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4. Simplify— . . 1 a? x+a • > x-2a ~ ~~ x l +2ax + ia' i ' (h \ I+X J- I ~~ X (0 -> 1+ vTTi +i - VT-x , s , _J> S I C 3 v 6 v (a-b)(a-c) + (b-a)(b-c) "+" <c-a)(c-b) 5. If unity be divided into two parts in the ratio of ,/3 : \/7, express the parts with rational denominators. 6. Extract the cube root of— a; 3 + 6aM + \ 5 ax 2 - 10« M _*£ tfx + % 7 «M - *J a". 7. Solve the equations— l \ L ,_i 1 , 1. V a v x-2 T x-5 - x-i + x-3' (b.) ax-\- by=m, 2bx — ay=n. (c.) Vx-b + Vx+b = Vix — 2a. (d.) x i -xy =2, 2xt+yz =9. 8. If 3 be added to both the numerator and the denominator of a certain fraction, its value will become f ; but, if 3 be subtracted from both, the value of the fraction will then be -J : what is the fraction ? 9. The depth of a rectangular box is equal to its breadth ; the sum of its length and breadth is a inches, and its diagonal is b inches : required the dimensions of the box.

Euclid. — For Class D, and for Junior Civil Service. Time allowed: 3 hours. [Optional.] 1. Define a right angle and an obtuse angle. Explain what is meant by the complement and the supplement of an angle. What is the greatest number of obtuse angles that a quadrilateral figure can have ? What is the greatest number of acute angles that it can have ? Explain why it cannot have more of either. 2. If at a point in a straight line two other straight lines, upon the opposite sides of it, make the adjacent angles together equal to two right angles, these two straight lines shall be in one and the same straight line. What name is given to the form of' proof adopted in this proposition ? In what circumstances is this form of proof usually resorted to ? 3. If a straight line fall upon two parallel straight lines it makes the alternate angles equal to one another, and the exterior angle equal to the interior and opposite upon the same side, and likewise the two interior angles upon the same side together equal to two right angles. What is meant by a converse proposition ? Enunciate the proposition or propositions of which this proposition is the converse. 4. Triangles upon the same base, and between the same parallels, are equal to one another. Show how to describe a parallelogram which shall be equal in area and in perimeter to a given triangle. 5. 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 that the squares described upon the two diagonals of a rhombus are together equal to four times the square described upon one of its sides. 6. 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 of the line between the points of section, is equal to the square of half the line. 7. 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. 8. Construct a rectangle equal to a given square, and having the difference of its adjacent sides equal to a given line.

Euclid, Books 1.-IV. — For Senior Civil Service. Time allowed: 3 hours. [Optional.] 1. Brove that any two sides of a triangle are together greater than the third. Show also that this proposition may be proved by bisecting the angle between the two sides which are to be taken together, and letting the bisector cut the third side. 2. Show how to divide a given straight line into any given number of equal parts. 3. Define a square, and show how to describe a square on a given finite straight line. 4. If a straight line be divided into any two parts, prove that the square on the whole line and on one part are together equal to twice the rectangle contained by the whole and that part together with the square on the other part. 5. Given an arc of a circle, show how you would proceed to find the centre, giving in full the necessary proof. 6. If two chords of a circle intersect in a point inside the circle, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other. 7. Prove that the three bisectors of the angles of a triangle meet in a point.