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Geometry.

Junior, Senior, and Higher Local.

Junior Work, Nos. 1-8 inclusive.

Senior Work, Nos. 5-12 inclusive.

Higher Local Work, Nos. 9-16 inclusive,

1. Give all definitions used in proving the propositions of Euclid, Book I.

2. The angles which one straight line makes with another straight line on one side of it, either are two right angles, or are together equal to two right angles.

3. If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, namely, either the sides adjacent to the equal angles, or sides which are opposite to equal angles in each; then shall the other sides be equal, each to each, and also the third angle of the one equal to the third angle of the other.

4. Two right-angled triangles have their hypothenuses equal, and a side of one equal to a side of the other. Show that they are equal in all respects.

5. If a straight line be bisected, and produced to any point, the square on the whole line thus produced, and the square on the part of it produced, are together double of the square on half the line bisected, and of the square on the line made up of the half and the part produced.

6. The sum of the squares on the sides of a parallelogram is equal to the sum of the squares on the diagonals.

7. Describe a square that shall be equal to a given rectilineal figure.

8. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles which this line makes with the line touching the circle shall be equal to the angles which are in the alternate segments of the circle.

9. Inscribe an equilateral and equiangular quindecagon in a given circle.

10. If a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of any triangle are together equal to two right angles.

11. The straight lines bisecting the angles at the base of an isosceles triangle meet the sides at D and E, show that DE is parallel to the base.

12. If a parallelogram and a triangle be on the same base and between the same parallels, the parallelogram shall be double of the triangle.

13. If a straight line be divided into two equal parts, and also into two unequal parts, the rectangle contained by the unequal parts, together with the square on the line between the points. of section, is equal to the square on half the line.

14. If a point be taken within a circle, from which there fall more than two equal lines to the circumference, that point is the centre of the circle.

15. Similar polygons may be divided into the same number of similar triangles, having the same ratio to one another that the polygons have, and the polygons are to one another in the duplicate ratio of their homologous sides.

16. From the same point in a given plane, there cannot be two straight lines at right angles to the plane, on the same side of it, and there can be but one perpendicular to a plane from a point without the plane.

1. Simplify

Algebra.

Junior, Senior, and Higher Local.

Junior Work, Nos. 1-10 inclusive.
Senior Work, Nos. 3 -12 inclusive.
Higher Local Work, Nos. 6-15 inclusive.

(a* - 2a2b2 + b) (a−b)
a2+2ab+ 62

2. Multiply together:

x2 − x + 1, x2 + x + 1, and x2 - x2 + 1.

3. Resolve into two factors:

(i) a1+b*; (ii) aa + a2b2 + ba.

4. Find a value of

which will make the expressions

4x3 7x+3 and 4x3 + 20x2 — 53x + 21 both vanish.

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5. Show that a quadratic equation has two, and only two roots. If p and q are the roots of ax2+ bx+c=0, find the value of p2 q2

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7. Interpret a", a-”, ao. If a'=b", show that (%) * =aï—1.

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9. Define variation. Two globes of gold whose radii are r and r', are melted into a single globe. Find its radius.

10. Insert n arithmetical, and n geometrical means between a and c. Sum to n terms the series in A.P., whose 7th term is 2r-1.

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12. Show that the nth coefficient in the expansion of (1—x)—”

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15. Prove that no rational integral algebraical formula can represent prime numbers only.

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Higher Mathematics.

Junior and Senior.

TRIGONOMETRY; STATICS; HYDROSTATICS.

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1. Find all the values of sin +(−1)T) where n is any

integer, positive, or negative.

2

Prove that

2. Find geometrically sin (A + B) and cos (A + B). Express sin (A + B) + sin (A— B) in the form of a product. cos A+ cos (120° — A) + cos (120° + A) = 0.

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5. Prove that the algebraic sum of the moments of two or more forces about a point in their plane is equal to the moment of their resultant, both for parallel and concurrent forces. Hence give conditions of equilibrium for any system of forces in one plane acting on a rigid body.

6. Define the term Centre of Gravity. Give a simple geometrical construction for finding that of a body of uniform density. Apply this to find the C. G. of (i) a straight line, (ii) a triangle, (iii) a parallelogram, (iv) a circle, (v) a sphere.

7. A uniform beam of length 2a rests against a vertical plane and over a peg at a distance h from the plane; show that if ✪ be the inclination of the beam to the vertical, sin 30 = ="

a

8. Define a fluid, and state its fundamental property. Distinguish between liquid and gaseous fluids.

9. What is the principle of the safety valve? Describe the hydrostatic bellows. Whenever the tube is one-eighth of an inch in diameter, and the weight rests on a circular disc one yard in diameter, find the weight which a pressure of 1 lb. on the water in the tube will sustain.

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