11. If from the vertical angle of a triangle a straight line be drawn perpendicular to the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circle described about the triangle.

Prove that the ratio of the diameter of the circle circumscribing a triangle to any one of the sides is equal to the ratio of half the rectangle of the two remaining sides to the area of the triangle.

ALGEBRA.

1. Find the values of (x − 10)2+ (x − 6)2 when x = 5, and when x=8+2√(-1). Explain why any real numerical value substituted for x in the above expres

sion will always give a positive result. If 5* find x.

2. Reduce to their simplest forms

=

1

625

(1) (x+a+b) (x − a − b ) (x + a − b) (x ·
b) (x+a-b) - a+b),
(x* — 4x3y + 6x3y2 — 4xy3 + y1) × (x2 + 2xy + y2)
(x2 - y2)2

4. If √{a+√(b)} = x+y, where (a) is a rational quantity, √(b) a quadratic surd, x and y one or both quadratic surds; prove √{a-√√(b)}=x-y. Extract the square root of 7-2 (10). Reduce

(3a — 2133 + 6 − 2*33 + 223a — 2o) × (3a + 2a1).

(3) (x − 3)3 + (x − 4)3 = 43 {(x − 3)2 – (x − 4)*}.

6. A clothier sold an army tailor 60 yards of scarlet cloth and 120 yards of coarse blue cloth for £49. 10s., and for £2. 5s. he sold 4 yards more of the blue cloth than of the scarlet cloth; what was the price per yard of each cloth?

7. Eliminate between the equations

x3- px2 + qx - r=0 and -y=0;

express the resulting equation free from surd quantities and arranged according to decreasing integral powers

of y.

8. Write down nth term of the arithmetical series of which the two first terms are a-3b, a-b, show that the sum of every pair of terms equidistant from the beginning and end of the series is invariable, and express the sum of (n) terms of the series.

The first term of an arithmetic series is 8, the common difference 4, and the sum of the series 108, find the number of terms and show how the negative value of (n) obtained from the formula is to be interpreted.

9. The condition that three quantities shall be in harmonical progression can be expressed by a property of proportion; state that property, and show that it is

The arithmetic mean between A and B is (1 + x2)2 and the harmonic mean (1-x), find the geometric mean; determine also A and B.

10. Assuming the general formula for the number of combinations of (n) things taken (r) at a time show that the formula will give the same number whether r = 4 or r = (n − 4), verify this when n is 10; also from the theory of combinations show that the product of any (n) consecutive integers is divisible by (1.2.3...n).

11. Assuming the form of the binomial theorem when the index is integral and positive, prove the theorem when the index is integral and negative. When the index (n) is positive, show that the 7th coefficient is greater than the (r+ 1)th coefficient if r be greater than

n+1
9
2

and when (n) is integral and negative,

show that the coefficients continually increase.

Find by the binomial theorem an approximate value

to (1013).

PURE MATHEMATICS.

1. Express (11) and 4-√(11) in the form of a continued fraction.

2. Find a solution in positive integers of the equa

tion 7x+11y = 245.

Find also the number of solutions admissible.

3. Given that

a2 = 1+a,x+a ̧x2+ax+ &c. ad infinitum.

4. Prove that log a" = m log a to any base. Given that log, 1.25890999912,

logo 1.25901000257.

10

Find 1(10) to 5 places of decimals.

5. Find the chance of throwing 6 with four counters, each of which is marked with 1 on one side and 2 on the other.

6. Prove that cos (A + B) = cos A cos B- sin A sin B. Find also sin 3A.

7. Prove that in any triangle ABC whose sides are a, b, and c,

a2 = b2+ c2 2bc cos A. Given b=2, c=3, cos A = . Find a, cos B, and

cos C.

8. If p be the greatest negative coefficient of an equation, prove that p+ 1 is a superior limit of the roots.

9. State how the derived function f'(x) may be employed to ascertain the existence of equal roots in the equation f(x) = 0.

10. Find the equation of a straight line referred to rectangular coordinates.

Find the equation of the line joining the origin to the point of intersection of the lines

2y−x+a=0, 2x+ya= 0.

11. Write down the equation of the tangent to the parabola y = 4ax at the point x', y'.

If this tangent intersect a second parabola having the same axis and vertex as the first but with a greater latus rectum, prove that the ordinate of the point of contact of the tangent with the first parabola is a harmonic mean between the ordinates of the points of intersection with the second parabola.

12. If P and D be the extremities of a pair of conjugate diameters of an ellipse, and 4 and

eccentric angles at those points, prove that o'

=

π

2

'the

+ $.

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

A. If Ρ prove that

be a prime number and a be prime to p,

is divisible by p.

B. Prove De Moivre's theorem and apply it to solve the equation

x-1=0.

C. If ABC be a triangle, and D, E, and F be the centres of the escribed circles, prove that the area of the triangle DEF is

R being the radius of the circumscribing circle.

D. Prove that the equation,

Ax2 + Bxy + Cy2 = 1,

represents an ellipse or hyperbola according as B2 < or > 4A C.

Find the eccentricity.

MIXED MATHEMATICS.

1. Assuming the parallelogram of forces for direction, prove it also for magnitude.

A picture is supported by a string passing through two smooth rings in the frame and over a smooth nail in the wall. If the string be lengthened will its tension be increased or diminished?