MATHEMATICAL EXAMINATION PAPERS FOR ADMISSION INTO Royal Military Academy, Woolwich, NOVEMBER, 1898. I. PART I. FIRST PAPER. [Great importance will be attached to accuracy.] 2. Find the value of £1875-225 crowns +375 guineas - 46 125 shillings. 3. At what rate of simple interest must £230,398. 6s. 8d. be invested in order that it may amount to £298,941. 16s. 9d. at the end of seven years? 4. Of a certain store of potatoes, II men would in 3 days consume all but 201 pounds, and 21 men would in 4 days consume all but 48 pounds; how many pounds of potatoes does the store contain? 5. If an integer which is less than 50 become a square number when 25 is added to it, show that it can be resolved into two factors whose difference is 10, and hence show that in such an integer the units digit is equal to the square of the tens digit; write down all such integers. 6. Find the value of 3(x+y+z)(yz + zx+xy) − x3 – 13 – 23 when x,y=}, z=3. x2+12+z2 −yz - zx — xy 7. Prove that the remainder which is left when x4+7x+3 is divided by x-1 is the same as the remainder which is left when x-6x+7 is divided by x-2. 8. In any division sum show that a common factor of the divisor and the dividend is a factor of any of the successive complete remainders. In finding the highest common factor of two expressions, 2xa − x3 − 3 was one of the divisors, and 5(x3 +3x+4) was the corresponding remainder. What was the highest common factor of the two expressions? IO. 5 22 I The mth and nth terms of an arithmetical progression are p, q. Find the first term and the common difference. The thirty-first term of a certain arithmetical progression exceeds the square of one-half of the fourth term by unity, and also exceeds twice the fourteenth term by unity; find its values. II. Prove that the number of combinations of ʼn things taken ⁄ at a time is n! You have seven envelopes directed to seven people, to four of whom you intend to send copies of a circular, the other three envelopes to be used for a different purpose. In how many incorrect ways can you put the four circulars in four of the seven envelopes ? 12. Assuming that the binomial theorem is true for a positive integral exponent, prove that it is true for a positive fractional exponent. Apply the binomial theorem to find the fifth root of 3126 to seven places of decimals. II. PART I. SECOND PAPER. N.B.-In questions on Geometry 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. In the absence of special directions to Candidates, any of the propositions within the limits prescribed for Examination may be used in the solution of problems and riders. I. Show that, if from the ends of the base of a triangle two right lines are drawn to any point within the triangle, the sum of these is less than the sum of the other two sides of the triangle, but they contain a greater angle. Show that, if any point inside a triangle is joined to the three vertices, the sum of the joining lines is less than the sum, and greater than half the sum, of the three sides of the triangle. 2. Show that, if the square on one side of a triangle is equal to the sum of the squares on the other two sides, the triangle is right-angled. Explain the meaning of the converse of a theorem. What is the converse of the theorem in this question? 3. Assuming Propositions 12 and 13 of Euclid, Bk. II., prove that the sum of the squares on the diagonals of any parallelogram is equal to the sum of the squares on the sides. 4. Prove that all angles contained in the same segment of a circle are equal, those in a semicircle being right angles. If any two circles be drawn each touching the three sides of a triangle, prove that the circle described on the line joining their centres as diameter passes through two vertices of the triangle. 5. Show how to draw a tangent to a circle from any given external point. Two points are given; one is to be the centre of a circle, and the tangent drawn to this circle from the other is to be of given length less than the distance between the given points; show how to draw the circle and the tangent. 6. Show how to describe a square about a given circle. Every parallelogram described about a circle must have all its sides equal. 7. Define similar triangles, and state (without proof) the relation between their areas. Give any method by which an equilateral triangle may be inscribed in a given triangle ABC, having one of its sides perpendicular to AB. 8. There is a piece of ground in the form of a trapezium, the lengths of the parallel sides of which are 20 and 34 yards, and the lengths of the other two sides 15 and 13 yards; find its area. 9. Prove that the area of the curved surface of a frustum of a cone is equal to the slant height multiplied by the perimeter of the mid section. IO. Find the number of cubic feet of earth removed per yard length from a cutting in level ground 12 feet in depth, the breadth of the base of the cutting being 15 feet, and the slopes of both sides 45°. I. III. PART I. THIRD PAPER. Prove that the angle subtended at the centre of a circle by an arc which is equal in length to the radius is an invariable angle, and explain what is meant by the circular measure of an angle. A circular wire of 3 inches radius is cut, and then bent so as to lie along the circumference of a hoop whose radius is 4 feet. Find the angle which it subtends at the centre of the hoop. 2. Define the cotangent and cosecant of an angle, and show that cot2A + I = cosec2A. Which is greater, the acute angle whose cotangent is, or the acute angle whose cosecant is ? 3. Obtain an expression for all the angles which have a given tangent. Find all the angles, lying between - 360° and +360°, which satisfy the equation 4. Prove geometrically, for the case in which A and B are two positive angles whose sum is less than a right angle, that sin (A+B) = sin A cos B+ cos A sin B. sin 34 Express sin 24 – sin A in terms of cos A. 541. 5. Prove the formulæ (1+cos A) tan2 = 1 - cos A 6. (sec A +2 sin A) (cosec A − 2 cos A) = 2 cos 2A cot 2A. Find the greatest angle of the triangle whose sides are 184, 425, and |