4. Find the nth term of, and sum the series—

(1.) 12 -83+16 etc.

(2.) 1 + 4 + 12 +32 +80 etc.

5. If of 80 persons born together one dies every year till all are dead, what is the chance of a person aged 36 reaching the age of 63 years?

How could you calculate the chance of a boy who is 15 years old surviving his father who is aged 40 ?

6. Show how to extract the square root of an expression of the form a + √b.

7. Give the 7th term of the expansion of (1-2) in its simplest forın; also sum

sin (A+B)=sin A cos B + cos A sin B,

when A and B are each greater than 45 and less than 90°.

9. The apparent diameter of the sun being 31', after looking at it for an instant, I glance at a house a mile away and see that the impression of the sun on the retina just extends from the top to the bottom of the house. Find approximately its height. 10. The sides of a triangle are 300 and 400 feet, and the included angle is 65°. Find the remaining angles. Given L cot 32° 30′ 10.19581, log 7.84510, L tan 12° 38' =9.35051, and L tan 12° 39'

9.35111.

=

11. Find the equation to the straight line which bisects the angle between two given straight lines.

How can the ambiguous sign be determined?

12. Find the equations to the tangents to the circle x + y2 c2, which pass through the point 4c, 3c, and the equation to the chord of contact.

13. Find the polar equation to a conic section, the focus being the pole, and the equation to the tangent to it at any point.

Find the locus of the foot of the perpendicular from the focus on the tangent.

14. Prove that the locus of a point the sum of whose distances from two given points is constant is such that the distance of any point in it from a fixed straight line bears a constant ratio to its distance from one of the given points.

PURE MATHEMATICS: SECOND PAPER.

Examiner.-Prof. C. NIVEN.

Students in naval architecture and marine engineering of first year.

1. Find the equation of a tangent to a parabola, and prove that it may be put into the form y = mx +

a m

Find the locus of the intersection of tangents which inclose a constant angle a.

2. Determine the equations of the lines which join the extremities of the major axis of an ellipse to the extremities of one of the latera recta, and find the poles of these lines.

3. Prove that the portion of the tangent to a hyperbola between the asymptoles is bisected at the point of contact.

Find the tangents to the ordinary and conjugate hyperbolas which pass respectively through the foci of the conjugate and ordinary hyperbolas, and show that, if these tangents be at right angles, the hyperbolas are rectangular.

4. Find the center, asymptotes, and eccentricity of the conic

(x-2y+1)(x + y − 2) = 3x-2y.

5. Investigate, directly from the definition, the differential co-efficient of x".

6. If the fraction f(x) take the form, when x = a, show how to find its true

value.

• (x)

7. Find the maxima and minima values of x'ya2-x2.

If lines be drawn through a point between two given straight lines, find that which cuts off the least area.

dy d'y dr' dr 9. Determine the perpendicular from the origin on the tangent to a given curve at any point.

8. If x and y are connected by the relation (x, y)=O), show how to find

(x2 + y2)1.

The perpendicular on the tangent to the curve (x2+ y2)2 = 2a2xy is · a2 10. Find the radius of curvature of the ellipse 4x2 + 11y248 at the point (1,2). 11. Explain what is meant by the envelope of a system of curves, and show how to find it.

Circles are drawn having their centers on the arc of a parabola and touching the tangent at the vertex; prove that the envelope of these circles is the given tangent and a certain circle.

13. Find an expression for the volume generated by the revolution of a plane curve round a line in its own plane.

Find the area between the curve √ +✓y=√a and the co-ordinate axes, and the volume generated by the revolution of this area about either. Find also the length of the curve between the axes.

PURE MATHEMATICS.

Examiner.-Prof. C. NIVEN.

Students in naval architecture and marine engineering of second and third years.

1. Draw the common tangents to two given circles.

If a quadrilateral circumscribe a circle, the sum of the angles which either pair of opposite sides subtend at the center is the same.

2. Find an expression for the radius of a circle inscribed in a triangle.

If r be the radius of the inscribed circle, and ri, ra, rз the radii of the escribed circles, prove that

rri+rara = bc.

3. Find the equation of the tangent to an ellipse at any point in the form y = mx + √ m2 a2 + b2.

Show that the locus of the intersection of tangents, to the ellipse

inclose a constant angle a, is

(x2 + y2 — a2 — b2 )2 = 4 cot2 a (b2x2 + a2y2 — a2b2).

x2

+

y'

= 1, which

4. For all rectangular transformations of co-ordinates in the equation Ax2 + 2Fxy + By2+2Ex+2Dy+ C = 0, the following functions of the coefficients remain unchanged, viz: AB, AB - F2.

Prove this, and investigate what is known about the curve when one of these expressions varies.

Find the locus of P such that, A and B being fixed points, PAB ~ 5. Show how to reduce the integral:

PBA = a.

6. Investigate an expression for the volume of a surface whose equation is given in polar co-ordinates.

By this, or any other method, find the volume cut off from the solid bounded by the surface of revolution ra (1+ cos 6), by the sphere r

7. Show how to differentiate an integral with regard to a constant contained in the subject of integration.

9. The equation (x2 + 2ax

2yvx2 + 2ax)dy + y(x + a)dx : =0 has an integrating factor of the form ø (x); find it, and solve the equation.

d-y

Solve the equation dx+ay=sin ax, by the method of the "Variation of Parameters."

d3y

Solve also the equation + a3y = x + sin ax.

11. Find the relation between a and b, that the ellipse ax2 + by2 =1 may envelop the parabola y2 = 4cx.

12. Investigate the theory of the number of constants in the solution of a system of ordinary simultaneous equations.

APPLIED MATHEMATICS: FIRST PAPER.

Examiner.-Prof. C. NIVEN.

Students in naval architecture and marine engineering of first year.

1. State the parallelogram of forces, and deduce from it that, if three forces acting on a particle be in equilibrium, each is proportional to the sine of the angle between the other two.

A small ring of given weight rests on the arc of a smooth circular hoop which is fixed in a vertical plane, being attached to the highest point by a string whose length is equal to the radius of the hoop; find the tension of the string and the pressure on the hoop.

2. Investigate the conditions of equilibrium of a particle acted on by any number of forces in one plane.

Three equal spheres rest in one vertical plane against each other, being suspended from a point by strings each equal to the radius of one of the spheres and attached to points in their surfaces. Find the tensions of the strings and the pressures between the spheres.

3. Define the center of gravity of a body, and prove that the center of gravity of a uniform triangular plate coincides with that of three equal particles placed at its angles.

Two uniform rods of the same substance and thickness, and of lengths 5 and 3 inches, are rigidly connected at one end A so as to be at right angles, and are suspended by a string so that the other ends are in the same horizontal plane. Show that the distance from A of the point on the longer rod at which the string is attached is 1.0225 inch.

4. Determine the resultant of a number of forces in one plane. Prove that the equation of the line of action of the resultant is

x'Z(Y) — y' E(X) = 2(xY — yX).

5. State the principal laws of friction which have been deduced from experiment. How can the coefficient of friction be found?

6. Determine the center of gravity

(i) of a sector of a circle.

(ii) of a segment of a circle.

7. Describe the different systems of pulleys, and find whether a weight of 15 lbs. will be able to support 2 cwt. in a system in which there are four movable pulleys, the strings around which are attached to a fixed beam, and each of which weighs one pound.

8. Explain the principle of the screw, and determine the mechanical advantage gained.

9. Explain the difference, in dynamics, between a ton and the weight of a ton. What are their numerical values on the foot-pound-second system? If the weight of a ton be the unit of force, and a minute and yard those of time and length, what will be the unit of mass?

10. Establish the equation v2 = V2+2gx for the motion of a falling body, and express it in the language of the science of energy.

A ball of 10 lbs. is dropped from a height of 289.8 feet, but, after falling half-way, it explodes into two equal parts, one of which is reduced by the explosion to rest. Find the subsequent motion of each part, and determine the kinetic energy developed by the explosion.

11. Find the range of a projectile on a horizontal plane passing through the point of projection.

A particle projected from a point, A, in the floor of a room returns to A after striking one of the opposite walls and the floor successively. Prove that if it strike the

wall at right angles and the floor once its elasticity, and that it strikes the floor half-way between the foot of the wall with half its original velocity and exactly opposite to its original direction.

12. Two balls, A, B, whose masses are as 2: 1, and which are moving in opposite directions, collide. If the first ball be brought to rest, and the coefficient of elasticity be, prove that their original velocities are as 7: 5.

APPLIED MATHEMATICS: SECOND PAPER.

Examiner.-T. S. ALDIS, Esq., M. A.

Students in naval architecture and marine engineering of first year.

1. Define fluid, vapor, gas.

Show that the pressure at any point in a fluid at rest is the same in every direction. 2. What do you mean by specific gravity? How would you compare the specific gravities of (a) two coins, (b) two samples of milk?

3. What is the center of pressure and the total resultant pressure? Find them in the case of an equilateral triangle one of whose sides (12 feet long) is on the surface and the opposite angular point 10 feet beneath it.

4. A hollow cylinder, a foot long, closed at the upper end, weighs as much as half the water it will hold. It is sunk, with the closed end uppermost, in a vertical position in water. How high will the water rise within it when the top is a foot below the surface? At what depth will it rest in equilibrium?

5. Explain how a ship can tack against the wind. A Chinese junk will run before the wind faster than an English ship. In what case, and why, will the ship outsail the junk?

6. In passing from the freezing to the boiling point, air expands .366 of its volume. A cubic foot of air at 60° F. (the barometer standing at 29 in.) weighs 527 grains. What will be the weight of a cubic foot of air when the thermometer is at 90° and the barometer at 28 in. ?

7. Determine the conditions of equilibrium of a floating body. Why is an ironclad with a low free-board specially unfitted to carry sail?

8. In Atwood's machine equal weights of 10 ozs. are suspended to the string which passes over the pulley and a bar of 1 oz. weight is placed across one. This, after falling through the space of a foot, passes through a ring which removes the 1 oz. weight. How far will the 10 oz. weight descend in the next minute?

9. Show how to calculate the space described in a given time under the action of a uniformly accelerating force, the motion being in a straight line.

A stone thrown down a rough board inclined at an angle of 30° neither gains nor loses velocity in its descent. What velocity will it gain by falling down the board (which is 20 feet long) when it is inclined at an angle of 60° ?

10. A particle revolves in an ellipse about a center of force in the focus. Calculate the law of attraction.

11. Explain the action of the governor of a steam-engine. Show how to calculate the position it will assume for a given number of revolutions per minute, neglecting all weights but those of the balls.

12. A smooth bead slides down the arc of a cycloid; determine the motion.

APPLIED MATHEMATICS.-FIRST PAPER.

Examiner.-T. S. ALDIS, Esq., M. A.

Students in naval architecture and marine engineering of second and third years.

1. State and prove, for direction, the principle of the parallelogram of forces, explaining clearly the assumptions you make.