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8. Solve the equations —■ 6x + 8 2a+3B_.| 1 1 _b-a ax+b bx+a abx ' 9. A composition of copper and tin containing 100 cubic inches weighs 5050z. How many ounces of each metal does it contain, supposing a cubic inch of copper to weigh s}oz. and a cubic inch of tin to weigh 4-jOz.'? Euclid. — For Class I) and Junior Civil Service. Time allowed: 3 hours. 1. Define a " polygon." What is meant by a regular polygon? What are the names given to regular three-sided and four-sided figures'? 2. Define " parallel straight lines " and a " parallelogram." Quote the axiom on which Euclid's treatment of parallel lines is based. What are the properties of parallel straight lines ? What are the conditions of the equality of parallelograms ? 3. To make a triangle of which the sides shall be equal to three given straight lines; but any two whatever of these must be greater than the third. Draw figures showing how the construction will fail if the limitation in the last clause of the enunciation be neglected. 4. The opposite sides and angles of a parallelogram are equal to one another, and the diameter bisects it—that is, divides it into two equal parts. "If a diagonal of a quadrilateral bisects it, the figure is a parallelogram." Is this true ? If not, amend it so as to make it the true converse of the latter part of the proposition given above, and prove it. 5. To a given straight line to apply a parallelogram which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle. 6. In any right-angled triangle, the square which is described upon the side subtending the right angle is equal to the squares described upon the sides which contain the right angle. Show how to find a square that shall be half of a given square. 7. If a straight line be divided into any two parts, the squares of the whole line and of one of the parts are equal to twice the rectangle contained by the whole and that part together with the square of the other part. Show that this is equivalent to the following: " The square on the difference of two straight lines is equal to the sum of the squares on the two straight lines diminished by twice the rectangle contained by the two straight lines." 8. In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square of the side subtending the obtuse angle is greater than the squares of the sides containing the obtuse angle by twice the rectangle contained by the side upon which, when produced, the perpendicular falls and the straight line intercepted without the triangle between the perpendicular and the obtuse angle. ABCD is a parallelogram having the angle ABC equal to an angle of an equilateral triangle : prove that BD 2 = BC 3 + CD 2 +BC-CD. Euclid, Books 1.-IV. — For Senior Civil Service. Time allowed: 3 hours. 1. Enunciate the various propositions in the First Book in which Euclid proves that two triangles are equal in all respects. If two triangles have two sides of the one equal to two sides of the other each to each, and an angle of one equal to the corresponding angle of the other, are the triangles necessarily equal in all respects ? State the various cases which may arise, and the conclusion in each case. 2. The opposite sides and angles of a parallelogram are equal, and its diagonals bisect one another. 3. Divide a given straight line into two parts such that the difference of their squares may be equal to a given square. 4. Describe a square equal to a given rectilineal figure. 5. Show how to draw a pair of tangents to a circle from an external point, and prove that they are equal to one another. Investigate the condition which must be satisfied for a quadrilateral to be such that a circle can be inscribed in it. 6. The opposite angles of a quadrilateral inscribed in a circle are together equal to two right angles. Prove this proposition, and state and prove its converse. 7. Inscribe a circle in a given triangle. If three circles be drawn each touching the side of a given triangle and the other two produced, show that the lines joining their centres pass through the angular points of the triangle. Trigonometry. — For Senior Civil Service. Time allowed : 3 hours. 1. Given the magnitude of an angle in degrees, show how to find its circular measure. A piece of string placed on the circumference of a circle whose radius is 25 inches subtends an angle of 30° at the centre : find the length of the string. 2. Define the tangent of an angle, and trace the changes in its value as the angle increases from zero to two right angles. If Sec x= ~zzh' fi an x mx - 3. Prove that Sec (A+ 180°)=-Sec A, and that Cot (A-90°)= -Tan A. 4. Prove the identities,' — (C^A + Sec A \ Cot A + Cot_A 1= g \Sm A Gosec A/ Tan A Tan 6A-Tan 4A = Tan 6A Tan 4A Tan 2A + Tan 2A.