Annual Examinations, 1901. SENIOR LEAVING OR HONOR MATRICULATION. EUCLID. (A. C. MCKAY, B.A. Examiners: A. ODELL. W. PRENDERGAST, B.A. 1. To divide a given straight line so that the rectangle contained by the whole line and one part shall be equal to the square on the other part. (Euc. II, 11.) 2. On a given straight line to describe a segment of a circle which shall contain an angle equal to a given angle. (Euc. III, 33.) 3. To circumscribe a square about a given circle. (Euc. IV, 7.) 4. To circumscribe a regular hexagon about a given circle. 5. State Euclid's test of equality of ratios. Apply the test to show that lines whose respective lengths are 3, 4, 5 and 6 inches, are not proportionals. 6. Parallelograms of the same altitude are to one another as their bases. (Euc. VI, 1.) 7. Equal parallelograms, which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional. (Euc. VI, 14.) 8. APB is any chord of a given circle, drawn through the fixed point P. On AB is described a semicircle, and PC is drawn perpendicular to APB, to meet the circumference of the semicircle in C. Show that the locus of C is a circle. 9. Through a given point within a circle draw, if possible, a chord which shall be divided at the point in the ratio of 2 to 1. 10. Show how to produce a given straight line so that the whole line produced shall be to the part produced in the duplicate ratio of the given line to the part produced. 3. (a) Write the equation whose roots are the reciprocals of the roots of (x+a)2+b(x+a)+m=0. (b) Find four linear factors of 2b2c2+2c2a2+2a2b2-a1bs-c1. (c) Show that the ratio of 999 to 997 is nearly the same as the ratio of 1005 to 997. 4. (a) Show how the whole number, N, may be expressed in the scale whose radix is r. (b) Prove that the difference of two integers, expressed in the ordinary scale and consisting of the same figures, is divisible by 9. 5. (a) If yoox when z is constant, and yo stant, prove that y ∞ when x and z are variable. problem exemplifying this principle. State a [OVER.] 7. (a) Prove that the number of combinations of n things rat a time is equal to the number of combinations of the same number of things taken n-r at a time. (b) How many different permutations may be formed of the letters in the word, arrange, taken all together? 8. (a) Find the sum of the coefficients in the expansion of (1+)10. (b) Show that the coefficient of an in the expansion of (1+x)2n is double the coefficient of an in the expansion of (1+x)2n-1. (c) If Co, C1, C2,..... C'n denote the coefficients of (1+x)", 9. (a) Find the discount on A dollars in t years at r per cent. per annum, compound interest being allowed. (b) Find the cash value of an annuity of A dollars deferred t years and to continue for T years, r being the interest on one dollar for one year. |