2. How many pounds of tea at 3s. 9d. a-pound must a merchant mix with 500 pounds of tea at 4s. 6d. a-pound, so that by selling the whole at 4s. 6d. a-pound, he shall gain 10 per cent. on the prime cost.

3. Given log 10 2 = 301030, find log 10 0000025. The logarithm of a number to a base 10 is 5, what is the logarithm of the number to a base 2, and what is the number itself.

Find by the aid of the logarithmic tables:

(1) {0004582}.

(2) A 4th proportional to (3)°; (1)',

6

7

State how the sixth root of a number may be extracted, by the application of the ordinary rules

If a, b, c, d, be in geometrical progression,

Prove (a-b+c-d)2 = ( a − b)2 + (c−d)2 + 2(b−c)2.

(2) x3-7x-6=0

3x+4x2-35x-12=0)

which have a common root.

Of what problem is the system of equations (1) the algebraical statement.

6. If (n) be a positive whole number, find the general term in the expansion of (a+b+c+d)",

explaining the conditions to which the different exponents are subject.

8

In the expansion of (1 + x + 2x2), find the coefficient of x. How many terms of the series precede and follow the term involving x3.

7. In equal circles, if the angles at the centres are equal, the circumferences also on which the equal angles stand shall be equal.

AB is the side of an equilateral triangle inscribed in a circle whose diameter is AOC and centre O.

Prove (1) The triangle BOC equilateral.

(2) AB2 3A02.

=

8. Find a third proportional to two given straight lines;

AB is the diameter of a circle, P any point in the circumference, from any point C in the diameter AB, draw CD perpendicular to AB meeting the circumference in F, and meeting AP and BP produced when necessary in D and E, prove CD a third proportional to CE and CF.

9. Given the cotangent of an angle, find its sine and secant,

find sin 5 A, and express generally the values of the angle A.

Prove

cos A+ cos 3A + cos 5A
sin A+ sin 3A+ sin 5A

=cot 3A.

10. When three sides of a triangle are given, investigate any one formula adapted to logarithmic computation, by which an angle of the triangle may generally be determined. Point out whether any peculiarities may exist in the elements of the triangle to make it undesirable in practice to apply the formula you have obtained.

Given the three sides of a triangle 508, 401, 299 feet, find the greatest angle of the triangle.

11. A and B are two stations a mile apart, A due north of B. At the same instant a balloon is seen from A to bear 60° 15' west of south, and as seen from B to bear 54° 30' west of north, also the angle of elevation of the balloon as seen from A at the same time was 35° 25′ 25′′, find the perpendicular height of the balloon above the horizontal plane passing through A and B.

12. In a parabola show that the abscissa varies as the square of the semi-ordinate to the axis. If a right cone be cut by a plane parallel to its slant side, the section is a parabola.

Find the area of a portion of a parabola included between two ordinates to the axis, one passing through the focus of the length of 12 feet, the other ordinate being 24 feet in length.

Afternoon Paper.

I. TODHUNTER, ESQ., M.A.

1. Find the cost of 7 cwt. 2 qrs. 14 lbs. at 17. 9s. 2d. per cwt.

2. Reduce 31. 11s. 9 d. to the decimal of 17.; and also to the decimal of 21. 10s.

3. The length of a room is 7 yards 1 foot 3 inches, the breadth is 5 yards 2 feet 9 inches, and the height 4 yards 6 inches. Find the expense of papering the walls, supposing the paper to be a yard broad and to cost 9d. per yard.

5. Find the term in the expansion of (1+x)" in powers of x by the Binomial Theorem, which is numerically the greatest.

1

For example, suppose n= − 12 and x=}·

5

6. Describe a circle about a given triangle.

A circle is described about a triangle ABC which has the side AB equal to the side AC. From A a line is drawn meeting the base of the triangle in D, and the circumference of the circle in E. Prove that the circle which passes through B, D, and E, touches AB.

7. The parallelograms about the diameter of a parallelogram are similar to the whole parallelogram and to one another.

ABCD is a parallelogram, and F any point in the diagonal AC. Through F a line is drawn parallel to AB, meeting AD in G and BC in H; and through F a line is drawn parallel to AD, meeting AB in E and DC in K. Shew that GK and EH produced will meet on AC produced.

8. Calculate by logarithms,

(2) The time in which a sum of money will be doubled at 34 per cent., compound interest. 9. If A, B, C are the angles of a triangle, show that

sin A+ sin B+ sin2 C-2 cos A cos B cos C=2.

The sum of the squares of the cosines of the angles of a triangle is equal to unity; shew that one of the angles is a right angle.

10. A, B, C are three objects at known distances apart; namely AB = 1056 yards, AC = 924 yards, and CB = 1716 yards. An observer places himself at a station D, from which C appears directly in front of A, and observes the angle CDB to be 14° 24'. Find the distance CD.

11. Assuming the expression for the volume of a pyramid, determine the volume of a frustum of a pyramid made by two parallel planes; having given, the distance between the two parallel planes, the area of the section of the pyramid made by one of the planes, and the lengths of two parallel edges of the frustum. Determine also the position of a third plane parallel to the two parallel planes which will bisect the frustum.

12. In a spherical triangle having given a = 70°, b=40°, and c = 38° 30', calculate numerically the angle A, assuming a suitable formula.

1. Explain the mechanical advantage of a single moveable pulley.

In a system of three moveable pullies, when the strings are vertical and each pulley hangs by a separate string, express the condition of equilibrium

B