If x,y are real quantities satisfying the equa14xy + 9y2 + 20x − 22y + 64 = 0, tion 4x2 5, and 4. If a, ẞ are the values of x which satisfy the 5. If n is one of the imaginary cube roots of unity, shew that 1+ w + w2 = 0, and that + + 1 a+b+c a + wb + w2c a + w2b + wc 6. When n + 1 figures of the square root of a number have been found by the ordinary method, prove that n more may be found by division, provided that the number is a perfect square whose square root contains 2n + 1 figures; but that if this be not the case, there may be an error in the last figure. 7. Prove that the geometrical progression 1 + x + x2 + to ni n2 nз .... .... n, terms is the product of the following r geometrical progressions, viz. :— 1 + x2 + x2n + to n, terms, .... 1 + x21n2 + x2n,n2 + to ng terms, &c., .... &c. 8. Describe the method of mathematical induction, and apply it to prove that the sum of the cubes of the first n natural numbers is equal to the square of the sum of the numbers. 9. Find the numbers of permutations of n things r at a time. Find the number of ways in which n ladies and n gentlemen may seat themselves at a round table so that no two ladies are together. 10. Prove the binomial theorem for a positive integral exponent. If a, be the coefficient of x in the expression of (1+x)", prove that a1 + a .... tor +1 terms = (− 1), (n − 1) (n − 2) . . . . (n − r) 11. Solve the equations (ii) ax2 + by2 = (a + b) (x2 — xy + y2) = a3 + b3. 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. If two quadrilaterals have three sides of the one equal respectively to three sides of the other taken in order, and have likewise the angles contained by those sides equal to one another, each to each, they are equal in all respects. 2. Find the locus of a point equidistant from two intersecting straight lines. How many points are there in a plane each of which is equidistant from two given unlimited straight lines as well as from two given points situated in that plane? 3. The sum of the squares on two sides of a triangle is double the sum of the square on half the base and on the line joining the vertex to the middle point of the base. The sum of the squares on the sides of any quadrilateral is greater than the sum of the squares on the diagonals by four times the square on the line joining the middle points of the diagonals. 4. If two circles meet in one point only that point is on the line joining their centres. Two circles touch externally; shew how to place a line of given length so that it shall pass through the point of contact and have its extremities on the circumferences of the circles. 5. The bisectors of the angles of a regular polygon meet in a point which is equidistant from all the vertices of the polygon and from all the sides. 6. If two straight lines are cut by a series of parallel straight lines, the intercepts on the one have to one another the same ratios as the corresponding intercepts on the other have. Three given straight lines meet in a point; draw a fourth straight line cutting them so that the intercepted segments may have given magnitudes. 7. Solve completely the equation sin20+ sin cos20+ cos 0. and that one of the angles X + Y ± Z is an odd multiple of two right angles. tan ẞ tan y tan y tan a + tan a tan ß = 1. 11. Shew how to solve a triangle having given the lengths of the straight lines drawn from the vertices to the middle points of the opposite sides. 12. O is the centre of the circumscribing circle of a triangle ABC, and AO, BO, CO meet the opposite sides in D,E,F. Shew that 1. Parse fully each word which is printed in italics in the following passages: "The remnant of people which hap to be reserved, are commonly ignorant and mountainous people, that can give no account of the time passed; so that the oblivion is all one, as if none had been left." D |