ABCD is a rectangle such that the square on AD is twice the square on AB. BE is drawn at right angles to AC to meet AC in E. Prove that AE is one-third of AC. 9. If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, prove that the triangles are similar.

10. Draw a straight line 5 inches long and divide it into three parts proportional to 37:46:53.

UNIVERSITY OF LONDON.

MATRICULATION EXAMINATIONS, JANUARY, 1916.

ARITHMETIC AND ALGEBRA.

Tuesday, January 11-Morning, 10 to 1.

1. Find x from the equation

x=635-1540X(.642)2.

If .642 denotes a number lying between .6415 and .6425, what are the extreme values of x? Give all the required values of x to three significant figures.

2. A consumer receives notice at the end of a quarter that the charge per unit of electricity supplied is to be raised p per cent. In the next quarter he succeeds in reducing the number of units consumed by q per cent. Give a formula for the percentage increase in the bill for the quarter.

If p=10, and the bill is decreased by 64 per cent, find q.

3. There are two kinds of floorcloth of the same pattern, one 27 inches wide costing 2s. 6d. per yard length, the other 45 inches wide costing 7s. 6d. per yard length. A lady requires 24 square yards of floorcloth altogether and buys as much as she can of the wider kind, but is determined not to spend more than £5. What length of the wider kind will she be able to buy?

7. (i) If 3x2+12xy+8y2+12x-10y-63-0, find y when x=7.

For what values of x is 27 −10x+x2 less than 12x-29-x2?

9. A sum of £1,000 is set aside at the end of each year and invested at 5 per cent compound interest. Find approximately the total sum accumulated at the end of 19 years, including the £1,000 set aside at the end of the nineteenth year. [Use which you require of the following:

(1.05)18-2.406619 (1.05)19 2.526950 (1.05)20-2.653298.]

10. A and B could between them type 4,500 pages of manuscript in 100 working hours, and undertook to do so. After 36 working hours B was replaced by C, and the first half of the task was completed in 54 working hours from the start. After 80 working hours from the start B returned, and all three just finished the task in time. Find the average number of pages typed in an hour by A, by B, and by C.

GEOMETRY.

Tuesday, January 11—Afternoon, 2.30 to 5.30.

1. Prove that, if two sides of a triangle are unequal, the angle opposite the greater side is greater than the angle opposite the less. Also, state and prove the converse of this theorem.

In an acute-angled triangle ABC in which AB is greater than AC, the lines drawn from B, C perpendicular to the opposite sides intersect in O. Prove that OB is greater than OC.

2. Prove that the diagonals of a rhombus bisect each other at right angles. The side BC of a rhombus ABCD is produced through C to a point E so that CE is greater than BC. The line ED is drawn and produced to cut CA produced in F, and F is joined to B. Prove that the angles BFA, EFA are equal. 3. Prove that parallelograms on the same base and between the same parallels are equal in area.

ABCD, AEFG are two parallelograms having a common point at A, and having the vertex E on BC, and the vertex D on FG. Prove that the parallelograms are equal in area.

4. In a right-angled triangle prove that the square on the hypotenuse is equal to the sum of the squares on the other two sides.

If two unequal right-angled triangles ABC, ADC are drawn on opposite sides of their common hypotenuse AC, and if AM, CN are drawn perpendicular to BD, cutting it in M, N, prove that BM2+BN2=DM2+DN2.

5. If a straight line AB is bisected at C and produced to any point D, prove that CD2=AD. BD+AC2.

Show how to find the position of D by a geometrical construction so that the rectangle AD. BD shall equal the square on AB.

6. Show, with proof, how to construct a square equal in area to a given triangle, illustrating your method by a well-drawn figure.

7. Prove that the angle at the center of a circle standing on a given arc is double any angle at the circumference standing on the same arc.

Prove that, if a square be described externally on the hypotenuse of a right-angled triangle, and the right angle be joined to the center of the square, the joining line will bisect the right angle.

8. The angles A, B of a cyclic quadrilateral ABCD are 110°, 85°, respectively, and AB subtends an angle of 65° at the point of intersection of the diagonals. Find the angles which the sides subtend at the center of the circle.

9. Prove that in equal circles (or the same circle) equal angles at the centers stand on chords which are equal.

Given the base BC and the vertical angle A of a triangle ABC, the angle A being acute; prove that if BM is drawn perpendicular to AC, cutting it in M, and if CN is drawn perpendicular to AB, cutting it in N, the line MN is of constant length.

10. If two chords AB, CD of a circle, on being produced, meet in a point P, that the rectangles PA. PB and PC. PD are equal.

prove

Two circles having a common chord AB cut a third circle, the chords of intersection with it being CD, EF, respectively. Prove that AB, CD, EF, produced if necessary, intersect in a common point.

MECHANICS.

Wednesday, January 12—Afternoon, 2.30 to 5.30.

1. The gravitational acceleration at the surface of the moon is approximately 5.4 feet per second. Calculate (a) the time taken by a projectile, starting vertically upward from the surface of the moon with a velocity of 120 feet per second, to return to its starting point, and (b) the maximum height reached.

How do these results compare with those upon the earth's surface?

2. Explain the variations of the force between the floor of a lift and the feet of a man standing in it during the upward and downward journeys from rest to rest.

3. Give exact definitions of the terms force, momentum, work, and power, and explain the connections between (a) force and momentum, (b) force and work, and (c) work and power.

4. Explain the principle by which the resultant of two forces not in the same straight line is determined.

The bob of a simple pendulum is deflected so that the string makes an angle of 30° with the vertical. It is then released. Calculate the direction and magnitude of the acceleration with which the bob begins to move.

5. Show how to calculate the magnitude and position of the resultant of a number of parallel forces acting in a plane.

Equal weights are situated at five of the angular points of a horizontally placed regular hexagon of side a. Find the line of action of the single force which would be in equilibrium with the weights.

6. Apply the principle of work to determine the mechanical advantage of a smooth plane inclined at an angle a to the horizontal, the load being raised by a horizontally applied force.

Show what would be the effect of friction on the mechanical advantage; and explain what is meant by the efficiency of a machine.

7. Define the density and the relative density of a substance.

Describe carefully how you would carry out-using a suitable bottle-a determination of the density of (a) a liquid, (b) a powder soluble in water, but insoluble in the liquid.

8. Explain how the pressure of the atmosphere can be accurately measured.

At the top of a mountain a mercury barometer reads 67.2 cm. What is the pressure in kilograms weight per sq. cm., given that mercury has 13.6 times the density of water.

APPENDIX B.

ENGLAND.

ENTRANCE SCHOLARSHIPS EXAMINATION PAPERS.

CAMBRIDGE UNIVERSITY.

EXAMINATION FOR SCHOLARSHIPS, EXHIBITIONS, AND SIZARSHIPS.

Trinity, Clare, Trinity Hall, Peterhouse, and Sidney Sussex.

December 7, 1910. 9-12.

1. The lines AB, BE, CF are drawn perpendicular to the sides BC, CA, AB of the triangle ABC, and EF, FD, DE are drawn to cut BC, CA, AB, respectively, in X, Y, Z. The tangents at A, B, C to the circle ABC cut BC, CA, AB, respectively, in P, Q, R, and L, M, N are the middle points of AP, BQ, CR, respectively. Prove that the six points X, Y, Z, L, M, N are on the radical axis of the circles ABC and DEF.

2. A, B are the points of contact of a common tangent of two given circles, and any line parallel to AB cuts one circle in P and the other in Q. Prove AP and BQ intersect on a fixed circle coaxial with the given circles.

3. Find three points X, Y, Z on the sides BC, CA, AB of the triangle ABC, such that YZ, ZX, XY will pass, respectively, through three given collinear points L, M, N.

Hence, or otherwise, find the points of contact of three given tangents to a conic having also given the pole of a given straight line.

5. Prove that, if a, ß, y are the three roots of the equation

then will

6. Prove that

x3-21x+35=0;

a2+2a-14 be equal to ẞ or to y.

cos 2a sin (B-y)+cos 2ẞ sin (y-a)+cos 2y sin (a−ẞ)

=4 sin † (8—7) sin † (7—α) sin (a−ẞ) {cos (ẞ+y)+cos (y+a)+cos (a+B)}. 7. The diameter AB of a circle is produced to C so that BC is equal to the radius OB of the circle; CD is drawn perpendicular to OBC and CD=BC. A point P is taken on the circle on the same side of AB as the point D, and such that AOP is half a right angle. Prove that, if Q is the point where PD cuts the circle again, ZBOQ is 1.001 radians, very nearly.

8. Prove that chords of the ellipse x2/a2+y2/b2—1=0 which subtend a right angle at a given point P of the ellipse intersect the normal at P in a point P', such that PP' is equal to 2abd/(a2+b2), where d is the semidiameter conjugate to CP.

Prove that as P varies the locus of P' is a similar ellipse, and that the normals to the ellipses at P and P', respectively, intersect either ellipse in four concyclic points.

9. Show that the locus of the intersection of normals at the extremities of chords of a parabola which pass through a fixed point is a parabola, and find the direction its axis.

10. Find the equation of the tangent at the point (p2, 1, p) on the conic aß-X2=0; and prove that, if the sides of a triangle touch the conic and two of its vertices are on the lines a-λ=0 and a-λ1⁄2ß=0, respectively, the locus of the third vertex is the conic

11. Show that if 0 and lie between 0 and π, a2 is less than 1, and (1-2a cos 0+a2) (1+2a cos ¢+a2)=(1−a2)2,

12. Prove that if the chord of curvature through the origin is 2rn/an−1 (n=1), for any point of a curve at distance r from the origin, then the radius of curvature is proportional to

n-1 a

Candidates are requested to attempt at least ONE question from each section of the paper, and not more than THREE in all.]

1. Write a short account of the method of projection in geometry. Include in this account the properties of a projected figure corresponding to (a) circles, (b) right angles, (c) a pair of equal angles, (d) middle points of lines, and (e) foci of conics, in the original figure, and illustrate your theory by stating in a form true for all conics the property that the angle at the center of a circle is double that at the circumference. 2. Prove that if m is prime to a the least positive remainders of the series of integers k+(m−1)a

k, k+a,

...

with respect to m are a permutation of the numbers of the series

where ø (m) is the number of integers less than m and prime to it.

Prove, also, that (m−1)!+1 is divisible by m if, and only if, m is a prime.
Show that if m is a prime and p<m,

(p−1)!(m−p)!+(−1)P ̄1=0(mod. m).

3. State and prove the leading propositions in the theory of determinants and indicate some applications of the theory.

4. Starting from the definition of a differential coefficient, develop methods and results which will enable you to differentiate any function obtained by combining exponential functions, circular functions, powers, and the inverses of these functions. 5. Establish formulæ for the curvature at any point of a plane curve, including the cases when the curve is defined (a) by a Cartesian equation, (b) by equations of the type x=(t), y=4(t), (c) by an intrinsic equation, (d) by a p, r equation, and (e)