E.—la.

2. Multiply a?*-l-2a? B +a? s —4a?—ll by a? 2 — %x +3, and find the value of k in order that x i — 5x 2 +4x—k may be divisible by 2a?+ 1 3. Find the factors of— (i.) 2<m? — 3cy — %ay+3cx (ii.) o? 2 +Ba? +15 (hi.) 5x 2 — 6xy—By* (iv.) 4«V-(a 2 -& 2 +c 2 ) 2 and calculate [(37655) 2 - (37649) 2 ] --24 4. A clock that gains three minutes a day is set right at 10 a.m. on Sunday. What is the true time when this clock indicates 4 p.m. on the following Wednesday ? 5. Solve the equations :— (i.) l(x+6)-ix = i(Bx-l) + l , n (3o? + 4z,=24 W Jsa?-6«/ = 2 (ill.) ■( „ . . ' ]^ + slzl5 lzl = 18 -5 x What do you know of the graphical method of solving simultaneous equations ? 6. If two angles of a triangle be equal to each other, then the sides that are opposite to the equal angles shall be equal to each other. Describe some simple method of calculating the height of a tree, and point out the propositions of Euclid that are involved in the method. 7. If the square described on one side of a triangle be equal to the sum of the squares described on the other two sides, then the angle contained by these two sides shall be a right angle. Prove that a triangle whose sides are 3, 4, and 5 feet long is right-angled. Mention any practical methods of drawing a right angle, and explain them by means of Euclid's propositions. 8. If a straight line is 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. 9. Through two given points, one in each of two parallel straight lines, draw two lines so as to form a rhombus with the two given lines.

Euclid. — For Class D. Time allowed: Three hours. 1. Distinguish between a plane angle and a plane rectilineal angle. Define a circle. Give Euclid's classification of triangles. 2. If two triangles have two sides of the one equal to two sides of the other, each to each, and have also the angles contained by those sides equal, the two triangles are equal in all respects. 3. If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles. 4. State and prove the properties of the angles made by two parallel straight lines with any third straight line intersecting them. 5. Show that parallelograms on equal bases and between the same parallels are equal in area. 6. What is meant by a parallelogram described about the diagonal of another parallelogram ? Prove that a parallelogram about a diagonal of a square is itself a square. 7. Prove that, if a straight line be divided into any two parts, the sum of the squares on the whole line and on one of the parts is equal to twice the rectangle contained by the whole line and that part together with the square on the other part. 8. Show how to construct a triangle, having given the perimeter and each of the angles at the base. 9. Show how to draw through a given point P a straight line such that the part of it intercepted between two given lines AB, AC may be bisected at P. 10. Prove that, in any triangle ABC, the sum of the sides AB, AC is greater than twice the median AX. 11. The triangle ABC has the angle BAC a right angle : prove that four times the sum of the squares on the medians BE and OF is equal to five times the square on BC.

Euclid and Trigonometry. — For Civil Service Senior. Time allowed : Three hours. 1. To a given straight line apply a parallelogram equal to a given triangle and having an angle equal to a given rectilineal angle. 2. Bisect a triangle by a line drawn from a given point in one of the sides. 3. The squares on the sides of a parallelogram are together equal to the squares on the diagonals. 4. If two straight lines cut one another in a circle the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other. 5. Describe a circle about a given triangle. A circle is described about an equilateral triangle ABC, and P is any point in the circumference on the side of BC remote from A : prove PA == PB+PC,

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