12. A weight W lbs. is raised from rest through a vertical height H feet in Tseconds, when it again comes to rest, by means of a chain which sustains a uniform tension for a part of the time and then becomes slack. Show that the weight ascends with acceleration

Further show that if the safe tension of the chain be P lbs. weight the time 7 must not be less than

VII. PART III. SECOND PAPER.

[Full marks will be given for about two-thirds of this paper.]

I. Show how to draw two tangents to a parabola from a given external point.

Prove that, if three tangents form a triangle PQR, and p is the point of contact of QR, and S the focus, then

Qp: Rp = PR. SQ : SR. PQ, the point p being between Q and R.

2. Prove that, if CP, CD are conjugate semi-diameters of an ellipse whose foci are S and S', then SP. S'P= CD2.

Prove also that, if the tangents at P and D meet in T, and if SP is produced to R so that PR = SP, then the triangles STR, SDT are similar.

3. Define the asymptotes of an hyperbola. Prove that, if the tangent at P meets one asymptote in 7, and the focal radius SP meets the same asymptote in U, the triangle STU is isosceles.

4. Find the angle between the two straight lines whose equations are y = mx + b and y = m'x+b'.

Form the equation of the straight line drawn through the point x = 1, y = 2 at right angles to the line x + 2y = 3.

5. Find the equation of the circle which has a given centre and radius. Prove that the locus of a point from which the tangents to two given circles are in a constant ratio is a circle.

6. Find the equation of the locus of the middle points of the chords of the parabola y2 = 4ax which make an angle @ with the axis of x.

Prove that the locus of the middle points of chords of y2 = 4ax, drawn through a given point, is a parabola, and explain the connexion between the two results.

7. Find the equation of the normal at any point (x', y') on the ellipse x2/a2+y2/b2 = I. Prove that the normals at the extremities of the chords lx/a + my/b = 1 and x/la+y/mb = I meet in a point, and find its co-ordinates.

I

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 μ' are the coefficients of friction at the upper and lower extremities.

10. 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 1.

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 θ of the plane to the horizontal is given by the equation

tan (a 0) = cot θ.

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 accelerationg.

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.]

Reduce to their simplest forms: ++, 2-, and 2-,

and divide the product of these three expressions by + - 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, 5 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?

10. 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 2n, 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.