10. Explain how probability is measured. A bag contains five sovereigns and ten shillings; if four coins be taken out at random, find the chance that two sovereigns and two shillings will be taken out.

11. Prove that an equation of an odd degree has at least one real root; and that an equation of an even degree with its last term negative has at least two real roots of contrary signs.

Can the equation

x+3x+4x2+5=0

have any real root?

12. Find the equation to a straight line referred to rectangular axes.

Determine the equation to the straight line which passes through the point (1, 2) and is perpendicular to the line y=x+1.

13. Define a parabola, and find its equation in the form y2 = 4ax.

Show that the equation to the tangent at the point (x', y') is

and that if p be the length of the perpendicular upon it from the vertex

p2 (a + x') = ax”.

14. Prove that the equations

Show that the ellipse x + 4y = 4, and the hyperbola x2 - 2y2 = 2, have the same foci; find the coordinates of their point of intersection, and draw figures of the two

curves.

MECHANICS AND HYDROSTATICS.

1. Define the terms resultant force and moment of a force about a point.

Assuming that the moments of two intersecting forces about every point in the line of action of their resultant are equal and opposite, prove the proposition of the parallelogram of forces both as to direction and magnitude.

2. If two equal particles be situated at the extremities of the diameter of a circle, each attracting with a force varying as the distance, prove that the resultant force on any particle on the circumference of the circle is the same for all positions of the particle, and always passes through the centre of the circle.

3. Prove that whenever two parallel forces can be replaced by a single resultant, the moments of the forces about any point in the line of action of the resultant are equal and opposite.

In what case is there no single resultant, and what is the system called in this case?

4. ACB is a bent lever, fulcrum C, and arms inclined at an angle of 135°. When a weight of 2 lbs. is suspended at A, a force of 1 lb. acting perpendicularly to the arm CB at B will keep AC horizontal, and if AC be lengthened and CB shortened, each by one foot, two weights of 2 lb. and 1lb. respectively suspended at A and B will keep BC horizontal; find the lengths of AC and CB.

5. Enunciate the principle of the centre of parallel forces, and apply it to find the centre of gravity of a triangle and of the curved surface of a cone.

If two triangles be formed by joining the middle points of alternate sides of any hexagon, prove that their centres of gravity are coincident.

6. State the laws of statical friction.

A particle of given weight is placed on a rough inclined plane, and attached to a string which passes over the highest horizontal line in the plane, and hangs vertically over it. Find the greatest and least weights which can be attached to the free end of the string consistently with equilibrium.

7. Define acceleration; and assuming that the acceleration of gravity is 32 when a foot and a second are the units of space and time respectively, find its numerical value when a yard and a minute are taken as the units.

8. Explain the proposition of the parallelogram of velocities.

If a railway train be moving at the rate of 20 miles an hour, and a heavy body fall from the roof to the floor of a carriage in one second, find the inclination of the path of the body to the horizon in fixed space at the instant before the end of its fall, the acceleration of gravity being 32 on the usual assumption as to units.

9. The path of a projectile in vacuo is a parabola. A bombshell on striking the ground bursts, scattering its fragments with a mean velocity v; find the area of ground covered by the fragments, assuming that it falls on a level surface. If it falls on a road running up a hill, find the greatest distances reached up and down the road respectively.

10. Find the conditions of equilibrium of a body floating in a fluid.

11. Define, and give equations for finding, the centre of pressure of a plane surface in contact with a heavy fluid at rest.

A cubical box has a heavy lid moveable about hinges on one of the edges. If the box be filled with water and placed on a rough inclined plane with the

line of hinges horizontal, find the inclination of the plane that the water may just begin to flow out.

The line of hinges being towards the upper end of the plane.

12. State the relation between the pressure, density, and temperature of a gas.

Explain the construction and action of the barometer.

If a tube of given length, and closed at one end, be filled with atmospheric air, and the open end be immersed to a given depth below the surface of mercury in a cistern, find how high the mercury will rise in the tube, the height of the mercurial barometer being h, and the temperature of the air in the tube being constant.

Any of the following questions may be substituted for an equal number of the above.

A. Define, and find the equations of the central axis of a given system of forces on a rigid body.

Ex. Three given forces acting along three edges of a cube, no two intersecting and no two parallel.

B. If the attached weight in No. 6 be intermediate between the greatest and least weights consistent with equilibrium, find the inclination of the string to a horizontal line in the plane when there is equilibrium; also find the direction in which the weight on the plane will begin to move if the attached weight be increased.

C. The resultant attraction of a uniform straight line upon any particle bisects the angle subtended by the line at the particle.

D. Prove the equation h'u' (du

forces.

+

u)

= P in central

E

PURE MATHEMATICS.

1. In any spherical triangle prove that the sines of the angles are proportional to the sines of the opposite sides.

2. Enunciate Napier's rules for the solution of rightangled spherical triangles, and prove them for the cases in which one of the sides containing the right angle is the middle part.

3. If a be the spherical excess of any spherical triangle, prove that

b

2

cot 2

cot + cos C

4. Find from definition the differential coefficients of

x" (n unrestricted) and of sinx.

5. Find the 7th differential coefficient of x"e" and

prove that if

mx

72

y" -ny + (n-1) x = a, then

day
dx2

6. Write down Taylor's expansion for f(x), with the remainder after n terms.

Expand ee to 4 terms, and find the first four terms of the expansion of y in terms of x in the equation x2 + y2 − 6x + 4y + 4 = 0.

7. State fully the conditions that f(x) may be a maximum or a minimum.

Apply to 3x + 8x3 + 6x2 + 3.

Upon the same circular base, and upon opposite sides of it, two right cones are constructed so that the volume of the whole figure thus formed may have a given value. Find the condition that the surface of the figure may be a minimum.