MATHEMATICAL EXAMINATION PAPERS FOR ADMISSION INTO Royal Military Academy, Woolwich, JUNE, 1896. OBLIGATORY EXAMINATION. I. EUCLID. [Ordinary abbreviations may be employed; but the method of proof must be geometrical. Proofs other than Euclid's must not violate Euclid's sequence of propositions. Great importance will be attached to accuracy.] I. ABC and DEF are two triangles. If the sides BC, CA, AB are equal to the sides EF, FD, DE respectively, prove that the angle BAC is equal to the angle EDF. 2. ABC is any triangle, D any point inside the triangle. Prove that BD and DC are together less than BA and AC, and that the angle BDC is greater than the angle BAC. The angle BAC is bisected by a straight line meeting BC in E, and P is any point on this straight line within the triangle. Prove that (i.) BA and AC are respectively greater than BE and EC; (ii.) BA and AC are respectively greater than BP and PC. 3. Write down Euclid's axiom with regard to parallel straight lines. OA, OB are two finite straight lines meeting at O. C and D are any two points in OA and OB respectively. Through C and D straight lines are drawn at right angles to OC and OD respectively. Prove that these straight lines, if produced indefinitely in both directions, must meet. 4. C is a point in a finite straight line AB. Prove that the square on AB is equal to the squares on AC and CB, together with twice the rectangle contained by AC and CB. Express this result algebraically. 5. ABC is an acute-angled triangle. AD is drawn perpendicular to BC. Prove that the squares on AB and BC are together equal to the square on AC and twice the rectangle contained by BC and BD. If M is the middle point of BC, prove that the difference of the squares on AB and AC is equal to twice the rectangle contained by BC and MD. 6. If a chord and a diameter of a circle intersect at right angles, prove that the diameter bisects the chord.. AB is a diameter of a circle. PQ is a chord not at right angles to AB. AM and BN are drawn perpendicular to PQ. Prove that PM = NQ. 7. If two circles touch one another externally, prove that the straight line which joins their centres passes through the point of contact. 8. Show how to draw tangents to a circle from a given point outside it. P and Q are points on a diameter of a circle produced, and are at equal distances from the centre. PR and QT are tangents from P and Q, the points Rand 7' lying on opposite sides of PQ. Prove that PRQT is a parallelogram. 9. Prove that angles in the same segment of a circle are equal. AB and CD are two intersecting chords of a circle. If the arcs AD and BC are together equal to the arcs DB and CA, prove that AB and CD are at right angles to one another. IO. Show how to describe a triangle whose sides shall touch a given circle and whose angles shall be equal to the angles of a given triangle. II. Show how to find a mean proportional between two given straight lines. 12. AB is a diameter of a circle, CD is a chord at right angles to it. If any chord AP drawn from A cuts CD in Q, prove that the rectangle contained by AP and AQ is constant for different positions of P. 3. In how many years will £1500 amount to £1781. 5s. od. at 24 per cent. per annum simple interest? 4. 5. of £53. 6. Find the value of 30875 of a mile in furlongs, poles, and yards. Express the sum of 7 guineas 3 crowns and 51 florins as a fraction What is the rental of an estate of 645 acres 3 roods 25 poles at £2. 11s. 4d. per acre? 7. Find the least common multiple of 555, 1221, and 2035. 8. Find the cost of a carpet, for a room 24 feet long and 173 feet wide, at 3s. 3d. per square yard, if a margin 2 feet wide be left uncovered. 9. A watch is offered for sale for £5. 15s. od.; and, if that price is reduced by 5 per cent., the dealer who is selling it will still make 94 per cent. profit: how much did the watch cost him? 10. State the rule for determining the remainder in a division sum when it is worked by dividing by two factors of the divisor successively. instead of by their product. As an example, reduce 107 lbs. to quarters by dividing by 4 and by 7; and explain why you do what you do to find how many pounds there are over. II. A man goes bankrupt with £1160 assets. His liabilities are the present value of three loans, at simple interest, amounting in all to £1800; one of them obtained 8 years ago at 4 per cent.; another, double of the first, 4 years ago at 5 per cent.; and the third, equal to the sum of the other two, a year ago at 8 per cent. How much does each of the three creditors lose? 12. A cistern 12 feet deep, of which the length is double the width, holds 21 tons of water: find the depth of another cistern which is 1% ft. shorter and 4 ft. wider than the other, and which holds the same quantity. One cub. ft. weighs 1000 ozs. 13. Taking the values of zinc and copper to be £17. 14s. oď. and £73. 15s. od. per ton respectively, find how much of each metal there will be in a mass compounded of the two, which weighs 14 cwt., and is worth £23. 125. Od. 14. A ship, steaming towards a port in a fog, fires a signal gun which is answered from the port, as soon as heard, by another gun; the report of the latter reaches the ship 25 seconds after the first gun was fired; the ship fires again immediately, is answered as before, and this time the reply is heard in 24 seconds. If the sound travels 1100 feet per second, at what rate must the ship be steaming? III. ALGEBRA. (Up to and including the Binomial Theorem, and the theory and use of Logarithms.) I. [N.B.-Great importance will be attached to accuracy.] Remove the brackets in 7a+6[b-5{c+4(b−3 (a+2c))}], and find its value when a = 2, b = 3, c = 1. 2. Prove that axbbxa, where a and b may have any positive integral values. Multiply together x-y, x+y, x2-xy+y2, x2+xy+y2. 3. Show that xn-an is divisible by x-a for all positive integral values of ??. x2 - 15x3+75x2 - 145x+84 and xa − 17x3+101x2 – 247x+210 have the same H.C.F. and the same L. C. M. as x1 − 13x3 +53x2 -83x+42 and x − 19x3 +131x2 - 389x+420. 5. In an examination a candidate takes five compulsory and two optional papers, makes the same marks on each paper, and gets fifty marks too few to pass. On a second attempt he increases his marks on each paper in the ratio of 63 to 50, and omits one of the optional papers, thus securing a hundred marks more than before and passing. If all the papers have the same maximum, find what number of marks is required to pass. |