5 = (a2 + b2 + c2 + bc + ca + ab). g + + c2 a2 b2 x + p x+9 x + r p 9 and shew that the roots are always real. 4. Rationalize the equation 5. Give the algebraical definition of proportion, and shew that if four quantities be proportional according to Euclid's definition then they are also proportional according to the algebraical definition. If (bcd+cda + dab + abc)2 = abcd (a+b+c+d)2 then it will be possible to arrange a, b, c, d so as to be proportionals. 6. Define a harmonical progression and insert a given number of harmonic means between two given quantities. If a1, A2, A3, an be in harmonical progression, then ... a2 An are also in harmonical progression. - 7. The difference between any number and the sum of its digits is divisible by r 1, where r is the radix of the scale in which the number is expressed. In the scale whose radix is r shew that the number (→2 — 1) (1⁄2” — 1) when divided by r-1 will give a quotient with the same digits in the reverse order. 8. Find, without assuming the formula for the number of permutations of n things r at a time, the number of combinations of n things r at a time. Find the number of ways in which mn things may be equally divided among n persons. 9. Prove the binomial theorem for a positive integral exponent. If a, be the coefficient of x" in the expansion of (1 + x)" prove that (i) 10. Solve the equations (x + a)2 + (x + b)a = 17 (a — b)*. GEOMETRY AND TRIGONOMETRY. The Board of Examiners. In the first six questions the symbol - must not be used; and the only abbreviation admitted for "the square described on the straight line AB" is "sq. on AB," and for "the rectangle contained by the straight lines AB, CD" is “rect. AB,CD.” 1. Define the locus of a point which satisfies a given condition. The side AB of a triangle ABC of constant area is given. Find the locus of G the intersection of straight lines drawn from A and B to bisect the sides opposite to them. 2. The sum of the four straight lines drawn from the intersection of the diagonals to the vertices of a quadrilateral is less than that of any other four straight lines similarly drawn from any other point. 3. Find the point in a given straight line the sum of whose distances from two given points on the same side of it is a minimum. 4. ABC is a triangle, B a right angle. Prove that the difference between AC and the sum of AB, BC is equal to the diameter of the inscribed circle. 5. ABC is a triangle. DE is parallel to BC and meets BA, CA produced in D, E respectively. Prove that DA: DBEA: EC. If an equilateral triangle be inscribed in a circle, and the adjacent arcs cut off by two of its sides be bisected, the straight line joining the points of bisection will be trisected by the sides. 6. In a right-angled triangle if a perpendicular be drawn from the right angle to the hypotenuse, the triangles on each side of it are similar to the triangle and to one another. State and prove the corollaries. Define triplicate ratio. If circles be described on the segments of the hypotenuse, they will cut off from the sides segments which will be in the triplicate ratio of the sides. 7. An angle of a regular pentagon is represented by the number 540. Find the angular unit in terms of the unit of circular measure. 8. Prove that cos x + cos y = 2 cos Find expressions for all the values of 0 which satisfy the equation cos 80-cos 50+ cos 30 = 1. 9. If d be the perpendicular drawn from the angle C of a triangle ABC to the opposite side, prove c.d a.b that sin C = and deduce the equation sin (0 + 4) = sin 0 cos + cos ✪ sin 4. 10. Define the logarithm of a given number to a given base. Given a table of logarithms of numbers to the base a, shew how to find the logarithms of the same numbers to base b. Adapt to logarithmic calculation the equation · cos x cos c. sin A. sin B-cos A. cos B. 11. Two chimneys on a horizontal plane are 120 feet apart. From their bases the altitude of one is double that of the other, but from a point midway between them their altitudes are complementary. Find their heights. 12. Find the radii of the inscribed and the escribed circles of a triangle. If A be the area of the triangle, A, that of the triangle formed by joining the points where the escribed circles touch its sides, A, that of the triangle formed by joining the centres of the escribed circles, prove that The Board of Examiners. 1. Parse fully each word in the following passages which is printed in italics; also make a note on any peculiarities in construction which you may remark: His mother was a witch, and one so strong D |