5a 16 5 13. A train A starts to go from P to Q, two stations 240 miles apart, and travels uniformly. An hour later another train B starts from P and after travelling for two hours comes to a point that A had passed 45 minutes previously. The pace of B is now increased by 5 miles an hour; and it overtakes A just on entering Q. Find the rates at which they started. TRIGONOMETRY. 1. Show that the angle subtended at the centre of a circle by an arc equal to the radius is invariable. Taking 22: 7 as the ratio of the circumference of a circle to its diameter, and assuming the diameter of the earth to be 8000 miles, find approximately the difference in latitude of two places, one of which is 100 miles north of the other. A 2. Find an expression for sin in terms (1) of sin A, (2) of cos A, and explain why more than one value appears. Find sin 15° and cos 165°. 4. Find an expression for all the angles which have a given tangent. Show that tan' (cot A) — tan-1 (tan A) π = nπ + 2A. 6. In a triangle ABC, A is a right angle and B = 60°. Show that 8. Find an expression for the area of a triangle. In an isosceles right-angled triangle a straight line is drawn from the middle point of one of the equal sides to the opposite angle. Show that it divides the angle into parts whose cotangents are 2 and 3. 9. Find an expression for the radius of the circle inscribed in a given triangle. Two sides of a triangle are a and 2a, and they include a right angle. If r, r, be the radii of the escribed circles touching these sides, show that r, r, = a2. |