JUNIOR FRESHMEN.

Mathematics.

A.

MR. WILLIAMSON.

1. Find the locus of a point such that the sum of the squares of its distances from four given points shall be equal to a given area.

2. Describe a regular hexagon which shall be equal to a given area. 3. Divide a triangle into five equal parts, by lines drawn parallel to one of its sides.

4. Calculate, to four decimal places, the area of the equilateral triangle whose side is 10.

5. Solve the simultaneous equations

vy -√a−x + √y-x=0

7. Determine the length of the side of a regular hexagon inscribed in a circle.

8. If a quadrilateral be inscribed in a circle, prove that the sum of the rectangles under the opposite sides, is equal to the rectangle under the diagonals, of the quadrilateral.

a. Hence prove the 47th Prop. Euc., Liber I.

9. If a circle touch two fixed circles the chord of contact passes through a fixed point on the line joining the centres of the fixed circles.

11. Determine the result of substituting for x in x + 12x value 1.22748; correct to three places of decimals.

12. Substitute x + 3 for x in the function

19x1 — 22x3- 35x2 – 16x – 2,

and arrange the result in powers of x.

17 the

MR. M'CAY.

13. On the base of a right-angled triangle ABC, AE is taken equal to BC, and BD equal to BC; prove DE2 = 24C. BC.

14. Describe a square, being given the difference between a diagonal and side.

15. Find the coefficient of x9 in (5a3 – 4x3)7.

17. Construct an isosceles triangle each of whose base angles shall be double the vertical angle.

18. If gold be beaten out so thin that an oz. av. will form a leaf of 20 sq. yds., how many of these leaves will make an inch thick, the weight of a cubic foot of gold being 10 cwt. 95 lbs.?

B.

MR. WILLIAMSON.

1. If A, B, C denote three given rectilinear figures; construct a figure similar to C, and having to it the ratio that A has to B.

2. Find the simplest value of the quantity

4+ √7 + √ 4-√7.

3. With three given points as centres, describe three circles, each of which shall touch the other two.

find the values of A, B, C in terms of m and n.

5. Find an expression for the area of a quadrilateral which is inscribable in a circle, in terms of the four sides, and express the radius of a circle in terms of the same lines.

6. Eliminate

MR. BURNSIDE.

between the equations

a tan (x + a) = b. tan (x + ẞ) = c. tan (x + y).

8. Inscribe in a given quadrilateral a quadrilateral similar to a given

one.

9. Expand

N I in the form of a continued fraction.

10. Find the sum of π terms of the series

11. A'B'C' are points on the sides of the triangle ABC and AA', BB', CC' meet in a point; prove that

where a, B, γ are alternate segments of the sides a, b, c.

12. Find a point within a triangle the sum of whose distances from the vertices shall be least.

13. Solve the equation

% – 5 = I2 + 8 và

14. If the tangents of the semi-angles of a triangle be in arithmetic progression, so also will the cosines of the angles be.

15. There are two vessels, one containing a gallons of wine and the other b gallons of water; show that a measure can be found of such a size (x) that after one interchange of a measure of wine for water, and a measure of water for wine, the mixture shall not be altered in strength by any further such exchanges.

2. Find an expression for the coefficient of xn in the expansion of

eax cos ẞx, in a series of ascending powers of x.

3. Being given the equation = y + 2y3; find, by the reversion of series, the expansion of y in a series of ascending powers of x.

4. Find two numbers whose difference multiplied by the difference of their squares is 160, and whose sum multiplied by the sum of their squares is 580.

5. The sum of five numbers in arithmetical progression is 35, and the sum of their squares is 285; find the numbers.

8. Given the lengths of the perpendiculars drawn from the vertices to the opposite sides, construct the triangle, and give a formula for its area in terms of the perpendiculars.

9. Prove that the sum of the binary combinations of m quantities in

geometric progression is

r

r + I

Sm, Sm-1, where is the common ratio

and Sp the sum of p terms of the series.

10. Prove the binomial theorem when n is a positive integer.

13. Give any proof of De Moivre's theorem.

14. A'B'C' are points on the sides of the triangle ABC, and

Express the ratio of the triangles ABC, A'B'C' in terms of m,

15. The problem to inscribe a triangle in a circle so that the sides may pass through given points is indeterminate when the given points are vertices of a triangle self-conjugate to the circle.

D.

MR. WILLIAMSON.

1. In a quadrilateral ABCD, if CA' be drawn parallel to DB and meeting AB produced in A', and if BD' be drawn parallel to AC meeting DC produced in D', prove that the line A'D' is parallel to AD.

2. Construct a circle touching two given circles, and bisecting the circumference of a third circle.

3. Prove that the area of any triangle is equal to the rectangle under the semi-perimeter of the triangle formed by joining the feet of the perpendiculars and the radius of the circumscribed circle.

4. Find the centre and the radius of the circle with respect to which the inverses of three given circles shall be each of given radius.

MR. BURNSIDE.

5. Distances OP, OQ are measured from the vertex of a triangle AOB of which base and vertical angle are given, along the internal bisector of that angle; prove that the loci of P and Q are circles when OP and OQ are the arithmetic and geometric means between the sides 40, OB of the triangle respectively.

6. Three fixed coaxal circles are intersected by every circle not of the system, in three pairs of concyclic points in involution ?

7. Show by inversion from the middle point of one of the sides of a triangle, that the “nine point circle” may be transformed into the second transverse common tangent to the inscribed circle and exscribed which touches this side internally.

8. Prove that the six distinct anharmonic ratios of four points A, B, C, D in a right line, namely,

(ABCD), (ACBD), (ADBC), (ABDC), (ACDB), (ADCB)

may be expressed as follows:

9. Four circles have a common point; draw a line through this point so as to be cut harmonically by the circles.

10. Given two pairs of points on a line, determine the magnitude and position of the segment cutting both harmonically.