8. Define the moment of a force about a line. A rigid uniform triangular board is supported by three equal vertical strings whose upper ends are attached to the corners of an equal triangle fixed in a horizontal plane, and a sphere of given weight is placed at a given point on the board. Determine the tensions of the strings. 9. Explain what is meant by the coefficient of statical friction between two bodies. A rigid uniform bar rests against a vertical wall and a horizontal floor, the vertical plane containing the bar being at right angles to the wall. Prove that, if the bar is on the point of slipping down, the angle a which it makes with the horizontal is given by the equation tan a = (1/μ' – μ) where μ and u' are the coefficients of friction at the upper and lower extremities. IO. A smooth sphere of mass m moving with velocity u impinges directly on a smooth sphere of mass m' at rest. Find the velocities of the spheres after impact, the coefficient of restitution being 4. Prove that, if I is the magnitude of the impulse between the spheres, the kinetic energy lost in the impact is § Iu. II. Prove that a particle moving freely under gravity describes a parabola. If the particle is projected at an elevation a and strikes an inclined plane through the point of projection at right angles, the inclination of the plane to the horizontal is given by the equation tan (a-0)=cot 0. 12. Two particles of equal mass are connected by an inelastic string, and held near to each other on a smooth horizontal table. A ring of mass equal to that of either particle is threaded on the string and hangs just over the edge of the table. Prove that, when the particles are let go, the ring descends with acceleration g. MATHEMATICAL EXAMINATION PAPERS FOR ADMISSION INTO Royal Military Academy, Woolwich, JUNE, 1899. I. I. PART I. FIRST PAPER. [Great importance will be attached to accuracy.] 201 Reduce to their simplest forms: +1+1, 2}−2%, and 24-1, and divide the product of these three expressions by +38-2. 2. A tradesman, having promised me 12 per cent. reduction from a bill for 16. 16s. 8d. takes off only 10 per cent. By what sum am I entitled to further diminish his bill? And how much per cent. is this of the account rendered? 3. Define the average of a set of numbers, and find the average of 213, 217, 199, 201, 208, 209, and 211. If the height of a number of men is measured; and the average of the first six is 5 ft. 5 in., of the next seven, ft. 5 in., and of the next eleven, 5 ft. 4 in.; what is the average height of the whole number of men to the nearest hundredth of an inch? 4. Two trains start at the same time, one from Liverpool to Manchester, and the other in the opposite direction, and running steadily complete the journey in 42 minutes and 56 minutes respectively. How long is it from the moment of starting before they meet? 5. A certain number is the product of 12 and another factor. The number is divisible by 16 and by 17. Show that the second factor is divisible by 17, but not necessarily by 16. State a general proposition of which this is a particular case. 6. Multiply together any three consecutive whole numbers, all greater than 100. Add to this product the middle number, and show (by extracting the cube root or otherwise) that the result is a perfect cube. If a is the smallest of the numbers, state and prove the result just given in a general form. 7. Divide x7-729x by x2-3x+9, and express the quotient in its simplest real factors. Also if the square of the first side of the above equation is equal to the second side (unaltered), find the two values of x; and explain why the first equation has only one solution and the second two. 9. Find the values of x and y in terms of a and b from the equations If the values of x and y which are to satisfy this equation are 50 and 51, what must be the values of a and b? IO. An arithmetical and geometrical progression have the same first term 10; and the common difference of the former series, which is 7, is equal to the common ratio of the latter series. Find if any of the numbers 3430, 3455, 3475 belong to either of these series, and, if so, to which. If -a and a are two terms in an arithmetic progression, and the number of intermediate terms is 27, find the value of the two terms which are numerically smallest. II. Find the number of permutations of four different things taken all together, giving your reasons in full. A boy, fresh from school, boasts that he is able to distinguish the four brands of champagne (of which he knows the names) in his father's cellar. Accordingly his father fills him a glass of each wine without allowing him to see the labels on the bottles, and requires him to name each variety. The boy gives the four names and gives each wrong. Determine in how many different ways he can do this, and in how many ways he could have named at least one right. I2. Prove the Binomial Theorem for a positive integral index. Employ it to show that, to the nearest pound, the interest on £10,000 for 10 years at 2 per cent. compound interest per annum is £2190. 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. Name and define the different four-sided figures of which Euclid takes note. If the middle points of every pair of adjacent sides of a rhombus be joined, what is the four-sided figure so formed? Prove that your answer is correct. 2. Through a given point draw a straight line parallel to a given straight line. Construct a triangle, having given one angle and the lengths of the perpendiculars from the other two angles on the opposite sides. 3. If a straight line be divided into two equal parts, and also into two unequal parts, prove that the rectangle contained by the two unequal parts, together with the square on the line between the points of section, is equal to the square on half the line. Compare the lengths of the two unequal parts, if the rectangle contained by them is eight times the square on the line between the points of section. 4. Draw a tangent to a circle from an external point. Given a circle and a straight line, find a point on the line such that, if tangents be drawn to the circle, the chord of contact will subtend a given angle at the circumference. Is it always possible to solve this problem? Give your reasons. 5. A chord AB of a circle is produced to C, so that BC is 7 inches in length. The tangent from C is one foot in length. What is the length of AB? |