MATHEMATICS. (2.)

1. If a straight line be divided into two equal, and also into two unequal parts, the squares of the two unequal parts are together double of the square of half the line and of the square of the line between the points of section.

Write down the corresponding algebraical formula.

2. The diameter is the greatest straight line in a circle: and of all others, that which is nearer to the centre is always greater than one more remote: and the greater is nearer to the centre than the less.

From a given point without a circle, draw a straight line, the part of which intercepted by the circle shall be equal to a given line not greater than the diameter of the circle.

3. Upon the same straight line and upon the same side of it, there cannot be two similar segments of circles, not coinciding with one another.

4. Inscribe an equilateral and equiangular pentagon in a given circle.

5. Define similar, and reciprocal rectilineal figures and prove that in right-angled triangles, the rectilineal figure described upon the side opposite to the right angle, is equal to the similar and similarly described figures upon the sides containing the right angle.

6. If the sides of a triangle be cut proportionally, the straight line which joins the points of section shall be parallel to the remaining side of the triangle.

Divide a triangle into four equal triangles.

7. Find the Arithmetic, Geometric, and Harmonic means between two given straight lines.

8. The angle subtended at the centre of a circle by an arc, which is equal in length to the radius, is invariable.

Express seven-sixteenths of a right angle in circular measure and in grades.

9. Trace the variations in sign and magnitude of cos A - sin A, as A varies from 0° to 180°.

10. Express the sine and cosine of the difference of two angles in terms of the sines and cosines of the angles.

Find sin 75°.

11. Prove the following formulæ :

(1.) Vers (A+B) vers (AB) = (cos A cos B)2.

12. Find the general value of A, when cos 2A =

13. Solve the triangle in which a

tan 54° 44' == 1.4140943

=

sin A.

=

18, b = 12,

c

=

=

tan 54° 45'

10, having given 1.4149673

tan 15° 48' = ⚫2829715.

and escribed circles of a triangle,
that
prove

tan 15° 47'

• 2826573

14. Find the radii of the inscribed
and if they be called r, r1, ጥ. 2, 13,

15. An equilateral triangle and a regular hexagon have the same perimeter show that the areas of their inscribed circles are as

4:9.

:

d

and also of 315 271, to two places of decimals: and show generally that if there be n figures in the root there cannot be more than 2n nor less than (2n - 1) figures in the number whose

root is to be extracted.

3. Prove the rule for finding the least common multiple of any two quantities, and simplify the following expressions:

4. Reduce the following expressions to their simplest form :

(3.) .PQQ &c. ad infinitum, where P and Q contain (p) and (q) digits respectively.

5. Solve the following equations:

6. A certain number, consisting of two figures, gives, when divided by the figure on the right, a quotient equal to 27, and remainder 2: but if divided by 9, it gives a quotient equal to three times the figure on the right, and a remainder 2.

the number.

7. Given log 37852

=

4.5787767, and log 37853

=

Find

4.5787882; find the number corresponding to the logarithm 6·5787836.

8. Find, very nearly, a fourth proportional to the sixth root of 9, the fourth root of 7, and the fifth root of 5. Also compare the values of these three quantities.

= ⚫47712

Given log 2 = ⚫30103 log 3
log 7 = .84510 log 155.6

= 2.19201

9. The outer wall of a circular stone tower, 96 feet high, is 2 feet thick, and the inner diameter is 6 feet: a winding stone staircase is built in it, the central column being 1 foot in diameter: each step is 8 inches high, and there is a headway of 8 feet between each step and the one directly above it. The central column and the surface of the steps are of dressed stone. Find the number of cubic feet of stone in the tower (including the outer wall) and its weight, one cubic foot weighing 168 lbs.: also the cost of dressing the stone at 15d. per superficial square

10. Cleopatra's Needle consists approximately of a frustum of a pyramid surmounted by a smaller pyramid. If the lower base were 7 feet square, and the upper one 4 feet square, the height of the frustum being 61 feet, and of the upper pyramid 7 feet: find its cubical contents, and its weight, if one cubic foot weighs 170 lbs.

11. Two sides of a triangular field containing an obtuse angle are 110 and 220 yards, find the length of the third side, that the field may contain exactly an acre.

12. Find at what rate simple interest a sum of money would amount in 4 years to the same as it would at 5 per cent. compound interest in the same time.

MATHEMATICS. (2.)

1. Show how to divide a straight line into two parts, so that the rectangle contained by the whole and one of the parts shall be equal to the square on the other part.

2. If a straight line drawn through the centre of a circle bisect a straight line in it which does not pass through the centre, it shall cut it at right angles; and conversely, if it cut it at right angles, it shall bisect it.

Two equal circles cut one another, the centre of each being on the circumference of the other. Show that the square on the common chord is three times the square on a radius.

3. The angle at the centre of a circle is double the angle at the circumference upon the same base.

4. In a circle the angle in a semicircle is a right angle; but the angle in a segment greater than a semicircle is less than a right angle, and the angle in a segment less than a semicircle is greater than a right angle.

Circles are described on the sides of a triangle as diameters. Show that their points of intersection all lie on the sides of the triangle.

5. About a given circle describe a triangle equiangular to a given triangle.

6. If the sides of two triangles, about each of their angles, be proportionals, the triangles shall be equiangular.

7. In a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another.

8. Describe the three different systems of units by means of which the magnitudes of angles are expressed.

Express in each of the three systems the angle between two adjacent sides of a regular hexagon.

9. Trace the changes in sign and magnitude of cot A as A changes from 90° to 270°.